back to indexStephen Wolfram: Fundamental Theory of Physics, Life, and the Universe | Lex Fridman Podcast #124
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The following is a conversation with Stephen Wolfram,
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his second time on the podcast.
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He's a computer scientist, mathematician,
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theoretical physicist, and the founder and CEO
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of Wolfram Research, a company behind Mathematica,
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Wolfram Alpha, Wolfram Language,
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and the new Wolfram Physics Project.
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He's the author of several books,
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including A New Kind of Science, and the new book,
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A Project to Find the Fundamental Theory of Physics.
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This second round of our conversation is primarily focused
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on this latter endeavor of searching for the physics
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of our universe in simple rules that do their work
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on hypergraphs and eventually generate the infrastructure
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from which space, time, and all of modern physics can emerge.
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Quick summary of the sponsors,
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SimpliSafe, Sunbasket, and Masterclass.
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Please check out these sponsors in the description
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to get a discount and to support this podcast.
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As a side note, let me say that to me,
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the idea that seemingly infinite complexity can arise
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from very simple rules and initial conditions
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is one of the most beautiful and important
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mathematical and philosophical mysteries in science.
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I find that both cellular automata
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and the hypergraph data structure
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that Stephen and team are currently working on
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to be the kind of simple, clear mathematical playground
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within which fundamental ideas about intelligence,
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consciousness, and the fundamental laws of physics
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can be further developed in totally new ways.
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In fact, I think I'll try to make a video or two
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about the most beautiful aspects of these models
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in the coming weeks, especially, I think,
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trying to describe how fellow curious minds like myself
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can jump in and explore them either just for fun
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or potentially for publication of new innovative research
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in math, computer science, and physics.
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But honestly, I think the emerging complexity
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in these hypergraphs can capture the imagination
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of everyone, even if you're someone
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who never really connected with mathematics.
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That's my hope, at least, to have these conversations
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that inspire everyone to look up to the skies
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and into our own minds in awe of our amazing universe.
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Let me also mention that this is the first time
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I ever recorded a podcast outdoors
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as a kind of experiment to see if this is an option
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in times of COVID.
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I'm sorry if the audio is not great.
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I did my best and promise to keep improving
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and learning as always.
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If you enjoy this thing, subscribe on YouTube,
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review it with Five Stars and Apple Podcast,
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follow on Spotify, support on Patreon,
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or connect with me on Twitter at Lex Friedman.
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As usual, I'll do a few minutes of ads now
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and no ads in the middle.
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I tried to make these interesting,
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but I do give you timestamps, so you're welcome to skip,
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but still, please do check out the sponsors
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by clicking the links in the description.
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It's the best way to support this podcast.
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Also, let me say, even though I'm talking way too much,
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that I did a survey and it seems like over 90% of people
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either enjoy these ad reads somehow magically
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or don't mind them, at least.
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That honestly just warms my heart
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that people are that supportive.
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This show is sponsored by SimpliSafe,
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a home security company.
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Go to SimpliSafe.com to get a free HD camera.
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It's simple, no contracts, 15 bucks a month, easy setup.
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Even I figured it out.
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I have it set up in my apartment.
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Of course, I also welcome intruders.
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One of my favorite movies is Leon or The Professional
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with Jean Reno, Gary Oldman,
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and the brilliant young Natalie Portman.
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If you haven't seen the movie,
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he's a hit man with a minimalist life that resembles my own.
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In fact, when I was younger, the idea of being a hit man
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or targeting evil in a skilled way,
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which is how I thought about it, really appealed to me.
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The skill of it, the planning, the craftsmanship.
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In another life, perhaps,
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if I didn't love engineering and science so much,
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I could see myself being something like a Navy SEAL.
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And in general, I love the idea of serving my country,
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of serving society by contributing my skill
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in some small way.
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Anyway, go to Simplisafe.com slash Lex
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to get a free HD camera and to support this podcast.
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They're a new sponsor, and this is a trial run,
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so you know what to do.
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This show is also sponsored by Sun Basket,
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a meal delivery service.
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Visit SunBasket.com slash Lex
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and use code LEX to get $30 off your order
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and to support this podcast.
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This is the last read of the trial they're doing,
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so this is the time to get them if you're considering it.
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And if you do, it'll help ensure
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that they decide to support this podcast long term.
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Their meals are healthy and delicious,
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a nice break from the minimalist meals of meat
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and vegetables that I usually eat.
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Maybe on a personal note,
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one of my favorite things to do is watch people cook,
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especially people who love cooking,
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and hang out with people over amazing meals.
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I still tend to be strict in my diet no matter what,
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even in fancy restaurants,
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but it brings me joy to see friends and family indulge
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something like a cake that has way too many calories
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or ice cream or whatever.
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My mom, in fact, for much of my life,
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made this cake called an anthill on my birthday
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that brings me a lot of joy and way too many calories.
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I was thinking of doing a video with my mom as she makes it.
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I thought it'd be a fun thing to do together.
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Anyway, go to SunBasket.com slash Lex and use code LEX.
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So they signed a longterm contract for this podcast.
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This show is also sponsored by Masterclass.
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Sign up at masterclass.com slash LEX.
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180 bucks a year, you get an all access pass
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to watch lessons from Chris Hadfield,
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and many more brilliant world experts.
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Masterclass has been a really special sponsor.
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They believe in this podcast in a way that gives me strength
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and motivation to take intellectual risks.
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I'm thinking of doing a few solo podcast episodes
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on difficult topics, especially in history,
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like the rise and fall of the Third Reich or Stalin, Putin,
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and many other difficult topics that I'm fascinated by.
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I have a worldview that seeks inspiring positive insights,
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even and perhaps especially from periods of tragedy and evil
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that perhaps some folks may find value in.
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If I can only learn to convey the ideas in my mind
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as clearly as I think them.
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I think deeply and rigorously and precisely,
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but to be honest, have trouble speaking in a way
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that reflects that rigor of thought.
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So it really does mean a lot, the love and support I get
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as I try to get better at this thing,
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at this talking thing.
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Anyway, go to masterclass.com slash LEX to get a discount
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and to support this podcast.
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And now finally, here's my conversation with Stephen Wolfram.
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You said that there are moments in history of physics
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and maybe mathematical physics or even mathematics
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where breakthroughs happen
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and then a flurry of progress follows.
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So if you look back through the history of physics,
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what moments stand out to you as important such breakthroughs
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where a flurry of progress follows?
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So the big famous one was 1920s,
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the invention of quantum mechanics,
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where in about five or 10 years,
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lots of stuff got figured out.
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That's now quantum mechanics.
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Can you mention the people involved?
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Yeah, it was kind of the Schrodinger, Heisenberg,
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Einstein had been a key figure, originally Planck,
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then Dirac was a little bit later.
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That was something that happened at that time,
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that's sort of before my time, right?
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In my time was in the 1970s,
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there was this sort of realization
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that quantum field theory was actually going to be useful
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in physics and QCD, quantum thermodynamics theory
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of quarks and gluons and so on was really getting started.
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And there was again, sort of big flurry of things
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happened then, I happened to be a teenager at that time
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and happened to be really involved in physics.
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And so I got to be part of that, which was really cool.
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Who were the key figures
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aside from your young selves at that time?
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You know, who won the Nobel Prize for QCD, okay?
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People, David Gross, Frank Wilczek, you know, David Politzer.
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The people who are the sort of the slightly older generation,
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Dick Feynman, Murray Gellman, people like that,
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who were Steve Weinberg, Gerhard Hoft, he's younger,
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he's in the younger group actually.
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But these are all, you know, characters who were involved.
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I mean, it's funny because those are all people
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who are kind of in my time and I know them
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and they don't seem like sort of historical,
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you know, iconic figures.
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They seem more like everyday characters, so to speak.
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And so it's always, you know, when you look at history
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from long afterwards, it always seems like
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everything happened instantly.
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And that's usually not the case.
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There was usually a long buildup,
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but usually there's, you know,
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there's some methodological thing happens
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and then there's a whole bunch of low hanging fruit
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And that usually lasts five or 10 years.
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You know, we see it today with machine learning
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and, you know, deep learning neural nets and so on.
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You know, methodological advance,
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things actually started working in, you know, 2011, 2012
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And, you know, there's been this sort of rapid
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picking of low hanging fruit, which is probably, you know,
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some significant fraction of the way done, so to speak.
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Do you think there's a key moment?
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Like if I had to really introspect,
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like what was the key moment
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for the deep learning, quote unquote, revolution?
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It's probably the AlexNet business.
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AlexNet with ImageNet.
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So is there something like that with physics
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where, so deep learning neural networks
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have been around for a long time.
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Absolutely, since the 1940s, yeah.
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There's a bunch of little pieces that came together
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and then all of a sudden everybody's eyes lit up.
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Like, wow, there's something here.
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Like even just looking at your own work,
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just your thinking about the universe,
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that there's simple rules can create complexity.
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You know, at which point was there a thing
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where your eyes light up?
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It's like, wait a minute, there's something here.
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Is it the very first idea
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or is it some moment along the line of implementations
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and experiments and so on?
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There's a couple of different stages to this.
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I mean, one is the think about the world computationally.
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Can we use programs instead of equations
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to make models of the world?
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That's something that I got interested in
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in the beginning of the 1980s.
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I did a bunch of computer experiments.
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When I first did them, I didn't really,
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I could see some significance to them,
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but it took me a few years to really say,
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wow, there's a big important phenomenon here
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that lets sort of complex things arise
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from very simple programs.
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That kind of happened back in 1984 or so.
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Then, you know, a bunch of other years go by,
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then I start actually doing a lot
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of much more systematic computer experiments and things
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and find out that the, you know,
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this phenomenon that I could only have said occurs
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in one particular case
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is actually something incredibly general.
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And then that led me to this thing called
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principle of computational equivalence.
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And that was a long story.
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And then, you know, as part of that process,
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I was like, okay, you can make simple programs,
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can make models of complicated things.
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What about the whole universe?
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That's our sort of ultimate example of a complicated thing.
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And so I got to thinking, you know,
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could we use these ideas to study fundamental physics?
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You know, I happen to know a lot about,
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you know, traditional fundamental physics.
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My first, you know, I had a bunch of ideas
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about how to do this in the early 1990s.
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I made a bunch of technical progress.
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I figured out a bunch of things
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I thought were pretty interesting.
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You know, I wrote about them back in 2002.
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With the new kind of science
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in the cellular automata world.
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And there's echoes in the cellular automata world
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with your new Wolfram physics project.
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We'll get to all that.
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Allow me to sort of romanticize a little more
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on the philosophy of science.
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So Thomas Kuhn, philosopher of science,
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describes that, you know, the progress in science
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is made with these paradigm shifts.
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And so to link on the sort of original line of discussion,
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do you agree with this view
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that there is revolutions in science
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that just kind of flip the table?
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What happens is it's a different way
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of thinking about things.
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It's a different methodology for studying things.
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And that opens stuff up.
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There's this idea of,
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he's a famous biographer,
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but I think it's called the innovators.
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There's a biographer of Steve Jobs, of Albert Einstein.
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He also wrote a book,
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I think it's called the innovators,
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where he discusses how a lot of the innovations
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in the history of computing has been done by groups.
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There's a complicated group dynamic going on,
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but there's also a romanticized notion
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that the individual is at the core of the revolution.
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Like where does your sense fall?
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Is ultimately like one person responsible
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for these revolutions that creates the spark
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or one or two, whatever,
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or is it just the big mush and mess and chaos
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of people interacting, of personalities interacting?
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I think it ends up being like many things,
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there's leadership and there ends up being,
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it's a lot easier for one person to have a crisp new idea
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than it is for a big committee to have a crisp new idea.
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And I think, but I think it can happen
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that you have a great idea,
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but the world isn't ready for it.
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And you can, I mean, this has happened to me plenty, right?
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It's, you have an idea, it's actually a pretty good idea,
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but things aren't ready,
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either you're not really ready for it,
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or the ambient world isn't ready for it.
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And it's hard to get the thing to get traction.
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It's kind of interesting.
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I mean, when I look at a new kind of science,
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you're now living inside the history,
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so you can't tell the story of these decades,
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but it seems like the new kind of science
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has not had the revolutionary impact
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I would think it might.
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Like, it feels like at some point, of course it might be,
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but it feels at some point people will return to that book
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and say, that was something special here.
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This was incredible.
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Or do you think that's already happened?
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Oh, yeah, it's happened, except that people aren't,
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the sort of the heroism of it may not be there,
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but what's happened is for 300 years,
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people basically said,
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if you want to make a model of things in the world,
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mathematical equations are the best place to go.
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Last 15 years, doesn't happen.
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New models that get made of things
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most often are made with programs, not with equations.
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Now, was that sort of going to happen anyway?
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Was that a consequence of my particular work
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and my particular book?
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It's hard to know for sure.
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I mean, I am always amazed at the amounts of feedback
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that I get from people where they say,
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oh, by the way, I started doing this whole line of research
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because I read your book, blah, blah, blah, blah, blah.
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It's like, well, can you tell that
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from the academic literature?
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Was there a chain of academic references?
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One of the interesting side effects of publishing
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in the way you did this tome
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is it serves as an education tool and an inspiration
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to hundreds of thousands, millions of people,
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but because it's not a single,
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it's not a chain of papers with spiffy titles,
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it doesn't create a splash of citations.
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It's had plenty of citations, but it's, you know,
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I think that people think of it as probably more,
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you know, conceptual inspiration than kind of a,
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you know, this is a line from here to here to here
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in our particular field.
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I think that the thing which I am disappointed by
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and which will eventually happen
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is this kind of study of the sort of pure computationalism,
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this kind of study of the abstract behavior
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of the computational universe.
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That should be a big thing that lots of people do.
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You mean in mathematics purely, almost like.
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It's like pure mathematics, but it isn't mathematics.
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But it isn't, it isn't.
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It's a new kind of mathematics.
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Is it a new title of the book?
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That's why the book is called that.
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Right, that's not coincidental.
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It's interesting that I haven't seen
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really rigorous investigation
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by thousands of people of this idea.
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I mean, you look at your competition around rule 30.
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I mean, that's fascinating.
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If you can say something.
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Is there some aspect of this thing that could be predicted?
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That's the fundamental question of science.
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Well, that has been a question of science.
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I think that is some people's view of what science is about
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and it's not clear that's the right view.
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In fact, as we live through this pandemic
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full of predictions and so on,
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it's an interesting moment to be pondering
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what science's actual role in those kinds of things is.
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Or you think it's possible that in science,
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clean, beautiful, simple prediction
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may not even be possible in real systems.
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That's the open question.
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I don't think it's open.
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I think that question is answered and the answer is no.
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The answer could be just humans are not smart enough yet.
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Like we don't have the tools yet.
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No, that's the whole point.
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I mean, that's sort of the big discovery
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of this principle of computational equivalence of mine.
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And this is something which is kind of a follow on
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to Gödel's theorem, to Turing's work
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on the halting problem, all these kinds of things.
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That there is this fundamental limitation
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built into science,
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this idea of computational irreducibility
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that says that even though you may know the rules
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by which something operates,
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that does not mean that you can readily sort of
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be smarter than it and jump ahead
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and figure out what it's going to do.
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Yes, but do you think there's a hope
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for pockets of computational reducibility?
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Computational reducibility.
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And then a set of tools and mathematics
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that help you discover such pockets.
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That's where we live is in the pockets of reducibility.
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That's why, and this is one of the things
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that sort of come out of this physics project
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and actually something that, again,
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I should have realized many years ago, but didn't,
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is it could very well be that everything about the world
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is computationally reducible and completely unpredictable.
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But in our experience of the world,
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there is at least some amount of prediction we can make.
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And that's because we have sort of chosen a slice of,
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probably talk about this in much more detail,
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but I mean, we've kind of chosen a slice
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of how to think about the universe
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in which we can kind of sample
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a certain amount of computational reducibility.
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And that's sort of where we exist.
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And it may not be the whole story of how the universe is,
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but it is the part of the universe that we care about
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and we sort of operate in.
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And that's, you know, in science,
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that's been sort of a very special case of that.
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That is science has chosen to talk a lot about places
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where there is this computational reducibility
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that it can find, you know,
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the motion of the planets can be more or less predicted.
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You know, something about the weather
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is much harder to predict.
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Something about, you know, other kinds of things
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that are much harder to predict.
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And it's, these are, but science has tended to,
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you know, concentrate itself on places
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where its methods have allowed successful prediction.
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So you think rule 30, if we could linger on it,
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because it's just such a beautiful, simple formulation
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of the essential concept underlying
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all the things we're talking about.
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Do you think there's pockets of reducibility
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Yes, that is the question of how big are they?
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What will they allow you to say?
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And that's, and figuring out where those pockets are,
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I mean, in a sense, that's the, that's sort of a,
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you know, that is an essential thing
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that one would like to do in science.
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But it's also, the important thing to realize
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that has not been, you know, is that science,
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if you just pick an arbitrary thing,
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you say, what's the answer to this question?
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That question may not be one
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that has a computationally reducible answer.
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That question, if you choose, you know,
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if you walk along the series of questions
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and you've got one that's reducible
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and you get to another one that's nearby
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and it's reducible too,
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if you stick to that kind of stick to the land,
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so to speak, then you can go down this chain
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of sort of reducible, answerable things.
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But if you just say, I'm just pick a question at random,
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I'm gonna have my computer pick a question at random.
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Most likely it's gonna be reducible.
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Most likely it will be reducible.
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And what we're thrown in the world, so to speak,
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we, you know, when we engineer things,
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we tend to engineer things to sort of keep
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in the zone of reducibility.
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When we're throwing things by the natural world,
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for example, not at all certain
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that we will be kept in this kind of zone of reducibility.
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Can we talk about this pandemic then?
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For a second, is a, so how do we,
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there's obviously huge amount of economic pain
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that people are feeling.
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There's a huge incentive and medical pain,
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health, just all kind of psychological.
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There's a huge incentive to figure this out,
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to walk along the trajectory of reducible, of reducibility.
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There's a lot of disparate data.
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You know, people understand generally how viruses spread,
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but it's very complicated
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because there's a lot of uncertainty.
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There's a, there could be a lot of variability also,
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like so many, obviously a nearly infinite number
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of variables that represent human interaction.
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And so you have to figure out,
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from the perspective of reducibility,
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figure out which variables are really important
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in this kind of, from an epidemiological perspective.
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So why aren't we, you kind of said
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that we're clearly failing.
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Well, I think it's a complicated thing.
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So, I mean, you know, when this pandemic started up,
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you know, I happened to be in the middle
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of being about to release this whole physics project thing,
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but I thought, you know.
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The timing is just cosmically absurd.
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A little bit bizarre, but you know,
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but I thought, you know,
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I should do the public service thing of, you know,
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trying to understand what I could about the pandemic.
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And, you know, we'd been curating data about it
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and all that kind of thing.
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But, you know, so I started looking at the data
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and started looking at modeling
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and I decided it's just really hard.
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You need to know a lot of stuff that we don't know
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about human interactions.
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It's actually clear now that there's a lot of stuff
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we didn't know about viruses
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and about the way immunity works and so on.
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And it's, you know, I think what will come out in the end
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is there's a certain amount of what happens
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that we just kind of have to trace each step
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and see what happens.
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There's a certain amount of stuff
link |
where there's going to be a big narrative
link |
about this happened because, you know, of T cell immunity.
link |
This could happen because there's this whole giant
link |
sort of field of asymptomatic viral stuff out there.
link |
You know, there will be a narrative
link |
and that narrative, whenever there's a narrative,
link |
that's kind of a sign of reducibility.
link |
But when you just say,
link |
let's from first principles figure out what's going on,
link |
then you can potentially be stuck
link |
in this kind of a mess of irreducibility
link |
where you just have to simulate each step
link |
and you can't do that unless you know details about,
link |
you know, human interaction networks
link |
and so on and so on and so on.
link |
The thing that has been very sort of frustrating to see
link |
is the mismatch between people's expectations
link |
about what science can deliver
link |
and what science can actually deliver, so to speak.
link |
Because people have this idea that, you know, it's science.
link |
So there must be a definite answer
link |
and we must be able to know that answer.
link |
And, you know, this is, it is both, you know,
link |
when you've, after you've played around
link |
with sort of little programs in the computational universe,
link |
you don't have that intuition anymore.
link |
You know, it's, I always, I'm always fond of saying,
link |
you know, the computational animals
link |
are always smarter than you are.
link |
That is, you know, you look at one of these things
link |
and it's like, it can't possibly do such and such a thing.
link |
Then you run it and it's like, wait a minute,
link |
it's doing that thing.
link |
How does that work?
link |
Okay, now I can go back and understand it.
link |
But that's the brave thing about science
link |
is that in the chaos of the irreducible universe,
link |
we nevertheless persist to find those pockets.
link |
That's kind of the whole point.
link |
That's like, you say that the limits of science,
link |
but that, you know, yes, it's highly limited,
link |
but there's a hope there.
link |
And like, there's so many questions I want to ask here.
link |
So one, you said narrative, which is really interesting.
link |
So obviously from a, at every level of society,
link |
you look at Twitter, everybody's constructing narratives
link |
about the pandemic, about not just the pandemic,
link |
but all the cultural tension that we're going through.
link |
So there's narratives,
link |
but they're not necessarily connected
link |
to the underlying reality of these systems.
link |
So our human narratives, I don't even know if they're,
link |
I don't like those pockets of reducibility
link |
because we're, it's like constructing things
link |
that are not actually representative of reality,
link |
and thereby not giving us like good solutions
link |
to how to predict the system.
link |
Look, it gets complicated because, you know,
link |
people want to say, explain the pandemic to me,
link |
explain what's going to happen.
link |
Yes, but also, can you explain it?
link |
Is there a story to tell?
link |
What already happened in the past?
link |
Yeah, or what's going to happen,
link |
but I mean, you know, it's similar to sort of
link |
explaining things in AI or in any computational system.
link |
It's like, you know, explain what happened.
link |
Well, it could just be this happened
link |
because of this detail and this detail and this detail,
link |
and a million details,
link |
and there isn't a big story to tell.
link |
There's no kind of big arc of the story that says,
link |
oh, it's because, you know, there's a viral field
link |
that has these properties
link |
and people start showing symptoms.
link |
You know, when the seasons change,
link |
people will show symptoms
link |
and people don't even understand, you know,
link |
seasonal variation of flu, for example.
link |
It's something where, you know,
link |
there could be a big story,
link |
or it could be just a zillion little details that mount up.
link |
See, but, okay, let's pretend that this pandemic,
link |
like the coronavirus, resembles something
link |
like the 1D rule 30 cellular automata, okay?
link |
So, I mean, that's how epidemiologists model virus spread.
link |
They sometimes use cellular automata, yes.
link |
Yeah, and okay, so you could say it's simplistic,
link |
but okay, let's say it's representative
link |
of actually what happens.
link |
You know, the dynamic of,
link |
it probably is closer to the hypergraph model.
link |
It's actually, that's another funny thing.
link |
As we were getting ready to release this physics project,
link |
we realized that a bunch of things we'd worked out
link |
about foliations of causal graphs and things
link |
were directly relevant to thinking about contact tracing.
link |
And interactions with cell phones and so on,
link |
which is really weird.
link |
But like, it just feels like,
link |
it feels like we should be able to get
link |
some beautiful core insight about the spread
link |
of this particular virus
link |
on the hypergraph of human civilization, right?
link |
I tried, I didn't manage to figure it out.
link |
But you're one person.
link |
Yeah, but I mean, I think actually it's a funny thing
link |
because it turns out the main model,
link |
you know, this SIR model,
link |
I only realized recently was invented by the grandfather
link |
of a good friend of mine from high school.
link |
So that was just a, you know, it's a weird thing, right?
link |
The question is, you know, okay, so you know,
link |
on this graph of how humans are connected,
link |
you know something about what happens
link |
if this happens and that happens.
link |
That graph is made in complicated ways
link |
that depends on all sorts of issues
link |
that where we don't have the data
link |
about how human society works well enough
link |
to be able to make that graph.
link |
There's actually, one of my kids did a study
link |
of sort of what happens on different kinds of graphs
link |
and how robust are the results, okay?
link |
His basic answer is there are a few general results
link |
that you can get that are quite robust.
link |
Like, you know, a small number of big gatherings
link |
is worse than a large number of small gatherings, okay?
link |
That's quite robust.
link |
But when you ask more detailed questions,
link |
it seemed like it just depends.
link |
It depends on details.
link |
In other words, it's kind of telling you in that case,
link |
you know, the irreducibility matters, so to speak.
link |
It's not, there's not gonna be this kind of one
link |
sort of master theorem that says,
link |
and therefore this is how things are gonna work.
link |
Yeah, but there's a certain kind of,
link |
from a graph perspective,
link |
the certain kind of dynamic to human interaction.
link |
So like large groups and small groups,
link |
I think it matters who the groups are.
link |
For example, you could imagine large,
link |
depends how you define large,
link |
but you can imagine groups of 30 people,
link |
as long as they are cliques or whatever.
link |
As long as the outgoing degree of that graph is small
link |
or something like that,
link |
like you can imagine some beautiful underlying rule
link |
of human dynamic interaction where I can still be happy,
link |
where I can have a conversation with you
link |
and a bunch of other people that mean a lot to me in my life
link |
and then stay away from the bigger, I don't know,
link |
not going to a Miley Cyrus concert or something like that
link |
and figuring out mathematically some nice.
link |
See, this is an interesting thing.
link |
So I mean, this is the question of what you're describing
link |
is kind of the problem of the many situations
link |
where you would like to get away
link |
from computational irreducibility.
link |
A classic one in physics is thermodynamics.
link |
The second law of thermodynamics,
link |
the law that says entropy tends to increase things
link |
that start orderly tend to get more disordered,
link |
or which is also the thing that says,
link |
given that you have a bunch of heat,
link |
it's hard, heat is the microscopic motion of molecules,
link |
it's hard to turn that heat into systematic mechanical work.
link |
It's hard to just take something being hot
link |
and turn that into, oh, all the atoms are gonna line up
link |
in the bar of metal and the piece of metal
link |
is gonna shoot in some direction.
link |
That's essentially the same problem
link |
as how do you go from this computationally irreducible
link |
mess of things happening
link |
and get something you want out of it.
link |
It's kind of mining, you're kind of,
link |
now, actually I've understood in recent years
link |
that the story of thermodynamics
link |
is actually precisely a story of computational irreducibility,
link |
but it is a, it is already an analogy.
link |
You can kind of see that as can you take the,
link |
what you're asking to do there
link |
is you're asking to go from the kind of,
link |
the mess of all these complicated human interactions
link |
and all this kind of computational processes going on
link |
and you say, I want to achieve
link |
this particular thing out of it.
link |
I want to kind of extract from the heat of what's happening.
link |
I want to kind of extract this useful piece
link |
of sort of mechanical work that I find helpful.
link |
Do you have a hope for the pandemic?
link |
So we'll talk about physics,
link |
but for the pandemic, can that be extracted?
link |
What's your intuition?
link |
The good news is the curves basically,
link |
for reasons we don't understand,
link |
the curves, the clearly measurable mortality curves
link |
and so on for the Northern Hemisphere have gone down.
link |
Yeah, but the bad news is that it could be a lot worse
link |
for future viruses.
link |
And what this pandemic revealed is we're highly unprepared
link |
for the discovery of the pockets of reducibility
link |
within a pandemic that's much more dangerous.
link |
Well, my guess is the specific risk of viral pandemics,
link |
you know, that the pure virology
link |
and immunology of the thing,
link |
this will cause that to advance to the point
link |
where this particular risk
link |
is probably considerably mitigated.
link |
But is the structure of modern society robust
link |
to all kinds of risks?
link |
Well, the answer is clearly no.
link |
And it's surprising to me the extent to which people,
link |
as I say, it's kind of scary actually
link |
how much people believe in science.
link |
That is people say, oh, you know,
link |
because the science says this, that and the other,
link |
we'll do this and this and this,
link |
even though from a sort of common sense point of view,
link |
it's a little bit crazy and people are not prepared
link |
and it doesn't really work in society
link |
as it is for people to say,
link |
well, actually we don't really know how the science works.
link |
People say, well, tell us what to do.
link |
Yeah, because then, yeah, what's the alternative?
link |
For the masses, it's difficult to sit,
link |
it's difficult to meditate on computational reducibility.
link |
It's difficult to sit,
link |
it's difficult to enjoy a good dinner meal
link |
while knowing that you know nothing about the world.
link |
Well, I think this is a place where, you know,
link |
this is what politicians and political leaders do
link |
for a living, so to speak,
link |
is you've got to make some decision about what to do.
link |
Tell some narrative that while amidst the mystery
link |
and knowing not much about the past or the future,
link |
still telling a narrative that somehow gives people hope
link |
that we know what the heck we're doing.
link |
Yeah, and get society through the issue.
link |
You know, even though, you know,
link |
the idea that we're just gonna, you know,
link |
sort of be able to get the definitive answer from science
link |
and it's gonna tell us exactly what to do.
link |
Unfortunately, you know, it's interesting
link |
because let me point out that if that was possible,
link |
if science could always tell us what to do,
link |
then in a sense, our, you know,
link |
that would be a big downer for our lives.
link |
If science could always tell us
link |
what the answer is gonna be,
link |
it's like, well, you know,
link |
it's kind of fun to live one's life
link |
and just sort of see what happens.
link |
If one could always just say,
link |
let me check my science.
link |
Oh, I know, you know,
link |
the result of everything is gonna be 42.
link |
I don't need to live my life and do what I do.
link |
It's just, we already know the answer.
link |
It's actually good news in a sense
link |
that there is this phenomenon
link |
of computational irreducibility
link |
that doesn't allow you to just sort of jump through time
link |
and say, this is the answer, so to speak.
link |
And that's, so that's a good thing.
link |
The bad thing is it doesn't allow you to jump through time
link |
and know what the answer is.
link |
Do you think we're gonna be okay as a human civilization?
link |
You said, we don't know.
link |
Do you think we'll prosper or destroy ourselves?
link |
No, I think that, you know,
link |
it'll be interesting to see, for example,
link |
with this, you know, pandemic,
link |
I, you know, to me, you know,
link |
when you look at like organizations, for example,
link |
you know, having some kind of perturbation,
link |
some kick to the system,
link |
usually the end result of that is actually quite good.
link |
You know, unless it kills the system,
link |
it's actually quite good usually.
link |
And I think in this case, you know, people,
link |
I mean, my impression, you know,
link |
it's a little weird for me because, you know,
link |
I've been a remote tech CEO for 30 years.
link |
It doesn't, you know, this is bizarrely, you know,
link |
and the fact that, you know, like this coming to see you here
link |
is the first time in six months that I've been like,
link |
you know, in a building other than my house, okay?
link |
So, you know, I'm a kind of ridiculous outlier
link |
in these kinds of things.
link |
But overall, your sense is when you shake up the system
link |
and throw in chaos that you challenge the system,
link |
we humans emerge better.
link |
Seems to be that way.
link |
I think that, you know, people, you know,
link |
my sort of vague impression is that people are sort of,
link |
you know, oh, what's actually important?
link |
You know, what is worth caring about and so on?
link |
And that seems to be something that perhaps is more,
link |
you know, emergent in this kind of situation.
link |
It's so fascinating that on the individual level,
link |
we have our own complex cognition.
link |
We have consciousness, we have intelligence,
link |
we're trying to figure out little puzzles.
link |
And then that somehow creates this graph
link |
of collective intelligence.
link |
Well, we figure out, and then you throw in these viruses
link |
of which there's millions different, you know,
link |
there's entire taxonomy and the viruses are thrown
link |
into the system of collective human intelligence.
link |
And when little humans figure out what to do about it,
link |
we get like, we tweet stuff about information.
link |
There's doctors as conspiracy theorists.
link |
And then we play with different information.
link |
I mean, the whole of it is fascinating.
link |
I am like you also very optimistic,
link |
but you said the computational reducibility.
link |
There's always a fear of the darkness
link |
of the uncertainty before us.
link |
Yeah, I know. And it's scary.
link |
I mean, the thing is, if you knew everything,
link |
it will be boring.
link |
And it would be, and then, and worse than boring,
link |
It would reveal the pointlessness, so to speak.
link |
And in a sense, the fact that there is
link |
this computational irreducibility,
link |
it's like as we live our lives, so to speak,
link |
something is being achieved.
link |
We're computing what our lives, you know,
link |
what happens in our lives.
link |
So the computational reducibility is kind of like,
link |
it gives the meaning to life.
link |
It is the meaning of life.
link |
Computational reducibility is the meaning of life.
link |
It gives it meaning, yes.
link |
I mean, it's what causes it to not be something
link |
where you can just say, you know,
link |
you went through all those steps to live your life,
link |
but we already knew what the answer was.
link |
Hold on one second.
link |
I'm going to use my handy Wolfram Alpha sunburn
link |
computation thing, so long as I can get network here.
link |
Oh, actually, you know what?
link |
It says sunburn unlikely.
link |
This is a QA moment.
link |
This is a good moment.
link |
Okay, well, let me just check what it thinks.
link |
See why it thinks that.
link |
It doesn't seem like my intuition.
link |
This is one of these cases where we can,
link |
the question is, do we trust the science
link |
or do we use common sense?
link |
The UV thing is cool.
link |
Yeah, yeah, well, we'll see.
link |
This is a QA moment, as I say.
link |
It's, do we trust the product?
link |
Yes, we trust the product, so.
link |
And then there'll be a data point either way.
link |
If I'm desperately sunburned,
link |
I will send in an angry feedback.
link |
Because we mentioned the concept so much
link |
and a lot of people know it,
link |
but can you say what computational reducibility is?
link |
The question is, if you think about things
link |
that happen as being computations,
link |
you think about some process in physics,
link |
something that you compute in mathematics, whatever else,
link |
it's a computation in the sense it has definite rules.
link |
You follow those rules.
link |
You follow them many steps and you get some result.
link |
So then the issue is,
link |
if you look at all these different kinds of computations
link |
whether they're computations
link |
that are happening in the natural world,
link |
whether they're happening in our brains,
link |
whether they're happening in our mathematics,
link |
the big question is, how do these computations compare?
link |
Is, are there dumb computations and smart computations
link |
or are they somehow all equivalent?
link |
And the thing that I kind of was sort of surprised to realize
link |
from a bunch of experiments that I did in the early nineties
link |
and now we have tons more evidence for it,
link |
this thing I call the principle of computational equivalence,
link |
which basically says, when one of these computations,
link |
one of these processes that follows rules,
link |
doesn't seem like it's doing something obviously simple,
link |
then it has reached the sort of equivalent level
link |
of computational sophistication of everything.
link |
So what does that mean?
link |
That means that, you might say, gosh,
link |
I'm studying this little tiny program on my computer.
link |
I'm studying this little thing in nature,
link |
but I have my brain
link |
and my brain is surely much smarter than that thing.
link |
I'm gonna be able to systematically outrun
link |
the computation that it does
link |
because I have a more sophisticated computation
link |
But what the principle of computational equivalence says
link |
is that doesn't work.
link |
Our brains are doing computations
link |
that are exactly equivalent to the kinds of computations
link |
that are being done in all these other sorts of systems.
link |
And so what consequences does that have?
link |
Well, it means that we can't systematically
link |
outrun these systems.
link |
These systems are computationally irreducible
link |
in the sense that there's no sort of shortcut
link |
that we can make that jumps to the answer.
link |
Now the general case.
link |
But the, so what has happened,
link |
what science has become used to doing
link |
is using the little sort of pockets
link |
of computational reducibility,
link |
which by the way are an inevitable consequence
link |
of computational irreducibility,
link |
that there have to be these pockets
link |
scattered around of computational reducibility
link |
to be able to find those particular cases
link |
where you can jump ahead.
link |
I mean, one thing sort of a little bit
link |
of a parable type thing that I think is fun to tell.
link |
If you look at ancient Babylon,
link |
they were trying to predict three kinds of things.
link |
They tried to predict where the planets would be,
link |
what the weather would be like,
link |
and who would win or lose a certain battle.
link |
And they had no idea which of these things
link |
would be more predictable than the other.
link |
And it turns out where the planets are
link |
is a piece of computational reducibility
link |
that 300 years ago or so we pretty much cracked.
link |
I mean, it's been technically difficult
link |
to get all the details right,
link |
but it's basically, we got that.
link |
Who's gonna win or lose the battle?
link |
No, we didn't crack that one.
link |
That one, that one, right.
link |
Game theorists are trying.
link |
Yes. And then the weather.
link |
It's kind of halfway on that one.
link |
Yeah, I think we're doing okay on that one.
link |
Long term climate, different story.
link |
But the weather, we're much closer on that.
link |
But do you think eventually we'll figure out the weather?
link |
So do you think eventually most think
link |
we'll figure out the local pockets in everything,
link |
essentially the local pockets of reducibility?
link |
No, I think that it's an interesting question,
link |
but I think that there is an infinite collection
link |
of these local pockets.
link |
We'll never run out of local pockets.
link |
And by the way, those local pockets
link |
are where we build engineering, for example.
link |
That's how we, if we want to have a predictable life,
link |
so to speak, then we have to build
link |
in these sort of pockets of reducibility.
link |
Otherwise, if we were sort of existing
link |
in this kind of irreducible world,
link |
we'd never be able to have definite things
link |
to know what's gonna happen.
link |
I have to say, I think one of the features,
link |
when we look at sort of today from the future, so to speak,
link |
I suspect one of the things where people will say
link |
I can't believe they didn't see that
link |
is stuff to do with the following kind of thing.
link |
So if we describe, oh, I don't know,
link |
something like heat, for instance,
link |
we say, oh, the air in here, it's this temperature,
link |
this pressure, that's as much as we can say.
link |
Otherwise, just a bunch of random molecules bouncing around.
link |
People will say, I just can't believe they didn't realize
link |
that there was all this detail
link |
and how all these molecules were bouncing around
link |
and they could make use of that.
link |
And actually, I realized there's a thing
link |
I realized last week, actually,
link |
was a thing that people say, one of the scenarios
link |
for the very long term history of our universe
link |
is a so called heat death of the universe,
link |
where basically everything just becomes
link |
thermodynamically boring.
link |
Everything's just this big kind of gas
link |
and thermal equilibrium.
link |
People say, that's a really bad outcome.
link |
But actually, it's not a really bad outcome.
link |
It's an outcome where there's all this computation going on
link |
and all those individual gas molecules
link |
are all bouncing around in very complicated ways
link |
doing this very elaborate computation.
link |
It just happens to be a computation that right now,
link |
we haven't found ways to understand.
link |
We haven't found ways, our brains haven't,
link |
and our mathematics and our science and so on,
link |
haven't found ways to tell an interesting story about that.
link |
It just looks boring to us.
link |
So you're saying there's a hopeful view
link |
of the heat death, quote unquote, of the universe
link |
where there's actual beautiful complexity going on.
link |
Similar to the kind of complexity we think of
link |
that creates rich experience in human life and life on Earth.
link |
So those little molecules interacting complex ways,
link |
that could be intelligence in that, there could be.
link |
I mean, this is what you learn from this principle.
link |
Wow, that's a hopeful message.
link |
I mean, this is what you kind of learn
link |
from this principle of computational equivalence.
link |
You learn it's both a message of sort of hope
link |
and a message of kind of, you know,
link |
you're not as special as you think you are, so to speak.
link |
I mean, because, you know, we imagine that
link |
with sort of all the things we do with human intelligence
link |
and all that kind of thing,
link |
and all of the stuff we've constructed in science,
link |
it's like, we're very special.
link |
But actually it turns out, well, no, we're not.
link |
We're just doing computations
link |
like things in nature do computations,
link |
like those gas molecules do computations,
link |
like the weather does computations.
link |
The only thing about the computations that we do
link |
that's really special is that we understand
link |
what they are, so to speak.
link |
In other words, we have a, you know,
link |
to us they're special because kind of,
link |
they're connected to our purposes,
link |
our ways of thinking about things and so on.
link |
And that's some, but so.
link |
That's very human centric.
link |
That's, we're just attached to this kind of thing.
link |
So let's talk a little bit of physics.
link |
Maybe let's ask the biggest question.
link |
What is a theory of everything in general?
link |
What does that mean?
link |
Yeah, so I mean, the question is,
link |
can we kind of reduce what has been physics
link |
as a something where we have to sort of pick away and say,
link |
do we roughly know how the world works
link |
to something where we have a complete formal theory
link |
where we say, if we were to run this program
link |
for long enough, we would reproduce everything,
link |
you know, down to the fact that we're having
link |
this conversation at this moment,
link |
et cetera, et cetera, et cetera.
link |
Any physical phenomena, any phenomena in this world?
link |
Any phenomenon in the universe.
link |
But the, you know, because of computational irreducibility,
link |
it's not, you know, that's not something where you say,
link |
okay, you've got the fundamental theory of everything.
link |
Then, you know, tell me whether, you know,
link |
lions are gonna eat tigers or something.
link |
You know, that's a, no, you have to run this thing
link |
for, you know, 10 to the 500 steps or something
link |
to know something like that, okay?
link |
So at some moment, potentially, you say,
link |
this is a rule and run this rule enough times
link |
and you will get the whole universe, right?
link |
That's what it means to kind of have
link |
a fundamental theory of physics as far as I'm concerned
link |
is you've got this rule.
link |
It's potentially quite simple.
link |
We don't know for sure it's simple,
link |
but we have various reasons to believe it might be simple.
link |
And then you say, okay, I'm showing you this rule.
link |
You just run it only 10 to the 500 times
link |
and you'll get everything.
link |
In other words, you've kind of reduced the problem
link |
of physics to a problem of mathematics, so to speak.
link |
It's like, it's as if, you know, you'd like,
link |
you generate the digits of pi.
link |
There's a definite procedure.
link |
You just generate them and it'd be the same thing
link |
if you have a fundamental theory of physics
link |
of the kind that I'm imagining, you know,
link |
you get this rule and you just run it out
link |
and you get everything that happens in the universe.
link |
So a theory of everything is a mathematical framework
link |
within which you can explain everything that happens
link |
in the universe, it's kind of in a unified way.
link |
It's not, there's a bunch of disparate modules of,
link |
does it feel like if you create a rule
link |
and we'll talk about the Wolfram physics model,
link |
which is fascinating, but if you have a simple set
link |
of rules with a data structure, like a hypergraph,
link |
does that feel like a satisfying theory of everything?
link |
Because then you really run up against the irreducibility,
link |
computational irreducibility.
link |
Right, so that's a really interesting question.
link |
So I, you know, what I thought was gonna happen
link |
is I thought we, you know, I thought we had a pretty good,
link |
I had a pretty good idea for what the structure
link |
of this sort of theory that sort of underneath space
link |
and time and so on might be like.
link |
And I thought, gosh, you know, in my lifetime,
link |
so to speak, we might be able to figure out what happens
link |
in the first 10 to the minus 100 seconds of the universe.
link |
And that would be cool, but it's pretty far away
link |
from anything that we can see today.
link |
And it will be hard to test whether that's right
link |
and so on and so on and so on.
link |
To my huge surprise, although it should have been obvious
link |
and it's embarrassing that it wasn't obvious to me,
link |
but to my huge surprise,
link |
we managed to get unbelievably much further than that.
link |
And basically what happened is that it turns out
link |
that even though there's this kind of bed
link |
of computational irreducibility,
link |
that sort of these, all these simple rules run into,
link |
there are certain pieces of computational reducibility
link |
that quite generically occur
link |
for large classes of these rules.
link |
And, and this is the really exciting thing
link |
as far as I'm concerned,
link |
the big pieces of computational reducibility
link |
are basically the pillars of 20th century physics.
link |
That's the amazing thing,
link |
that general relativity and quantum field theory
link |
is sort of the pillars of 20th century physics
link |
turn out to be precisely the stuff you can say.
link |
There's a lot you can't say,
link |
there's a lot that's kind of at this irreducible level
link |
where you kind of don't know what's going to happen,
link |
you have to run it, you know,
link |
you can't run it within our universe,
link |
et cetera, et cetera, et cetera, et cetera.
link |
But the thing is there are things you can say
link |
and the things you can say turn out to be very beautifully
link |
exactly the structure that was found
link |
in 20th century physics,
link |
namely general relativity and quantum mechanics.
link |
And general relativity and quantum mechanics
link |
are these pockets of reducibility that we think of as,
link |
that 20th century physics
link |
is essentially pockets of reducibility.
link |
And then it is incredibly surprising
link |
that any kind of model that's generative
link |
from simple rules would have such pockets.
link |
Yeah, well, I think what's surprising
link |
is we didn't know where those things came from.
link |
It's like general relativity,
link |
it's a very nice mathematically elegant theory.
link |
You know, quantum mechanics, why is it true?
link |
What we realized is that from this,
link |
that these theories are generic
link |
to a huge class of systems
link |
that have these particular
link |
very unstructured underlying rules.
link |
And that's the thing that is sort of remarkable
link |
and that's the thing to me
link |
that's just, it's really beautiful.
link |
I mean, it's, and the thing that's even more beautiful
link |
is that it turns out that, you know,
link |
people have been struggling for a long time.
link |
You know, how does general relativity theory of gravity
link |
relate to quantum mechanics?
link |
They seem to have all kinds of incompatibilities.
link |
It turns out what we realized is
link |
at some level they are the same theory.
link |
And that's just, it's just great as far as I'm concerned.
link |
So maybe like taking a little step back
link |
from your perspective, not from the low,
link |
not from the beautiful hypergraph,
link |
well, from physics model perspective,
link |
but from the perspective of 20th century physics,
link |
what is general relativity?
link |
What is quantum mechanics?
link |
How do you think about these two theories
link |
from the context of the theory of everything?
link |
Like just even definition.
link |
Yeah, yeah, yeah, right.
link |
So I mean, you know, a little bit of history of physics,
link |
So, I mean the, you know, okay,
link |
very, very quick history of this, right?
link |
So, I mean, you know, physics, you know,
link |
in ancient Greek times, people basically said,
link |
we can just figure out how the world works.
link |
As you know, we're philosophers,
link |
we're gonna figure out how the world works.
link |
You know, some philosophers thought there were atoms.
link |
Some philosophers thought there were,
link |
you know, continuous flows of things.
link |
People had different ideas about how the world works.
link |
And they tried to just say,
link |
we're gonna construct this idea of how the world works.
link |
They didn't really have sort of notions
link |
of doing experiments and so on quite the same way
link |
as developed later.
link |
So that was sort of an early tradition
link |
for thinking about sort of models of the world.
link |
Then by the time of 1600s, time of Galileo and then Newton,
link |
sort of the big idea there was, you know,
link |
title of Newton's book, you know, Principia Mathematica,
link |
mathematical principles of natural philosophy.
link |
We can use mathematics to understand natural philosophy,
link |
to understand things about the way the world works.
link |
And so that then led to this kind of idea that, you know,
link |
we can write down a mathematical equation
link |
and have that represent how the world works.
link |
So Newton's one of his most famous ones
link |
is his universal law of gravity,
link |
inverse square law of gravity
link |
that allowed him to compute all sorts of features
link |
of the planets and so on.
link |
Although some of them he got wrong
link |
and it took another hundred years
link |
for people to actually be able to do the math
link |
to the level that was needed.
link |
But so that had been this sort of tradition
link |
was we write down these mathematical equations.
link |
We don't really know where these equations come from.
link |
We write them down.
link |
Then we figure out, we work out the consequences
link |
and we say, yes, that agrees with what we actually observe
link |
in astronomy or something like this.
link |
So that tradition continued.
link |
And then the first of these two
link |
sort of great 20th century innovations was,
link |
well, the history is actually a little bit more complicated,
link |
but let's say that there were two,
link |
quantum mechanics and general relativity.
link |
Quantum mechanics kind of 1900
link |
was kind of the very early stuff done by Planck
link |
that led to the idea of photons, particles of light.
link |
But let's take general relativity first.
link |
One feature of the story is that special relativity
link |
thing Einstein invented in 1905
link |
was something which surprisingly
link |
was a kind of logically invented theory.
link |
It was not a theory where it was something where
link |
given these ideas that were sort of axiomatically
link |
thought to be true about the world,
link |
it followed that such and such a thing would be the case.
link |
It was a little bit different
link |
from the kind of methodological structure
link |
of some existing theories in the more recent times,
link |
where it's just been, we write down an equation
link |
and we find out that it works.
link |
So what happened there.
link |
So there's some reasoning about the light.
link |
The basic idea was the speed of light
link |
appears to be constant.
link |
Even if you're traveling very fast,
link |
you shine a flashlight, the light will come out.
link |
Even if you're going at half the speed of light,
link |
the light doesn't come out of your flashlight
link |
at one and a half times the speed of light.
link |
It's still just the speed of light.
link |
And to make that work,
link |
you have to change your view of how space and time work
link |
to be able to account for the fact
link |
that when you're going faster,
link |
it appears that length is foreshortened
link |
and time is dilated and things like this.
link |
And that's special relativity.
link |
That's special relativity.
link |
So then Einstein went on with sort of
link |
vaguely similar kinds of thinking.
link |
In 1915, invented general relativity,
link |
which is the theory of gravity.
link |
And the basic point of general relativity
link |
is it's a theory that says,
link |
when there is mass in space, space is curved.
link |
And what does that mean?
link |
Usually you think of what's the shortest distance
link |
between two points.
link |
Like ordinarily on a plane in space, it's a straight line.
link |
Photons, light goes in straight lines.
link |
Well, then the question is,
link |
is if you have a curved surface,
link |
a straight line is no longer straight.
link |
On the surface of the earth,
link |
the shortest distance between two points is a great circle.
link |
So, you know, Einstein's observation was
link |
maybe the physical structure of space
link |
is such that space is curved.
link |
So the shortest distance between two points,
link |
the path, the straight line in quotes,
link |
won't be straight anymore.
link |
And in particular, if a photon is, you know,
link |
traveling near the sun or something,
link |
or if a particle is going,
link |
something is traveling near the sun,
link |
maybe the shortest path will be one
link |
that is something which looks curved to us
link |
because it seems curved to us
link |
because space has been deformed by the presence of mass
link |
associated with that massive object.
link |
So the kind of the idea there is,
link |
think of the structure of space
link |
as being a dynamical changing kind of thing.
link |
But then what Einstein did
link |
was he wrote down these differential equations
link |
that basically represented the curvature of space
link |
and its response to the presence of mass and energy.
link |
And that ultimately is connected to the force of gravity,
link |
which is one of the forces that seems to,
link |
based on its strength,
link |
operate on a different scale than some of the other forces.
link |
So it operates in a scale that's very large.
link |
What happens there is just this curvature of space,
link |
which causes, you know, the paths of objects to be deflected.
link |
That's what gravity does.
link |
It causes the paths of objects to be deflected.
link |
And this is an explanation for gravity, so to speak.
link |
And the surprise is that from 1915 until today,
link |
everything that we've measured about gravity
link |
precisely agrees with general relativity.
link |
And that, you know, it wasn't clear black holes
link |
were sort of a predict,
link |
well, actually the expansion of the universe
link |
was an early potential prediction,
link |
although Einstein tried to sort of patch up his equations
link |
to make it not cause the universe to expand,
link |
because it was kind of so obvious
link |
the universe wasn't expanding.
link |
And, you know, it turns out it was expanding
link |
and he should have just trusted the equations.
link |
And that's a lesson for those of us
link |
interested in making fundamental theories of physics
link |
is you should trust your theory and not try and patch it
link |
because of something that you think might be the case
link |
that might turn out not to be the case.
link |
Even if the theory says something crazy is happening.
link |
Like the universe is expanding.
link |
Like the universe is expanding, right, which is,
link |
but, you know, then it took until the 1940s,
link |
probably even really until the 1960s,
link |
until people understood that black holes
link |
were a consequence of general relativity and so on.
link |
But that's, you know, the big surprise has been
link |
that so far this theory of gravity has perfectly agreed
link |
with, you know, these collisions of black holes
link |
seen by their gravitational waves, you know,
link |
it all just works.
link |
So that's been kind of one pillar of the story of physics
link |
it's mathematically complicated to work out
link |
the consequences of general relativity,
link |
but it's not, there's no, I mean,
link |
and some things are kind of squiggly and complicated.
link |
Like people believe, you know, energy is conserved.
link |
Okay, well, energy conservation doesn't really work
link |
in general activity in the same way as it ordinarily does.
link |
And it's all a big mathematical story
link |
of how you actually nail down something that is definitive
link |
that you can talk about it and not specific to the,
link |
you know, reference frames you're operating in
link |
and so on and so on and so on.
link |
But fundamentally, general relativity is a straight shot
link |
in the sense that you have this theory,
link |
you work out its consequences.
link |
And that theory is useful in terms of basic science
link |
and trying to understand the way black holes work,
link |
the way the creation of galaxies work,
link |
sort of all of these kinds of cosmological things,
link |
understanding what happened, like you said, at the Big Bang.
link |
Yeah. Like all those kinds of,
link |
well, no, not at the Big Bang actually, right?
link |
Well, features of the expansion of the universe, yes.
link |
I mean, and there are lots of details
link |
where we don't quite know how it's working, you know,
link |
is there, you know, where's the dark matter,
link |
is there dark energy, you know, et cetera, et cetera, et cetera.
link |
But fundamentally, the, you know,
link |
the testable features of general relativity,
link |
it all works very beautifully.
link |
And it's in a sense, it is mathematically sophisticated,
link |
but it is not conceptually hard to understand in some sense.
link |
Okay. So that's general relativity.
link |
And what's its friendly neighbor, like you said,
link |
there's two theories, quantum mechanics.
link |
Right. So quantum mechanics,
link |
the sort of the way that that originated was,
link |
one question was, is the world continuous or is it discrete?
link |
You know, in ancient Greek times,
link |
people have been debating this.
link |
People debated it, you know, throughout history.
link |
Is light made of waves?
link |
Is it continuous? Is it discrete?
link |
Is it made of particles, corpuscles, whatever.
link |
You know, what had become clear in the 1800s is that atoms,
link |
that, you know, materials are made of discrete atoms.
link |
You know, when you take some water,
link |
the water is not a continuous fluid,
link |
even though it seems like a continuous fluid
link |
to us at our scale.
link |
But if you say, let's look at it,
link |
smaller and smaller and smaller and smaller scale,
link |
eventually you get down to these, you know,
link |
these molecules and then atoms.
link |
It's made of discrete things.
link |
The question is sort of how important is this discreteness?
link |
Just what's discrete, what's not discrete?
link |
Is energy discrete?
link |
Is, you know, what's discrete, what's not?
link |
Does it have mass?
link |
Those kinds of questions.
link |
Yeah, yeah, right.
link |
Well, there's a question, I mean, for example,
link |
is mass discrete is an interesting question,
link |
which is now something we can address.
link |
But, you know, what happened in the coming up to the 1920s,
link |
there was this kind of mathematical theory developed
link |
that could explain certain kinds of discreteness
link |
in particularly in features of atoms and so on.
link |
And, you know, what developed was this mathematical theory
link |
that was the theory of quantum mechanics,
link |
theory of wave functions, Schrodinger's equation,
link |
That's a mathematical theory that allows you to calculate
link |
lots of features of the microscopic world,
link |
lots of things about how atoms work,
link |
et cetera, et cetera, et cetera.
link |
Now, the calculations all work just great.
link |
The question of what does it really mean
link |
is a complicated question.
link |
Now, I mean, to just explain a little bit historically,
link |
the, you know, the early calculations of things like atoms
link |
worked great in 1920s, 1930s and so on.
link |
There was always a problem.
link |
There were, in quantum field theory,
link |
which is a theory of, in quantum mechanics,
link |
you're dealing with a certain number of electrons
link |
and you fix the number of electrons.
link |
You say, I'm dealing with a two electron thing.
link |
In quantum field theory,
link |
you allow for particles being created and destroyed.
link |
So you can emit a photon that didn't exist before.
link |
You can absorb a photon, things like that.
link |
That's a more complicated,
link |
mathematically complicated theory.
link |
And it had all kinds of mathematical issues
link |
and all kinds of infinities that cropped up.
link |
And it was finally figured out more or less
link |
how to get rid of those.
link |
But there were only certain ways of doing the calculations
link |
and those didn't work for atomic nuclei among other things.
link |
And that led to a lot of development up until the 1960s
link |
of alternative ideas for how one could understand
link |
what was happening in atomic nuclei, et cetera,
link |
et cetera, et cetera.
link |
End result, in the end,
link |
the kind of most quotes obvious mathematical structure
link |
of quantum field theory seems to work.
link |
Although it's mathematically difficult to deal with,
link |
but you can calculate all kinds of things.
link |
You can calculate to a dozen decimal places,
link |
certain things, you can measure them.
link |
It's all beautiful.
link |
The underlying fabric is the model
link |
of that particular theory is fields.
link |
Like you keep saying fields.
link |
Those are quantum fields.
link |
Those are different from classical fields.
link |
A field is something like you say,
link |
like you say the temperature field in this room.
link |
It's like there is a value of temperature
link |
at every point around the room.
link |
That's some, or you can say the wind field
link |
would be the vector direction of the wind at every point.
link |
Yes, and that's a classical field.
link |
The quantum field is a much more
link |
mathematically elaborate kind of thing.
link |
And I should explain that one of the pictures
link |
of quantum mechanics that's really important is,
link |
in classical physics, one believes
link |
that sort of definite things happen in the world.
link |
You pick up a ball, you throw it,
link |
the ball goes in a definite trajectory
link |
that has certain equations of motion.
link |
It goes in a parabola, whatever else.
link |
In quantum mechanics, the picture is
link |
definite things don't happen.
link |
Instead, sort of what happens is this whole
link |
sort of structure of all many different paths being followed
link |
and we can calculate certain aspects of what happens,
link |
certain probabilities of different outcomes and so on.
link |
And you say, well, what really happened?
link |
What's really going on?
link |
What's the sort of, what's the underlying,
link |
what's the underlying story?
link |
How do we turn this mathematical theory
link |
that we can calculate things with
link |
into something that we can really understand
link |
and have a narrative about?
link |
And that's been really, really hard for quantum mechanics.
link |
My friend, Dick Feynman, always used to say,
link |
nobody understands quantum mechanics,
link |
even though he'd made his whole career
link |
out of calculating things about quantum mechanics.
link |
And so it's a little bit.
link |
Nevertheless, it's what the quantum field theory is very,
link |
very accurate at predicting a lot of the physical phenomena.
link |
But there are things about it, it has certain,
link |
when we apply it, the standard model of particle physics,
link |
for example, we, which we apply to calculate
link |
all kinds of things that works really well.
link |
And you say, well, it has certain parameters.
link |
It has a whole bunch of parameters actually.
link |
You say, why is the, why does the muon particle exist?
link |
Why is it 206 times the mass of the electron?
link |
We don't know, no idea.
link |
But so the standard model of physics is one of the models
link |
that's very accurate for describing
link |
three of the fundamental forces of physics.
link |
And it's looking at the world of the very small.
link |
And then there's back to the neighbor of gravity,
link |
of general relativity.
link |
So, and then in the context of a theory of everything,
link |
what's traditionally the task of the unification
link |
of these theories?
link |
And why is it hard?
link |
The issue is you try to use the methods
link |
of quantum field theory to talk about gravity
link |
and it doesn't work.
link |
Just like there are photons of light.
link |
So there are gravitons,
link |
which are sort of the particles of gravity.
link |
And when you try and compute sort of the properties
link |
of the particles of gravity,
link |
the kind of mathematical tricks that get used
link |
in working things out in quantum field theory don't work.
link |
And that's, so that's been a sort of fundamental issue.
link |
And when you think about black holes,
link |
which are a place where sort of the structure of space
link |
is, you know, has sort of rapid variation
link |
and you get kind of quantum effects mixed in
link |
with effects from general relativity,
link |
things get very complicated
link |
and there are apparent paradoxes and things like that.
link |
And people have, you know,
link |
there've been a bunch of mathematical developments
link |
in physics over the last, I don't know, 30 years or so,
link |
which have kind of picked away at those kinds of issues
link |
and got hints about how things might work.
link |
But it hasn't been, you know,
link |
and the other thing to realize is,
link |
as far as physics is concerned,
link |
it's just like here's general relativity,
link |
here's quantum field theory, you know, be happy.
link |
Yeah, so do you think there's a quantization of gravity,
link |
so quantum gravity, what do you think of efforts
link |
that people have tried to, yeah,
link |
what do you think in general of the efforts
link |
of the physics community to try to unify these laws?
link |
So I think what's, it's interesting.
link |
I mean, I would have said something very different
link |
before what's happened with our physics project.
link |
I mean, you know, the remarkable thing is
link |
what we've been able to do is to make
link |
from this very simple, structurally simple,
link |
underlying set of ideas,
link |
we've been able to build this, you know,
link |
very elaborate structure that's both very abstract
link |
and very sort of mathematically rich.
link |
And the big surprise, as far as I'm concerned,
link |
is that it touches many of the ideas that people have had.
link |
So in other words, things like string theory and so on,
link |
twister theory, it's like the, you know,
link |
we might've thought, I had thought we're out on a prong,
link |
we're building something that's computational,
link |
it's completely different from what other people have done.
link |
But actually it seems like what we've done
link |
is to provide essentially the machine code that, you know,
link |
these things are various features
link |
of domain specific languages, so to speak,
link |
that talk about various aspects of this machine code.
link |
And I think this is something that to me is very exciting
link |
because it allows one both for us to provide
link |
sort of a new foundation for what's been thought about there
link |
and for all the work that's been done in those areas
link |
to give us, you know, more momentum
link |
to be able to figure out what's going on.
link |
Now, you know, people have sort of hoped,
link |
oh, we're just gonna be able to get, you know,
link |
string theory to just answer everything.
link |
That hasn't worked out.
link |
And I think we now kind of can see a little bit about
link |
just sort of how far away certain kinds of things are
link |
from being able to explain things.
link |
Some things, one of the big surprises to me,
link |
actually I literally just got a message
link |
about one aspect of this is the, you know,
link |
it's turning out to be easier.
link |
I mean, this project has been so much easier
link |
than I could ever imagine it would be.
link |
That is, I thought we would be, you know,
link |
just about able to understand
link |
the first 10 to the minus 100 seconds of the universe.
link |
And, you know, it would be a hundred years
link |
before we get much further than that.
link |
It's just turned out, it actually wasn't that hard.
link |
I mean, we're not finished, but, you know.
link |
So you're seeing echoes of all the disparate theories
link |
of physics in this framework.
link |
I mean, it's a very interesting, you know,
link |
sort of history of science like phenomenon.
link |
I mean, the best analogy that I can see
link |
is what happened with the early days
link |
of computability and computation theory.
link |
You know, Turing machines were invented in 1936.
link |
People sort of understand computation
link |
in terms of Turing machines,
link |
but actually there had been preexisting theories
link |
of computation, combinators, general recursive functions,
link |
Lambda calculus, things like this.
link |
But people hadn't, those hadn't been concrete enough
link |
that people could really wrap their arms around them
link |
and understand what was going on.
link |
And I think what we're gonna see in this case
link |
is that a bunch of these mathematical theories,
link |
including some very,
link |
I mean, one of the things that's really interesting
link |
is one of the most abstract things
link |
that's come out of sort of mathematics,
link |
higher category theory, things about infinity group voids,
link |
things like this, which to me always just seemed
link |
like they were floating off into the stratosphere,
link |
ionosphere of mathematics, turn out to be things
link |
which our sort of theory anchors down
link |
to something fairly definite and says are super relevant
link |
to the way that we can understand how physics works.
link |
By the way, I just threw a hat on.
link |
You've said that with this metaphor analogy
link |
that the theory of everything is a big mountain
link |
and you have a sense that however far we are up the mountain,
link |
that the Wolfram physics model view of the universe
link |
is at least the right mountain.
link |
We're the right mountain, yes, without question.
link |
Which aspect of it is the right mountain?
link |
So for example, I mean, so there's so many aspects
link |
to just the way of the Wolfram physics project,
link |
the way it approaches the world that's clean, crisp,
link |
and unique and powerful, so there's a discreet nature to it,
link |
there's a hypergraph, there's a computational nature,
link |
there's a generative aspect, you start from nothing,
link |
you generate everything, do you think the actual model
link |
is actually a really good one,
link |
or do you think this general principle
link |
from simplicity generating complexity is the right,
link |
like what aspect of the mountain is the correct?
link |
Yeah, right, I think that the kind of the meta idea
link |
about using simple computational systems to do things,
link |
that's the ultimate big paradigm
link |
that is sort of super important.
link |
The details of the particular model are very nice and clean
link |
and allow one to actually understand what's going on.
link |
They are not unique, and in fact, we know that.
link |
We know that there's a very, very, very, very,
link |
there's a large number of different ways
link |
to describe essentially the same thing.
link |
I mean, I can describe things in terms of hypergraphs,
link |
I can describe them in terms of higher category theory,
link |
I can describe them in a bunch of different ways.
link |
They are in some sense all the same thing,
link |
but our sort of story about what's going on
link |
and the kind of cultural mathematical resonances
link |
are a bit different.
link |
And I think it's perhaps worth sort of saying a little bit
link |
about kind of the foundational ideas
link |
of these models and things.
link |
Great, so can you maybe, can we like rewind?
link |
We've talked about it a little bit,
link |
but can you say like what the central idea is
link |
of the Wolfram Physics Project?
link |
Right, so the question is we're interested
link |
in finding sort of simple computational rule
link |
that describes our whole universe.
link |
Can we just pause on that?
link |
It's just so beautiful, that's such a beautiful idea
link |
that we can generate our universe
link |
from a data structure, a simple structure,
link |
simple set of rules, and we can generate our entire universe.
link |
Yes, that's the idea. That's awe inspiring.
link |
Right, but so the question is how do you actualize that?
link |
What might this rule be like?
link |
And so one thing you quickly realize is
link |
if you're gonna pack everything about our universe
link |
into this tiny rule, not much that we are familiar with
link |
in our universe will be obvious in that rule.
link |
So you don't get to fit all these parameters of the universe,
link |
all these features of, you know, this is how space works,
link |
this is how time works, et cetera, et cetera, et cetera.
link |
You don't get to fit that all in.
link |
It all has to be sort of packed in to this thing,
link |
something much smaller, much more basic,
link |
much lower level machine code, so to speak, than that.
link |
And all the stuff that we're familiar with
link |
has to kind of emerge from the operation.
link |
So the rule in itself,
link |
because of the computational reducibility,
link |
is not gonna tell you the story.
link |
It's not gonna give you the answer to,
link |
it's not gonna let you predict
link |
what you're gonna have for lunch tomorrow,
link |
and it's not going to let you predict
link |
basically anything about your life, about the universe.
link |
Right, and you're not going to be able to see in that rule,
link |
oh, there's the three
link |
for the number of dimensions of space and so on.
link |
That's not gonna be there.
link |
Spacetime is not going to be obviously.
link |
Right, so the question is then,
link |
what is the universe made of?
link |
That's a basic question.
link |
And we've had some assumptions
link |
about what the universe is made of
link |
for the last few thousand years
link |
that I think in some cases just turn out not to be right.
link |
And the most important assumption
link |
is that space is a continuous thing.
link |
That is that you can, if you say,
link |
let's pick a point in space.
link |
We're gonna do geometry.
link |
We're gonna pick a point.
link |
We can pick a point absolutely anywhere in space.
link |
Precise numbers we can specify of where that point is.
link |
In fact, Euclid who kind of wrote down
link |
the original kind of axiomatization of geometry
link |
back in 300 BC or so,
link |
his very first definition, he says,
link |
a point is that which has no part.
link |
A point is this indivisible infinitesimal thing.
link |
Okay, so we might've said that about material objects.
link |
We might've said that about water, for example.
link |
We might've said water is a continuous thing
link |
that we can just pick any point we want in some water,
link |
but actually we know it isn't true.
link |
We know that water is made of molecules that are discrete.
link |
And so the question, one fundamental question
link |
is what is space made of?
link |
And so one of the things that's sort of a starting point
link |
for what I've done is to think of space as a discrete thing,
link |
to think of there being sort of atoms of space
link |
just as there are atoms of material things,
link |
although very different kinds of atoms.
link |
And by the way, I mean, this idea,
link |
you know, there were ancient Greek philosophers
link |
who had this idea.
link |
There were, you know, Einstein actually thought
link |
this is probably how things would work out.
link |
I mean, he said, you know, repeatedly he thought
link |
that's the way it would work out.
link |
We don't have the mathematical tools in our time,
link |
which was 1940s, 1950s and so on to explore this.
link |
Like the way he thought,
link |
you mean that there is something very, very small
link |
and discrete that's underlying space.
link |
And that means that, so, you know, the mathematical theory,
link |
mathematical theories in physics assume that space
link |
can be described just as a continuous thing.
link |
You can just pick coordinates
link |
and the coordinates can have any values.
link |
And that's how you define space.
link |
Space is this just sort of background sort of theater
link |
on which the universe operates.
link |
But can we draw a distinction between space
link |
as a thing that could be described by three values,
link |
coordinates, and how you're,
link |
are you using the word space more generally when you say?
link |
No, I'm just talking about space
link |
as in what we experience in the universe.
link |
So that you think this 3D aspect of it is fundamental.
link |
No, I don't think that 3D is fundamental at all, actually.
link |
I think that the thing that has been assumed
link |
is that space is this continuous thing
link |
where you can just describe it by,
link |
let's say three numbers, for instance.
link |
But most important thing about that
link |
is that you can describe it by precise numbers
link |
because you can pick any point in space
link |
and you can talk about motions,
link |
any infinitesimal motion in space.
link |
And that's what continuous means.
link |
That's what continuous means.
link |
That's what, you know, Newton invented calculus
link |
to describe these kind of continuous small variations
link |
That was, that's kind of a fundamental idea
link |
from Euclid on that's been a fundamental idea about space.
link |
Is that right or wrong?
link |
It's right at the level of our experience most of the time.
link |
It's not right at the level of the machine code,
link |
Yeah, of the simulation.
link |
They're the very lowest level of the fabric of the universe,
link |
at least under the Wolfram physics model
link |
is your senses is discrete.
link |
So now what does that mean?
link |
So it means what is space then?
link |
So in models, the basic idea is you say
link |
there are these sort of atoms of space.
link |
They're these points that represent,
link |
you know, represent places in space,
link |
but they're just discrete points.
link |
And the only thing we know about them
link |
is how they're connected to each other.
link |
We don't know where they are.
link |
They don't have coordinates.
link |
We don't get to say this is a position, such and such.
link |
It's just, here's a big bag of points.
link |
Like in our universe,
link |
there might be 10 to the 100 of these points.
link |
And all we know is this point is connected
link |
to this other point.
link |
So it's like, you know,
link |
all we have is the friend network, so to speak.
link |
We don't have, you know, people's, you know,
link |
physical addresses.
link |
All we have is the friend network of these points.
link |
The underlying nature of reality is kind of like a Facebook.
link |
We don't know their location, but we have the friends.
link |
Yeah, yeah, right.
link |
We know which point is connected to which other points.
link |
And that's all we know.
link |
And so you might say, well,
link |
how on earth can you get something
link |
which is like our experience of, you know,
link |
what seems like continuous space?
link |
Well, the answer is,
link |
by the time you have 10 to the 100 of these things,
link |
those connections can work in such a way
link |
that on a large scale,
link |
it will seem to be like continuous space
link |
in let's say three dimensions
link |
or some other number of dimensions
link |
or 2.6 dimensions or whatever else.
link |
Because they're much, much, much, much larger.
link |
So like the number of relationships here we're talking about
link |
is just a humongous amount.
link |
So the kind of thing you're talking about
link |
is very, very, very small relative
link |
to our experience of daily life.
link |
Right, so I mean, you know,
link |
we don't know exactly the size,
link |
but maybe 10 to the minus,
link |
maybe around 10 to the minus 100 meters.
link |
So, you know, the size of, to give a comparison,
link |
the size of a proton is 10 to the minus 15 meters.
link |
And so this is something incredibly tiny compared to that.
link |
And the idea that from that would emerge
link |
the experience of continuous space is mind blowing.
link |
Well, what's your intuition why that's possible?
link |
Like, first of all, I mean, we'll get into it,
link |
but I don't know if we will
link |
through the medium of conversation,
link |
but the construct of hypergraphs is just beautiful.
link |
Cellular automata are beautiful.
link |
We'll talk about it.
link |
But this thing about, you know,
link |
continuity arising from discrete systems
link |
is in today's world is actually not so surprising.
link |
I mean, you know, your average computer screen, right?
link |
Every computer screen is made of discrete pixels.
link |
Yet we have the, you know,
link |
we have the idea that we're seeing
link |
these continuous pictures.
link |
I mean, it's, you know,
link |
the fact that on a large scale,
link |
continuity can arise from lots of discrete elements.
link |
This is at some level unsurprising now.
link |
Wait, wait, wait, wait, wait, wait.
link |
But the pixels have a very definitive structure
link |
of neighbors on a computer screen.
link |
There's no concept of spatial,
link |
of space inherent in the underlying fabric of reality.
link |
Right, right, right.
link |
So the point is that, but there are cases where there are.
link |
So for example, let's just imagine you have a square grid.
link |
Okay, and at every point on the grid,
link |
you have one of these atoms of space
link |
and it's connected to four other,
link |
four other atoms of space on the, you know,
link |
Northeast, Southwest corners, right?
link |
There you have something where if you zoom out from that,
link |
it's like a computer screen.
link |
Yeah, so the relationship creates the spatial,
link |
like the relationship creates a constraint,
link |
which then in an emergent sense creates a like,
link |
yeah, like basically a spatial coordinate for that thing.
link |
Even though the individual point doesn't have a space.
link |
Even though the individual point doesn't know anything,
link |
it just knows what its neighbors are.
link |
On a large scale, it can be described by saying,
link |
oh, it looks like it's a, you know,
link |
this grid is zoomed out grid.
link |
You can say, well, you can describe these different points
link |
by saying they have certain positions,
link |
coordinates, et cetera.
link |
Now, in the sort of real setup,
link |
it's more complicated than that.
link |
It isn't just a square grid or something.
link |
It's something much more dynamic and complicated,
link |
which we'll talk about.
link |
But so, you know, the first idea,
link |
the first key idea is, you know,
link |
what's the universe made of?
link |
It's made of atoms of space basically
link |
with these connections between them.
link |
What kind of connections do they have?
link |
Well, so the simplest kind of thing you might say is,
link |
we've got something like a graph
link |
where every atom of space,
link |
where we have these edges that go between,
link |
these connections that go between atoms of space.
link |
We're not saying how long these edges are.
link |
We're just saying there is a connection
link |
from this place, from this atom to this atom.
link |
Just a quick pause,
link |
because there's a lot of very people that listen to this.
link |
Just to clarify, because I did a poll actually,
link |
what do you think a graph is a long time ago?
link |
And it's kind of funny how few people
link |
know the term graph outside of computer science.
link |
Let's call it a network.
link |
I think that's it.
link |
Let's call it a network is better.
link |
So, but every time, I like the word graph though.
link |
So let's define, let's just say that a graph
link |
will use terms nodes and edges maybe.
link |
And it's just the nodes represent some abstract entity
link |
and then the edges represent relationships
link |
between those entities.
link |
So that's what a graph says.
link |
Sorry, so there you go.
link |
So that's the basic structure.
link |
That is the simplest case of a basic structure.
link |
Actually, it tends to be better to think about hypergraphs.
link |
So a hypergraph is just, instead of saying
link |
there are connections between pairs of things,
link |
we say there are connections between any number of things.
link |
So there might be ternary edges.
link |
So instead of just having two points
link |
are connected by an edge,
link |
you say three points are all associated with a hyperedge,
link |
are all connected by a hyperedge.
link |
That's just, at some level, that's a detail.
link |
It's a detail that happens to make the, for me,
link |
sort of in the history of this project,
link |
the realization that you could do things that way
link |
broke out of certain kinds of arbitrariness
link |
that I felt that there was in the model
link |
before I had seen how this worked.
link |
I mean, a hypergraph can be mapped to a graph.
link |
It's just a convenient representation.
link |
Mathematical speaking.
link |
That's correct. That's correct.
link |
But so then, so, okay, so the first question,
link |
the first idea of these models of ours is
link |
space is made of these connected sort of atoms of space.
link |
The next idea is space is all there is.
link |
There's nothing except for this space.
link |
So in traditional ideas in physics,
link |
people have said there's space, it's kind of a background.
link |
And then there's matter, all these particles, electrons,
link |
all these other things, which exist in space, right?
link |
But in this model, one of the key ideas is
link |
there's nothing except space.
link |
So in other words, everything that exists in the universe
link |
is a feature of this hypergraph.
link |
So how can that possibly be?
link |
Well, the way that works is
link |
that there are certain structures in this hypergraph
link |
where you say that little twisty knotted thing,
link |
we don't know exactly how this works yet,
link |
but we have sort of idea about how it works mathematically.
link |
This sort of twisted knotted thing,
link |
that's the core of an electron.
link |
This thing over there that has this different form,
link |
that's something else.
link |
So the different peculiarities of the structure
link |
of this graph are the very things
link |
that we think of as the particles inside the space,
link |
but in fact, it's just a property of the space.
link |
Mind blowing, first of all, that it's mind blowing,
link |
and we'll probably talk in its simplicity and beauty.
link |
Yes, I think it's very beautiful.
link |
I mean, this is, I'm...
link |
But okay, but that's space,
link |
and then there's another concept
link |
we didn't really kind of mention,
link |
but you think it of computation as a transformation.
link |
Let's talk about time in a second.
link |
Let's just, I mean, on the subject of space,
link |
there's this question of kind of what,
link |
there's this idea, there is this hypergraph,
link |
it represents space,
link |
and it represents everything that's in space.
link |
The features of that hypergraph,
link |
you can say certain features in this part we do know,
link |
certain features of the hypergraph
link |
represent the presence of energy, for example,
link |
or the presence of mass or momentum,
link |
and we know what the features of the hypergraph
link |
that represent those things are,
link |
but it's all just the same hypergraph.
link |
So one thing you might ask is,
link |
you know, if you just look at this hypergraph and you say,
link |
and we're gonna talk about sort of what the hypergraph does,
link |
but if you say, you know,
link |
how much of what's going on in this hypergraph
link |
is things we know and care about,
link |
like particles and atoms and electrons
link |
and all this kind of thing,
link |
and how much is just the background of space?
link |
So it turns out, so far as in one rough estimate of this,
link |
everything that we care about in the universe
link |
is only one part in 10 to the 120
link |
of what's actually going on.
link |
The vast majority of what's happening
link |
is purely things that maintain the structure of space.
link |
That, in other words, that the things that are
link |
the features of space that are the things
link |
that we consider notable,
link |
like the presence of particles and so on,
link |
that's a tiny little piece of froth
link |
on the top of all this activity
link |
that mostly is just intended to,
link |
you know, mostly, I can't say intended,
link |
there's no intention here,
link |
that just maintains the structure of space.
link |
Let me load that in.
link |
It just makes me feel so good as a human being.
link |
To be the froth on the one in a 10 to the 120
link |
or something of, well.
link |
And also just humbling how,
link |
in this mathematical framework,
link |
how much work needs to be done
link |
on the infrastructure of our universe.
link |
Right, to maintain the infrastructure of our universe
link |
We are merely writing a little tiny things
link |
on top of that infrastructure.
link |
But you were just starting to talk a little bit about,
link |
we talked about space,
link |
that represents all the stuff that's in the universe.
link |
The question is, what does that stuff do?
link |
And for that, we have to start talking about time
link |
and what is time and so on.
link |
And, you know, one of the basic idea of this model
link |
is time is the progression of computation.
link |
So in other words, we have a structure of space
link |
and there is a rule that says
link |
how that structure of space will change.
link |
And it's the application,
link |
the repeated application of that rule
link |
that defines the progress of time.
link |
And what does the rule look like
link |
in the space of hypergraphs?
link |
Right, so what the rule says is something like,
link |
if you have a little tiny piece of hypergraph
link |
that looks like this,
link |
then it will be transformed into a piece of hypergraph
link |
that looks like this.
link |
So that's all it says.
link |
It says you pick up these elements of space
link |
and you can think of these edges,
link |
these hyper edges as being relations
link |
between elements in space.
link |
You might pick up these two relations
link |
between elements in space.
link |
And we're not saying where those elements are
link |
but every time there's a certain arrangement
link |
of elements in space,
link |
then arrangement in the sense of the way they're connected,
link |
then we transform it into some other arrangement.
link |
So there's a little tiny pattern
link |
and you transform it into another little pattern.
link |
And then because of this,
link |
I mean, again, it's kind of similar to cellular automata
link |
in that like on paper, the rule looks like super simple.
link |
It's like, yeah, okay.
link |
Yeah, right, from this, the universe can be born.
link |
But like once you start applying it,
link |
beautiful structure starts being,
link |
potentially can be created.
link |
And what you're doing is you're applying that rule
link |
to different parts,
link |
like anytime you match it within the hypergraph.
link |
And then one of the like incredibly beautiful
link |
and interesting things to think about
link |
is the order in which you apply that rule,
link |
because that pattern appears all over the place.
link |
Right, so this is a big complicated thing,
link |
very hard to wrap one's brain around, okay?
link |
So you say the rule is every time you see this little pattern
link |
transform it in this way.
link |
But yet, as you look around the space
link |
that represents the universe,
link |
there may be zillions of places
link |
where that little pattern occurs.
link |
So what it says is just do this,
link |
apply this rule wherever you feel like.
link |
And what is extremely non trivial is,
link |
well, okay, so this is happening sort of
link |
in computer science terms, sort of asynchronously,
link |
you're just doing it wherever you feel like doing it.
link |
And the only constraint is
link |
that if you're going to apply the rule somewhere,
link |
the things to which you apply the rule,
link |
the little elements to which you apply the rule,
link |
if they have to be,
link |
okay, well, you can think of each application of the rule
link |
as being kind of an event that happens in the universe.
link |
And the input to an event has to be ready
link |
for the event to occur.
link |
That is, if one event occurred,
link |
if one transformation occurred,
link |
and it produced a particular atom of space,
link |
then that atom of space has to already exist
link |
before another transformation that's going to apply
link |
to that atom of space can occur.
link |
So that's like the prerequisite for the event.
link |
That's right, that's right.
link |
So that defines a kind of,
link |
this sort of set of causal relationships between events.
link |
It says, this event has to have happened before this event.
link |
But that's not a very limiting constraint.
link |
And what's interesting...
link |
You still get the zillion,
link |
that's a technical term, options.
link |
But, okay, so this is where things get a little bit more
link |
But they're mind blowing, so...
link |
Right, but so what happens is,
link |
so the first thing you might say is,
link |
you know, let's...
link |
Well, okay, so this question about the freedom
link |
of which event you do when.
link |
Well, let me sort of state an answer and then explain it.
link |
Okay, the validity of special relativity
link |
is a consequence of the fact that in some sense,
link |
it doesn't matter in what order you do
link |
these underlying things, so long as they respect
link |
this kind of set of causal relationships.
link |
And that's the part that's in a certain sense
link |
is a really important one,
link |
but the fact that it sometimes doesn't matter,
link |
I don't know what to...
link |
That's another, like, beautiful thing.
link |
Well, okay, so there's this idea
link |
of what I call causal invariance.
link |
Causal invariance, exactly.
link |
Really, really powerful idea.
link |
Right, it's a powerful idea,
link |
which has actually arisen in different forms
link |
many times in the history of mathematics,
link |
mathematical logic, even computer science,
link |
has many different names.
link |
I mean, our particular version of it
link |
is a little bit tighter than other versions,
link |
but it's basically the same idea.
link |
Here's how to think about that idea.
link |
So imagine that...
link |
Well, let's talk about it in terms of math for a second.
link |
Let's say you're doing algebra and you're told,
link |
you know, multiply out this series of polynomials
link |
that are multiplied together, okay?
link |
You say, well, which order should I do that in?
link |
Say, well, do I multiply the third one by the fourth one
link |
and then do it by the first one?
link |
Or do I do the fifth one by the sixth one and then do that?
link |
Well, it turns out it doesn't matter.
link |
You can multiply them out in any order,
link |
you'll always get the same answer.
link |
That's a property...
link |
If you think about kind of making a kind of network
link |
that represents in what order you do things,
link |
you'll get different orders
link |
for different ways of multiplying things out,
link |
but you'll always get the same answer.
link |
Same thing if you...
link |
Let's say you're sorting.
link |
You've got a bunch of A's and B's.
link |
They're in random, some random order,
link |
you know, BAA, BBBAA, whatever.
link |
And you have a little rule that says,
link |
every time you see BA, flip it around to AB, okay?
link |
Eventually you apply that rule enough times,
link |
you'll have sorted the string
link |
so that it's all the A's first and then all the B's.
link |
Again, there are many different orders
link |
in which you can do that to many different sort of places
link |
where you can apply that update.
link |
In the end, you'll always get the string sorted the same way.
link |
I know with sorting the string, it sounds obvious.
link |
That's to me surprising
link |
that there is in complicated systems,
link |
obviously with a string,
link |
but in a hypergraph that the application of the rule,
link |
asynchronous rule can lead to the same results sometimes.
link |
Yes, yes, that is not obvious.
link |
And it was something that, you know,
link |
I sort of discovered that idea for these kinds of systems
link |
and back in the 1990s.
link |
And for various reasons, I was not satisfied
link |
by how sort of fragile finding that particular property was.
link |
And let me just make another point,
link |
which is that it turns out that even if the underlying rule
link |
does not have this property of causal invariance,
link |
it can turn out that every observation
link |
made by observers of the rule can,
link |
they can impose what amounts to causal invariance
link |
We can explain that.
link |
It's a little bit more complicated.
link |
I mean, technically that has to do with this idea
link |
of completions, which is something that comes up
link |
in term rewriting systems,
link |
automated theorem proving systems and so on.
link |
But let's ignore that for a second.
link |
We can come to that later.
link |
But is it useful to talk about observation?
link |
So there's some concept of causal invariance
link |
as you apply these rules in an asynchronous way,
link |
you can think of those transformations as events.
link |
So there's this hypergraph that represents space
link |
and all of these events happening in the space
link |
and the graph grows in interesting complicated ways.
link |
And eventually the froth arises of what we experience
link |
as human existence.
link |
That's some version of the picture,
link |
but let's explain a little bit more.
link |
What's a little more detail like?
link |
Well, so one thing that is sort of surprising
link |
in this theory is one of the sort of achievements
link |
of 20th century physics was kind of bringing
link |
space and time together.
link |
That was, you know, special relativity.
link |
People talk about space time, this sort of unified thing
link |
where space and time kind of a mixed
link |
and there's a nice mathematical formalism
link |
that in which, you know, space and time sort of appear
link |
as part of the space time continuum,
link |
the space time, you know, four vectors and things like this.
link |
You know, we talk about time as the fourth dimension
link |
and all these kinds of things.
link |
It's, you know, and it seems like the theory of relativity
link |
sort of says space and time are fundamentally
link |
the same kind of thing.
link |
So one of the things that took a while to understand
link |
in this approach of mine is that in my kind of approach,
link |
space and time are really not fundamentally
link |
the same kind of thing.
link |
Space is the extension of this hypergraph.
link |
Time is the kind of progress of this inexorable computation
link |
of these rules getting applied to the hypergraph.
link |
So it's, they seem like very different kinds of things.
link |
And so that at first seems like
link |
how can that possibly be right?
link |
How can that possibly be Lorentz invariant?
link |
That's the term for things being, you know,
link |
following the rules of special relativity.
link |
Well, it turns out that when you have causal invariants
link |
that, and let's see, we can, it's worth explaining
link |
a little bit how this works.
link |
It's a little bit elaborate,
link |
but the basic point is that even though space and time
link |
sort of come from very different places,
link |
it turns out that the rules of sort of space time
link |
that special relativity talks about come out of this model
link |
when you're looking at large enough systems.
link |
So a way to think about this, you know,
link |
in terms of when you're looking at large enough systems,
link |
the part of that story is when you look at some fluid
link |
like water, for example, there are equations
link |
that govern the flow of water.
link |
Those equations are things that apply on a large scale.
link |
If you look at the individual molecules,
link |
they don't know anything about those equations.
link |
It's just the sort of the large scale effect
link |
of those molecules turns out to follow those equations.
link |
And it's the same kind of thing happening in our models.
link |
I know this might be a small point,
link |
but it might be a very big one.
link |
We've been talking about space and time
link |
at the lowest level of the model, which is space.
link |
The hypergraph time is the evolution of this hypergraph.
link |
But there's also space time that we think about
link |
and general relativity for your special relativity.
link |
Like how do you go from the lowest source code
link |
of space and time as we're talking about
link |
to the more traditional terminology of space and time?
link |
So the key thing is this thing we call the causal graph.
link |
So the causal graph is the graph
link |
of causal relationships between events.
link |
So every one of these little updating events,
link |
every one of these little transformations
link |
of the hypergraph happens somewhere in the hypergraph,
link |
happens at some stage in the computation.
link |
That event has a causal relationship to other events
link |
in the sense that if another event needs as its input,
link |
the output from the first event,
link |
there will be a causal relationship
link |
of the future event will depend on the past event.
link |
So you can say it has a causal connection.
link |
And so you can make this graph
link |
of causal relationships between events.
link |
That graph of causal relationships,
link |
causal invariance implies that that graph is unique.
link |
It doesn't matter even though you think,
link |
oh, I'm, let's say we were sorting a string, for example,
link |
I did that particular transposition of characters
link |
at this time, then I did that one, then I did this one.
link |
Turns out if you look at the network of connections
link |
between those updating events, that network is the same.
link |
It's the, if you were to, the structure.
link |
So in other words, if you were to draw that,
link |
if you were to put that network on a picture
link |
of where you're doing all the updating,
link |
the places where you put the nodes of the network
link |
will be different, but the way the nodes are connected
link |
will always be the same.
link |
So, but the causal graph is, I don't know,
link |
it's kind of an observation, it's not enforced,
link |
it's just emergent from a set of events.
link |
It's a feature of, okay, so what it is is.
link |
The characteristic, I guess, of the way events happen.
link |
Right, it's an event can't happen
link |
until its input is ready.
link |
And so that creates this network of causal relationships.
link |
And that's the causal graph.
link |
And the thing that the next thing to realize is,
link |
okay, we, when you're going to observe
link |
what happens in the universe,
link |
you have to sort of make sense of this causal graph.
link |
So, and you are an observer who yourself
link |
is part of this causal graph.
link |
And so that means, so let me give you an example
link |
of how that works.
link |
So imagine we have a really weird theory of physics
link |
of the world where it says this updating process,
link |
there's only gonna be one update at every moment in time.
link |
And there's just gonna be like a Turing machine.
link |
It has a little head that runs around
link |
and just is always just updating one thing at a time.
link |
So you say, I have a theory of physics
link |
and the theory of physics says,
link |
there's just this one little place where things get updated.
link |
You say, that's completely crazy because,
link |
it's plainly obvious that things are being updated
link |
sort of at the same time.
link |
Async obviously, yeah, at the same time, yeah.
link |
But the fact is that the thing is that if I'm talking to you
link |
and you seem to be being updated as I'm being updated,
link |
but if there's just this one little head
link |
that's running around updating things,
link |
I will not know whether you've been updated or not
link |
until I'm updated.
link |
So in other words, draw this causal graph
link |
of the causal relationship between the updatings in you
link |
and the updatings in me,
link |
it'll still be the same causal graph,
link |
whether even though the underlying sort of story
link |
of what happens is, oh, there's just this one little thing
link |
and it goes and updates in different places in the universe.
link |
So is that clear or is that a hypothesis?
link |
Is that clear that there's a unique causal graph?
link |
If there's causal invariance, there's unique causal graph.
link |
So it's okay to think of what we're talking about
link |
as a hypergraph and the operations on it
link |
as a kind of touring machine with a single head,
link |
like a single guy running around updating stuff.
link |
Is that safe to intuitively think of it this way?
link |
Let me think about that for a second.
link |
I think there's nothing, it doesn't matter.
link |
I mean, you can say, okay, there is one,
link |
the reason I'm pausing for a second is that I'm wondering,
link |
well, when you say running around,
link |
depends how far it jumps every time it runs.
link |
Yeah, yeah, that's right.
link |
But I mean like one operation at a time.
link |
Yeah, you can think of it as one operation at a time.
link |
It's easier for the human brain to think of it that way
link |
as opposed to simultaneous.
link |
Well, maybe it's not, okay, but the thing is
link |
that's not how we experience the world.
link |
What we experience is we look around,
link |
everything seems to be happening
link |
at successive moments in time everywhere in space.
link |
That is the, and that's partly a feature
link |
of our particular construction.
link |
I mean, that is the speed of light is really fast
link |
compared to, you know, we look around, you know,
link |
I can see maybe a hundred feet away right now.
link |
You know, it's the, my brain does not process very much
link |
in the time it takes light to go a hundred feet.
link |
The brain operates at a scale of hundreds of milliseconds
link |
or something like that, I don't know.
link |
And speed of light is much faster.
link |
Right, you know, light goes,
link |
in a billionth of a second light has gone afoot.
link |
So it goes a billion feet every second.
link |
There's certain moments through this conversation
link |
where I imagine the absurdity of the fact
link |
that there's two descendants of apes modeled by a hypergraph
link |
that are communicating with each other
link |
and experiencing this whole thing
link |
as a real time simultaneous update with,
link |
I'm taking in photons from you right now,
link |
but there's something much, much deeper going on here.
link |
Right, it does have a.
link |
It's paralyzing sometimes to just.
link |
Right, no, I mean, you know, it's a, you know.
link |
As a small little tangent, I just remembered
link |
that we're talking about,
link |
I mean, about the fabric of reality.
link |
Right, so we've got this causal graph
link |
that represents the sort of causal relationships
link |
between all these events in the universe.
link |
That causal graph kind of is a representation of space time,
link |
but our experience of it requires
link |
that we pick reference frames.
link |
This is kind of a key idea.
link |
Einstein had this idea that what that means is
link |
we have to say, what are we going to pick
link |
as being the sort of what we define
link |
as simultaneous moments in time?
link |
So for example, we can say, you know,
link |
how do we set our clocks?
link |
You know, if we've got a spacecraft landing on Mars,
link |
you know, do we say that, you know,
link |
what time is it landing at?
link |
Was it, you know, even though there's a 20 minute
link |
speed of light delay or something, you know,
link |
what time do we say it landed at?
link |
How do we set up sort of time coordinates for the world?
link |
And that turns out to be that there's kind of
link |
this arbitrariness to how we set these reference frames
link |
that defines sort of what counts as simultaneous.
link |
And what is the essence of special relativity
link |
is to think about reference frames going at different speeds
link |
and to think about sort of how they assign,
link |
what counts as space, what counts as time and so on.
link |
That's all a bit technical, but the basic bottom line is
link |
that this causal invariance property,
link |
that means that it's always the same causal graph,
link |
independent of how you slice it with these reference frames,
link |
you'll always sort of see the same physical processes go on.
link |
And that's basically why special relativity works.
link |
So there's something like special relativity,
link |
like everything around space and time
link |
that fits this idea of the causal graph.
link |
Right, well, you know, one way to think about it is
link |
given that you have a basic structure
link |
that just involves updating things in these,
link |
you know, connected updates and looking at
link |
the causal relationships between connected updates,
link |
that's enough when you unravel the consequences of that,
link |
that together with the fact that there are lots
link |
of these things and that you can take a continuum limit
link |
and so on implies special relativity.
link |
And so that, it's kind of not a big deal
link |
because it's kind of a, you know,
link |
it was completely unobvious when you started off
link |
with saying, we've got this graph,
link |
it's being updated in time, et cetera, et cetera, et cetera,
link |
that just looks like nothing to do with special relativity.
link |
And yet you get that.
link |
And what, I mean, then the thing,
link |
I mean, this was stuff that I figured out back in the 1990s.
link |
The next big thing you get is general relativity.
link |
And so in this hypergraph,
link |
the sort of limiting structure,
link |
when you have a very big hypergraph,
link |
you can think of as being just like, you know,
link |
water seems continuous on a large scale.
link |
So this hypergraph seems continuous on a large scale.
link |
One question is, you know,
link |
how many dimensions of space does it correspond to?
link |
So one question you can ask is,
link |
if you've just got a bunch of points
link |
and they're connected together,
link |
how do you deduce what effective dimension of space
link |
that bundle of points corresponds to?
link |
And that's pretty easy to explain.
link |
So basically if you say you've got a point
link |
and you look at how many neighbors does that point have?
link |
Okay, imagine it's on a square grid.
link |
Then it'll have four neighbors.
link |
Go another level out.
link |
How many neighbors do you get then?
link |
What you realize is as you go more and more levels out,
link |
as you go more and more distance on the graph out,
link |
you're capturing something which is essentially a circle
link |
in two dimensions so that, you know,
link |
the number of the area of a circle is pi R squared.
link |
So it's the number of points that you get to
link |
goes up like the distance you've gone squared.
link |
And in general, in D dimensional space,
link |
it's R to the power D.
link |
It's the number of points you get to
link |
if you go R steps on the graph grows like
link |
the number of steps you go to the power of the dimension.
link |
And that's a way that you can estimate
link |
the effective dimension of one of these graphs.
link |
So what does that grow to?
link |
So how does the dimension grow?
link |
There's a, I mean, obviously the visual aspect
link |
of these hypergraphs,
link |
they're often visualized in three dimensions.
link |
So there's a certain kind of structure,
link |
like you said, there's, I mean, a circle, a sphere,
link |
there's a planar aspect to it,
link |
to this graph to where it kind of,
link |
it almost starts creating a surface,
link |
like a complicated surface, but a surface.
link |
So how does that connect to effective dimension?
link |
Okay, so if you can lay out the graph
link |
in such a way that the points in the graph that,
link |
you know, the points that are neighbors on the graph
link |
are neighbors as you lay them out,
link |
and you can do that in two dimensions,
link |
then it's gonna approximate a two dimensional thing.
link |
If you can't do that in two dimensions,
link |
if everything would have to fold over a lot
link |
in two dimensions,
link |
then it's not approximating a two dimensional thing.
link |
Maybe you can lay it out in three dimensions.
link |
Maybe you have to lay it out in five dimensions
link |
to have it be the case
link |
that it sort of smoothly lays out like that.
link |
Well, but okay, so I apologize
link |
for the different tangent questions,
link |
but you know, there's an infinity number of possible rules.
link |
So we have to look for rules
link |
that create the kind of structures
link |
that are reminiscent for,
link |
that have echoes of the different physics theories in them.
link |
So what kind of rules,
link |
is there something simple to be said
link |
about the kind of rules that you have found beautiful,
link |
that you have found powerful?
link |
Right, so I mean, what, you know,
link |
one of the features of computational irreducibility is,
link |
it's very, you can't say in advance,
link |
what's gonna happen with any particular,
link |
you can't say, I'm gonna pick these rules
link |
from this part of rule space, so to speak,
link |
because they're gonna be the ones that are gonna work.
link |
That's, you can make some statements along those lines,
link |
but you can't generally say that.
link |
Now, you know, the state of what we've been able to do
link |
is, you know, different properties of the universe,
link |
like dimensionality, you know, integer dimensionality,
link |
features of other features of quantum mechanics,
link |
At this point, what we've got is,
link |
we've got rules that any one of those features,
link |
we can get a rule that has that feature.
link |
We don't have the sort of, the final,
link |
here's a rule which has all of these features,
link |
we do not have that yet.
link |
So if I were to try to summarize
link |
the Wolfram physics project, which is, you know,
link |
something that's been in your brain for a long time,
link |
but really has just exploded in activity,
link |
you know, only just months ago.
link |
So it's an evolving thing, and next week,
link |
I'll try to publish this conversation
link |
as quickly as possible, because by the time it's published,
link |
already new things will probably have come out.
link |
So if I were to summarize it,
link |
we've talked about the basics of,
link |
there's a hypergraph that represents space,
link |
there is transformations in that hypergraph
link |
that represents time.
link |
The progress of time.
link |
The progress of time, there's a causal graph
link |
that's a characteristic of this,
link |
and the basic process of science,
link |
of, yeah, of science within the Wolfram physics model
link |
is to try different rules and see which properties
link |
of physics that we know of, known physical theories,
link |
are, appear within the graphs that emerge from that rule.
link |
That's what I thought it was going to be.
link |
It turns out we can do a lot better than that.
link |
It turns out that using kind of mathematical ideas,
link |
we can say, and computational ideas,
link |
we can make general statements,
link |
and those general statements turn out to correspond
link |
to things that we know from 20th century physics.
link |
In other words, the idea of you just try a bunch of rules
link |
and see what they do,
link |
that's what I thought we were gonna have to do.
link |
But in fact, we can say, given causal invariance
link |
and computational irreducibility, we can derive,
link |
and this is where it gets really pretty interesting,
link |
we can derive special relativity,
link |
we can derive general relativity,
link |
we can derive quantum mechanics.
link |
And that's where things really start to get exciting,
link |
is, you know, it wasn't at all obvious to me
link |
that even if we were completely correct,
link |
and even if we had, you know, this is the rule,
link |
you know, even if we found the rule,
link |
to be able to say, yes, it corresponds
link |
to things we already know,
link |
I did not expect that to be the case.
link |
So for somebody who is a simple mind
link |
and definitely not a physicist, not even close,
link |
what does derivation mean in this case?
link |
Okay, so let me, this is an interesting question.
link |
Okay, so there's, so one thing...
link |
In the context of computational irreducibility.
link |
Yeah, yeah, right, right.
link |
So what you have to do, let me go back to, again,
link |
the mundane example of fluids and water
link |
and things like that, right?
link |
So you have a bunch of molecules bouncing around.
link |
You can say, just as a piece of mathematics,
link |
I happen to do this from cellular automata
link |
back in the mid 1980s, you can say,
link |
just as a matter of mathematics,
link |
you can say the continuum limit
link |
of these little molecules bouncing around
link |
is the Navier Stokes equations.
link |
That's just a piece of mathematics.
link |
It's not, it doesn't rely on...
link |
You have to make certain assumptions
link |
that you have to say there's enough randomness
link |
in the way the molecules bounce around
link |
that certain statistical averages work,
link |
et cetera, et cetera, et cetera.
link |
Okay, it is a very similar derivation
link |
to derive, for example, the Einstein equations.
link |
Okay, so the way that works, roughly,
link |
the Einstein equations are about curvature of space.
link |
Curvature of space, I talked about sort of
link |
how you can figure out dimension of space.
link |
There's a similar kind of way of figuring out
link |
if you just sort of say, you know,
link |
you're making a larger and larger ball
link |
or larger and larger, if you draw a circle
link |
on the surface of the earth, for example,
link |
you might think the area of a circle is pi r squared,
link |
but on the surface of the earth,
link |
because it's a sphere, it's not flat,
link |
the area of a circle isn't precisely pi r squared.
link |
As the circle gets bigger, the area is slightly smaller
link |
than you would expect from the formula pi r squared
link |
as a little correction term that depends on the ratio
link |
of the size of the circle to the radius of the earth.
link |
Okay, so it's the same basic thing,
link |
allows you to measure from one of these hypergraphs
link |
what is its effective curvature.
link |
So the little piece of mathematics
link |
that explains special general relativity
link |
can map nicely to describe fundamental property
link |
of the hypergraphs, the curvature of the hypergraphs.
link |
So special relativity is about the relationship
link |
General relativity is about curvature
link |
and this space represented by this hypergraph.
link |
So what is the curvature of a hypergraph?
link |
Okay, so first I have to explain,
link |
what we're explaining is,
link |
first thing you have to have is a notion of dimension.
link |
You don't get to talk about curvature of things.
link |
If you say, oh, it's a curved line,
link |
but I don't know what a line is yet.
link |
Yeah, what is the dimension of a hypergraph then?
link |
From where, we've talked about effective dimension, but...
link |
Right, that's what this is about.
link |
What this is about is, you have your hypergraph,
link |
it's got a trillion nodes in it.
link |
What is it roughly like?
link |
Is it roughly like a grid, a two dimensional grid?
link |
Is it roughly like all those nodes are arranged online?
link |
What's it roughly like?
link |
And there's a pretty simple mathematical way
link |
to estimate that by just looking at this thing
link |
I was describing, this sort of the size of a ball
link |
that you construct in the hypergraph.
link |
That's a, you just measure that,
link |
you can just compute it on a computer for a given hypergraph
link |
and you can say, oh, this thing is wiggling around,
link |
but it's roughly corresponds to two or something like that,
link |
or roughly corresponds to 2.6 or whatever.
link |
So that's how you have a notion of dimension
link |
in these hypergraphs.
link |
Curvature is something a little bit beyond that.
link |
If you look at how the size of this ball increases
link |
as you increase its radius,
link |
curvature is a correction
link |
to the size increase associated with dimension.
link |
It's a sort of a second order term
link |
in determining the size.
link |
Just like the area of a circle is roughly pi R squared.
link |
So it goes up like R squared.
link |
The two is because it's in two dimensions,
link |
but when that circle is drawn on a big sphere,
link |
the actual formula is pi R squared times one minus
link |
R squared over A squared and some coefficient.
link |
So in other words, there's a correction to,
link |
and that correction term, that gives you curvature.
link |
And that correction term
link |
is what makes this hypergraph correspond,
link |
have the potential to correspond to curved space.
link |
Now, the next question is, is that curvature,
link |
is the way that curvature works
link |
the way that Einstein's equations for general relativity,
link |
is it the way they say it should work?
link |
And the answer is yes.
link |
And so how does that work?
link |
The calculation of the curvature of this hypergraph
link |
for some set of rules?
link |
No, it doesn't matter what the rules are.
link |
So long as they have causal invariance
link |
and computational irreducibility,
link |
and they lead to finite dimensional space,
link |
noninfinite dimensional space.
link |
Noninfinite dimensional.
link |
It can grow infinitely,
link |
but it can't be infinite dimensional.
link |
So what is a infinitely dimensional hypergraph look like?
link |
So that means, for example, so in a tree,
link |
you start from one root of the tree,
link |
it doubles, doubles again, doubles again, doubles again.
link |
And that means if you ask the question,
link |
starting from a given point,
link |
how many points do you get to?
link |
Remember, in like a circle,
link |
you get to R squared, the two there.
link |
On a tree, you get to, for example, two to the R.
link |
It's exponential dimensional, so to speak,
link |
or infinite dimensional.
link |
Do you have a sense of, in the space of all possible rules,
link |
how many lead to infinitely dimensional hypergraphs?
link |
Is that an important thing to know?
link |
Yes, it's an important thing to know.
link |
I would love to know the answer to that.
link |
But it gets a little bit more complicated
link |
because, for example, it's very possibly the case
link |
that in our physical universe,
link |
that the universe started infinite dimensional.
link |
And it only, as the Big Bang,
link |
it was very likely infinite dimensional.
link |
And as the universe sort of expanded and cooled,
link |
its dimension gradually went down.
link |
And so one of the bizarre possibilities,
link |
which actually there are experiments you can do
link |
to try and look at this,
link |
the universe can have dimension fluctuations.
link |
So in other words,
link |
we think we live in a three dimensional universe,
link |
but actually there may be places
link |
where it's actually 3.01 dimensional,
link |
or where it's 2.99 dimensional.
link |
And it may be that in the very early universe,
link |
it was actually infinite dimensional,
link |
and it's only a late stage phenomenon
link |
that we end up getting three dimensional space.
link |
But from your perspective of the hypergraph,
link |
one of the underlying assumptions you kind of implied,
link |
but you have a sense, a hope set of assumptions
link |
that the rules that underlie our universe,
link |
or the rule that underlies our universe is static.
link |
Is that one of the assumptions
link |
you're currently operating under?
link |
Yes, but there's a footnote to that,
link |
which we should get to,
link |
because it requires a few more steps.
link |
Well, actually then, let's backtrack to the curvature,
link |
because we're talking about as long as it's finite dimensional.
link |
Finite dimensional computational irreducibility
link |
and causal invariance,
link |
then it follows that the large scale structure
link |
will follow Einstein's equations.
link |
And now let me again, qualify that a little bit more,
link |
there's a little bit more complexity to it.
link |
The, okay, so Einstein's equations in their simplest form
link |
apply to the vacuum, no matter, just the vacuum.
link |
And they say, in particular, what they say is,
link |
if you have, so there's this term GD6,
link |
that's a term that means shortest path,
link |
comes from measuring the shortest paths on the Earth.
link |
So you look at a bunch of, a bundle of GD6,
link |
a bunch of shortest paths,
link |
it's like the paths that photons
link |
would take between two points.
link |
Then the statement of Einstein's equations,
link |
it's basically a statement about a certain the,
link |
that as you look at a bundle of GD6,
link |
the structure of space has to be such that,
link |
although the cross sectional area of this bundle may,
link |
although the actual shape of the cross section may change,
link |
the cross sectional area does not.
link |
That's a version, that's the most simple minded version
link |
of R mu nu minus a half R G mu nu equals zero,
link |
which is the more mathematical version
link |
of Einstein's equations.
link |
It's a statement of the thing called the Ritchie tensor
link |
That's Einstein's equations for the vacuum.
link |
Okay, so we get that as a result of this model,
link |
but footnote, big footnote,
link |
because all the matter in the universe
link |
is the stuff we actually care about.
link |
The vacuum is not stuff we care about.
link |
So the question is, how does matter come into this?
link |
And for that, you have to understand what energy is
link |
And one of the things that we realized, you know,
link |
late last year was that there's a very simple interpretation
link |
of energy in these models, okay?
link |
And energy is basically, well, intuitively,
link |
it's the amount of activity in these hypergraphs
link |
and the way that that remains over time.
link |
So a little bit more formally,
link |
you can think about this causal graph
link |
as having these edges that represent causal relationships.
link |
You can think about, oh boy,
link |
there's one more concept that we didn't get to.
link |
It's the notion of space like hypersurfaces.
link |
So this is not as scary as it sounds.
link |
It's a common notion in general activity.
link |
The notion is you are defining what is a possibly,
link |
where in space time might be a particular moment in time.
link |
So in other words, what is a consistent set of places
link |
where you can say, this is happening now, so to speak.
link |
And you make the series of sort of slices
link |
through the space time, through this causal graph
link |
to represent sort of what we consider
link |
to be successive moments in time.
link |
It's somewhat arbitrary because you can deform that
link |
if you're going at a different speed in a special activity,
link |
you tip those things, there are different kinds
link |
of deformations, but only certain deformations
link |
are allowed by the structure of the causal graph.
link |
Anyway, be that as it may, the basic point is
link |
there is a way of figuring out,
link |
you say, what is the energy associated
link |
with what's going on in this hypergraph?
link |
And the answer is there is a precise definition of that.
link |
And it is the formal way to say it is,
link |
it's the flux of causal edges
link |
through space like hypersurfaces.
link |
The slightly less formal way to say it,
link |
it's basically the amount of activity.
link |
See, the reason it gets tricky is you might say
link |
it's the amount of activity per unit volume
link |
in this hypergraph, but you haven't defined what volume is.
link |
So it's a little bit, you have to be a little more careful.
link |
But this hypersurface gives some more formalism to that.
link |
Yeah, yeah, it gives a way to connect that.
link |
But intuitive, we should think about as the just activity.
link |
Right, so the amount of activity that kind of remains
link |
in one place in the hypergraph corresponds to energy.
link |
The amount of activity that is kind of where an activity here
link |
affects an activity somewhere else,
link |
corresponds to momentum.
link |
And so one of the things that's kind of cool
link |
is that I'm trying to think about
link |
how to say this intuitively.
link |
The mathematics is easy,
link |
but the intuitive version, I'm not sure.
link |
But basically the way that things sort of stay
link |
in the same place and have activity
link |
is associated with rest mass.
link |
And so one of the things that you get to derive
link |
is E equals MC squared.
link |
That is a consequence of this interpretation of energy
link |
in terms of the way the causal graph works,
link |
which is the whole thing is sort of a consequence
link |
of this whole story about updates and hypergraphs and so on.
link |
So can you linger on that a little bit?
link |
How do we get E equals MC squared?
link |
So where does the mass come from?
link |
I mean, is there an intuitive, it's okay.
link |
First of all, you're pretty deep
link |
in the mathematical explorations of this thing right now.
link |
We're in a very, we're in a flux currently.
link |
So maybe you haven't even had time
link |
to think about intuitive explanations, but.
link |
Yeah, I mean, this one is, look, roughly what's happening,
link |
that derivation is actually rather easy.
link |
And everybody, and I've been saying
link |
we should pay more attention to this derivation
link |
because it's such, you know,
link |
cause people care about this one.
link |
But everybody says, it's just easy.
link |
So there's some concept of energy
link |
that can be intuitively thought of as the activity,
link |
the flux, the level of changes that are occurring
link |
based on the transformations within a certain volume,
link |
however the heck do you find the volume.
link |
Okay, so, and then mass.
link |
Well, mass is associated with kind of the energy
link |
that does not cause you to,
link |
that does not somehow propagate through time.
link |
Yeah, I mean, one of the things that was not obvious
link |
in the usual formulation of special relativity
link |
is that space and time are connected in a certain way.
link |
Energy and momentum are also connected in a certain way.
link |
The fact that the connection of energy to momentum
link |
is analogous to the connection to space
link |
between space and time
link |
is not self evident in ordinary relativity.
link |
It is a consequence of this, of the way this model works.
link |
It's an intrinsic consequence of the way this model works.
link |
And it's all to do with that,
link |
with unraveling that connection
link |
that ends up giving you this relationship
link |
between energy and, well, it's energy, momentum, mass,
link |
they're all connected.
link |
And so like, that's hence the general relativity.
link |
You have a sense that it appears to be baked in
link |
to the fundamental properties
link |
of the way these hypergraphs are evolved.
link |
Well, I didn't yet get to,
link |
so I got as far as special relativity and equals MC squared.
link |
The one last step is, in general relativity,
link |
the final connection is energy and mass
link |
cause curvature in space.
link |
And that's something that when you understand
link |
this interpretation of energy,
link |
and you kind of understand the correspondence
link |
to curvature and hypergraphs,
link |
then you can finally sort of, the big final answer is,
link |
you derive the full version of Einstein's equations
link |
for space, time and matter.
link |
Is that, have you, that last piece with curvature,
link |
have, is that, have you arrived there yet?
link |
Oh yeah, we're there, yes.
link |
And here's the way that we,
link |
here's how we're really, really going to know
link |
we've arrived, okay?
link |
So, you know, we have the mathematical derivation,
link |
it's all fine, but, you know,
link |
mathematical derivations, okay.
link |
So one thing that's sort of a,
link |
you know, we're taking this limit
link |
of what happens when you, the limit,
link |
you have to look at things which are large
link |
compared to the size of an elementary length,
link |
small compared to the whole size of the universe,
link |
large compared to certain kinds of fluctuations,
link |
There's a, there's a, there's a tower
link |
of many, many of these mathematical limits
link |
that have to be taken.
link |
So if you're a pure mathematician saying,
link |
where's the precise proof?
link |
It's like, well, there are all these limits,
link |
we can, you know, we can try each one of them
link |
computationally and we could say, yeah, it really works,
link |
but the formal mathematics is really hard to do.
link |
I mean, for example, in the case of deriving
link |
the equations of fluid dynamics from molecular dynamics,
link |
that derivation has never been done.
link |
There is no rigorous version of that derivation.
link |
So, so that could be.
link |
Because you can't do the limits?
link |
Yeah, because you can't do the limits.
link |
But so the limits allow you to try to describe
link |
something general about the system
link |
and very, very particular kinds of limits that you need
link |
to take with these very.
link |
Right, and the limits will definitely work
link |
the way we think they work.
link |
And we can do all kinds of computer experiments.
link |
It's just a hard derivation.
link |
Yeah, it's just, it's just the mathematical structure
link |
kind of, you know, ends up running right into
link |
computational irreducibility.
link |
And you end up with a bunch of, a bunch of difficulty there.
link |
But here's the way that we're getting really confident
link |
that we know completely what we're talking about,
link |
which is when people study things like black hole mergers,
link |
using Einstein's equations, what do they actually do?
link |
Well, they actually use Mathematica or a whole bunch
link |
to analyze the equations and so on.
link |
But in the end, they do numerical relativity,
link |
which means they take these nice mathematical equations
link |
and they break them down so that they can run them
link |
And they break them down into something
link |
which is actually a discrete approximation
link |
to these equations.
link |
Then they run them on a computer, they get results.
link |
Then you look at the gravitational waves
link |
and you see if they match, okay?
link |
It turns out that our model gives you a direct way
link |
to do numerical relativity.
link |
So in other words, instead of saying,
link |
you start from these continuum equations from Einstein,
link |
you break them down into these discrete things,
link |
you run them on a computer,
link |
you say, we're doing it the other way around.
link |
We're starting from these discrete things
link |
that come from our model.
link |
And we're just running big versions on the computer.
link |
And, you know, what we're saying is,
link |
and this is how things will work.
link |
So the way I'm calling this is proof by compilation,
link |
so to speak, that is, in other words,
link |
you're taking something where, you know,
link |
we've got this description of a black hole system.
link |
And what we're doing is we're showing that the, you know,
link |
what we get by just running our model agrees
link |
with what you would get by doing the computation
link |
from the Einstein equations.
link |
As a small tangent or actually a very big tangent,
link |
but proof by compilation is a beautiful concept.
link |
In a sense, the way of doing physics with this model
link |
is by running it or compiling it.
link |
And have you thought about,
link |
and these things can be very large,
link |
is there a totally new possibilities of computing hardware
link |
and computing software,
link |
which allows you to perform this kind of compilation?
link |
Well, algorithms, software, hardware.
link |
So first comment is these models seem to give one
link |
a lot of intuition about distributed computing,
link |
a lot of different intuition about how to think
link |
about parallel computation.
link |
And that particularly comes from the quantum mechanics
link |
side of things, which we didn't talk about much yet.
link |
But the question of what, you know,
link |
given our current computer hardware,
link |
how can we most efficiently simulate things?
link |
That's actually partly a story of the model itself,
link |
because the model itself has deep parallelism in it.
link |
The ways that we are simulating it,
link |
we're just starting to be able to use that deep parallelism
link |
to be able to be more efficient
link |
in the way that we simulate things.
link |
But in fact, the structure of the model itself
link |
allows us to think about parallel computation
link |
in different ways.
link |
And one of my realizations is that, you know,
link |
so it's very hard to get in your brain
link |
how you deal with parallel computation.
link |
And you're always worrying about, you know,
link |
if multiple things can happen on different computers
link |
at different times, oh, what happens
link |
if this thing happens before that thing?
link |
And we've really got, you know,
link |
we have these race conditions where something can race
link |
to get to the answer before another thing.
link |
And you get all tangled up because you don't know
link |
which thing is gonna come in first.
link |
And usually when you do parallel computing,
link |
there's a big obsession to lock things down
link |
to the point where you've had locks and mutexes
link |
and God knows what else,
link |
where you've arranged it so that there can only be
link |
one sequence of things that can happen.
link |
So you don't have to think about
link |
all the different kinds of things that can happen.
link |
Well, in these models, physics is throwing us into,
link |
forcing us to think about all these possible things
link |
But these models together with what we know from physics
link |
is giving us new ways to think about
link |
all possible things happening,
link |
about all these different things happening in parallel.
link |
And so I'm guessing...
link |
They have built in protection for some of the parallelism.
link |
Well, causal invariance is the built in protection.
link |
Causal invariance is what means that
link |
even though things happen in different orders,
link |
it doesn't matter in the end.
link |
As a person who struggled with concurrent programming
link |
with all the basic concepts of concurrent programming,
link |
that if there could be built up
link |
a strong mathematical framework for causal invariance,
link |
that's so liberating.
link |
And that could be not just liberating,
link |
but really powerful for massively distributed computation.
link |
No, I mean, what's eventual consistency
link |
in distributed databases
link |
is essentially the causal invariance idea.
link |
But have you thought about,
link |
like really large simulations?
link |
Yeah. I mean, I'm also thinking about,
link |
look, the fact is I've spent much of my life
link |
as a language designer, right?
link |
So I can't possibly not think about,
link |
what does this mean for designing languages
link |
for parallel computation?
link |
In fact, another thing that's one of these...
link |
I'm always embarrassed at how long it's taken me
link |
to figure stuff out.
link |
But back in the 1980s,
link |
I worked on trying to make up languages
link |
for parallel computation.
link |
I thought about doing graph rewriting.
link |
I thought about doing these kinds of things,
link |
but I couldn't see how to actually make the connections
link |
to actually do something useful.
link |
I think now physics is kind of showing us
link |
how to make those things useful.
link |
And so my guess is that in time,
link |
we'll be talking about, we do parallel programming.
link |
We'll be talking about programming
link |
in a certain reference frame,
link |
just as we think about thinking about physics
link |
in a certain reference frame.
link |
It's a certain coordination of what's going on.
link |
We say, we're gonna program in this reference frame.
link |
Oh, let's change the reference frame
link |
to this reference frame.
link |
And then our program will seem different
link |
and we'll have a different way to think about it.
link |
But it's still the same program underneath.
link |
So let me ask on this topic,
link |
cause I put out that I'm talking to you.
link |
I got way more questions than I can deal with,
link |
but what pops to mind is a question somebody asked
link |
on Reddit I think is, please ask Dr. Wolfram,
link |
what are the specs of the computer running the universe?
link |
So we're talking about specs of hardware and software
link |
for simulations of a large scale thing.
link |
What about a scale that is comparative
link |
to something that eventually leads
link |
to the two of us talking and about?
link |
Right, right, right.
link |
So actually I did try to estimate that.
link |
And we actually have to go a couple more stages
link |
before we can really get to that answer
link |
because we're talking about this thing.
link |
This is what happens when you build these abstract systems
link |
and you're trying to explain the universe,
link |
they're quite a number of levels deep, so to speak.
link |
You mean conceptually or like literally?
link |
Cause you're talking about small objects
link |
and there's 10 to the 120 something.
link |
It is conceptually deep.
link |
And one of the things that's happening sort of structurally
link |
in this project is, you know, there were ideas,
link |
there's another layer of ideas,
link |
there's another layer of ideas
link |
to get to the different things that correspond to physics.
link |
They're just different layers of ideas.
link |
And they are, you know, it's actually probably,
link |
if anything, getting harder to explain this project
link |
cause I'm realizing that the fraction of way through
link |
that I am so far and explaining this to you is less than,
link |
than, you know, it might be because we know more now,
link |
you know, every week basically we know a little bit more.
link |
Those are just layers on the initial fundamental structure.
link |
Yes, but the layers are, you know,
link |
you might be asking me, you know,
link |
how do we get the difference between fermions and bosons,
link |
the difference between particles
link |
that can be all in the same state
link |
and particles that exclude each other, okay.
link |
Last three days, we've kind of figured that out.
link |
But, and it's very interesting.
link |
And those are some kind of properties at a certain level,
link |
layer of abstraction on the graph.
link |
And there's, but the layers of abstraction are kind of,
link |
they're compounding.
link |
So it's difficult, but...
link |
But the specs nevertheless remain the same.
link |
Okay, the specs underneath.
link |
So I have an estimate.
link |
So the question is, what are the units?
link |
So we've got these different fundamental constants
link |
So one of them is the speed of light, which is the...
link |
So the thing that's always the same
link |
in all these different ways of thinking about the universe
link |
is the notion of time, because time is computation.
link |
And so there's an elementary time,
link |
which is sort of the amount of time that we ascribe
link |
to elapsing in a single computational step.
link |
So that's the elementary time.
link |
So then there's an elementary...
link |
That's a parameter or whatever.
link |
That's a constant.
link |
It's whatever we define it to be,
link |
because I mean, we don't, you know...
link |
I mean, it's all relative, right?
link |
It doesn't matter.
link |
Yes, it doesn't matter what it is,
link |
because we could be, it could be slower.
link |
It's just a number which we use to convert that
link |
to seconds, so to speak,
link |
because we are experiencing things
link |
and we say this amount of time has elapsed, so to speak.
link |
But we're within this thing.
link |
So it doesn't matter, right?
link |
But what does matter is the ratio,
link |
what we can, the ratio of the spatial distance
link |
and this hypergraph to this moment of time.
link |
Again, that's an arbitrary thing,
link |
but we measure that in meters per second, for example,
link |
and that ratio is the speed of light.
link |
So the ratio of the elementary distance
link |
to the elementary time is the speed of light, okay?
link |
And so there's another,
link |
there are two other levels of this, okay?
link |
So there is a thing which we can talk about,
link |
which is the maximum entanglement speed,
link |
which is a thing that happens at another level
link |
in this whole sort of story
link |
of how these things get constructed.
link |
That's a sort of maximum speed in quantum,
link |
in the space of quantum states.
link |
Just as the speed of light
link |
is a maximum speed in physical space,
link |
this is a maximum speed in the space of quantum states.
link |
There's another level which is associated
link |
with what we call ruleal space,
link |
which is another one of these maximum speeds.
link |
We'll get to this.
link |
So these are limitations on the system
link |
that are able to capture the kind of physical universe
link |
The quantum mechanical.
link |
There are inevitable features of having a rule
link |
that has only a finite amount of information in the rule.
link |
So long as you have a rule that only involves
link |
a bounded amount, a limited amount of,
link |
only involving a limited number of elements,
link |
limited number of relations,
link |
it is inevitable that there are these speed constraints.
link |
We knew about the one for speed of light.
link |
We didn't know about the one for maximum entanglement speed,
link |
which is actually something that is possibly measurable,
link |
particularly in black hole systems and things like this.
link |
Anyway, this is long, long story short.
link |
You're asking what the processing specs of the universe,
link |
of the sort of computation of the universe.
link |
There's a question of even what are the units
link |
of some of these measurements, okay?
link |
So the units I'm using are Wolfram language instructions
link |
Because you gotta have some,
link |
what computation are you doing?
link |
There gotta be some kind of frame of reference.
link |
So, because it turns out in the end,
link |
there will be, there's sort of an arbitrariness
link |
in the language that you use to describe the universe.
link |
So in those terms, I think it's like 10 to the 500,
link |
Wolfram language operations per second, I think,
link |
is the, I think it's of that order.
link |
You know, basically.
link |
So that's the scale of the computation.
link |
What about memory?
link |
If there's an interesting thing to say
link |
about storage and memory.
link |
Well, there's a question of how many sort of atoms
link |
of space might there be?
link |
You know, maybe 10 to the 400.
link |
We don't know exactly how to estimate these numbers.
link |
I mean, this is based on some, I would say,
link |
somewhat rickety way of estimating things.
link |
You know, when there start to be able to be experiments done,
link |
if we're lucky, there will be experiments
link |
that can actually nail down some of these numbers.
link |
And because of computation reducibility,
link |
there's not much hope for very efficient compression,
link |
like very efficient representation
link |
of this atom space? Good question.
link |
I mean, there's probably certain things, you know,
link |
the fact that we can deduce anything,
link |
okay, the question is how deep does the reducibility go?
link |
And I keep on being surprised
link |
that it's a lot deeper than I thought.
link |
Okay, and so one of the things is that,
link |
that there's a question of sort of how much
link |
of the whole of physics do we have to be able to get
link |
in order to explain certain kinds of phenomena?
link |
Like for example, if we want to study quantum interference,
link |
do we have to know what an electron is?
link |
Turns out I thought we did, turns out we don't.
link |
I thought to know what energy is,
link |
we would have to know what electrons were.
link |
So you get a lot of really powerful shortcuts.
link |
There's a bunch of sort of bulk information about the world.
link |
The thing that I'm excited about last few days, okay,
link |
is the idea of fermions versus bosons, fundamental idea
link |
that I mean, it's the reason we have matter
link |
that doesn't just self destruct,
link |
is because of the exclusion principle
link |
that means that two electrons can never be
link |
in the same quantum state.
link |
Is it useful for us to maybe first talk
link |
about how quantum mechanics fits
link |
into the Wolfram physics model?
link |
So we talked about general relativity.
link |
Now, what have you found from quantum mechanics
link |
within and outside of the Wolfram physics?
link |
Right, so I mean, the key idea of quantum mechanics
link |
that sort of the typical interpretation
link |
is classical physics says a definite thing happens.
link |
Quantum physics says there's this whole set of paths
link |
of things that might happen.
link |
And we are just observing some overall probability
link |
of how those paths work.
link |
Okay, so when you think about our hypergraphs
link |
and all these little updates that are going on,
link |
there's a very remarkable thing to realize,
link |
which is if you say, well,
link |
which particular sequence of updates should you do?
link |
Say, well, it's not really defined.
link |
You can do any of a whole collection
link |
of possible sequences of updates.
link |
Okay, that set of possible sequences of updates
link |
defines yet another kind of graph
link |
that we call a multiway graph.
link |
And a multiway graph just is a graph
link |
where at every node, there is a choice
link |
of several different possible things that could happen.
link |
So for example, you go this way, you go that way.
link |
Those are two different edges in the multiway graph.
link |
And you're building up the set of possibilities.
link |
So actually, like, for example, I just made the one,
link |
the multiway graph for tic tac toe, okay?
link |
So tic tac toe, you start off with some board
link |
that, you know, is everything is blank,
link |
and then somebody can put down an X somewhere,
link |
an O somewhere, and then there are different possibilities.
link |
At each stage, there are different possibilities.
link |
And so you build up this multiway graph
link |
of all those possibilities.
link |
Now notice that even in tic tac toe,
link |
you have the feature that there can be something
link |
where you have two different things that happen
link |
and then those branches merge
link |
because you end up with the same shape,
link |
you know, the same configuration of the board,
link |
even though you got there in two different ways.
link |
So the thing that's sort of an inevitable feature
link |
of our models is that just like quantum mechanics suggests,
link |
definite things don't happen.
link |
Instead, you get this whole multiway graph
link |
of all these possibilities.
link |
Okay, so then the question is, so, okay,
link |
so that's sort of a picture of what's going on.
link |
Now you say, okay, well, quantum mechanics
link |
has all these features of, you know,
link |
all this mathematical structure and so on.
link |
How do you get that mathematical structure?
link |
Okay, a couple of things to say.
link |
So quantum mechanics is actually, in a sense,
link |
two different theories glued together.
link |
Quantum mechanics is the theory
link |
of how quantum amplitudes work
link |
that more or less give you the probabilities
link |
of things happening.
link |
And it's the theory of quantum measurement,
link |
which is the theory of how we actually
link |
conclude definite things.
link |
Because the mathematics just gives you
link |
these quantum amplitudes, which are more or less
link |
probabilities of things happening,
link |
but yet we actually observe definite things in the world.
link |
Quantum measurement has always been a bit mysterious.
link |
It's always been something where people just say,
link |
well, the mathematics says this,
link |
but then you do a measurement,
link |
and there are philosophical arguments
link |
about what the measurement is.
link |
But it's not something where there's a theory
link |
of the measurement.
link |
Somebody on Reddit also asked,
link |
please ask Stephen to tell his story
link |
of the double slit experiment.
link |
Okay, yeah, I can.
link |
Is that, does that make sense?
link |
Oh yeah, it makes sense.
link |
Absolutely makes sense.
link |
Why, is this like a good way to discuss?
link |
Let me go, let me explain a couple of things first.
link |
So the structure of quantum mechanics
link |
is mathematically quite complicated.
link |
One of the features, let's see,
link |
well, how to describe this.
link |
Okay, so first point is there's this multiway graph
link |
of all these different paths of things
link |
that can happen in the world.
link |
And the important point is that these,
link |
you can have branchings and you can have mergings.
link |
Okay, so this property turns out causal invariance
link |
is the statement that the number of mergings
link |
is equal to the number of branchings.
link |
So in other words, every time there's a branch,
link |
eventually there will also be a merge.
link |
In other words, every time there were two possibilities
link |
for what might've happened, eventually those will merge.
link |
Beautiful concept by the way, but yeah, yeah, yeah.
link |
So that idea, okay, so then, so that's one thing
link |
and that's closely related to the sort of objectivity
link |
in quantum mechanics.
link |
The fact that we believe definite things happen,
link |
it's because although there are all these different paths,
link |
in some sense, because of causal invariance,
link |
they all imply the same thing.
link |
I'm cheating a little bit in saying that,
link |
but that's roughly the essence of what's going on.
link |
Okay, next thing to think about
link |
is you have this multiway graph,
link |
it has all these different possible things
link |
that are happening.
link |
Now we ask, this multiway graph
link |
is sort of evolving with time.
link |
Over time, it's branching, it's merging,
link |
it's doing all these things, okay?
link |
Question we can ask is if we slice it at a particular time,
link |
And that slice represents in a sense,
link |
something to do with the state of the universe
link |
at a particular time.
link |
So in other words, we've got this multiway graph
link |
of all these possibilities,
link |
and then we're asking, okay, we take the slice,
link |
this slice represents, okay,
link |
each of these different paths
link |
corresponds to a different quantum possibility
link |
for what's happening.
link |
When we take the slice, we're saying,
link |
what are the set of quantum possibilities
link |
that exist at a particular time?
link |
And when you say slice, you slice the graph
link |
and then there's a bunch of leaves.
link |
A bunch of leaves.
link |
Those represent the state of things.
link |
Right, but then, okay, so the important thing
link |
that you are quickly picking up on
link |
is that what matters is kind of
link |
how these leaves are related to each other.
link |
So a good way to tell how leaves are related
link |
is just to say on the step before
link |
do they have a common ancestor?
link |
So two leaves might be,
link |
they might have just branched from one thing
link |
or they might be far away,
link |
way far apart in this graph
link |
where to get to a common ancestor,
link |
maybe you have to go all the way back
link |
to the beginning of the graph,
link |
all the way back to the beginning.
link |
So there's some kind of measure of distance.
link |
Right, but what you get is by making the slice,
link |
we call it branchial space, the space of branches.
link |
And in this branchial space,
link |
you have a graph that represents the relationships
link |
between these quantum states in branchial space.
link |
You have this notion of distance in branchial space.
link |
It's connected to quantum entanglement.
link |
Yes, yes, it's basically,
link |
the distance in branchial space
link |
is kind of an entanglement distance.
link |
That's a very nice model.
link |
Right, it is very nice, it's very beautiful.
link |
I mean, it's so clean.
link |
I mean, it's really, and it tells one,
link |
okay, so anyway, so then this branchial space
link |
has this sort of map of the entanglements
link |
between quantum states.
link |
So in physical space, we have,
link |
so you can say, take, let's say the causal graph,
link |
and we can slice that at a particular time,
link |
and then we get this map
link |
of how things are laid out in physical space.
link |
When we do the same kind of thing,
link |
there's a thing called the multiway causal graph,
link |
which is the analog of a causal graph
link |
for the multiway system.
link |
We slice that, we get essentially the relationships
link |
between things, not in physical space,
link |
but in the space of quantum states.
link |
It's like which quantum state
link |
is similar to which other quantum state.
link |
Okay, so now I think next thing to say
link |
is just to mention how quantum measurement works.
link |
So quantum measurement has to do with reference frames
link |
in branchial space.
link |
So, okay, so measurement in physical space,
link |
it matters whether how we assign spatial position
link |
and how we define coordinates in space and time.
link |
And that's how we make measurements in ordinary space.
link |
Are we making a measurement based on us sitting still here?
link |
Are we traveling at half the speed of light
link |
and making measurements that way?
link |
These are different reference frames
link |
in which we're making our measurements.
link |
And the relationship between different events
link |
and different points in space and time
link |
will be different depending on what reference frame we're in.
link |
Okay, so then we have this idea
link |
of quantum observation frames,
link |
which are the analog of reference frames,
link |
but in branchial space.
link |
And so what happens is what we realize
link |
is that a quantum measurement is the observer
link |
is sort of arbitrarily determining this reference frame.
link |
The observer is saying, I'm going to understand the world
link |
by saying that space and time are coordinated this way.
link |
I'm gonna understand the world by saying
link |
that quantum states and time are coordinated in this way.
link |
And essentially what happens is
link |
that the process of quantum measurement
link |
is a process of deciding how you slice up
link |
this multiway system in these quantum observation frames.
link |
So in a sense, the observer, the way the observer enters
link |
is by their choice of these quantum observation frames.
link |
And what happens is that the observer,
link |
because, okay, this is again,
link |
another stack of other concepts, but anyway,
link |
because the observer is computationally bounded,
link |
there is a limit to the type of quantum observation frames
link |
that they can construct.
link |
Interesting, okay, so there's some constraints,
link |
some limit on the choice of observation frames.
link |
Right, and by the way, I just want to mention
link |
that there's a, I mean, it's bizarre,
link |
but there's a hierarchy of these things.
link |
So in thermodynamics,
link |
the fact that we believe entropy increases,
link |
we believe things get more disordered,
link |
is a consequence of the fact
link |
that we can't track each individual molecule.
link |
If we could track every single molecule,
link |
we could run every movie in reverse, so to speak,
link |
and we would not see that things are getting more disordered.
link |
But it's because we are computationally bounded,
link |
we can only look at these big blobs
link |
of what all these molecules collectively do,
link |
that we think that things are,
link |
that we describe it in terms of entropy increasing
link |
And it's the same phenomenon, basically,
link |
and also a consequence of computational irreducibility
link |
that causes us to basically be forced to conclude
link |
that definite things happen in the world,
link |
even though there's this quantum,
link |
this set of all these different quantum processes
link |
that are going on.
link |
So, I mean, I'm skipping a little bit,
link |
but that's a rough picture.
link |
And in the evolution of the Wolfram Physics Project,
link |
where do you feel we stand on some of the puzzles
link |
that are along the way?
link |
See, you're skipping along a bunch of stuff.
link |
It's amazing how much these things are unraveling.
link |
I mean, you know, these things, look,
link |
it used to be the case that I would agree with Dick Feynman,
link |
nobody understands quantum mechanics, including me, okay?
link |
I'm getting to the point where I think
link |
I actually understand quantum mechanics.
link |
My exercise, okay, is can I explain quantum mechanics
link |
for real at the level of kind of middle school
link |
And I'm getting closer, it's getting there.
link |
I'm not quite there, I've tried it a few times,
link |
and I realized that there are things
link |
where I have to start talking about
link |
elaborate mathematical concepts and so on.
link |
But I think, and you've got to realize
link |
that it's not self evident that we can explain
link |
at an intuitively graspable level,
link |
something which, about the way the universe works,
link |
the universe wasn't built for our understanding,
link |
But I think then, okay, so another important idea
link |
is this idea of branchial space, which I mentioned,
link |
this sort of space of quantum states.
link |
It is, okay, so I mentioned Einstein's equations
link |
describing the effect of mass and energy
link |
on trajectories of particles, on GD6.
link |
The curvature of physical space is associated
link |
with the presence of energy,
link |
according to Einstein's equations, okay?
link |
So it turns out that, rather amazingly,
link |
the same thing is true in branchial space.
link |
So it turns out the presence of energy
link |
or more accurately Lagrangian density,
link |
which is a kind of relativistic invariant version of energy,
link |
the presence of that causes essentially deflection of GD6
link |
in this branchial space, okay?
link |
So you might say, so what?
link |
Well, it turns out that the sort of the best formulation
link |
we have of quantum mechanics,
link |
this Feynman path integral,
link |
is a thing that describes quantum processes
link |
in terms of mathematics that can be interpreted as,
link |
well, in quantum mechanics, the big thing
link |
is you get these quantum amplitudes,
link |
which are complex numbers that represent,
link |
when you combine them together,
link |
represent probabilities of things happening.
link |
And so the big story has been,
link |
how do you derive these quantum amplitudes?
link |
And people think these quantum amplitudes,
link |
they have a complex number,
link |
has a real part and an imaginary part.
link |
You can also think of it as a magnitude and a phase.
link |
And people have sort of thought these quantum amplitudes
link |
have magnitude and phase, and you compute those together.
link |
Turns out that the magnitude and the phase
link |
come from completely different places.
link |
The magnitude comes, okay, so how do you compute things
link |
in quantum mechanics?
link |
Roughly, I'm telling you, I'm getting there
link |
to be able to do this at a middle school level,
link |
but I'm not there yet.
link |
Roughly what happens is you're asking,
link |
does this state in quantum mechanics
link |
evolve to this other state in quantum mechanics?
link |
And you can think about that like a particle traveling
link |
or something traveling through physical space,
link |
but instead it's traveling through branchial space.
link |
And so what's happening is, does this quantum state evolve
link |
to this other quantum state?
link |
It's like saying, does this object move
link |
from this place in space to this other place in space?
link |
Okay, now the way that these quantum amplitudes
link |
characterize kind of to what extent the thing
link |
will successfully reach some particular point
link |
in branchial space, just like in physical space,
link |
you could say, oh, it had a certain velocity
link |
and it went in this direction.
link |
In branchial space, there's a similar kind of concept.
link |
Is there a nice way to visualize for me now
link |
mentally branchial space?
link |
It's just, you have this hypergraph,
link |
sorry, you have this multiway graph.
link |
It's this big branching thing, branching and merging thing.
link |
But I mean, like moving through that space,
link |
I'm just trying to understand what that looks like.
link |
You know, that space is probably exponential dimensional,
link |
which makes it again, another can of worms
link |
in understanding what's going on.
link |
That space as in an ordinary space,
link |
this hypergraph, the spatial hypergraph
link |
limits to something which is like a manifold,
link |
like something like three dimensional space.
link |
Almost certainly the multiway graph limits
link |
to a Hilbert space, which is something that,
link |
I mean, it's just a weird exponential dimensional space.
link |
And by the way, you can ask, I mean,
link |
there are much weirder things that go on.
link |
For example, one of the things I've been interested in
link |
is the expansion of the universe in branchial space.
link |
So we know the universe is expanding in physical space,
link |
but the universe is probably also expanding
link |
in branchial space.
link |
So that means the number of quantum states
link |
of the universe is increasing with time.
link |
The diameter of the thing is growing.
link |
Right, so that means that the,
link |
and by the way, this is related
link |
to whether quantum computing can ever work.
link |
Okay, so let me explain why.
link |
So let's talk about, okay, so first of all,
link |
just to finish the thought about quantum amplitudes,
link |
that the incredibly beautiful thing,
link |
but I'm just very excited about this.
link |
The fine path integral is this formula.
link |
It says that the amplitude, the quantum amplitude
link |
is E to the I S over H bar,
link |
where S is the thing called the action.
link |
And it, okay, so that can be thought of
link |
as representing a deflection of the angle
link |
of this path in the multiway graph.
link |
So it's a deflection of a geodesic in the multiway path
link |
that is caused by this thing called the action,
link |
which is essentially associated with energy, okay?
link |
And so this is a deflection of a path in branchial space
link |
that is described by this path integral,
link |
which is the thing that is the mathematical essence
link |
of quantum mechanics.
link |
Turns out that deflection is,
link |
the deflection of geodesics in branchial space
link |
follows the exact same mathematical setup
link |
as the deflection of geodesics in physical space,
link |
except the deflection of geodesics in physical space
link |
is described with Einstein's equations.
link |
The deflection of geodesics in branchial space
link |
is defined by the Feynman path integral,
link |
and they are the same.
link |
In other words, they are mathematically the same.
link |
So that means that general relativity
link |
is a story of essentially motion in physical space.
link |
Quantum mechanics is a story of essentially motion
link |
in branchial space.
link |
And the underlying equation for those two things,
link |
although it's presented differently
link |
because one's interested in different things
link |
in branchial space than physical space,
link |
but the underlying equation is the same.
link |
So in other words, it's just these two theories,
link |
which are those two sort of pillars
link |
of 20th century physics,
link |
which have seemed to be off in different directions,
link |
are actually facets of the exact same theory.
link |
That's exciting to see where that evolves
link |
and exciting that that just is there.
link |
Right, I mean, to me,
link |
look, having spent some part of my early life
link |
working in the context of these theories
link |
of 20th century physics,
link |
it's, they just, they seem so different.
link |
And the fact that they're really the same
link |
is just really amazing.
link |
Actually, you mentioned double slit experiment, okay?
link |
So the double slit experiment
link |
is an interference phenomenon where you say there are,
link |
you can have a photon or an electron,
link |
and you say there are these two slits
link |
that could have gone through either one,
link |
but there is this interference pattern
link |
where there's destructive interference,
link |
where you might've said in classical physics,
link |
oh, well, if there are two slits,
link |
then there's a better chance
link |
that it gets through one or the other of them.
link |
But in quantum mechanics,
link |
there's this phenomenon of destructive interference
link |
that means that even though there are two slits,
link |
two can lead to nothing,
link |
as opposed to two leading to more
link |
than, for example, one slit.
link |
And what happens in this model,
link |
and we've just been understanding this
link |
in the last few weeks, actually,
link |
is that what essentially happens
link |
is that the double slit experiment
link |
is a story of the interface
link |
between branchial space and physical space.
link |
And what's essentially happening
link |
is that the destructive interference
link |
is the result of the two possible paths
link |
associated with photons going through those two slits
link |
winding up at opposite ends of branchial space.
link |
And so that's why there's sort of nothing there
link |
when you look at it,
link |
is because these two different sort of branches
link |
couldn't get merged together
link |
to produce something that you can measure
link |
in physical space.
link |
Is there a lot to be understood about branchial space?
link |
I guess, mathematically speaking.
link |
Yes, it's a very beautiful mathematical thing.
link |
And it's very, I mean, by the way,
link |
this whole theory is just amazingly rich
link |
in terms of the mathematics that it says should exist.
link |
Okay, so for example,
link |
calculus is a story of infinitesimal change
link |
in integer dimensional space,
link |
one dimensional, two dimensional, three dimensional space.
link |
We need a theory of infinitesimal change
link |
in fractional dimensional and dynamic dimensional space.
link |
No such theory exists.
link |
So there's tools of mathematics that are needed here.
link |
And this is a motivation for that actually.
link |
Right, and there are indications
link |
and we can do computer experiments
link |
and we can see how it's gonna come out,
link |
but we need to, the actual mathematics doesn't exist.
link |
And in branchial space, it's actually even worse.
link |
There's even more sort of layers of mathematics that are,
link |
we can see how it works roughly
link |
by doing computer experiments,
link |
but to really understand it,
link |
we need more sort of mathematical sophistication.
link |
So quantum computers.
link |
Okay, so the basic idea of quantum computers,
link |
the promise of quantum computers
link |
is quantum mechanics does things in parallel.
link |
And so you can sort of intrinsically do computations
link |
And somehow that can be much more efficient
link |
than just doing them one after another.
link |
And I actually worked on quantum computing a bit
link |
with Dick Feynman back in 1981, two, three,
link |
that kind of timeframe.
link |
It's a fascinating image.
link |
You and Feynman working on quantum computers.
link |
Well, we tried to work,
link |
the big thing we tried to do was invent a randomness chip
link |
that would generate randomness at a high speed
link |
using quantum mechanics.
link |
And the discovery that that wasn't really possible
link |
was part of the story of,
link |
we never really wrote anything about it.
link |
I think maybe he wrote some stuff,
link |
but we didn't write stuff about what we figured out
link |
about sort of the fact that it really seemed like
link |
the measurement process in quantum mechanics
link |
was a serious damper on what was possible to do
link |
in sort of the possible advantages of quantum mechanics
link |
But anyway, so the sort of the promise of quantum computing
link |
is let's say you're trying to factor an integer.
link |
Well, you can, instead of,
link |
when you factor an integer, you might say,
link |
well, does this factor work?
link |
Does this factor work?
link |
Does this factor work?
link |
In ordinary computing,
link |
it seems like we pretty much just have to try
link |
all these different factors,
link |
kind of one after another.
link |
But in quantum mechanics, you might have the idea,
link |
oh, you can just sort of have the physics,
link |
try all of them in parallel, okay?
link |
And there's this algorithm, Shor's algorithm,
link |
according to the formalism of quantum mechanics,
link |
to do everything in parallel
link |
and to do it much faster than you can on a classical computer.
link |
Okay, the only little footnote is
link |
you have to figure out what the answer is.
link |
You have to measure the result.
link |
So the quantum mechanics internally has figured out
link |
all these different branches,
link |
but then you have to pull all these branches together
link |
to say, and the classical answer is this, okay?
link |
The standard theory of quantum mechanics
link |
does not tell you how to do that.
link |
It tells you how the branching works,
link |
but it doesn't tell you the process
link |
of corralling all these things together.
link |
And that process, which intuitively you can see
link |
is gonna be kind of tricky,
link |
but our model actually does tell you
link |
how that process of pulling things together works.
link |
And the answer seems to be, we're not absolutely sure.
link |
We've only got to two times three so far
link |
which is kind of in this factorization
link |
in quantum computers.
link |
But we can, what seems to be the case
link |
is that the advantage you get from the parallelization
link |
from quantum mechanics is lost
link |
from the amount that you have to spend
link |
pulling together all those parallel threads
link |
to get to a classical answer at the end.
link |
Now, that phenomenon is not unrelated
link |
to various decoherence phenomena
link |
that are seen in practical quantum computers and so on.
link |
I mean, I should say as a very practical point,
link |
I mean, it's like, should people stop bothering
link |
to do quantum computing research?
link |
No, because what they're really doing
link |
is they're trying to use physics
link |
to get to a new level of what's possible in computing.
link |
And that's a completely valid activity.
link |
Whether you can really put, you know,
link |
whether you can say,
link |
oh, you can solve an NP complete problem.
link |
You can reduce exponential time to polynomial time.
link |
You know, we're not sure.
link |
And I'm suspecting the answer is no,
link |
but that's not relevant to the practical speed ups
link |
you can get by using different kinds of technologies,
link |
different kinds of physics to do basic computing.
link |
But you're saying, I mean,
link |
some of the models you're playing with,
link |
the indication is that to get all the sheep back together
link |
and, you know, to corral everything together,
link |
to get the actual solution to the algorithm is...
link |
You lose all the...
link |
You lose all of the...
link |
By the way, I mean, so again, this question,
link |
do we actually know what we're talking about
link |
about quantum computing and so on?
link |
So again, we're doing proof by compilation.
link |
So we have a quantum computing framework
link |
in Wolfram language,
link |
and which is, you know,
link |
a standard quantum computing framework
link |
that represents things in terms of the standard,
link |
you know, formalism of quantum mechanics.
link |
And we have a compiler that simply compiles
link |
the representation of quantum gates into multiway systems.
link |
So, and in fact, the message that I got
link |
was from somebody who's working on the project
link |
who has managed to compile one of the sort of
link |
a core formalism based on category theory
link |
and core quantum formalism into multiway systems.
link |
When you say multiway system, these multiway graphs?
link |
So you're compiling...
link |
Yeah, okay, that's awesome.
link |
And then you can do all kinds of experiments
link |
on that multiway graph.
link |
Right, but the point is that what we're saying is
link |
the thing we've got this representation
link |
of let's say Shor's algorithm
link |
in terms of standard quantum gates.
link |
And it's just a pure matter of sort of computation
link |
to just say that is equivalent.
link |
We will get the same result as running this multiway system.
link |
Can you do complexity analysis on that multiway system?
link |
Well, that's what we've been trying to do, yes.
link |
We're getting there.
link |
We haven't done that yet.
link |
I mean, there's a pretty good indication
link |
of how that's gonna work out.
link |
We've done, as I say, our computer experiments.
link |
We've unimpressively gotten to about two times three
link |
in terms of factorization,
link |
which is kind of about how far people have got
link |
with physical quantum computers as well.
link |
But yes, we will be able to do...
link |
We definitely will be able to do complexity analysis
link |
and we will be able to know.
link |
So the one remaining hope for quantum computing
link |
really, really working at this formal level
link |
of quantum brand exponential stuff being done
link |
in polynomial time and so on.
link |
The one hope, which is very bizarre,
link |
is that you can kind of piggyback
link |
on the expansion of branchial space.
link |
So here's how that might work.
link |
So you think, you know, energy conservation,
link |
standard thing in high school physics,
link |
energy is conserved, right?
link |
But now you imagine, you think about energy
link |
in the context of cosmology
link |
and the context of the whole universe.
link |
It's a much more complicated story.
link |
The expansion of the universe kind of violates
link |
energy conservation.
link |
And so for example, if you imagine you've got two galaxies,
link |
they're receding from each other very quickly.
link |
They've got two big central black holes.
link |
You connect a spring between these two central black holes.
link |
Not easy to do in practice,
link |
but let's imagine you could do it.
link |
Now that spring is being pulled apart.
link |
It's getting more potential energy in the spring
link |
as a result of the expansion of the universe.
link |
So in a sense, you are piggybacking on the expansion
link |
that exists in the universe
link |
and the sort of violation of energy conservation
link |
that's associated with that cosmological expansion
link |
to essentially get energy.
link |
You're essentially building a perpetual motion machine
link |
by using the expansion of the universe.
link |
And that is a physical version of that.
link |
It is conceivable that the same thing can be done
link |
in branchial space to essentially mine the expansion
link |
of the universe in branchial space
link |
as a way to get sort of quantum computing for free,
link |
so to speak, just from the expansion of the universe
link |
in branchial space.
link |
Now, the physical space version is kind of absurd
link |
and involves springs between black holes and so on.
link |
It's conceivable that the branchial space version
link |
and that it's actually something you can reach
link |
with physical things you can build in labs and so on.
link |
We don't know yet.
link |
Okay, so like you were saying,
link |
the branch of space might be expanding
link |
and there might be something that could be exploited.
link |
Right, in the same kind of way
link |
that you can exploit that expansion of the universe
link |
in principle, in physical space.
link |
You just have like a glimmer of hope.
link |
Right, I think that the,
link |
look, I think the real answer is going to be
link |
that for practical purposes,
link |
the official brand that says you can do exponential things
link |
in polynomial time is probably not gonna work.
link |
For people curious to kind of learn more,
link |
so this is more like, it's not middle school,
link |
we're gonna go to elementary school for a second.
link |
Maybe middle school, let's go to middle school.
link |
So if I were to try to maybe write a pamphlet
link |
of like Wolfram physics project for dummies,
link |
AKA for me, or maybe make a video on the basics,
link |
but not just the basics of the physics project,
link |
but the basics plus the most beautiful central ideas.
link |
How would you go about doing that?
link |
Could you help me out a little bit?
link |
Yeah, yeah, I mean, as a really practical matter,
link |
we have this kind of visual summary picture that we made,
link |
which I think is a pretty good,
link |
when I've tried to explain this to people
link |
and it's a pretty good place to start.
link |
As you got this rule, you apply the rule,
link |
you're building up this big hypergraph,
link |
you've got all these possibilities,
link |
you're kind of thinking about that
link |
in terms of quantum mechanics.
link |
I mean, that's a decent place to start.
link |
So basically the things we've talked about,
link |
which is space represented as a hypergraph,
link |
transformation of that space is kind of time.
link |
Structure of that space,
link |
the curvature of that space has gravity.
link |
That can be explained without going anywhere
link |
near quantum mechanics.
link |
I would say that's actually easier to explain
link |
than special relativity.
link |
Oh, so going into general, so go into curvature.
link |
Yeah, I mean, special relativity,
link |
I think it's a little bit elaborate to explain.
link |
And honestly, you only care about it
link |
if you know about special relativity,
link |
if you know how special relativity
link |
is ordinarily derived and so on.
link |
So general relativity is easier.
link |
And then what about quantum?
link |
What's the easiest way to reveal...
link |
I think the basic point is just this.
link |
This fact that there are all these different branches,
link |
that there's this kind of map of how the branches work.
link |
And that, I mean, I think actually the recent things
link |
that we have about the double slit experiment
link |
are pretty good, because you can actually see this.
link |
You can see how the double slit phenomenon arises
link |
from just features of these graphs.
link |
Now, having said that,
link |
there is a little bit of sleight of hand there
link |
because the true story of the way
link |
that double slit thing works
link |
depends on the coordination of branchial space
link |
that, for example, in our internal team,
link |
there is still a vigorous battle going on
link |
about how that works.
link |
And what's becoming clear is...
link |
I mean, what's becoming clear
link |
is that it's mathematically really quite interesting.
link |
I mean, that is that there's a...
link |
It involves essentially putting space filling curves.
link |
You'll basically have a thing
link |
which is naturally two dimensional,
link |
and you're sort of mapping it into one dimension
link |
with a space filling curve.
link |
And it's like, why is it this space filling curve
link |
and another space filling curve?
link |
And that becomes a story about Riemann surfaces and things,
link |
and it's quite elaborate.
link |
But there's a more, a little bit sleight of hand way
link |
of doing it where it's surprisingly direct.
link |
So a question that might be difficult to answer,
link |
but for several levels of people,
link |
could you give me advice on how we can learn more?
link |
Specifically, there is people that are completely outside
link |
and just curious and are captivated
link |
by the beauty of hypergraphs, actually.
link |
So people that just wanna explore, play around with this.
link |
Second level is people from, say, people like me
link |
who somehow got a PhD in computer science,
link |
but are not physicists.
link |
But fundamentally, the work you're doing
link |
is of computational nature.
link |
So it feels very accessible.
link |
So what can a person like that do to learn enough physics
link |
or not to be able to, one, explore the beauty of it,
link |
and two, the final level of contribute something
link |
of a level of even publishable,
link |
like strong, interesting ideas.
link |
So at all those layers, complete beginner,
link |
a CS person, and the CS person that wants to publish.
link |
I mean, I think that, I've written a bunch of stuff,
link |
a person called Jonathan Gorod,
link |
who's been a key person working on this project,
link |
has also written a bunch of stuff.
link |
And some other people started writing things too.
link |
And he's a physicist.
link |
Well, he's, I would say, a mathematical physicist.
link |
Mathematical physicist.
link |
He's pretty mathematically sophisticated.
link |
He regularly outmathematicizes me.
link |
Yeah, strong mathematical physicist.
link |
Yeah, I looked at some of the papers.
link |
Right, but so, I mean,
link |
I wrote this kind of original announcement blog post
link |
about this project, which people seem to have found.
link |
I've been really happy, actually, that people who,
link |
people seem to have grokked key points from that,
link |
much deeper key points, people seem to have grokked
link |
than I thought they would grokk.
link |
And that's a kind of a long blog post
link |
that explains some of the things we talked about,
link |
like the hypergraph and the basic rules.
link |
And I don't, does it, I forget,
link |
it doesn't have any quantum mechanics in here.
link |
But we know a little bit more since that blog post
link |
that probably clarifies,
link |
but that blog post does a pretty decent job.
link |
And, you know, talking about things like, again,
link |
something we didn't mention,
link |
the fact that the uncertainty principle
link |
is a consequence of curvature in branchial space.
link |
How much physics should a person know
link |
to be able to understand the beauty of this framework
link |
and to contribute something novel?
link |
Okay, so I think that those are different questions.
link |
So, I mean, I think that the, why does this work?
link |
Why does this make any sense?
link |
To really know that,
link |
you have to know a fair amount of physics, okay?
link |
And for example, have a decent understanding.
link |
When you say, why does this work?
link |
You're referring to the connection between this model
link |
and general relativity, for example.
link |
You have to understand something about general relativity.
link |
There's also a side of this where just
link |
as the pure mathematical framework is fascinating.
link |
If you throw the physics out completely.
link |
Then it's quite accessible to, I mean, you know,
link |
I wrote this sort of long technical introduction
link |
to the project, which seems to have been very accessible
link |
to people who are, you know, who understand computation
link |
and formal abstract ideas, but are not specialists
link |
in physics or other kinds of things.
link |
I mean, the thing with the physics part of it is,
link |
you know, there's both a way of thinking
link |
and literally a mathematical formalism.
link |
I mean, it's like, you know,
link |
to know that we get the Einstein equations,
link |
to know we get the energy momentum tensor,
link |
you kind of have to know what the energy momentum tensor is.
link |
And that's physics.
link |
I mean, that's kind of graduate level physics basically.
link |
And so that, you know, making that final connection
link |
is requires some depth of physics knowledge.
link |
I mean, that's the unfortunate thing,
link |
the difference in machine learning and physics
link |
in the 21st century.
link |
Is it really out of reach of a year or two worth of study?
link |
No, you could get it in a year or two,
link |
but you can't get it in a month.
link |
So, but it doesn't require necessarily like 15 years.
link |
And in fact, a lot of what has happened with this project
link |
makes a lot of this stuff much more accessible.
link |
There are things where it has been quite difficult
link |
to explain what's going on.
link |
And it requires much more, you know,
link |
having the concreteness of being able to do simulations,
link |
knowing that this thing that you might've thought
link |
was just an analogy is really actually what's going on,
link |
makes one feel much more secure
link |
about just sort of saying, this is how this works.
link |
And I think it will be, you know,
link |
the, I'm hoping the textbooks of the future,
link |
the physics textbooks of the future,
link |
there will be a certain compression.
link |
There will be things that used to be
link |
very much more elaborate because for example,
link |
even doing continuous mathematics
link |
versus this discrete mathematics,
link |
that, you know, to know how things work
link |
in continuous mathematics,
link |
you have to be talking about stuff
link |
and waving your hands about things.
link |
Whereas with discrete, the discrete version,
link |
it's just like, here is a picture.
link |
This is how it works.
link |
And there's no, oh, do we get the limit right?
link |
Did this, you know, did this thing that is of,
link |
you know, zero, you know, measure zero object,
link |
you know, interact with this thing in the right way.
link |
You don't have to have that whole discussion.
link |
It's just like, here's a picture, you know,
link |
this is what it does.
link |
And, you know, you can, then it takes more effort to say,
link |
what does it do in the limit when the picture gets very big?
link |
But you can do experiments
link |
to build up an intuition actually.
link |
And you can get sort of core intuition for what's going on.
link |
Now, in terms of contributing to this, the, you know,
link |
I would say that the study of the computational universe
link |
and how all these programs work
link |
in the computational universe,
link |
there's just an unbelievable amount to do there.
link |
And it is very close to the surface.
link |
That is, you know, high school kids,
link |
you can do experiments.
link |
It's not, you know, and you can discover things.
link |
I mean, you know, we, you can discover stuff about,
link |
I don't know, like this thing about expansion
link |
of branchial space.
link |
That's an absolutely accessible thing to look at.
link |
Now, you know, the main issue with doing these things
link |
is not, there isn't a lot of technical depth difficulty
link |
The actual doing of the experiments, you know,
link |
all the code is all on our website to do all these things.
link |
The real thing is sort of the judgment
link |
of what's the right experiment to do.
link |
How do you interpret what you see?
link |
That's the part that, you know,
link |
people will do amazing things with.
link |
And that's the part that, but,
link |
but it isn't like you have to have done 10 years of study
link |
to get to the point where you can do the experiments.
link |
That's a cool thing you can do experiments day one,
link |
That's the amazing thing about,
link |
and you've actually put the tools out there.
link |
There's still, I would say, maybe you can correct me.
link |
It feels like there's a huge number of log hanging fruit
link |
on the mathematical side, at least not the physics side,
link |
No, there's, look on the, on the, okay.
link |
On the physics side, we are,
link |
we're definitely in harvesting mode, you know.
link |
Of which, which fruit, the low hanging ones or?
link |
The low hanging ones, yeah, right.
link |
I mean, basically here's the thing.
link |
There's a certain list of, you know,
link |
here are the effects in quantum mechanics.
link |
Here are the effects in general activity.
link |
It's just like industrial harvesting.
link |
It's like, can we get this one, this one, this one,
link |
this one, this one?
link |
And the thing that's really, you know,
link |
interesting and satisfying, and it's like, you know,
link |
is one climbing the right mountain?
link |
Does one have the right model?
link |
The thing that's just amazing is, you know,
link |
we keep on like, are we going to get this one?
link |
How hard is this one?
link |
It's like, oh, you know, it looks really hard.
link |
It looks really hard.
link |
Oh, actually we can get it.
link |
And you're, you're continually surprised.
link |
I mean, it seems like I've been following your progress.
link |
It's kind of exciting.
link |
All the, in harvesting mode,
link |
all the things you're picking up along the way.
link |
No, I mean, it's, it's the thing that is,
link |
I keep on thinking it's going to be more difficult
link |
Now that's a, you know, that's a, who knows what,
link |
I mean, the one thing, so the, the, the,
link |
the thing that's been a, was a big thing
link |
that I think we're, we're pretty close to.
link |
I mean, I can give you a little bit of the roadmap.
link |
It's sort of interesting to see, it's like,
link |
what are particles?
link |
What are things like electrons?
link |
How do they really work?
link |
Are you close to get like, what, what's a,
link |
are you close to trying to understand like the atom,
link |
the electrons, neutrons, protons?
link |
Okay, so this is, this is the stack.
link |
So the first thing we want to understand is
link |
the quantization of spin.
link |
So particles, they, they kind of spin,
link |
they have a certain angular momentum,
link |
that angular momentum,
link |
even though the masses of particles are all over the place,
link |
you know, the electron has a mass of 0.511 MeV,
link |
but you know, the proton is 938 MeV, et cetera, et cetera,
link |
et cetera, they're all kind of random numbers.
link |
The, the spins of all these particles
link |
are either integers or half integers.
link |
And that's a fact that was discovered in the 1920s, I guess.
link |
The, I think that we are close to understanding
link |
why spin is quantized.
link |
And that's a, and it, it appears to be
link |
a quite elaborate mathematical story
link |
about homotopic groups in twister space
link |
and all kinds of things.
link |
But bottom line is that seems within reach.
link |
And that's, that's a big deal
link |
because that's a very core feature of understanding
link |
how particles work in quantum mechanics.
link |
Another core feature is this difference between particles
link |
that obey the exclusion principle and sort of stay apart,
link |
that leads to the stability of matter and things like that,
link |
and particles that love to get together
link |
and be in the same state, things like photons,
link |
that, and that's what leads to phenomena like lasers,
link |
where you can get sort of coherently
link |
everything in the same state.
link |
That difference is the particles of integer spin
link |
are bosons like to get together in the same state,
link |
the particles of half integer spin are fermions,
link |
like electrons that they tend to stay apart.
link |
And so the question is, can we get that in our models?
link |
And, oh, just the last few days, I think we made,
link |
I mean, I think the story of,
link |
I mean, it's one of these things where we're really close.
link |
Is this connected fermions and bosons?
link |
So this was what happens is what seems to happen, okay?
link |
It's, you know, subject to revision in the next few days.
link |
But what seems to be the case is that
link |
bosons are associated with essentially
link |
merging in multiway graphs,
link |
and fermions are associated with branching
link |
in multiway graphs.
link |
And that essentially the exclusion principle
link |
is the fact that in branchial space,
link |
things have a certain extent in branchial space
link |
that in which things are being sort of forced apart
link |
in branchial space, whereas the case of bosons,
link |
they get, they come together in branchial space.
link |
And the real question is, can we explain the relationship
link |
between that and these things called spinners,
link |
which are the representation of half integer spin particles
link |
that have this weird feature that usually when you go
link |
around 360 degree rotation,
link |
you get back to where you started from.
link |
But for a spinner, you don't get back
link |
to where you started from.
link |
It takes 720 degrees of rotation to get back
link |
to where you started from.
link |
And we are just, it feels like we are,
link |
we're just incredibly close to actually having that,
link |
understanding how that works.
link |
And it turns out, it looks like,
link |
my current speculation is that it's as simple
link |
as the directed hypergraphs versus undirected hypergraphs,
link |
the relationship between spinners and vectors.
link |
So, which is just interesting.
link |
Yeah, that would be interesting if these are all these kind
link |
of nice properties of this multi way graphs of branching
link |
Spinners have been very mysterious.
link |
And if that's what they turn out to be,
link |
there's going to be an easy explanation
link |
of what's going on.
link |
Directive versus undirective.
link |
It's just, and that's why there's only two different cases.
link |
It's why are spinners important in quantum mechanics?
link |
Can you just give a...
link |
Yeah, so spinners are important because they are,
link |
they're the representation of electrons
link |
which have half an inch of spin.
link |
They are, the wave functions of electrons are spinners.
link |
Just like the wave functions of photons are vectors,
link |
the wave functions of electrons are spinners.
link |
And they have this property that when you rotate
link |
by 360 degrees, they come back to minus one of themselves
link |
and take 720 degrees to get back to the original value.
link |
And they are a consequence of,
link |
we usually think of rotation in space as being,
link |
when you have this notion of rotational invariance
link |
and rotational invariance, as we ordinarily experience it,
link |
doesn't have the feature.
link |
If you go through 360 degrees,
link |
you go back to where you started from,
link |
but that's not true for electrons.
link |
And so that's why understanding how that works is important.
link |
Yeah, I've been playing with Mobius Strip
link |
quite a bit lately, just for fun.
link |
It adds some funk, it has the same kind of funky properties.
link |
Yes, right, exactly.
link |
You can have this so called belt trick,
link |
which is this way of taking an extended object
link |
and you can see properties like spinners
link |
with that kind of extended object that...
link |
Yeah, it would be very cool if there's,
link |
it somehow connects the directive versus undirective.
link |
I think that's what it's gonna be.
link |
I think it's gonna be as simple as that, but we'll see.
link |
I mean, this is the thing that,
link |
this is the big sort of bizarre surprise is that,
link |
because I learned physics as probably, let's say,
link |
let's say a fifth generation in the sense that,
link |
if you go back to the 1920s and so on,
link |
there were the people who were originating
link |
quantum mechanics and so on.
link |
Maybe it's a little less than that.
link |
Maybe I was like a third generation or something.
link |
I don't know, but the people from whom I learned physics
link |
were the people who had been students of the students
link |
of the people who originated
link |
the current understanding of physics.
link |
And we're now at probably the seventh generation
link |
of physicists or something
link |
from the early days of 20th century physics.
link |
And whenever a field gets that many generations deep,
link |
it seems the foundations seem quite inaccessible.
link |
And they seem, it seems like
link |
you can't possibly understand that.
link |
We've gone through seven academic generations
link |
and that's been, you know, that's been this thing
link |
that's been difficult to understand for that long.
link |
It just can't be that simple.
link |
But in a sense, maybe that journey takes you
link |
to a simple explanation that was there all along.
link |
That's the whole. Right, right, right.
link |
I mean, you know, and the thing for me personally,
link |
the thing that's been quite interesting is, you know,
link |
I didn't expect this project to work in this way.
link |
And I, you know, but I had this sort of weird piece
link |
of personal history that I used to be a physicist
link |
and I used to do all this stuff.
link |
And I know, you know, the standard canon of physics,
link |
I knew it very well.
link |
And, you know, but then I'd been working
link |
on this kind of computational paradigm
link |
for basically 40 years.
link |
And the fact that, you know, I'm sort of now coming back
link |
to, you know, trying to apply that in physics,
link |
it kind of felt like that journey was necessary.
link |
Was this, when did you first try to play with a hypergraph?
link |
So what happened is,
link |
yeah, so what I had was, okay, so this is again,
link |
you know, one always feels dumb after the fact.
link |
It's obvious after the fact.
link |
But so back in the early 1990s,
link |
I realized that using graphs
link |
as a sort of underlying thing underneath space and time
link |
was going to be a useful thing to do.
link |
I figured out about multiway systems.
link |
I figured out the things about general relativity
link |
I'd figured out by the end of the 1990s.
link |
But I always felt there was a certain inelegance
link |
because I was using these graphs
link |
and there were certain constraints on these graphs
link |
that seemed like they were kind of awkward.
link |
It was kind of like, you can pick,
link |
it's like you couldn't pick any rule.
link |
It was like pick any number, but the number has to be prime.
link |
It was kind of like you couldn't,
link |
it was kind of an awkward special constraint.
link |
I had these trivalent graphs,
link |
graphs with just three connections from every node.
link |
Okay, so, but I discovered a bunch of stuff with that.
link |
And I thought it was kind of inelegant.
link |
And, you know, the other piece of sort of personal history
link |
is obviously I spent my life
link |
as a computational language designer.
link |
And so the story of computational language design
link |
is a story of how do you take all these random ideas
link |
in the world and kind of grind them down
link |
into something that is computationally
link |
as simple as possible.
link |
And so, you know, I've been very interested
link |
in kind of simple computational frameworks
link |
for representing things and have, you know,
link |
ridiculous amounts of experience in trying to do that.
link |
And actually all of those trajectories of your life
link |
kind of came together.
link |
So you make it sound like you could have come up
link |
with everything you're working on now decades ago,
link |
Look, two things slowed me down.
link |
I mean, one thing that slowed me down was
link |
I couldn't figure out how to make it elegant.
link |
And that turns out hypergraphs were the key to that.
link |
And that I figured out about less than two years ago now.
link |
And the other, I mean, I think,
link |
so that was sort of a key thing.
link |
Well, okay, so the real embarrassment of this project, okay,
link |
is that the final structure that we have
link |
that is the foundation for this project
link |
is basically a kind of an idealized version,
link |
a formalized version of the exact same structure
link |
that I've used to build computational languages
link |
for more than 40 years.
link |
But it took me, but I didn't realize that.
link |
And there yet may be others.
link |
So we're focused on physics now,
link |
but I mean, that's what the new kind of science was about.
link |
Same kind of stuff.
link |
And this, in terms of mathematically,
link |
well, the beauty of it.
link |
So there could be entire other kind of objects
link |
that are useful for,
link |
like we're not talking about, you know,
link |
machine learning, for example.
link |
Maybe there's other variants of the hypergraph
link |
that are very useful for reasoning.
link |
Well, we'll see whether the multiway graph
link |
or machine learning system is interesting.
link |
Let's leave it at that.
link |
That's conversation number three.
link |
That's, we're not gonna go there right now, but.
link |
One of the things you've mentioned
link |
is the space of all possible rules
link |
that we kind of discussed a little bit.
link |
That, you know, that could be, I guess,
link |
the set of possible rules is infinite.
link |
Well, so here's the big sort of one of the conundrums
link |
that I'm kind of trying to deal with is,
link |
let's say we think we found the rule for the universe
link |
and we say, here it is.
link |
You know, write it down.
link |
It's a little tiny thing.
link |
And then we say, gosh, that's really weird.
link |
Why did we get that one?
link |
And then we're in this whole situation
link |
because let's say it's fairly simple.
link |
How did we come up the winners
link |
getting one of the simple possible universe rules?
link |
Why didn't we get what some incredibly complicated rule?
link |
Why do we get one of the simpler ones?
link |
And that's a thing which, you know,
link |
in the history of science, you know,
link |
the whole sort of story of Copernicus and so on was,
link |
you know, we used to think the earth
link |
was the center of the universe,
link |
but now we find out it's not.
link |
And we're actually just in some, you know,
link |
random corner of some random galaxy
link |
out in this big universe, there's nothing special about us.
link |
So if we get, you know, universe number 317
link |
out of all the infinite number of possibilities,
link |
how do we get something that small and simple?
link |
Right, so I was very confused by this.
link |
And it's like, what are we going to say about this?
link |
How are we going to explain this?
link |
And I thought it was, might be one of these things
link |
where you just, you know, you can get it to the threshold,
link |
and then you find out its rule number, such and such,
link |
and you just have no idea why it's like that.
link |
Okay, so then I realized
link |
it's actually more bizarre than that, okay?
link |
So we talked about multiway graphs.
link |
We talked about this idea that
link |
you take these underlying transformation rules
link |
on these hypergraphs, and you apply them
link |
wherever the rule can apply, you apply it.
link |
And that makes this whole multiway graph of possibilities.
link |
Okay, so let's go a little bit weirder.
link |
Let's say that at every place,
link |
not only do you apply a particular rule
link |
in all possible ways it can apply,
link |
but you apply all possible rules
link |
in all possible ways they can apply.
link |
As you say, that's just crazy.
link |
That's way too complicated.
link |
You're never going to be able to conclude anything.
link |
Okay, however, turns out that...
link |
Don't tell me there's some kind of invariance.
link |
So what happens is...
link |
And that would be amazing.
link |
Right, so this thing that you get
link |
is this kind of ruleal multiway graph,
link |
this multiway graph that is a branching of rules
link |
as well as a branching of possible applications of rules.
link |
This thing has causal invariance.
link |
It's an inevitable feature that it shows causal invariance.
link |
And that means that you can take different reference frames,
link |
different ways of slicing this thing,
link |
and they will all in some sense be equivalent.
link |
If you make the right translation, they will be equivalent.
link |
So, okay, so the basic point here is...
link |
If that's true, that would be beautiful.
link |
It is true, and it is beautiful.
link |
It's not just an intuition, there is some...
link |
No, no, no, there's real mathematics behind this,
link |
Okay, so here's where it comes in.
link |
Yeah, that's amazing.
link |
Right, so by the way, I mean,
link |
the mathematics it's connected to
link |
is the mathematics of higher category theory
link |
and group voids and things like this,
link |
which I've always been afraid of,
link |
but now I'm finally wrapping my arms around it.
link |
But it's also related to...
link |
It also relates to computational complexity theory.
link |
It's also deeply related to the P versus NP problem
link |
and other things like this.
link |
Again, it seems completely bizarre
link |
that these things are connected,
link |
but here's why it's connected.
link |
This space of all possible...
link |
Okay, so a Turing machine, very simple model of computation.
link |
You know, you just got this tape
link |
where you write down, you know, ones and zeros
link |
or something on the tape,
link |
and you have this rule that says, you know,
link |
you change the number,
link |
you move the head on the tape, et cetera.
link |
You have a definite rule for doing that.
link |
A deterministic Turing machine
link |
just does that deterministically.
link |
Given the configuration of the tape,
link |
it will always do the same thing.
link |
A non deterministic Turing machine
link |
can have different choices that it makes at every step.
link |
And so, you know, you know this stuff,
link |
you probably teach this stuff.
link |
It, you know, so a non deterministic Turing machine
link |
has the set of branching possibilities,
link |
which is in fact, one of these multiway graphs.
link |
And in fact, if you say,
link |
imagine the extremely non deterministic Turing machine,
link |
the Turing machine that can just do,
link |
that takes any possible rule at each step,
link |
that is this real multiway graph.
link |
The set of possible histories
link |
of that extreme non deterministic Turing machine
link |
is a Rulio multiway graph.
link |
And you're, what term are you using?
link |
It's a weird word.
link |
Yeah, it's a weird word, right?
link |
Rulio multiway graph.
link |
Okay, so this, so that.
link |
I'm trying to think of,
link |
I'm trying to think of the space of rules.
link |
So these are basic transformations.
link |
So in a Turing machine,
link |
it's like it says, move left, move, you know,
link |
if it's a one, if it's a black square under the head,
link |
move left and right to green square.
link |
That's a very basic rule,
link |
but I'm trying to see the rules on the hypergraphs,
link |
how rich of the programs can they be?
link |
Or do they all ultimately just map into something simple?
link |
Yeah, they're all, I mean, hypergraphs,
link |
that's another layer of complexity on this whole thing.
link |
You can think about these in transformations of hypergraphs,
link |
but Turing machines are a little bit simpler.
link |
You just think of it Turing machines, okay.
link |
Right, they're a little bit simpler.
link |
So if you look at these extreme
link |
non deterministic Turing machines,
link |
you're mapping out all the possible non deterministic paths
link |
that the Turing machine can follow.
link |
And if you ask the question, can you reach, okay,
link |
so a deterministic Turing machine follows a single path.
link |
The non deterministic Turing machine fills out
link |
this whole sort of ball of possibilities.
link |
And so then the P versus MP problem
link |
ends up being questions about,
link |
and we haven't completely figured out
link |
all the details of this,
link |
but it's basically has to do with questions
link |
about the growth of that ball relative
link |
to what happens with individual paths and so on.
link |
So essentially there's a geometrization
link |
of the P versus MP problem that comes out of this.
link |
That's a sideshow, okay.
link |
The main event here is the statement
link |
that you can look at this multiway graph
link |
where the branches correspond
link |
not just to different applications of a single rule,
link |
but to different applications of different rules, okay.
link |
And that then that when you say,
link |
I'm going to be an observer embedded in that system
link |
and I'm going to try and make sense
link |
of what's going on in the system.
link |
And to do that, I essentially am picking a reference frame
link |
and that turns out to be, well, okay.
link |
So the way this comes out essentially
link |
is the reference frame you pick
link |
is the rule that you infer is what's going on
link |
in the universe, even though all possible rules
link |
are being run, although all those possible rules
link |
are in a sense giving the same answer
link |
because of causal invariance.
link |
But what you see could be completely different.
link |
If you pick different reference frames,
link |
you essentially have a different description language
link |
for describing the universe.
link |
Okay, so what does this really mean in practice?
link |
So imagine there's us.
link |
We think about the universe in terms of space and time
link |
and we have various kinds of description models and so on.
link |
Now let's imagine the friendly aliens, for example, right?
link |
How do they describe their universe?
link |
Well, you know, our description of the universe
link |
probably is affected by the fact that, you know,
link |
we are about the size we are, you know,
link |
a meter ish tall, so to speak.
link |
We have brain processing speeds,
link |
we're about the speeds we have.
link |
We're not the size of planets, for example,
link |
where the speed of light really would matter.
link |
You know, in our everyday life,
link |
the speed of light doesn't really matter.
link |
Everything can be, you know,
link |
the fact that the speed of light is finite is irrelevant.
link |
It could as well be infinite.
link |
We wouldn't make any difference.
link |
You know, it affects the ping times on the internet.
link |
That's about the level of how we notice the speed of light.
link |
In our sort of everyday existence,
link |
we don't really notice it.
link |
And so we have a way of describing the universe
link |
that's based on our sensory, you know, our senses,
link |
these days also on the mathematics we've constructed
link |
and so on, but the realization is
link |
it's not the only way to do it.
link |
There will be completely, utterly incoherent descriptions
link |
of the universe, which correspond
link |
to different reference frames in this sort of ruleal space.
link |
In the ruleal space, that's fascinating.
link |
So we have some kind of reference frame
link |
in this ruleal space, and from that.
link |
That's why we are attributing this rule to the universe.
link |
So in other words, when we say,
link |
why is it this rule and not another,
link |
the answer is just, you know,
link |
shine the light back on us, so to speak.
link |
It's because of the reference frame that we've picked
link |
in our way of understanding what's happening
link |
in this sort of space of all possible rules and so on.
link |
But also in the space from this reference frame,
link |
because of the ruleal, the invariance,
link |
that simple, that the rule on which the universe,
link |
with which you can run the universe,
link |
might as well be simple.
link |
Yes, yes, but okay, so here's another point.
link |
So this is, again, these are a little bit mind twisting
link |
in some ways, but the, okay, another thing that's sort of,
link |
we know from computation is this idea
link |
of computation universality.
link |
The fact that given that we have a program
link |
that runs on one kind of computer, we can as well,
link |
you know, we can convert it to run
link |
on any other kind of computer.
link |
We can emulate one kind of computer with another.
link |
So that might lead you to say, well,
link |
you think you have the rule for the universe,
link |
but you might as well be running it on a Turing machine
link |
because we know we can emulate any computational rule
link |
on any kind of machine.
link |
And that's essentially the same thing
link |
that's being said here.
link |
That is that what we're doing is we're saying
link |
these different interpretations of physics correspond
link |
to essentially running physics
link |
on different underlying, you know,
link |
thinking about the physics as running in different
link |
with different underlying rules
link |
as if different underlying computers were running them.
link |
And, but because of computation universality
link |
or more accurately, because of this principle
link |
of computational equivalence thing of mine,
link |
there's that they are,
link |
these things are ultimately equivalent.
link |
So the only thing that is the ultimate fact
link |
about the universe, the ultimate fact that doesn't depend
link |
on any of these, you know, we don't have to talk
link |
about specific rules, et cetera, et cetera, et cetera.
link |
The ultimate fact is the universe is computational
link |
and it is the things that happen in the universe
link |
are the kinds of computations that the principle
link |
of computational equivalence says should happen.
link |
Now that might sound like you're not really saying
link |
anything there, but you are because you can,
link |
you could in principle have a hyper computer
link |
that things that take an ordinary computer
link |
an infinite time to do the hyper computer can just say,
link |
oh, I know the answer.
link |
It's this immediately.
link |
What this is saying is the universe is not a hyper computer.
link |
It's not simpler than a,
link |
an ordinary Turing machine type computer.
link |
It's exactly like an ordinary Turing machine type computer.
link |
And so that's the, that's in the end,
link |
the sort of net net conclusion is that's the thing
link |
that is the sort of the hard immovable fact
link |
about the universe.
link |
That's sort of the fundamental principle of the universe
link |
is that it is computational and not hyper computational
link |
and not sort of infra computational.
link |
It is this level of computational ability
link |
and it's, it kind of has,
link |
and that's sort of the, the, the core fact, but now,
link |
you know, this, this idea that you can have these different
link |
kind of a rule reference frames,
link |
these different description languages for the universe.
link |
It makes me, you know, I used to think, okay, you know,
link |
imagine the aliens,
link |
imagine the extraterrestrial intelligence thing, you know,
link |
at least they experienced the same physics.
link |
And now I've realized it isn't true.
link |
They could have a different rule frame.
link |
That's fascinating.
link |
That they can end up with a, a, a,
link |
a description of the universe that is utterly,
link |
utterly incoherent with ours.
link |
And that's also interesting in terms of how we think about,
link |
well, intelligence, the nature of intelligence and so on.
link |
You know, I'm, I'm fond of the quote, you know,
link |
the weather has a mind of its own because these are,
link |
you know, these are sort of computationally that,
link |
that system is computationally equivalent to the system
link |
that is our brains and so on.
link |
And what's different is we don't have a way to understand,
link |
you know, what the weather is trying to do, so to speak.
link |
We have a story about what's happening in our brains.
link |
We don't have a sort of connection
link |
to what's happening there.
link |
So we actually, it's funny,
link |
last time we talked maybe over a year ago,
link |
we talked about how it was more based on your work
link |
We talked about how would we communicate
link |
with alien intelligences.
link |
Can you maybe comment on how we might,
link |
how the Wolfram Physics Project changed your view,
link |
how we might be able to communicate
link |
with alien intelligence?
link |
Like if they showed up,
link |
is it possible that because of our comprehension
link |
of the physics of the world might be completely different,
link |
we would just not be able to communicate at all?
link |
Here's the thing, you know, intelligence is everywhere.
link |
The fact this idea that there's this notion of,
link |
oh, there's gonna be this amazing
link |
extraterrestrial intelligence
link |
and it's gonna be this unique thing.
link |
It's just not true.
link |
It's the same thing.
link |
You know, I think people will realize this
link |
about the time when people decide
link |
that artificial intelligences are kind of
link |
just natural things that are like human intelligences.
link |
They'll realize that extraterrestrial intelligences
link |
or intelligences associated with physical systems
link |
and so on, it's all the same kind of thing.
link |
It's ultimately computation.
link |
It's all the same.
link |
It's all just computation.
link |
And the issue is, can you, are you sort of inside it?
link |
Are you thinking about it?
link |
Do you have sort of a story you're telling yourself
link |
And you know, the weather could have a story
link |
it's telling itself about what it's doing.
link |
We just, it's utterly incoherent with the stories
link |
that we tell ourselves based on how our brains work.
link |
I mean, ultimately it must be a question
link |
whether we can align.
link |
Align with the kind of intelligence.
link |
Right, right, right.
link |
So there's a systematic way of doing it.
link |
Right, so the question is in the space
link |
of all possible intelligences,
link |
what's the, how do you think about the distance
link |
between description languages
link |
for one intelligence versus another?
link |
And needless to say, I have thought about this
link |
and you know, I don't have a great answer yet,
link |
but I think that's a thing
link |
where there will be things that can be said
link |
and there'll be things that where you can sort of
link |
start to characterize, you know,
link |
what is the translation distance between this,
link |
you know, version of the universe
link |
or this kind of set of computational rules
link |
and this other one.
link |
In fact, okay, so this is a, you know,
link |
there's this idea of algorithmic information theory.
link |
There's this question of sort of what is the,
link |
when you have something,
link |
what is the sort of shortest description you can make of it
link |
where that description could be saying,
link |
run this program to get the thing, right?
link |
So I'm pretty sure that there will be a physicalization
link |
of the idea of algorithmic information
link |
and that, okay, this is again, a little bit bizarre,
link |
but so I mentioned that there's the speed of light,
link |
maximum speed of information transmission in physical space.
link |
There's a maximum speed of information transmission
link |
in branchial space, which is a maximum entanglement speed.
link |
There's a maximum speed of information transmission
link |
in ruleal space, which is,
link |
has to do with a maximum speed of translation
link |
between different description languages.
link |
And again, I'm not fully wrapped my brain around this one.
link |
Yeah, that one just blows my mind to think about that,
link |
but that starts getting closer to the, yeah,
link |
the intelligence. It's kind of a physicalization.
link |
Right, and it's also a physicalization
link |
of algorithmic information.
link |
And I think there's probably a connection between,
link |
I mean, there's probably a connection
link |
between the notion of energy and some of these things,
link |
which again, I hadn't seen all this coming.
link |
I've always been a little bit resistant
link |
to the idea of connecting physical energy
link |
to things in computation theory,
link |
but I think that's probably coming.
link |
And that's what essentially at the core
link |
with the physics project is
link |
that you're connecting information theory with physics.
link |
Yeah, it's computation.
link |
Computation with our physical universe.
link |
I mean, the fact that our physical universe is,
link |
right, that we can think of it as a computation
link |
and that we can have discussions like,
link |
the theory of the physical universe
link |
is the same kind of a theory as the P versus MP problem
link |
and so on is really, I think that's really interesting.
link |
And the fact that, well, okay,
link |
so this kind of brings me to one more thing
link |
that I have to in terms of this sort of unification
link |
of different ideas, which is metamathematics.
link |
Yeah, let's talk about that.
link |
You mentioned that earlier.
link |
What the heck is metamathematics and...
link |
Okay, so here's what, okay.
link |
So what is mathematics?
link |
Mathematics, sort of at a lowest level,
link |
one thinks of mathematics as you have certain axioms.
link |
You say things like X plus Y is the same as Y plus X.
link |
That's an axiom about addition.
link |
And then you say, we've got these axioms
link |
and from these axioms, we derive all these theorems
link |
that fill up the literature of mathematics.
link |
The activity of mathematicians
link |
is to derive all these theorems.
link |
Actually, the axioms of mathematics are very small.
link |
You can fit, when I did my new kind of science book,
link |
I fit all of the standard axioms of mathematics
link |
on basically a page and a half.
link |
It's like a very simple rule
link |
from which all of mathematics arises.
link |
The way it works though is a little different
link |
from the way things work in sort of a computation
link |
because in mathematics, what you're interested in
link |
is a proof and the proof says,
link |
from here, you can use, from this expression, for example,
link |
you can use these axioms to get to this other expression.
link |
So that proves these two things are equal.
link |
Okay, so we can begin to see how this has been going to work.
link |
What's gonna happen is there are paths
link |
in metamathematical space.
link |
So what happens is each, two different ways to look at it.
link |
You can just look at it as mathematical expressions
link |
or you can look at it as mathematical statements,
link |
postulates or something.
link |
But either way, you think of these things
link |
and they are connected by these axioms.
link |
So in other words, you have some fact
link |
or you have some expression, you apply this axiom,
link |
you get some other expression.
link |
And in general, given some expression,
link |
there may be many possible different expressions
link |
You basically build up a multiway graph
link |
and a proof is a path through the multiway graph
link |
that goes from one thing to another thing.
link |
The path tells you how did you get from one thing
link |
to the other thing.
link |
It's the story of how you got from this to that.
link |
The theorem is the thing at one end
link |
is equal to the thing at the other end.
link |
The proof is the path you go down
link |
to get from one thing to the other.
link |
You mentioned that Gödel's incompleteness theorem
link |
fits naturally there.
link |
Yeah, so what happens there is that the Gödel's theorem
link |
is basically saying that there are paths of infinite length.
link |
That is that there's no upper bound.
link |
If you know these two things,
link |
you say, I'm trying to get from here to here,
link |
how long do I have to go?
link |
You say, well, I've looked at all the paths of length 10.
link |
Somebody says, that's not good enough.
link |
That path might be of length a billion.
link |
And there's no upper bound on how long that path is.
link |
And that's what leads to the incompleteness theorem.
link |
So I mean, the thing that is kind of an emerging idea
link |
is you can start asking,
link |
what's the analog of Einstein's equations
link |
in metamathematical space?
link |
What's the analog of a black hole
link |
in metamathematical space?
link |
What's the hope of this?
link |
So yeah, it's fascinating to model all the mathematics
link |
So here's what it is.
link |
This is mathematics in bulk.
link |
So human mathematicians have made a few million theorems.
link |
They've published a few million theorems.
link |
But imagine the infinite future of mathematics.
link |
Apply something to mathematics
link |
that mathematics likes to apply to other things.
link |
What is the limit of the infinite future of mathematics?
link |
What does it look like?
link |
What is the continuum limit of mathematics?
link |
What is the, as you just fill in
link |
more and more and more theorems,
link |
what does it look like?
link |
How does, what kinds of conclusions can you make?
link |
So for example, one thing I've just been doing
link |
So Euclid, very impressive.
link |
He had 10 axioms, he derived 465 theorems, okay?
link |
His book, you know,
link |
that was the sort of defining book of mathematics
link |
So you can actually map out,
link |
and I actually did this 20 years ago,
link |
but I've done it more seriously now.
link |
You can map out the theorem dependency
link |
of those 465 theorems.
link |
So from the axioms, you grow this graph,
link |
it's actually a multiway graph,
link |
of how all these theorems get proved from other theorems.
link |
And so you can ask questions about, you know,
link |
well, you can ask things like,
link |
what's the hardest theorem in Euclid?
link |
The answer is, the hardest theorem
link |
is that there are five platonic solids.
link |
That turns out to be the hardest theorem in Euclid.
link |
That's actually his last theorem in all his books.
link |
What's the hardness, the distance you have to travel?
link |
Yeah, let's say it's 33 steps from the,
link |
the longest path in the graph is 33 steps.
link |
So that's the, there's a 33 step path you have to follow
link |
to go from the axioms, according to Euclid's proofs,
link |
to the statement there are five platonic solids.
link |
So, okay, so then the question is,
link |
in, what does it mean if you have this map?
link |
Okay, so in a sense, this metamathematical space
link |
is the infrastructural space of all possible theorems
link |
that you could prove in mathematics.
link |
That's the geometry of metamathematics.
link |
There's also the geography of mathematics.
link |
That is, where did people choose to live in space?
link |
And that's what, for example,
link |
exploring the sort of empirical metamathematics
link |
that Euclid is doing.
link |
You could put each individual, like, human mathematician,
link |
you can embed them into that space.
link |
I mean, they kind of live.
link |
They represent a path in the space.
link |
The things they do.
link |
Maybe a set of paths.
link |
So like a set of axioms that are chosen.
link |
Right, so for example,
link |
here's an example of a thing that I realized.
link |
So one of the surprising things about,
link |
well, there are two surprising facts about math.
link |
One is that it's hard,
link |
and the other is that it's doable, okay?
link |
So first question is, why is math hard?
link |
You know, you've got these axioms.
link |
They're very small.
link |
Why can't you just solve every problem in math easily?
link |
Yeah, it's just logic.
link |
Well, logic happens to be a particular special case
link |
that does have certain simplicity to it.
link |
But general mathematics, even arithmetic,
link |
already doesn't have the simplicity that logic has.
link |
So why is it hard?
link |
Because of computational irreducibility.
link |
Because what happens is, to know what's true,
link |
and this is this whole story about the path
link |
you have to follow and how long is the path,
link |
and Gödel's theorem is the statement
link |
that the path is not a bounded length,
link |
but the fact that the path is not always compressible
link |
to something tiny is a story of computational irreducibility.
link |
So that's why math is hard.
link |
Now, the next question is, why is math doable?
link |
Because it might be the case that most things you care about
link |
don't have finite length paths.
link |
Most things you care about might be things
link |
where you get lost in the sea of computational irreducibility
link |
and worse, undecidability.
link |
That is, there's just no finite length path
link |
that gets you there.
link |
Why is mathematics doable?
link |
Gödel proved his incompleteness theorem in 1931.
link |
Most working mathematicians don't really care about it.
link |
They just go ahead and do mathematics,
link |
even though it could be that the questions they're asking
link |
It could have been that Fermat's last theorem
link |
It turned out it had a proof.
link |
It's a long, complicated proof.
link |
The twin prime conjecture might be undecidable.
link |
The Riemann hypothesis might be undecidable.
link |
These things might be, the axioms of mathematics
link |
might not be strong enough to reach those statements.
link |
It might be the case that depending on what axioms
link |
you choose, you can either say that's true
link |
or that's not true.
link |
And by the way, from Fermat's last theorem,
link |
there could be a shorter path.
link |
Yeah, so the notion of geodesics in metamathematical space
link |
is the notion of shortest proofs in metamathematical space.
link |
And that's a, you know, human mathematicians
link |
do not find shortest paths,
link |
nor do automated theorem provers.
link |
But the fact, and by the way, the, I mean,
link |
this stuff is so bizarrely connected.
link |
I mean, if you're into automated theorem proving,
link |
there are these so called critical pair lemmas
link |
and automated theorem proving.
link |
Those are precisely the branch pairs in our,
link |
that in multiway graphs.
link |
Let me just finish on the why mathematics is doable.
link |
Oh yes, the second part.
link |
So you know why it's hard, why is it doable?
link |
Right, why do we not just get lost
link |
in undecidability all the time?
link |
So, and here's another fact,
link |
is in doing computer experiments
link |
and doing experimental mathematics,
link |
you do get lost in that way.
link |
When you just say, I'm picking a random integer equation.
link |
How do I, does it have a solution or not?
link |
And you just pick it at random
link |
without any human sort of path getting there.
link |
Often, it's really, really hard.
link |
It's really hard to answer those questions.
link |
We just pick them at random from the space of possibilities.
link |
But what I think is happening is,
link |
and that's a case where you just fell off
link |
into this ocean of sort of irreducibility and so on.
link |
What's happening is human mathematics
link |
is a story of building a path.
link |
You started off, you're always building out
link |
on this path where you are proving things.
link |
You've got this proof trajectory
link |
and you're basically, the human mathematics
link |
is the sort of the exploration of the world
link |
along this proof trajectory, so to speak.
link |
You're not just parachuting in from anywhere.
link |
You're following Lewis and Clark or whatever.
link |
You're actually going, doing the path.
link |
And the fact that you are constrained to go along that path
link |
is the reason you don't end up with,
link |
every so often you'll see a little piece of undecidability
link |
and you'll avoid that part of the path.
link |
But that's basically the story of why human mathematics
link |
has seemed to be doable.
link |
It's a story of exploring these paths
link |
that are by their nature,
link |
they have been constructed to be paths that can be followed.
link |
And so you can follow them further.
link |
Now, why is this relevant to anything?
link |
So, okay, so here's my belief.
link |
The fact that human mathematics works that way
link |
is I think there's some sort of connections
link |
between the way that observers work in physics
link |
and the way that the axiom systems of mathematics are set up
link |
to make mathematics be doable in that kind of way.
link |
And so, in other words, in particular,
link |
I think there is an analog of causal invariance,
link |
which I think is, and this is again,
link |
it's sort of the upper reaches of mathematics
link |
and stuff that it's a thing,
link |
there's this thing called homotopy type theory,
link |
which is an abstract, it's came out of category theory,
link |
and it's sort of an abstraction of mathematics.
link |
Mathematics itself is an abstraction,
link |
but it's an abstraction of the abstraction of mathematics.
link |
And there is the thing called the univalence axiom,
link |
which is a sort of a key axiom in that set of ideas.
link |
And I'm pretty sure the univalence axiom
link |
is equivalent to causal invariance.
link |
What was the term you used again?
link |
Is that something for somebody like me accessible?
link |
There's a statement of it that's fairly accessible.
link |
I mean, the statement of it is,
link |
basically it says things which are equivalent
link |
can be considered to be identical.
link |
Yeah, it's in higher category.
link |
Okay, so it's a, but I mean,
link |
the thing just to give a sketch of how that works.
link |
So category theory is an attempt to idealize,
link |
it's an attempt to sort of have a formal theory
link |
of mathematics that is at a sort of higher level
link |
It's where you just think about these mathematical objects
link |
and these categories of objects and these morphisms,
link |
these connections between categories.
link |
Okay, so it turns out the morphisms and categories,
link |
at least weak categories,
link |
are very much like the paths in our hypergraphs and things.
link |
And it turns out, again, this is where it all gets crazy.
link |
I mean, the fact that these things are connected
link |
So category theory, our causal graphs
link |
are like second order category theory.
link |
And it turns out you can take the limits
link |
of infinite order category theory.
link |
So just give roughly the idea.
link |
This is a roughly explainable idea.
link |
So a mathematical proof will be a path
link |
that says you can get from this thing to this other thing.
link |
And here's the path that you get from this thing
link |
to this other thing.
link |
But in general, there may be many paths,
link |
many proofs that get you many different paths
link |
that all successfully go from this thing
link |
to this other thing, okay?
link |
Now you can define a higher order proof,
link |
which is a proof of the equivalence of those proofs.
link |
Okay, so you're saying there's a...
link |
A path between those proofs essentially.
link |
Yes, a path between the paths, okay?
link |
And so you do that.
link |
That's the sort of second order thing.
link |
That path between the paths is essentially related
link |
to our causal graphs.
link |
Then you can take the limit.
link |
The path between path, between path, between path.
link |
The infinite limit.
link |
That infinite limit turns out to be
link |
our Rulial Multiway System.
link |
Yeah, the Rulial Multiway System,
link |
that's a fascinating, both in the physics world
link |
and as you're saying now, that's fast.
link |
I'm not sure I've loaded it in completely, but...
link |
Well, I'm not sure I have either.
link |
And it may be one of these things where,
link |
in another five years or something, it's like,
link |
it was obvious, but I didn't see it.
link |
No, but the thing which is sort of interesting to me
link |
is that there's sort of an upper reach of mathematics,
link |
of the abstraction of mathematics.
link |
This thing, there's this mathematician called Grothendieck
link |
who's generally viewed as being sort of one
link |
of the most abstract,
link |
sort of creator of the most abstract mathematics
link |
of 1970s ish timeframe.
link |
And one of the things that he constructed was this thing
link |
he called the Infinity Grupoid.
link |
And he has this sort of hypothesis
link |
about the inevitable appearance of geometry
link |
from essentially logic in the structure of this thing.
link |
Well, it turns out this Rulial Multiway System
link |
is the Infinity Grupoid.
link |
So it's this limiting object.
link |
And this is an instance of that limiting object.
link |
So what to me is, I mean, again,
link |
I've been always afraid of this kind of mathematics
link |
because it seemed incomprehensibly abstract to me.
link |
But what I'm sort of excited about with this
link |
is that we've sort of concretified the way
link |
that you can reach this kind of mathematics,
link |
which makes it, well, both seem more relevant
link |
and also the fact that I don't yet know exactly
link |
what mileage we're gonna get from using
link |
the sort of the apparatus that's been built
link |
in those areas of mathematics to analyze what we're doing.
link |
But the thing that's.
link |
So using mathematics to understand what you're doing
link |
and using what you're doing computationally
link |
to understand that.
link |
Right, so for example,
link |
the understanding of metamathematical space,
link |
one of the reasons I really want to do that
link |
is because I want to understand quantum mechanics better.
link |
And that, what you see,
link |
we live that kind of the multiway graph of mathematics
link |
because we actually know this is a theorem we've heard of.
link |
This is another one we've heard of.
link |
We can actually say these are actual things in the world
link |
that we relate to,
link |
which we can't really do as readily for the physics case.
link |
And so it's kind of a way to help my intuition.
link |
It's also, there are bizarre things
link |
like what's the analog of Einstein's equations
link |
in metamathematical space?
link |
What's the analog of a black hole?
link |
It turns out it looks like not completely sure yet,
link |
but there's this notion of nonconstructive proofs
link |
And I think those relate to,
link |
well, actually they relate to things
link |
related to event horizons.
link |
So the fact that you can take ideas from physics
link |
like event horizons.
link |
And map them into the same kind of space, metamath.
link |
So do you think there'll be,
link |
do you think you might stumble upon
link |
some breakthrough ideas in theorem proving?
link |
Like for, from the other direction?
link |
No, I mean, what's really nice is that we are using,
link |
so this absolutely directly maps to theorem proving.
link |
So pods and multiway graphs,
link |
that's what a theorem prover is trying to do.
link |
But I also mean like automated theorem.
link |
That's what, right.
link |
So the finding of pods, the finding of shortest pods
link |
or finding of pods at all
link |
is what automated theorem provers do.
link |
And actually what we've been doing.
link |
So we've actually been using automated theorem proving
link |
both in the physics project to prove things
link |
and using that as a way to understand multiway graphs.
link |
And because what an automated theorem prover is doing
link |
is it's trying to find a path through a multiway graph
link |
and its critical pair lemmas
link |
are precisely little stubs of branch pairs
link |
going off into branchial space.
link |
And that's, I mean, it's really weird.
link |
You know, we have these visualizations in Wolfram language
link |
of proof graphs from our automated theorem proving system.
link |
And they look reminiscent of.
link |
Well, it's just bizarre
link |
because we made these up a few years ago
link |
and they have these little triangle things
link |
and they are, we didn't quite get it right.
link |
We didn't quite get the analogy perfectly right,
link |
but it's very close.
link |
You know, just to say,
link |
in terms of how these things are connected.
link |
So there's another bizarre connection
link |
that I have to mention because which is,
link |
which again, we don't fully know,
link |
but it's a connection to something else
link |
you might not have thought was in the slightest
link |
but connected, which is distributed blockchain like things.
link |
Now you might figure out that that's,
link |
you would figure out that that's connected
link |
because it's a story of distributed computing.
link |
And the issue, you know, with the blockchain,
link |
you're saying there's going to be this one ledger
link |
that globally says, this is what happened in the world.
link |
But that's a bad deal.
link |
If you've got all these different transactions
link |
that are happening and you know,
link |
this transaction in country A
link |
doesn't have to be reconciled with the transaction
link |
in country B, at least not for a while.
link |
And that story is just like what happens
link |
with our causal graphs.
link |
That whole reconciliation thing is just like
link |
what happens with light cones and all this kind of thing.
link |
That's where the causal awareness comes into play.
link |
I mean, that's, you know,
link |
most of your conversations are about physics,
link |
but it's kind of funny that this probably
link |
and possibly might have even bigger impact
link |
and revolutionary ideas and totally other disciplines.
link |
Right, well, you see, yeah, right.
link |
So the question is, why is that happening, right?
link |
And the reason it's happening,
link |
I've thought about this obviously,
link |
because I like to think about these meta questions of,
link |
you know, what's happening is this model that we have
link |
is an incredibly minimal model.
link |
And once you have an incredibly minimal model,
link |
and this happened with cellular automata as well,
link |
cellular automata are an incredibly minimal model.
link |
And so it's inevitable that it gets you,
link |
it's sort of an upstream thing
link |
that gets used in lots of different places.
link |
And it's like, you know, the fact that it gets used,
link |
you know, cellular automata is sort of a minimal model
link |
of let's say road traffic flow or something.
link |
And they're also a minimal model of something in,
link |
you know, chemistry,
link |
and they're also a minimal model of something
link |
in epidemiology, right?
link |
It's because they're such a simple model that they can,
link |
that they apply to all these different things.
link |
Similarly, this model that we have with the physics project
link |
is another, cellular automata are a minimal model
link |
of parallel, of basically of parallel computation
link |
where you've defined space and time.
link |
These models are minimal models
link |
where you have not defined space and time.
link |
And they have been very hard to understand in the past,
link |
but the, I think the,
link |
perhaps the most important breakthrough there
link |
is the realization that these are models of physics.
link |
And therefore that you can use everything
link |
that's been developed in physics
link |
to get intuition about how things like that work.
link |
And that's why you can potentially use ideas from physics
link |
to get intuition about how to do parallel computing.
link |
And because the underlying model is the same.
link |
But we have all of this achievement in physics.
link |
I mean, you know, you might say,
link |
oh, you've come up with the fundamental theory of physics
link |
that throws out what people have done in physics before.
link |
Well, it doesn't, but also the real power
link |
is to use what's been done before in physics
link |
to apply it in these other places.
link |
This kind of brings up,
link |
I know you probably don't particularly love commenting
link |
on the work of others,
link |
but let me bring up a couple of personalities
link |
just because it's fun and people are curious about it.
link |
So there's Sabine Hassenfelder.
link |
I don't know if you're familiar with her.
link |
She wrote this book that I need to read,
link |
but I forget what the title is,
link |
but it's Beauty Leads Us Astray in Physics
link |
is a subtitle or something like that.
link |
Which so much about what we're talking about now,
link |
like this simplification,
link |
to us humans seems to be beautiful.
link |
Like there's a certain intuition with physicists,
link |
with people that a simple theory,
link |
like this reducibility,
link |
pockets of reducibility is the ultimate goal.
link |
And I think what she tries to argue is no,
link |
we just need to come up with theories
link |
that are just really good at predicting physical phenomena.
link |
It's okay to have a bunch of disparate theories
link |
as opposed to trying to chase this beautiful theory
link |
of everything is the ultimate beautiful theory,
link |
What's your response to that?
link |
Well, so what you're quoting,
link |
I don't know the Sabine Hassenfelder's,
link |
exactly what she said,
link |
but I mean that you're quoting the title of her book.
link |
Let me respond to what you were describing,
link |
which may or may not have nothing to do with
link |
what Sabine Hassenfelder says or thinks.
link |
Sorry for misquoting.
link |
But I mean, the question is,
link |
is beauty a guide to whether something is correct?
link |
Which is kind of also the story of Occam's razor.
link |
If you've got a bunch of different explanations of things,
link |
is the thing that is the simplest explanation
link |
likely to be the correct explanation?
link |
And there are situations where that's true
link |
and there are situations where it isn't true.
link |
Sometimes in human systems, it is true
link |
because people have kind of,
link |
in evolutionary systems, sometimes it's true
link |
because it's sort of been kicked
link |
to the point where it's minimized.
link |
But in physics, does Occam's razor work?
link |
Is there a simple, quotes, beautiful explanation for things
link |
or is it a big mess?
link |
We don't intrinsically know.
link |
I think that the, I wouldn't,
link |
before I worked on the project in recent times,
link |
I would have said,
link |
we do not know how complicated
link |
the rule for the universe will be.
link |
And I would have said, the one thing we know,
link |
which is a fundamental fact about science,
link |
that's the thing that makes science possible,
link |
is that there is order in the universe.
link |
I mean, early theologians would have used that
link |
as an argument for the existence of God
link |
because it's like, why is there order in the universe?
link |
Why doesn't every single particle in the universe
link |
just do its own thing?
link |
Something must be making there be order in the universe.
link |
We, in the sort of early theology point of view,
link |
that's the role of God is to do that, so to speak.
link |
In our, we might say,
link |
it's the role of a formal theory to do that.
link |
And then the question is,
link |
but how simple should that theory be?
link |
And should that theory be one that,
link |
where I think the point is, if it's simple,
link |
it's almost inevitably somewhat beautiful
link |
in the sense that, because all the stuff that we see
link |
has to fit into this little tiny theory.
link |
And the way it does that has to be,
link |
it depends on your notion of beauty,
link |
but I mean, for me, the sort of the surprising
link |
connectivity of it is, at least in my aesthetic,
link |
that's something that responds to my aesthetic.
link |
But the question is, I mean,
link |
you're a fascinating person in the sense that
link |
you're at once talking about computational,
link |
the fundamental computational reducibility of the universe,
link |
and on the other hand,
link |
trying to come up with a theory of everything,
link |
which simply describes the,
link |
the simple origins of that computational reducibility.
link |
I mean, both of those things are kind of,
link |
it's paralyzing to think that we can't make any sense
link |
of the universe in the general case,
link |
but it's hopeful to think like,
link |
one, we can think of a rule
link |
and that generates this whole complexity,
link |
and two, we can find pockets of reducibility
link |
that are powerful for everyday life
link |
to do different kinds of predictions.
link |
I suppose Sabine wants to find,
link |
focus on the finding of small pockets of reducibility
link |
versus the theory of everything.
link |
You know, it's a funny thing because,
link |
you know, a bunch of people have started working
link |
on this physics project,
link |
people who are physicists, basically,
link |
and it is really a fascinating sociological phenomenon
link |
because what, you know,
link |
when I was working on this before in the 1990s,
link |
you know, wrote it up, put it,
link |
it's 100 pages of this 1200 page book
link |
that I wrote, New Kind of Science,
link |
is, you know, 100 pages of that is about physics,
link |
but I saw it at that time,
link |
not as a pinnacle achievement,
link |
but rather as a use case, so to speak.
link |
I mean, my main point was this new kind of science,
link |
and it's like, you can apply it to biology,
link |
you can apply it to, you know, other kinds of physics,
link |
you can apply it to fundamental physics,
link |
it's just an application, so to speak,
link |
it's not the core thing.
link |
But then, you know, one of the things that was interesting
link |
with that book was, you know,
link |
book comes out, lots of people think it's pretty interesting
link |
and lots of people start using what it has
link |
in different kinds of fields.
link |
The one field where there was sort of a heavy pitchforking
link |
was from my friends, the fundamental physics people,
link |
which was, it's like, no,
link |
this can't possibly be right.
link |
And, you know, it's like, you know,
link |
if what you're doing is right,
link |
it'll overturn 50 years of what we've been doing.
link |
And it's like, no, it won't, was what I was saying.
link |
And it's like, but, you know, for a while,
link |
when I started, you know, I was going to go on back in 2002,
link |
well, 2004, actually, I was going to go on
link |
working on this project.
link |
And I actually stopped,
link |
partly because it's like, why am I, you know,
link |
this is like, I've been in business a long time, right?
link |
I'm building a product for a target market
link |
that doesn't want the product.
link |
Why work, yeah, yeah, why work against the,
link |
swim against the current or whatever.
link |
Right, but you see what's happened,
link |
which is sort of interesting is that,
link |
so a couple of things happened and it was like,
link |
you know, it was like, I don't want to do this project
link |
because I can do so many other things,
link |
which I'm really interested in where, you know,
link |
people say, great, thanks for those tools.
link |
Thanks for those ideas, et cetera.
link |
Whereas, you know, if you're dealing with kind of a,
link |
you know, a sort of a structure where people are saying,
link |
no, no, we don't want this new stuff.
link |
We don't need any new stuff.
link |
We're really fine with what we're doing.
link |
Yeah, there's like literally like, I don't know,
link |
millions of people who are thankful for Wolfram Alpha.
link |
A bunch of people wrote to me, how thankful,
link |
they are a different crowd
link |
than the theoretical physics community, perhaps.
link |
Yeah, well, but you know,
link |
the theoretical physics community
link |
pretty much uniformly uses Wolfram language
link |
and Mathematica, right?
link |
And so it's kind of like, you know, and that's,
link |
but the thing is what happens, you know,
link |
this is what happens, mature fields do not, you know,
link |
it's like, we're doing what we're doing.
link |
We have the methods that we have
link |
and we're just fine here.
link |
Now what's happened in the last 18 years or so,
link |
I think there's a couple of things have happened.
link |
First of all, the hope that, you know,
link |
string theory or whatever would deliver
link |
the fundamental theory of physics,
link |
that hope has disappeared.
link |
That the, another thing that's happened
link |
is the sort of the interest in computation around physics
link |
has been greatly enhanced
link |
by the whole quantum information,
link |
quantum computing story.
link |
People, you know, the idea there might be something
link |
sort of computational related to physics
link |
has somehow grown.
link |
And I think, you know, it's sort of interesting.
link |
I mean, right now, if we say, you know,
link |
it's like, if you're like,
link |
who else is trying to come up
link |
with the fundamental theory of physics?
link |
It's like, there aren't professional,
link |
no professional physicists, no professional physicists.
link |
What are your, I mean, you've talked with him,
link |
but just as a matter of personalities,
link |
cause it's a beautiful story.
link |
What are your thoughts about Eric Weinstein's work?
link |
You know, I think his, I mean,
link |
he did a PhD thesis in mathematical physics at Harvard.
link |
He's a mathematical physicist.
link |
And, you know, it seems like it's kind of,
link |
you know, it's in that framework.
link |
And it's kind of like,
link |
I'm not sure how much further it's got than his PhD thesis,
link |
which was 20 years ago or something.
link |
And I think that, you know, the, you know,
link |
it's a fairly specific piece of mathematical physics.
link |
That's quite nice.
link |
What trajectory do you hope it takes?
link |
Well, I think in his particular case,
link |
I mean, from what I understand,
link |
which is not everything at all,
link |
but, you know, I think I know the rough tradition,
link |
at least what he's operating in is sort of theory of gauge theories.
link |
Gauge theories, yeah.
link |
Local gauge invariants and so on.
link |
Okay, we are very close to understanding
link |
how local gauge invariants works in our models.
link |
And it's very beautiful.
link |
And, you know, does some of the mathematical structure
link |
that he's enthusiastic about fit?
link |
Quite possibly, yes.
link |
So there might be a possibility of trying to understand
link |
how those things fit, how gauge theory fits.
link |
I mean, the question is, you know,
link |
so there are a couple of things
link |
one might try to get in the world.
link |
So for example, it's like,
link |
can we get three dimensions of space?
link |
We haven't managed to get that yet.
link |
Gauge theory, the standard model of particle physics says,
link |
but it's SU3 cross SU2 cross U1.
link |
Those are the designations of these Lie groups.
link |
It doesn't, but anyway,
link |
so those are sort of representations
link |
of symmetries of the theory.
link |
And so, you know, it is conceivable
link |
that it is generically true.
link |
Okay, so all those are subgroups of a group called E8,
link |
which is a weird, exceptional Lie group, okay?
link |
It is conceivable, I don't know whether it's the case,
link |
that that will be generic in these models,
link |
that it will be generic,
link |
that the gauge invariance of the model has this property,
link |
just as things like general relativity,
link |
which corresponds to the thing called general covariance,
link |
which is another gauge like invariance.
link |
It could conceivably be the case
link |
that the kind of local gauge invariance
link |
that we see in particle physics is somehow generic.
link |
And that would be a, you know,
link |
the thing that's really cool, I think, you know,
link |
sociologically, although this hasn't really hit yet,
link |
is that all of these different things,
link |
all these different things people have been working on
link |
in these, in some cases,
link |
quite abstruse areas of mathematical physics,
link |
an awful lot of them seem to tie into what we're doing.
link |
And, you know, it might not be that way.
link |
That's a beautiful thing, I think.
link |
I mean, but the reason Eric Weinstein is important
link |
is to the point that you mentioned before,
link |
which is, it's strange that the theory of everything
link |
is not at the core of the passion, the dream,
link |
the focus, the funding of the physics community.
link |
It's too hard and people gave up.
link |
I mean, basically what happened is ancient Greece,
link |
people thought we're nearly there.
link |
You know, the world is made of platonic solids.
link |
It's, you know, water is a tetrahedron or something.
link |
We're almost there, okay?
link |
Long period of time where people were like,
link |
no, we don't know how it works.
link |
You know, time of Newton, you know, we're almost there.
link |
Everything is gravitation.
link |
You know, time of Faraday and Maxwell, we're almost there.
link |
Everything is fields, everything is the ether, you know?
link |
And the whole time we're making big progress though.
link |
Oh yes, absolutely.
link |
But the fundamental theory of physics is almost a footnote
link |
because it's like, it's the machine code.
link |
It's like we're operating in the high level languages.
link |
You know, that's what we really care about.
link |
That's what's relevant for our everyday physics.
link |
You talked about different centuries
link |
and the 21st century will be everything is computation.
link |
If that takes us all the way, we don't know,
link |
but it might take us pretty far.
link |
Yes, right, that's right.
link |
And I, but I think the point is that it's like, you know,
link |
if you're doing biology, you might say,
link |
how can you not be really interested in the origin of life
link |
and the definition of life?
link |
Well, it's irrelevant.
link |
You know, you're studying the properties of some virus.
link |
It doesn't matter, you know, where, you know,
link |
you're operating at some much higher level.
link |
And it's the same, what's happening with physics is,
link |
I was sort of surprised actually.
link |
I was sort of mapping out this history of people's efforts
link |
to understand the fundamental theory of physics.
link |
And it's remarkable how little has been done on this question.
link |
And it's, you know, because, you know,
link |
there've been times when there's been bursts of enthusiasm.
link |
Oh, we're almost there.
link |
And then it decays and people just say,
link |
oh, it's too hard, but it's not relevant anyway.
link |
And I think that the thing that, you know,
link |
so the question of, you know, one question is,
link |
why does anybody, why should anybody care, right?
link |
Why should anybody care
link |
what the fundamental theory of physics is?
link |
I think it's intellectually interesting,
link |
but what will be the sort of,
link |
what will be the impact of this?
link |
What, I mean, this is the key question.
link |
What do you think will happen
link |
if we figure out the fundamental theory of physics?
link |
Outside of the intellectual curiosity of us.
link |
Okay, so here's my best guess, okay?
link |
So if you look at the history of science,
link |
I think a very interesting analogy is Copernicus.
link |
Okay, so what did Copernicus do?
link |
There'd been this Ptolemaic system
link |
for working out the motion of planets.
link |
It did pretty well.
link |
It used epicycles, et cetera, et cetera, et cetera.
link |
It had all this computational ways
link |
of working out where planets will be.
link |
When we work out where planets are today,
link |
we're basically using epicycles.
link |
But Copernicus had this different way of formulating things
link |
in which he said, you know,
link |
and the earth is going around the sun,
link |
and that had a consequence.
link |
The consequence was you can use this mathematical theory
link |
to conclude something which is absolutely not
link |
what we can tell from common sense, right?
link |
So it's like, trust the mathematics, trust the science, okay?
link |
Now fast forward 400 years,
link |
and now we're in this pandemic,
link |
and it's kind of like everybody thinks the science
link |
will figure out everything.
link |
It's like from the science,
link |
we can just figure out what to do.
link |
We can figure out everything.
link |
That was before Copernicus.
link |
Nobody would have thought if the science says something
link |
that doesn't agree with our everyday experience,
link |
where we just have to compute the science
link |
and then figure out what to do,
link |
people would say that's completely crazy.
link |
And so your sense is,
link |
once we figure out the framework of computation
link |
that can basically do any,
link |
understand the fabric of reality,
link |
we'll be able to derive totally counterintuitive things.
link |
No, the point I think is the following.
link |
That right now, you know,
link |
I talk about computational irreducibility.
link |
People, you know, I was very proud
link |
that I managed to get the term computational irreducibility
link |
into the congressional record last year.
link |
That's right, by the way,
link |
that's a whole nother topic we could talk about.
link |
Fascinating. Different topic.
link |
But Tim, in any case, you know,
link |
but so computational irreducibility
link |
is one of these sort of concepts
link |
that I think is important in understanding
link |
lots of things in the world.
link |
But the question is, it's only important
link |
if you believe the world is fundamentally computational.
link |
But if you know the fundamental theory of physics
link |
and it's fundamentally computational,
link |
then you've rooted the whole thing.
link |
That is, you know the world is computational.
link |
And while you can discuss whether, you know,
link |
it's not the case that people would say,
link |
well, you have this whole computational irreducibility,
link |
all these features of computation.
link |
We don't care about those
link |
because after all the world isn't computational,
link |
But if you know, you know, base, base, base thing,
link |
physics is computational,
link |
then you know that that stuff is, you know,
link |
that that's kind of the grounding for that stuff.
link |
Just as in a sense Copernicus was the grounding
link |
for the idea that you could figure out something
link |
with math and science
link |
that was not what you would intuitively think
link |
So now we've got to this point where, for example,
link |
we say, you know, once we have the idea
link |
that computation is the foundational thing
link |
that explains our whole universe,
link |
then we have to say, well, what does it mean
link |
Like it means there's computational irreducibility.
link |
That means science is limited in certain ways.
link |
That means this, that means that.
link |
But the fact that we have that grounding means that,
link |
you know, and I think, for example, for Copernicus,
link |
for instance, the implications of his work
link |
on the set of mathematics of astronomy were cool,
link |
but they involved a very small number of people.
link |
The implications of his work for sort of the philosophy
link |
of how you think about things were vast
link |
and involved, you know, everybody more or less.
link |
But do you think, so that's actually the way scientists
link |
and people see the world around us.
link |
So it has a huge impact in that sense.
link |
Do you think it might have an impact more directly
link |
to engineering derivations from physics,
link |
like propulsion systems, our ability to colonize the world?
link |
Like, for example, okay, this is like sci fi,
link |
but if you understand the computational nature, say,
link |
of the different forces of physics, you know,
link |
there's a notion of being able to warp gravity,
link |
Yeah, can we make warp drive?
link |
So like, would we be able to, will, you know,
link |
will like Elon Musk start paying attention?
link |
Like it's awfully costly to launch these rockets.
link |
Do you think we'll be able to, yeah, create warp drive?
link |
And, you know, I set myself some homework.
link |
I agreed to give a talk at some NASA workshop
link |
in a few weeks about faster than light travel.
link |
So I haven't figured it out yet, but no, but.
link |
You got two weeks.
link |
But do you think that kind of understanding
link |
of fundamental theory of physics can lead
link |
to those engineering breakthroughs?
link |
Okay, I think it's far away, but I'm not certain.
link |
I mean, you know, this is the thing that,
link |
I set myself an exercise when gravity waves,
link |
gravitational waves were discovered, right?
link |
I set myself the exercise of what would black hole
link |
technology look like?
link |
In other words, right now, you know,
link |
black holes are far away.
link |
They're, you know, how on earth can we do things with them?
link |
But just imagine that we could get, you know,
link |
pet black holes right in our backyard.
link |
You know, what kind of technology could we build with them?
link |
I got a certain distance, not that far,
link |
but I think in, you know, so there are ideas, you know,
link |
I have this, one of the weirder ideas is things
link |
I'm calling space tunnels,
link |
which are higher dimensional pieces of space time,
link |
where basically you can, you know,
link |
in our three dimensional space,
link |
there might be a five dimensional, you know,
link |
region, which actually will appear as a white hole
link |
at one end and a black hole at the other end,
link |
you know, who knows whether they exist.
link |
And then the questions, another one,
link |
okay, this is another crazy one,
link |
is the thing that I'm calling a vacuum cleaner, okay?
link |
So, I mentioned that, you know,
link |
there's all this activity in the universe,
link |
which is maintaining the structure of space.
link |
And that leads to a certain energy density
link |
effectively in space.
link |
And so the question, in fact, dark energy
link |
is a story of essentially negative mass
link |
produced by the absence of energy
link |
you thought would be there, so to speak.
link |
And we don't know exactly how it works
link |
in either our model or the physical universe,
link |
but this notion of a vacuum cleaner is a thing where,
link |
you know, you have all these things
link |
that are maintaining the structure of space,
link |
but what if you could clean out some of that stuff
link |
that's maintaining the structure of space
link |
and make a simpler vacuum somewhere?
link |
You know, what would that do?
link |
A totally different kind of vacuum.
link |
Right, and that would lead to negative energy density,
link |
which would need to, so gravity is usually
link |
a purely attractive force, but negative mass
link |
would lead to repulsive gravity
link |
and lead to all kinds of weird things.
link |
Now, can it be done in our universe?
link |
You know, my immediate thought is no,
link |
but you know, the fact is that, okay, so here's the thing.
link |
Well, once you understand the fact,
link |
because you're saying like, at this level of abstraction,
link |
can we reach to the lower levels and mess with it?
link |
Once you understand the levels, I think you can start to.
link |
I know, and I'm, you know, I have to say
link |
that this reminds me of people telling one years ago
link |
that, you know, you'll never transmit data
link |
over a copper wire at more than 1,000,
link |
you know, 1,000 board or something, right?
link |
And this is, why did that not happen?
link |
You know, why do we have this much,
link |
much faster data transmission?
link |
Because we've understood many more of the details
link |
of what's actually going on.
link |
And it's the same exact story here.
link |
And it's the same, you know, I think that this,
link |
as I say, I think one of the features of sort of,
link |
one of the things about our time
link |
that will seem incredibly naive in the future
link |
is the belief that, you know, things like heat
link |
is just random motion of molecules,
link |
that it's just throw up your hands, it's just random.
link |
We can't say anything about it.
link |
That will seem naive.
link |
Yeah, at the heat death of the universe,
link |
those particles would be laughing at us humans thinking.
link |
That life is not beautiful.
link |
I'll have a whole civilization, you know.
link |
Humans used to think they're special
link |
with their little brains.
link |
Well, right, but also, and they used to think
link |
that this would just be random and uninteresting.
link |
But that's, but so this question about whether you can,
link |
you know, mess with the underlying structure
link |
and how you find a way to mess with the underlying structure,
link |
that's a, you know, I have to say, you know,
link |
my immediate thing is, boy, that seems really hard,
link |
but then, and you know,
link |
possibly computational irreducibility will bite you,
link |
but then there's always some path
link |
of computational reducibility.
link |
And that path of computational reducibility
link |
is the engineering invention that has to be made.
link |
Those little pockets can have huge engineering impact.
link |
Right, and I think that that's right.
link |
And I mean, we live in, you know, we make use of so many
link |
And the fact is, you know, I, you know, this is, yes,
link |
it's a, you know, it's one of these things where,
link |
where, you know, I'm a person who likes to figure out ideas
link |
and so on, and the sort of tests of my level of imagination,
link |
And so a couple of places where there's sort of serious
link |
humility in terms of my level of imagination,
link |
one is this thing about different reference frames
link |
for understanding the universe,
link |
where like, imagine the physics of the aliens,
link |
what will it be like?
link |
And I'm like, that's really hard.
link |
I don't know, you know?
link |
And I mean, I think that...
link |
But once you have the framework in place,
link |
you can at least reason about the things you don't know,
link |
maybe can't know, or like, it's too hard for you to know,
link |
but then the mathematics can, that's exactly it,
link |
allow you to reach beyond where you can reason about.
link |
So I'm, you know, I'm trying to not have, you know,
link |
if you think back to Alan Turing, for example,
link |
and, you know, when he invented Turing machines, you know,
link |
and imagining what computers would end up doing,
link |
You know, and it's...
link |
It's very difficult.
link |
It's difficult, right.
link |
And it's, and I mean, this thing...
link |
Made a few reasonable predictions,
link |
but most of it, he couldn't predict, possibly.
link |
By the time, by 1950, he was making reasonable predictions
link |
about some things.
link |
But not the 30s, yeah.
link |
Right, not when he first, you know, conceptualized,
link |
you know, and he conceptualized universal computing
link |
for a very specific mathematical reason
link |
that wasn't as general.
link |
But yes, it's a good sort of exercise in humility
link |
to realize that it's kind of like,
link |
it's really hard to figure these things out.
link |
The engineering of the universe,
link |
if we know how the universe works, how can we engineer it?
link |
That's such a beautiful vision.
link |
That's such a beautiful vision.
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By the way, I have to mention one more thing,
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which is the ultimate question from physics is,
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okay, so we have this abstract model of the universe.
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Why does the universe exist at all, right?
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So, you know, we might say there is a formal model
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that if you run this model, you get the universe,
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or the model gives you, you know, a model of the universe,
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right, you run this mathematical thing
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and the mathematics unfolds in the way
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that corresponds to the universe.
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But the question is, why was that actualized?
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Why does the actual universe actually exist?
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And so this is another one of these humility
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and it's like, can you figure this out?
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I have a guess, okay, about the answer to that.
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And my guess is somewhat unsatisfying,
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but my guess is that it's a little bit similar
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to Gödel's second incompleteness theorem,
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which is the statement that from within,
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as an axiomatic theory like piano arithmetic,
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you cannot from within that theory
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prove the consistency of the theory.
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So my guess is that for entities within the universe,
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there is no finite determination that can be made
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of the statement the universe exists
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is essentially undecidable to any entity
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that is embedded in the universe.
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Within that universe, how does that make you feel?
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Does that put you at peace that it's impossible,
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or is it really ultimately frustrating?
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Well, I think it just says that it's not a kind of question
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that, you know, there are things that it is reasonable.
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I mean, there's kinds of, you know,
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you can talk about hyper computation as well.
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You can say, imagine there was a hyper computer,
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here's what it would do.
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So okay, great, it would be lovely to have a hyper computer,
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but unfortunately we can't make it in the universe.
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Like it would be lovely to answer this,
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but unfortunately we can't do it in the universe.
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And you know, this is all we have, so to speak.
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And I think it's really just a statement.
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It's sort of, in the end, it'll be a kind of a logical,
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logically inevitable statement, I think.
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I think it will be something where it is,
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as you understand what it means to have,
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what it means to have a sort of predicate of existence
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and what it means to have these kinds of things,
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it will sort of be inevitable that this has to be the case,
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that from within that universe, you can't establish
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the reason for its existence, so to speak.
link |
You can't prove that it exists and so on.
link |
And nevertheless, because of computational reducibility,
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the future is ultimately not predictable, full of mystery,
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and that's what makes life worth living.
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Right, I mean, right.
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And you know, it's funny for me,
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because as a pure sort of human being doing what I do,
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it's, you know, like I'm interested in people,
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I like sort of the whole human experience, so to speak.
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And yet, it's a little bit weird when I'm thinking,
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you know, it's all hypergraphs down there,
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and it's all just.
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Hypergraphs all the way down.
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It's like turtles all the way down.
link |
Yeah, yeah, right.
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And it's kind of, you know, to me, it is a funny thing,
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because every so often I get this, you know,
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as I'm thinking about, I think we've really gotten,
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you know, we've really figured out kind of the essence
link |
of how physics works, and I'm like thinking to myself,
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you know, here's this physical thing,
link |
and I'm like, you know,
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this feels like a very definite thing.
link |
How can it be the case that this is just
link |
some rule or reference frame of, you know,
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this infinite creature that is so abstract and so on?
link |
And I kind of, it is a, it's a funny sort of feeling
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that, you know, we are, we're sort of, it's like,
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in the end, it's just sort of,
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we're just happy we're just humans type thing.
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And it's kind of like, but we're making,
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we make things as, it's not like we're just a tiny speck.
link |
We are, in a sense, the, we are more important
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by virtue of the fact that, in a sense,
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it's not like there's, there is no ultimate, you know,
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it's like, we're important because,
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because, you know, we're here, so to speak,
link |
and we're not, it's not like there's a thing
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where we're saying, you know, we are just but one
link |
sort of intelligence out of all these other intelligences.
link |
And so, you know, ultimately there'll be
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the super intelligence, which is all of these put together
link |
and they'll be very different from us.
link |
No, it's actually going to be equivalent to us.
link |
And the thing that makes us a sort of special
link |
is just the details of us, so to speak.
link |
It's not something where we can say,
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oh, there's this other thing, you know,
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just, you think humans are cool,
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just wait until you've seen this.
link |
You know, it's going to be much more impressive.
link |
Well, no, it's all going to be
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kind of computationally equivalent.
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And the thing that, you know, it's not going to be,
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oh, this thing is amazingly much more impressive
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and amazingly much more meaningful, let's say.
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I mean, that's the...
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And the symbolism of this particular moment.
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So this has been one of the,
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one of the favorite conversations I've ever had, Stephen.
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It's a huge honor to talk to you,
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to talk about a topic like this for four plus hours
link |
on the fundamental theory of physics.
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And yet we're just two finite descendants of apes
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that have to end this conversation
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because darkness have come upon us.
link |
Right, and we're going to get bitten by mosquitoes
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and all kinds of terrible things.
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The symbolism of that,
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we're talking about the most basic fabric of reality
link |
and having to end because of the fact that things end.
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It's tragic and beautiful, Stephen.
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Thank you so much.
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I can't wait to see what you do in the next couple of days
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and next week, a month.
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We're all watching with excitement.
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Thank you so much.
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Thanks for listening to this conversation
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with Stephen Wolfram.
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And thank you to our sponsors,
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Please check out our sponsors in the description
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If you enjoy this thing, subscribe on YouTube,
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And now let me leave you with some words
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from Richard Feynman.
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Physics isn't the most important thing, love is.
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Thank you for listening and hope to see you next time.