Stephen Wolfram: Complexity and the Fabric of Reality
Guests: Stephen Wolfram
Full transcript · Read the full conversation in order, with its original context.
Contents27 chapters
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The following is a conversation with Stephen Wolfram,
his third time on the podcast.
He's a computer scientist, mathematician,
theoretical physicist, and the founder of Wolfram Research,
a company behind Mathematica, Wolfram Alpha,
Wolfram Language, and the new Wolfram Physics Project.
This conversation is a wild, technical rollercoaster ride
through topics of complexity, mathematics, physics,
computing, and consciousness.
I think this is what this podcast is becoming, a wild ride.
Some episodes are about physics, some about robots,
some are about war and power,
some are about the human condition
and our search for meaning,
and some are just what the comedian Tim Dillon calls fun.
This is the Lex Friedman Podcast.
To support it, please check out the sponsors
in the description.
And now, here's my conversation with Stephen Wolfram.
Almost 20 years ago, you published "A New Kind of Science,"
where you presented a study of complexity
and an approach for modeling of complex systems.
So, let us return again to the core idea of complexity.
What is complexity?
- I don't know, I think that's not
the most interesting question.
It's like, you know, if you ask a biologist, what is life?
- Yeah.
- That's not the question they care the most about.
What I was interested in is how does something
that we would usually identify as complexity
arise in nature?
And I got interested in that question like 50 years ago,
which is really embarrassingly a long time ago.
And, you know, I was, you know,
how does snowflakes get to have complicated forms?
How do galaxies get to have complicated shapes?
How does, you know, how do living systems get produced?
Things like that.
And the question is, what's the sort of underlying
scientific basis for those kinds of things?
And the thing that I was at first very surprised by,
'cause I've been doing physics and particle physics
and fancy mathematical physics and so on.
And it's like, I know all this fancy stuff.
I should be able to solve this sort of basic
science question.
And I couldn't, this was like early,
maybe 1980-ish timeframe.
And it's like, okay, what can one do to understand
the sort of basic secret that nature seems to have?
'Cause it seems like nature, you know,
you look around in the natural world,
it's full of incredibly complicated forms.
You look at sort of most engineered kinds of things,
for instance, they tend to be, you know,
we got sort of circles and lines and things like this.
The question is, what secret does nature have
that lets it make all this complexity
that we in doing engineering, for example,
don't naturally seem to have?
And so that was the kind of the thing
that I got interested in.
And then the question was, you know,
could I understand that with things
like mathematical physics?
Well, it didn't work very well.
So then I got to thinking about, okay,
is there some other way to try to understand this?
And then the question was,
if you're going to look at some system in nature,
how do you make a model for that system,
for what that system does?
So, you know, a model is some abstract representation
of the system, some formal representation of the system.
What is the raw material
that you can make that model out of?
And so what I realized was, well, actually,
programs are really good source of raw material
for making models of things.
And, you know, in terms of my personal history,
the, to me, that seemed really obvious.
And the reason it seemed really obvious
is because I just spent several years building
this big piece of software that was sort of a predecessor
to Mathematica and Morphine Language,
thing called SMP, Symbolic Manipulation Program,
which was something that had this idea
of starting from just these computational primitives
and building up everything one had to build up.
And so kind of the notion of, well,
let's just try and make models
by starting from computational primitives
and seeing what we can build up.
That seemed like a totally obvious thing to do.
In retrospect, it might not have been externally
quite so obvious, but it was obvious to me at the time,
given the path that I happened to have been on.
So, you know, so that got me into this question
of let's use programs to model what happens in nature.
And the question then is, well, what kind of programs?
And, you know, we're used to programs
that you write for some particular purpose
and it's a big, long piece of code
and it does some specific thing.
But what I got interested in was, okay,
if you just go out into the sort of computational universe
of possible programs, you say,
take the simplest program you can imagine, what does it do?
And so I started studying these things
called cellular automata.
Actually, I didn't know at first
they were called cellular automata,
but I found that out subsequently.
But it's just a line of cells, you know,
each one is black or white,
and it's just some rule that says the color of the cell
is determined by the color that it had on the previous step
and its two neighbors on the previous step.
And I had initially thought that, you know,
sufficiently simple setup
is not gonna do anything interesting.
It's always gonna be simple, no complexity,
simple rule, simple behavior.
Okay, but then I actually ran the computer experiment,
which is pretty easy to do.
I mean, it probably took a few hours originally.
And the results were not what I'd expected at all.
Now, needless to say,
in the way that science actually works,
the results that I got had a lot of unexpected things
which I thought were really interesting,
but the really strongest result,
which was already right there in the printouts I made,
I didn't really understand for a couple more years.
So it was not, you know,
the compressed version of the story is
you run the experiment
and you immediately see what's going on,
but I wasn't smart enough to do that, so to speak.
But the big thing is,
even with very simple rules of that type,
sort of the minimal, tiniest program,
sort of the one line program or something,
it's possible to get very complicated behavior.
My favorite example is this thing called rule 30,
which is a particular cellular automaton rule.
You just start it off from one black cell
and it makes this really complicated pattern.
And so that, for me, was sort of a critical discovery
that then kind of said, playing back onto, you know,
how does nature make complexity?
I sort of realized that might be how it does it.
That might be kind of the secret that it's using
is that in this kind of computational universe
of possible programs,
it's actually pretty easy to get programs
where even though the program is simple,
the behavior when you run the program is not simple at all.
And that was, so for me,
that was the kind of the story of kind of how,
that was sort of the indication that one had got an idea
of what the sort of secret that nature uses
to make complexity
and how complexity can be made in other places.
Now, if you say, what is complexity?
You know, it's, complexity is,
it's not easy to tell what's going on.
That's the informal version of what is complexity.
- But there is something going on.
- But there's a rule. - But it's not easy
to know what. - Right.
- Well, no, the rules can generate just randomness, right?
- Well, that's not obvious.
In other words-- - That's not obvious,
that's right. - It's not obvious at all.
And it wasn't what I expected.
It's not what people's intuition had been
and has been for, you know, for a long time.
That is, one might think you have a rule,
you can tell there's a rule behind it.
I mean, it's just like, you know,
the early, you know, robots in science fiction movies,
right, you can tell it's a robot
'cause it does simple things, right?
Turns out that isn't actually the right story,
but it's not obvious that isn't the right story
'cause people assume simple rules, simple behavior.
And that, the sort of the key discovery
about the computational universe is that isn't true.
And that discovery goes very deep
and relates to all kinds of things
that I've spent years and years studying.
But, you know, in the end,
the sort of the what is complexity is,
well, you can't easily tell what it's going to do.
You could just run the rule and see what happens,
but you can't just say, oh, you know, show me the rule.
Great, now I know what's gonna happen.
And, you know, the key phenomenon around that
§02Transcript · 08:05–16:19
is this thing I call computational irreducibility.
This fact that in something like rule 30,
you might say, well, what's it gonna do
after a million steps?
Well, you can run it for a million steps
and just do what it does to find out,
but you can't compress that, you can't reduce that
and say, I'm gonna be able to jump ahead and say,
this is what it's gonna do after a million steps,
but I don't have to go through anything
like that computational effort.
- By the way, has anybody succeeded at that?
Do you had to challenge a competition
for predicting the middle column of rule 30?
- Indeed.
- Anybody?
- A number of people have sent things in
and sort of people are picking away at it,
but it's hard.
I mean, I've been actually even proving
that the center column of rule 30 doesn't repeat.
That's something I think might be doable, okay?
- Mathematically proving.
- Yes, and so that's analogous to a similar kind of thing
is like the digits of pi,
which are also generated in this very deterministic way.
And so a question is how random are the digits of pi?
For example, does every, first of all,
do the digits of pi ever repeat?
Well, we know they don't.
This was proved in the 1800s
that pi is not a rational number.
So that means only rational numbers
have digit sequences that repeat.
So we know the digits of pi don't repeat.
So now the question is, does, you know,
zero, one, two, three, or whatever,
do all the digits base 10 or base two,
or however you work it out,
do they all occur with equal frequency?
Nobody knows.
That's far away from what can be understood mathematically
at this point.
And that's kind of, but I'm even looking for step one,
which is prove that the center column doesn't repeat
and then prove other things about it,
like equidistribution of equal numbers of zeros and ones.
And those are things which I, you know,
I kind of set up this little prize thing
because I thought those were not too out of range.
Those are things which are within, you know,
a modest amount of time,
it's conceivable that those could be done.
They're not far away from what current mathematics
might allow.
They'll require a bunch of cleverness
and hopefully some interesting new ideas
that, you know, will be useful other places.
- But you started in 1980 with this idea
before I think you realized, you know,
this idea of programs.
You thought that there might be some kind of
thermodynamic like randomness,
and then the complexity comes from a clever filter
that you kind of, like, I don't know,
spaghetti or something.
You filter the randomness and outcomes complexity,
which is an interesting intuition.
How do we know that's not actually what's happening?
So just because you were then able to develop,
look, you don't need this like incredible randomness.
You can just have very simple, predictable,
initial conditions and predictable rules.
And then from that emerged complexity.
Still, there might be some systems
where it's filtering randomness in the inputs.
- Well, the point is, when you have quotes
randomness in the input,
that means there's all kinds of information in the input.
And in a sense, what you get out
will be maybe just something close to what you put in.
Like people are very, in dynamical systems theory,
sort of big area of mathematics that developed
from the early 1900s and really got big in the 1980s.
You know, an example of what people study there a lot,
and it's popular version is chaos theory.
And an example of what people study a lot
is the shift map, which is basically taking 2X mod one,
the fractional part of 2X,
which is basically just taking digits in binary
and shifting them to the left.
So at every step, you get to see,
if you say, how big is this number that I got out?
Well, the most important digit in that number
is whatever ended up at the left-hand end.
But now if you start off from an arbitrary random number,
which is quotes randomly chosen,
so all its digits are random,
then when you run that sort of chaos theory shift map,
all that you get out is just whatever you put in.
You just get to see what you,
it's not obvious that you would excavate
all of those digits.
And if you're, for example, making a theory,
I don't know, fluid mechanics, for example,
if there was that phenomenon in fluid mechanics,
then the equations of fluid mechanics can't be right.
Because what that would be saying is,
the equations of that it matters to the fluid,
what happens in the fluid at the level of the,
you know, millionth digit of the initial conditions,
which is far below the point at which you're hitting
kind of sizes of molecules and things like that.
So it's kind of almost explaining,
if that phenomenon is an important thing,
it's kind of telling you that the, you know,
fluid dynamics, which describes fluids as continuous media
and so on, isn't really right.
But so, you know, so this idea that, you know,
there's a, it's a tricky thing,
because as soon as you put randomness in,
you have to know, you know, what,
how much of what's coming out is what you put in,
versus how much is actually something
that's being generated.
And what's really nice about these systems
where you just have very simple initial conditions,
and where you get random stuff out,
or seemingly random stuff out,
is you don't have that issue.
You don't have to argue about,
was there something complicated put in?
Because it's plainly obvious there wasn't.
Now, as a practical matter in doing experiments,
the big thing is, if the thing you see
is complex and reproducible,
then it didn't come from just filtering some,
quotes, "randomness from the outside world."
It has to be something that is intrinsically made,
because it wouldn't otherwise be, I mean, you know,
it could be the case that you set things up,
and it's always the same each time,
and you say, well, it's kind of the same,
but it's not random each time,
because it's kind of the definition of it being random,
is it was kind of picked at random each time, so to speak.
- So is it possible to for sure know
that our universe does not,
at the fundamental level, have randomness?
Is it possible to conclusively say,
there's no randomness at the bottom?
- Well, it's an interesting question.
I mean, you know, science, natural science,
is an inductive business, right?
You observe a bunch of things, and you say,
can we fit these together?
What is our hypothesis for what's going on?
The thing that I think I can say fairly definitively
is at this point, we understand enough
about fundamental physics that there is,
if there was sort of an extra dice being thrown,
it's something that doesn't need to be there.
We can get what we see without that.
Now, you know, could you add that in
as an extra little featureoid, you know,
without breaking the universe?
Probably, but in fact, almost certainly yes.
But is it necessary for understanding the universe?
No, and I think actually from a more fundamental
point of view, it's, I think I might be able to argue.
So one of the things that I've been interested in,
I've been pretty surprised that I've had anything
sentient to say about, is the question of,
why does the universe exist?
I didn't think that was a question that I would,
you know, I thought that was a far out there
metaphysical kind of thing.
Even the philosophers have stayed away
from that question for the most part.
It's so such a kind of, you know,
difficult to address question.
But I actually think to my great surprise
that from our physics project and so on,
that it is possible to actually address that question
and explain why the universe exists.
And I kind of have a suspicion, I've not thought it through.
I kind of have a suspicion that that explanation
will eventually show you that in no meaningful sense,
can there be randomness underneath the universe?
That is that if there is, it's something
that is necessarily irrelevant
to our perception of the universe.
That is that it could be there, but it doesn't matter.
Because in a sense, we've already, you know,
whatever it would do, whatever extra thing it would add
is not relevant to our perception of what's going on.
- So why does the universe exist?
How does the irrelevance of randomness
connect to the big why question of the universe?
- So, okay, so I mean, why does the universe exist?
§03Transcript · 16:22–24:06
Well, let's see.
- And is this the only universe we got?
- It's the only one.
About that, I'm pretty sure.
- So you may be, which one,
which of these topics is better to enter first?
Why does the universe exist?
And why you think it's the only one that exists?
- Well, I think they're very closely related.
- Okay. - Okay?
So I mean, the first thing, let's see.
I mean, this why does the universe exist question
is built on top of all these things
that we've been figuring out about fundamental physics.
'Cause if you wanna know why the universe exists,
you kind of have to know what the universe is made of.
And I think the, well, let me describe a little bit
about the why does the universe exist question.
So the main issue is,
let's say you have a model for the universe.
And you say, I've got this program or something,
and you run it and you make the universe.
Now you say, well, how do you actually,
why is that program actually running?
And people say, you've got this program
that makes the universe, what computer is it running on?
Right?
What does it mean?
What actualizes something?
You know, two plus two equals four,
but that's different from saying there's two,
a pile of two rocks, another pile of two rocks,
and somebody moves them together and makes four,
so to speak.
And so what is it that kind of turns it
from being just this formal thing
to being something that is actualized?
Okay, so there we have to start thinking about,
well, what do we actually know
about what's going on in the universe?
Well, we are observers of this universe,
but confusingly enough, we're part of this universe.
So in a sense, if we say,
what do we know about what's going on in the universe?
Well, what we know is what sort of our consciousness records
about what's going on in the universe.
- And consciousness is part of the fabric of the universe,
so we're in it.
- Yes, we're in it.
And maybe I should start off
by saying something about the consciousness story,
because that's--
- Yes, maybe we should begin even before that,
at the very base layer of the Wolfram Physics Project.
Maybe you can give a broad overview, once again,
really quick about this hypergraph model.
- Yes. - And also,
what is it, a year and a half ago,
since you've brought this project to the world,
what is the status update?
What are all the beautiful ideas you have come across?
What are the interesting things you can sort of mention?
- I mean, it's a frigging Cambrian explosion.
I mean, it's crazy.
I mean, there are all these things,
which I've kind of wondered about for years,
and suddenly, there's actually a way to think about them.
And I really did not see,
I mean, the real strength of what's happened,
I absolutely did not see coming.
And the real strength of it is,
we've got this model for physics,
but it turns out it's a foundational kind of model
that's a different kind of computation-like model,
that I'm kind of calling the sort of
multi-computational model.
And that that kind of model is applicable,
not only to physics,
but also to lots of other kinds of things.
And one reason that's extremely powerful
is because physics has been very successful.
So we know a lot, based on what we figured out in physics.
And if we know that the same model governs physics,
and governs, I don't know, economics,
linguistics, immunology, whatever,
we know that the same kind of model governs those things,
we can start using things
that we've successfully discovered in physics,
and applying those intuitions in all these other areas.
And that's pretty exciting, and very surprising to me.
And in fact, it's kind of like,
in the original story of, sort of,
you go and you explain why is there complexity
in the natural world,
then you realize, well, there's all this complexity,
there's all this computational irreducibility,
you know, there's a lot we can't know
about what's going to happen.
It's kind of a very confusing thing for people who say,
you know, science has nailed everything down,
we're gonna, you know,
based on science, we can know everything.
Well, actually, there's this computational irreducibility
thing right in the middle of that,
thrown up by science, so to speak.
And then the question is,
well, given computational irreducibility,
how can we actually figure out anything
about what happens in the world?
Why aren't we, why are we able to predict anything?
Why are we able to sort of operate in the world?
And the answer is that we sort of live
in these slices of computational reusability
that exist in this kind of ocean
of computational irreducibility.
And it turns out that it seems
that it's a very fundamental feature
of the kind of model that seems to operate in physics
and perhaps in a lot of these other areas,
that there are these particular slices
of computational reducibility that are relevant to us.
And those are the things that both allow us
to operate in the world
and not just have everything be completely unpredictable,
but they're also things that potentially give us
what amount to sort of physics-like laws
in all these other areas.
So that's been sort of an exciting thing,
but I would say that in general for our project,
it's been going spectacularly well.
I mean, you know, it's very, honestly,
it wasn't something I expected to happen in my lifetime.
I mean, it's, you know, it's something where it's,
and in fact, one of the things about it,
some of the things that we've discovered
are things where I was pretty sure
that wasn't how things worked and turns out I'm wrong.
And, you know, in a major area in meta-mathematics,
I'd be realizing that something I've long believed,
we can talk about it later,
that just really isn't right.
But I think that the thing that,
so what's happened with the physics project,
I mean, you know, it's a,
it can explain a little bit about how the model works,
but basically--
- We can maybe ask you the following question.
So it's easy through words
to describe how cellular automata works.
You've explained this.
And it's the fundamental mechanism
by which you, in your book,
a new kind of science,
explored the idea of complexity
and how to do science in this world
of reducible islands and irreducible,
generally irreducibility.
Okay, so how does the model of hypergraphs
differ from cellular automata?
And how does the idea of multi-computation differ?
Like maybe that's a way to describe it.
- Right, we're, yeah, right.
This is a, you know, my life is,
like all of our lives,
something of a story of computational irreducibility.
And, you know, it's been going for a few years now.
So it's always a challenge
to kind of find these appropriate pockets of reducibility.
But let me see what I can do.
So, I mean, first of all,
let's talk about physics, first of all.
And, you know, a key observation,
the starting point of our physics project
is things about what is space?
What is the universe made of?
And, you know, ever since Euclid,
people just sort of say space is just this thing
where you can put things at any position you want.
And they're just points,
and they're just geometrical things
that you can just arbitrarily put
at different coordinate positions.
Sort of the first thing in our physics project
is the idea that space is made of something.
Just like water is made of molecules,
space is made of kind of atoms of space.
And the only thing we can say about these atoms of space
is they have some identity.
It's this atom as opposed to this atom.
And, you know, you could give them,
if you were a computer person,
you'd give them UUIDs or something.
(Luke laughs)
- Yes.
- And that's all there is to say about them, so to speak.
And then all we know about these atoms of space
is how they relate to each other.
So we say these three atoms of space
are associated with each other in some relation.
So you can think about that as, you know,
§04Transcript · 24:08–32:19
what atom of space is friends
with what other atom of space?
You can build this essentially friend network
of the atoms of space.
And the sort of starting point of our physics project
is that's what our universe is.
It's a giant friend network of the atoms of space.
And so how can that possibly represent our universe?
Well, it's like in something like water,
you know, there are molecules bouncing around,
but on a large scale, that produces fluid flow
and we have fluid vortices
and we have all of these phenomena
that are sort of the emergent phenomena
from that underlying kind of collection of molecules
bouncing around.
And by the way, it's important
that that collection of molecules bouncing around
have this phenomenon of computational irreducibility.
That's actually what leads
to the second law of thermodynamics among other things.
And that leads to the sort of randomness
of the underlying behavior,
which is what gives you something,
which on a large scale
seems like it's a smooth, continuous type of thing.
And so, okay, so first thing is space is made of something.
It's made of all these atoms of space
connected together in this network.
And then everything that we experience
is sort of features of that structure of space.
So, you know, when we have an electron or something
or a photon, it's some kind of tangle
in the structure of space,
much like kind of a vortex in a fluid
would be just this thing that is, you know,
it can actually, the vortex can move around,
it can involve different molecules in the fluid,
but the vortex still stays there.
- And if you zoom out enough,
the vortex looks like an atom itself,
like a basic element.
- Yes.
- So there's the levels of abstraction.
If you squint and kind of blur things out,
it looks like at every level of abstraction,
you can define what is a basic individual entity.
- Yes, but, you know, in this model,
there's a bottom level.
- Yeah.
- You know, there's an elementary length,
maybe 10 to the minus 100 meters, let's say,
which is really small, you know,
a proton is 10 to the minus 15 meters.
The smallest we've ever been able to sort of see
with a particle accelerator
is around 10 to the minus 21 meters.
So, you know, if we don't know precisely
what the correct scale is,
but it's perhaps over the order of 10 to the minus 100 meters
so it's pretty small.
And, but that's the end, that's what things are made of.
- What's your intuition
where the 10 to the minus 100 comes from?
What's your intuition about this scale?
- Well, okay, so there's a calculation
which I consider to be somewhat rickety, okay?
Which has to do with comparing,
so there are various fundamental constants.
There's the speed of light,
the speed of light, once you know the elementary time,
the speed of light tells you the conversion
from the elementary time to the elementary length.
Then there's the question of how do you convert
to the elementary energy?
And how do you convert to, between other things?
And the various constants we know,
we know the speed of light,
we know the gravitational constant,
we know Planck's constant in quantum mechanics,
those are the three important ones.
And we actually know some other things,
we know things like the size of the universe,
the Hubble constant, things like that.
And essentially this calculation of the elementary length
comes from looking at the sort of combination of those,
okay, so the most obvious thing,
people have sort of assumed that quantum gravity
happens at this thing, the Planck scale,
10 to the minus 34 meters,
which is the sort of the combination of Planck's constant
and the gravitational constant, the speed of light,
that gives you that kind of length.
Turns out in our model, there is an additional parameter,
which is essentially the number of simultaneous threads
of execution of the universe,
which is essentially the number of sort of
independent quantum processes that are going on.
And that number, let's see if I remember that number,
that number is 10 to the 170, I think,
and so it's a big number,
but that number then connects,
sort of modifies what you might think
from all these Planck units
to give you the things we're giving.
And there's been sort of a mystery actually,
and there's a more technical physics thing,
that the Planck mass, the Planck energy,
Planck energy is actually surprisingly big.
The Planck length is tiny, 10 to the minus 34 meters,
the Planck time, 10 to the minus 43 meters, I think,
that seconds, I think,
but the Planck energy is like the energy
of a lightning strike, okay, which is pretty weird.
In our models, the actual elementary energy
is that divided by the number of sort of
simultaneous quantum threads,
and it ends up being really small too.
And that sort of explains that mystery
that's been around for a while
about how Planck units work.
But whether that precise estimate is right,
we don't know yet.
I mean, that's one of the things
that's sort of been a thing we've been pretty interested in
is how do you see through,
how do you make a gravitational microscope
that can kind of see through to the atoms of space?
You know, how do you get, in fluid flow, for example,
if you go to hypersonic flow or something,
you know, you've got a Mark 20,
you know, space plane or something,
it really matters that there are individual molecules
hitting the space plane, not a continuous fluid.
The question is, what is the analogous kind of,
what is the analog of hypersonic flow
for things about the structure of space-time?
And it looks like a rapidly rotating black hole, right,
at the sort of critical rotation rate,
it looks as if that's a case where essentially
the structure of space-time is just about to fall apart.
And you may be able to kind of see the evidence
of sort of discrete elements, you know,
you may be able to kind of see there
the sort of gravitational microscope
of actually seeing these discrete elements of space.
And there may be some effect in, for example,
gravitational waves produced by rapidly rotating black hole
that in which one could actually see some phenomenon
where one can say, yes,
these don't come out the way one would expect
based on having a continuous structure of space-time,
that is it something where you can kind of see through
to the discrete structure?
We don't know that yet.
- So can you maybe elaborate a little bit deeper
how a microscope that can see to 10 to the minus 100,
how rotating black holes and presumably
the detailed accurate detection of gravitational waves
from such black holes can reveal the discreteness of space?
- Okay, first thing is what is a black hole?
Actually, we need to go a little bit further
in the story of what space-time is
'cause I explained a little bit about what space is,
but I didn't talk about what time is.
And that's sort of important
in understanding space-time, so to speak.
- And your sense is both space and time
in the story are discrete.
- Absolutely, absolutely.
But it's a complicated story.
And needless to say.
- Well, it's simple at the bottom.
- It's very simple at the bottom.
It's very, in the end, it's simple but deeply abstract.
And something that is simple in conception,
but kind of wrapping one's head around what's going on
is pretty hard.
But so first of all, we have this,
so I've described these kind of atoms of space
and their connections.
You can think about these things as a hypergraph,
a graph is just, you connect nodes to nodes,
but a hypergraph, you can have sort of not just friends,
individual friends to friends,
but you can have these triplets of friends or whatever else.
And so we're just saying,
and that's just the relations between atoms of space
are the hyper edges of the hypergraph.
And so we got some big collection of these atoms of space,
maybe 10 to the 400 or something in our universe.
And that's the structure of space.
That's an every feature of what we experience in the world
is a feature of that hypergraph, that spatial hypergraph.
So then the question is,
well, what does that spatial hypergraph do?
Well, the idea is that there are rules
that update that spatial hypergraph.
And in a cellular automaton,
you've just got this line of cells
§05Transcript · 32:21–40:39
and you just say at every step, at every time step,
you've got fixed time steps, fixed array of cells.
At every step, every cell gets updated
according to a certain rule.
And that's kind of the, that's the way it works.
Now in this hypergraph,
it's sort of vaguely the same kind of thing.
We say, every time you see a little piece of hypergraph
that looks like this, update it to one that looks like this.
So it's just keep rewriting this hypergraph.
Every time you see something that looks like that,
anywhere in the universe, it gets rewritten.
Now, one thing that's tricky about that,
which we'll come to is this multi-computational idea,
which has to do with, you're not saying
in some kind of lockstep way,
do this one, then this one, then this one.
It's just whenever you see one you can do,
you can go ahead and do it.
And that leads one not to have a single thread of time
in the universe.
Because if you knew which one to do,
you just say, okay, we do this one,
then we do this one, then we do this one.
But if you say, just do whichever one you feel like,
you end up with these multiple threads of time,
these kind of multiple histories of the universe,
depending on which order you happen to do
the things you could do in.
- So it's fundamentally asynchronous and parallel.
- Yes, yes.
- Which is very uncomfortable for the human brain
that seeks for things to be sequential and synchronous.
- Right, well, I think that this is part
of the story of consciousness,
is I think the key aspect of consciousness
that is important for parsing the universe
is this point that we have a single thread of experience.
We have a memory of what happened in the past,
we can say something, predict something about the future,
but there's a single thread of experience.
And it's not obvious it should work that way.
I mean, we've got 100 billion neurons in our brains
and they're all firing at all kinds of different ways,
but yet our experience is that there is the single thread
of time that goes along.
And I think that one of the things I've kind of realized
with a lot more clarity in the last year
is the fact that we conclude that the universe
has the laws it has as a consequence of the fact
that we have consciousness the way we have consciousness.
And so the fact, so I mean, just to go on
with kind of the basic setup,
it's so we've got this spatial hypergraph,
it's got all these atoms of space,
they're getting these little clumps of atoms of space
are getting turned into other clumps of atoms of space,
and that's happening everywhere in the universe all the time.
And so one thing that's a little bit weird
is there's nothing permanent in the universe.
The universe is getting rewritten everywhere all the time.
And if it wasn't getting rewritten,
space wouldn't be knitted together.
That is space would just fall apart.
There wouldn't be any way in which we could say
this part of space is next to this part of space.
You know, one of the things that I was,
people were confused about back in antiquity,
the ancient Greek philosophers and so on,
is how does motion work?
You know, how can it be the case that you can take a thing
that we can walk around and it's still us
when we walked a foot forward, so to speak.
And in a sense with our models, that's again a question
because it's a different set of atoms of space.
When I move my hand,
it's moving into a different set of atoms of space.
It's having to be recreated.
It's not the thing itself is not there.
It's being continuously recreated all the time.
Now it's a little bit like waves in an ocean,
you know, vortices in a fluid,
which again, the actual molecules that exist in those
are not what define the identity of the thing.
But it's a little bit, you know,
this idea that there can be pure motion,
that it is even possible for an object
to just move around in the universe and not change.
It's not self-evident that such a thing should be possible.
And that is part of our perception of the universe
is that we parse those aspects of the universe
where things like pure motion are possible.
Now, pure motion, even in general relativity,
the theory of gravity,
pure motion is a little bit of a complicated thing.
I mean, if you imagine your average, you know,
teacup or something approaching a black hole,
it is deformed and distorted by the structure of space time.
And to say, you know, is it really pure motion?
Is it that same teacup that's the same shape?
Well, it's a bit of a complicated story.
And this is a more extreme version of that.
So anyway, the thing that's happening is we've got space,
we've got this notion of time.
So time is this kind of this rewriting of the hypergraph.
And one of the things that's important about that,
time is this sort of computational irreducible process.
There's something, you know, time is not something where,
in kind of the mathematical view of time,
tends to be time is just a coordinate.
We can, you know, slide a slider, turn a knob,
and we'll change the time that we've got in this equation.
But in this picture of time, that's not how it works at all.
Time is this inexorable, irreducible
kind of set of computations that go on,
that go from where we are now to the future.
But so the thing, and one of the things that is, again,
something one sort of has to break out of
is your average trained physicist like me says,
you know, space and time are the same kind of thing.
They're related by, you know, the Poincaré group
and Lorentz transformations and relativity
and all these kinds of things.
And, you know, space and time, you know,
there are all these kind of sort of folk stories
you can tell about why space and time
are the same kind of thing.
In this model, they're fundamentally
not the same kind of thing.
Space is this kind of sort of connections
between these atoms of space.
Time is this computational process.
So the thing that the first sort of surprising thing
is, well, it turns out you get relativity anyway.
And the reason that happens,
there are a few bits and pieces here,
which one has to understand,
but the fundamental point is,
if you are an observer embedded in the system
that are part of this whole story
of things getting updated in this way and that,
there's sort of a limit to what you can tell
about what's going on.
And really in the end, the only thing you can tell
is what are the causal relationships between events.
So an event in this sort of,
an elementary event is a little piece of hypergraph
got rewritten.
And that means a few hyper edges of the hypergraph
were consumed by the event
and you produce some other hyper edges.
And that's an elementary event.
And so then the question is what we can tell
is kind of what the network of causal relationships
between elementary events is.
That's the ultimate thing, the causal graph of the universe.
And it turns out that,
well, there's this property of causal invariance
that is true of a bunch of these models.
And I think is inevitably true for a variety of reasons
that makes it be the case that it doesn't matter
kind of if you are sort of saying,
well, I've got this hypergraph
and I can rewrite this piece here and this piece here
and I do them all in different orders.
When you construct the causal graph
for each of those orders that you choose to do things in,
you'll end up with the same causal graph.
And so that's essentially why,
well, that's in the end why relativity works.
It's why our perception of space and time
is as having this kind of connection
that relativity says they should have.
And that's kind of how that works.
- I think I'm missing a little piece.
If we can go there again,
you said the fact that the observer is embedded
in this hypergraph, what's missing?
What is the observer not able to state
about this universe of space and time?
- So if you look from the outside,
you can say, oh, I see this particular place was updated
and then this one was updated
and I'm seeing which order things were updated in.
But the observer embedded in the universe
doesn't know which order things were updated in
because until they've been updated,
they have no idea what else happened.
So the only thing they know
is the set of causal relationships.
Let me give an extreme example.
Let's imagine that the universe is a Turing machine.
Turing machines have just this one update head,
which does something
and otherwise the Turing machine just does nothing.
And the Turing machine works by having this head
move around and do its updating,
just where the head happens to be.
The question is, could the universe be a Turing machine?
Could the universe just have a single updating head
§06Transcript · 40:42–48:59
that's just zipping around all over the place?
You say, that's crazy because I'm talking to you,
you seem to be updating, I'm updating, et cetera.
But the thing is, there's no way to know that
because if there was just this head moving around,
it's like, okay, it updates me,
but you're completely frozen at that point.
Until the head has come over and updated you,
you have no idea what happened to me.
And so if you sort of unravel that argument,
you realize the only thing we actually can tell
is what the network of causal relationships
between the things that happened were.
We don't get to know from some sort of outside
sort of God's eye view of the thing.
We don't get to know what sort of
from the outside what happened.
We only get to know sort of what the set of relationships
between the things that happened actually were.
- Yeah, but if I somehow record like a trace
of this, I guess would be called multi-computation,
can't I then look back in the-
- Where do you record the trace?
- Some, you place throughout the universe,
like throughout like a log that records
in my own pocket in this hypergraph,
can't I like realizing that I'm getting an outdated picture,
can't I record-
- See, the problem is,
and this is where things start getting very entangled
in terms of what one understands.
The problem is that any such recording device
is itself part of the universe.
So you don't get to say, you never get to say,
let's go outside the universe and go do this.
And that's why, I mean, lots of the features of this model
and the way things work end up being a result of that.
- So, but what, I guess from on a human level,
what is the cost you're paying?
What are you missing from not getting
an updated picture all the time?
Okay, I understand what you're just saying.
- Yeah, yeah, right.
- But like what, like how does consciousness
emerge from that?
Like how, like what are the limitations of that observer?
I understand you're getting a delayed picture.
- Well, there's a, okay,
so there's a bunch of limitations of the observer, I think.
Maybe just explain something about quantum mechanics,
'cause that maybe is an extreme version
of some of these issues,
which helps to kind of motivate
why one should sort of think things through
a little bit more carefully.
So one feature of this, okay,
so in standard physics, like high school physics,
you learn the equations of motion for a ball.
And it says, you throw the ball this angle, this velocity,
things will move in this way,
and there's a definite answer, right?
The story, the key story of quantum mechanics
is there aren't definite answers to where does the ball go?
There's kind of this whole sort of bundle of possible paths.
And all we say we know from quantum mechanics
is certain probabilities for where the ball will end up.
Okay, so that's kind of the core idea of quantum mechanics.
So in our models, you, quantum mechanics
is not some kind of plugin add-on type thing.
You absolutely cannot get away from quantum mechanics,
because as you think about updating this hypergraph,
there isn't just one sequence of things,
one definite sequence of things that can happen.
There are all these different possible update sequences
that can occur.
You could do this piece of the hypergraph now,
and then this one later, and et cetera, et cetera, et cetera.
All those different paths of history
correspond to these quantum paths in quantum mechanics,
these different possible quantum histories.
And one of the things that's kind of surprising about it
is they branch, you know,
there can be a certain state of the universe,
and it could do this, or it could do that,
but they can also merge.
There can be two states of the universe,
which their next state,
the next state they produce is the same for both of them.
And that process of branching and merging
is kind of critical,
and the idea that they can be merging is critical,
and somewhat non-trivial for these hypergraphs,
because there's a whole graph isomorphism story,
and there's a whole very elaborate set of mathematics.
- And that's where the causal invariance comes in?
- Yes, among other things, right, yes.
But so then what happens is that what one's seeing,
okay, so we've got this thing,
it's branching, it's merging, et cetera, et cetera, et cetera.
Okay, so now the question is, how do we perceive that?
Why don't we notice that the universe
is branching and merging?
Why is it the case that we just think
a definite set of things happen?
Well, the answer is we are embedded in that universe,
and our brains are branching and merging too.
And so what quantum mechanics becomes a story of
is how does a branching brain perceive a branching universe?
And the key thing is, as soon as you say,
"I think definite things happen in the universe,"
that means you are essentially conflating
lots of different parts of history.
You're saying, "Actually, as far as I'm concerned,
"because I'm convinced that definite things happen
"in the universe, all these parts of history
"must be equivalent."
Now, it's not obvious that that would be
a consistent thing to do.
It might be you say, "All these parts of history
"are equivalent, but by golly, moments later,
"that would be a completely inconsistent point of view.
"Everything would have gone to hell in different ways."
The fact that that doesn't happen is,
well, that's a consequence of this causal invariance thing,
but that's, and the fact that that does happen a little bit
is what causes little quantum effects.
And that, if that didn't happen at all,
there wouldn't be anything that sort of
is like quantum mechanics.
It would be, quantum mechanics is kind of like in this,
in this kind of this bundle of paths,
it's a little bit like what happens
in statistical mechanics and fluid mechanics, whatever,
that most of the time, you just see this continuous fluid.
You just see the world just progressing
in this kind of way that's like this continuous fluid.
But every so often, if you look at the exact right
experiment, you can start seeing,
well, actually, it's made of these molecules
where they might go that way or they might go this way,
and that's kind of quantum effects.
And so, that's, so this kind of idea of
where we're sort of embedded in the universe,
this branching brain is perceiving this branching universe,
and that ends up being sort of a story of quantum mechanics.
That's part of the whole picture of what's going on.
But I think, I mean, to come back to sort of
where does conscious, what is the story of consciousness?
So, in the universe, we've got, you know,
whatever it is, 10 to the 400 atoms of space,
they're all doing these complicated things.
It's all a big, complicated, irreducible computation.
The question is, what do we perceive from all of that?
And the answer is that we are parsing the universe
in a particular way.
Let me again go back to the gas molecules analogy.
You know, in the gas in this room,
there are molecules bouncing around
in all kinds of complicated patterns, but we don't care.
All we notice is there's, you know,
the gas laws are satisfied,
maybe there's some fluid dynamics.
These are kind of features of that assembly of molecules
that we notice, and there are lots of details
we don't notice.
- When you say we, do you mean the tools of physics,
or do you mean literally the human brain
and its perception system?
- Well, okay, so the human brain is where it starts,
but we've built a bunch of instruments
to do a bit better than the human brain,
but they still have many of the same kinds of ideas.
You know, they're cameras, and they're pressure sensors,
and they're these kinds of things.
They're not, you know, at this point,
we don't know how to make fundamentally, qualitatively
different sensory devices.
- Right, so it's always just an extension
of the conscious experience.
- Or our sensory experience.
- Sensory experience, but sensory experience
is somehow intricately tied to consciousness.
- Right, well, so one question is,
when we are looking at all these molecules in the gas,
and there might be 10th to 20th molecules
in some little box or something,
it's like, what do we notice about those molecules?
So one thing that we can say is,
we don't notice that much.
We are, you know, we are computationally bounded observers.
We can't go in and say, okay,
I'm the 10th to 20th molecules,
and I know that I can sort of decrypt their motions,
and I can figure out this and that.
It's like, I'm just going to say,
what's the average density of molecules?
And so one key feature of us
is that we are computationally bounded,
and that when you are looking at a universe,
which is full of computation,
§07Transcript · 49:00–57:41
and doing huge amounts of computation,
but we are computationally bounded,
there's only certain things about that universe
that we're going to be sensitive to.
We're not going to be figuring out
what all the atoms of space are doing,
because we're just computationally bounded observers,
and we are only sampling these small set of features.
So I think the two defining features of consciousness,
and I would say that the sort of the preamble to this is,
for years, as I've talked about sort of computation
and fundamental features of physics and science,
people ask me, so what about consciousness?
And I, for years, I've said,
I have nothing to say about consciousness.
And I've kind of told this story,
you talk about intelligence, you talk about life.
These are both features where you say,
what's the abstract definition of life?
We don't really know the abstract definition.
We know the one for life on Earth,
it's got RNA, it's got cell membranes,
it's got all this kind of stuff.
Similarly for intelligence,
we know the human definition of intelligence,
but what is intelligence abstractly?
We don't really know.
And so what I've long believed is that
sort of the abstract definition of intelligence
is just computational sophistication.
That is, that as soon as you can be
computationally sophisticated,
that's kind of the abstract version,
the generalized version of intelligence.
So then the question is, what about consciousness?
And what I sort of realized is that consciousness
is actually a step down from intelligence.
That is, that you might think,
oh, consciousness is the top of the pile,
but actually I don't think it is.
I think that there's this notion
of kind of computational sophistication,
which is the generalized intelligence,
but consciousness has two limitations, I think.
One of them is computational boundedness.
That is, that we're only perceiving
a sort of computationally bounded view of the universe.
And the other is this idea of a single thread of time.
That is, that we, and in fact, we know,
neurophysiologically, our brains go to some trouble
to give us this one thread of attention, so to speak.
And it isn't the case that in all the neurons in our brains,
that at least in our conscious,
note the correspondence of language,
in our conscious experience,
we just have this single thread of attention,
single thread of perception.
And maybe there's something unconscious
that's bubbling around that's the kind of
almost the quantum version of what's happening
in our brain, so to speak.
We've got the classical flow
of what we are mostly thinking about, so to speak.
But there's this kind of bubbling around of other paths
that is all those other neurons that didn't make it
to be part of our sort of conscious stream of experience.
- So in that sense, intelligence
as computational sophistication is much broader
than the computational constraints
which consciousness operates under,
and also the sequential thing.
- Yes, right. - Like the notion of time.
That's kind of interesting,
but then the follow-up question is like,
okay, starting to get a sense of what is intelligence,
and how does that connect to our human brain?
'Cause you're saying intelligence is almost like a fabric,
like what, we like plug into it or something?
Like our consciousness plugs into it?
- Yeah, I mean, the intelligence, I think, the core,
I mean, intelligence at some level is just a word,
but we are asking what is the notion of intelligence
as we generalize it beyond the bounds of humans,
beyond the bounds of even the AIs
that we humans have built and so on, what is intelligence?
Is the weather, people say the weather
has a mind of its own, what does that mean?
Can the weather be intelligent?
- Yeah, what does agency have to do with intelligence here?
So is intelligence just like your conception of computation?
Is just intelligence, is the capacity
to perform computation in a sea of?
- Yeah, I think so.
I mean, I think that's right,
and I think that this question of is it for a purpose,
that quickly degenerates into a horrible philosophical mess
because whenever you say,
did the weather do that for a purpose?
- Yeah. - Right?
Well, yes, it did, it was trying to move a bunch of hot air
from the equator to the poles or something,
that's its purpose.
But why, 'cause I seem to be equally as dumb today
as I was yesterday, so there's some persistence,
like a consistency over time
that the intelligence I plugged into.
So it seems like there's a hard constraint
between the amount of computation
I can perform in my consciousness.
Like they seem to be really closely connected somehow.
- Well, I think the point is that the thing
that gives you kind of the ability
to have kind of conscious intelligence,
you can have kind of this, okay,
so one thing is we don't know intelligences
other than the ones that are very much like us, right?
And the ones that are very much like us,
I think have this feature of single thread of time,
bounded, computationally bounded.
Now, that, but you also need computational sophistication.
Having a single thread of time
and being computationally bounded,
you could just be a clock going tick-tock, you know,
that would satisfy those conditions.
But the fact that we have this sort of irreducible,
you know, computational ability,
that's an important feature.
That's the sort of the bedrock
on which we can construct the things we construct.
Now, the fact that we have this experience of the world
that has a single thread of time
and computational boundedness,
the thing that I sort of realized is
it's that that causes us to deduce
from this irreducible mess of what's going on
in the physical world,
the laws of physics that we think exist.
So in other words, if we say,
why do we believe that there is, you know,
continuous space, let's say,
why do we believe that gravity works the way it does?
Well, in principle,
we could be kind of parsing details of the universe
that were, you know, that,
okay, the analogy is, again,
with the statistical mechanics and molecules in a box,
we could be sensitive to every little detail
of the swirling around of those molecules.
And we could say, what really matters is the,
you know, the wiggle effect.
- Yes. - That is, you know,
that is something that we humans just never noticed
because it's some weird thing that happens
when there are 15 collisions of air molecules
and this happens and that happens.
- We just see the pure motion of a ball moving about.
- Right. - Why do we see that?
- Right, and the point is that what seems to be the case
is that the things that,
if we say, given this sort of hypergraph that's updating
and all the details about all the sort of,
sort of atoms of space and what they do,
and we say, how do we slice that
to what we can be sensitive to?
What seems to be the case is that as soon as we assume,
you know, computational boundedness,
single thread of time, that leads us to general relativity.
In other words, we can't avoid that.
That's the way that we will parse the universe.
Given those constraints, we parse the universe
according to those particular,
in such a way that we say the aggregate reducible,
sort of pocket of computational reducibility
that we slice out of this kind of whole
computationally irreducible ocean of behavior
is just this one that corresponds to general relativity.
- Yeah, but we don't perceive general relativity.
- Well, we do if we do fancy experiments.
- So you're saying, so perceive really does mean the full.
- We drop something.
That's a great example of general relativity in action.
The gravity force. - No, but like,
what's the difference in that and Newtonian mechanics?
I mean, what? - Oh, it doesn't,
this is, when I say general relativity,
that's just the-- - You mean gravity.
- The Uber theory, so to speak.
I mean, Newtonian gravity is just the approximation
that we can make, you know,
on the earth and things like that.
So this is, you know, the phenomenon of gravity
is one that is a consequence of, you know,
we would perceive something very different from gravity.
So the way to understand that is when we think about,
okay, so we make up reference frames
with which we parse what's happening in space and time.
So in other words, one of the things that we do
is we say as time progresses,
everywhere in space, something happens at a particular time
and then we go to the next time
§08Transcript · 57:42–1:05:43
and we say this is what space is like at the next time,
this is what space is like at the next time.
That's, it's, the reason we are used to doing that
is because, you know, when we look around,
we might see, you know, 10, 100 meters away.
The time it takes light to travel that distance
is really short compared to the time it takes our brains
to know what happened.
So as far as our brains are concerned,
we are parsing the universe in this,
there is a moment in time, it's all of space.
There's a moment in time, it's all of space.
You know, if we were the size of planets or something,
we would have a different perception
'cause the speed of light
would be much more important to us.
We wouldn't have this perception
that things happen progressively in time,
everywhere in space.
And so that's an important kind of constraint.
And the reason that we kind of parse the universe
in the way that causes us to say gravity works
the way it does is because we're doing things
like deciding that we can say the universe exists,
space has a definite structure.
There is a moment in time,
space has this definite structure.
We move to the next moment in time,
space has another structure.
That kind of setup is what lets us kind of deduce,
kind of parse the universe in such a way that we say
gravity works the way it does.
- So that kind of reference frame,
is that the illusion of that,
is that you're saying that's somehow useful
for consciousness?
- That's what consciousness does.
Because in a sense, what consciousness is doing is,
it's insisting that the universe is kind of sequentialized.
And it is not allowing the possibility that,
oh, there are these multiple threads of time
and they're all flowing differently.
It's like saying, no, everything is happening
in this one thread of experience that we have.
- And that illusion of that one thread of experience
cannot happen at the planetary scale.
So are you saying typical human,
are you saying we are at a human level is special
here for consciousness?
- Well, for our kind of consciousness,
if we existed at a scale close to the elementary length,
for example, then our perception of the universe
will be absurdly different.
- Okay, so, but this makes consciousness seem like
a weird side effect to this particular scale.
And so who cares?
I mean, so consciousness is not that special.
- I think, look, I think that a very interesting question is
which I've certainly thought a little bit about,
is what can you imagine?
What is the sort of factoring of something,
what are some other possible ways
you could exist, so to speak?
And if you were a photon, if you were sort of,
some kind of thing that was some kind of intelligence
represented in terms of photons.
For example, the photons we receive
in the cosmic microwave background,
those photons, as far as they're concerned,
the universe just started.
They were emitted 100,000 years
after the beginning of the universe.
They've been traveling at the speed of light.
Time stayed still for them, and then they just arrived
and we just detected them.
So for them, the universe just started.
And that's a different perception of,
that has implications for a very different
perception of time.
- They don't have that single thread
that seems to be really important
for being able to tell a heck of a good story.
So we humans-- - Yes.
- Tell a story. - We can tell a story.
Right, we can tell a story.
- What other kind of stories can you tell?
So a photon is a really boring story.
- Yeah, I mean, so that's a,
I don't know if they're a boring story,
but I think it's, you know,
I've been wondering about this
and I've been asking friends of mine
who are science fiction writers and things,
have you written stuff about this?
And I've got one example,
the great collection of books from my friend,
Rudy Rooker, which were, which I have to say,
they're books about, that are very informed
by a bunch of science that I've done.
And the thing that I really loved about them is,
you know, in the first chapter of the book,
the Earth is consumed by these things he called Nantes,
which are nano, nanobot type things.
- Nice.
- And it's, so, you know, so the Earth is gone
in the first, but then it comes back.
But then-- - Spoiler alert.
- Yeah, right, that was only a micro spoiler.
It's only chapter one.
- Okay, good.
- It's, but the thing that is not a real spoiler alert
because it's such a complicated concept,
but in the end, the Earth is saved by this thing
called the principle of computational equivalence,
which is a kind of a core scientific idea of mine.
And I was just like, like thrilled.
I don't read fiction books very often.
And I was just thrilled I get to the end of this.
And it's like, oh my gosh, you know,
everything is saved by this sort of
deep scientific principle.
- Can you maybe elaborate how the principle
of computational equivalence can save a planet?
- That would be a terrible spoiler for real.
- That would be a spoiler, okay.
- Yeah, yeah, but no, but let me say what
the principle of computational equivalence is.
So the question is, you have a system, you have some rule.
You can think of its behavior
as corresponding to a computation.
The question is, how sophisticated is that computation?
The statement of the principle of computational equivalence
is as soon as it's not obviously simple,
it will be as sophisticated as anything.
And so that has the implication that, you know, rule 30,
you know, our brains, other things in physics,
they're all ultimately equivalent
in the computations they can do.
And that's what leads to this
computational irreducibility idea,
because the reason we don't get to jump ahead,
you know, and out think rule 30,
is because we're just computationally equivalent to rule 30.
So we're kind of just both just running computations
that are the same sort of raw,
the same level of computation, so to speak.
So that's kind of the idea there.
And the question, I mean, it's like,
you know, in the science fiction version would be,
okay, somebody says, we just need more servers,
get us more servers.
The way to get even more servers is turn the whole planet
into a bunch of micro servers.
And that's where it starts.
And so the question of, you know,
computational equivalence,
principle of computational equivalence is,
well, actually, you don't need to build those custom servers.
Actually, you can just use natural computation
to compute things, so to speak.
You can use nature to compute.
You don't need to have done all that engineering.
And it's kind of feels a little disappointing
that you say, we're going to build all these servers.
We're going to do all these things.
We're going to make, you know,
maybe we're going to have human consciousness
uploaded into, you know, some elaborate digital environment.
And then you look at that thing and you say,
it's got electrons moving around, just like in a rock.
And then you say, well, what's the difference?
And the principle of computational equivalence says,
there isn't at some level a fundamental, you know,
you can't say mathematically,
there's a fundamental difference between the rock
that is the future of human consciousness
and the rock that's just a rock.
Now, what I've sort of realized
with this kind of consciousness thing is
there is an aspect of this that seems to be more special
that isn't, and for example,
something I haven't really teased apart properly
is when it comes to something like the weather
and the weather having a mind of its own or whatever,
or your average, you know, pulsar magnetosphere
acting like a sort of intelligent thing,
how does that relate to, you know,
how is that entity related to the kind of consciousness
that we have and sort of what would the world look like,
you know, to the weather?
If we think about the weather as a mind,
what will it perceive?
What will its laws of physics be?
I don't really know.
- 'Cause it's very parallel.
- It's very parallel among other things.
And it's not obvious.
I mean, this is a really kind of mind bending thing
'cause we've got to try and imagine where, you know,
we've got to try and imagine a parsing of the universe
§09Transcript · 1:05:46–1:14:05
different from the one we have.
And by the way, when we think about
extraterrestrial intelligence and so on,
I think that's kind of the key thing is, you know,
we have always assumed, I've always assumed,
okay, the extraterrestrials,
at least they have the same physics.
We all live in the same universe.
They've got the same physics.
But actually that's not really right
because the extraterrestrials
could have a completely different way of parsing
that the universe.
So it's as if, you know, there could be for all we know,
right here in this room, you know,
in the details of the motion of these gas molecules,
there could be an amazing intelligence that we were like,
but we have no way of,
we're not parsing the universe in the same way.
If only we could parse the universe in the right way,
you know, immediately this amazing thing that's going on
and this, you know, huge culture that's developed
and all that kind of thing would be obvious to us.
But it's not because we have our particular way
of parsing the universe.
- Would that thing also have an agency?
I don't know the right word to use,
but something like consciousness,
but a different kind of consciousness?
- I think it's a question of just what you mean by the word,
because I think that the, you know,
this notion of consciousness and the, okay,
so some people think of consciousness
as sort of a key aspect of it is that we feel that,
the sort of a feeling of that we exist in some way,
that we have this intrinsic feeling about ourselves.
You know, I suspect that any of these things
would also have an intrinsic feeling about themselves.
I've been sort of trying to think recently
about constructing an experiment
about what if you were just a piece of a cellular automaton,
let's say, you know,
what would your feeling about yourself actually be?
And, you know, can we put ourselves in the shoes,
in the cells of the cellular automaton, so to speak?
Can we get ourselves close enough to that,
that we could have a sense of what the world would be like
if you were operating in that way?
And it's a little difficult because, you know,
you have to not only think about what are you perceiving,
but also what's actually going on in your brain,
and our brains do what they actually do.
And they don't, it's, you know,
I think there might be some experiments that are possible
with, you know, neural nets and so on,
where you can have something where you can at least see
in detail what's happening inside the system.
And I've been sort of one of my projects to think about is,
is there a way of kind of getting a sense
kind of from inside the system
about what its view of the world is and how it, you know,
can we make a bridge?
See, the main issue is this, where, you know,
it's a sort of philosophically difficult thing,
because it's like, we do what we do.
We understand ourselves, at least to some extent.
- We humans understand ourselves.
- That's correct.
And, but yet, okay, so what are we trying to do,
for example, when we are trying to make a model of physics,
what are we actually trying to do?
Because, you know, you say, well,
can we work out what the universe does?
Well, of course we can, we just watch the universe.
The universe does what it does.
But what we're trying to do when we make a model of physics
is we're trying to get to the point
where we can tell a story to ourselves
that we understand that is also a representation
of what the universe does.
So it's this kind of, you know,
can we make a bridge between what we humans
can understand in our minds and what the universe does?
And in a sense, you know, a large part of my kind of life
efforts have been devoted to making computational language,
which kind of is a bridge between what is possible
in the computational universe
and what we humans can conceptualize and think about.
In a sense, what, you know, when I built Wolfram Language
and our whole sort of computational language story,
it's all about how do you take sort of raw computation
and this ocean of computational possibility
and how do we sort of represent pieces of it
in a way that we humans can understand
and that map onto things that we care about doing.
And in a sense, when you add physics,
you're adding this other piece where we can,
you know, mediated by a computer,
can we get physics to the point where we humans
can understand something about what's happening in it?
And when we talk about an alien intelligence,
it's kind of the same story.
It's like, is there a way of mapping what's happening there
onto something that we humans can understand?
And, you know, physics in some sense is like our exhibit one
of the story of alien intelligence.
It's a, you know, it's an alien intelligence in some sense.
And what we're doing in making a model of physics
is mapping that onto something that we understand.
And I think, you know, a lot of these other things
that I've recently been kind of studying,
whether it's molecular biology, other kinds of things,
which we can talk about a bit,
those are other cases where we're in a sense
trying to again, make that bridge
between what we humans understand
and sort of the natural language
of that sort of alien intelligence in some sense.
- When you're talking about,
just to backtrack a little bit,
about cellular automata being able to,
what's it like to be a cellular automata
in the way that's equivalent
to what is it like to be a conscious human being?
How do you approach that?
So is it looking at some subset of the cellular automata,
asking questions of that subset,
like how the world is perceived,
how you as that subset,
like for that local pocket of computation,
what are you able to say about the broader cellular automata?
And that somehow then can give you a sense
of how to step outside of that cellular automata.
- Right, but the tricky part is that that little subset,
it's what it's doing is it has a view of itself.
And the question is, how do you get inside it?
It's like, when we with humans, right?
It's like, we can't get inside each other's consciousness.
That doesn't really even make sense.
It's like, there is an experience
that somebody is having,
but you can perceive things from the outside,
but sort of getting inside it,
it doesn't quite make sense.
And for me, these sort of philosophical issues
and this one I have not untangled.
So let's be, for me,
the thing that has been really interesting
in thinking through some of these things is,
when it comes to questions about consciousness
or whatever else, it's like, when I can run a program
and actually see pictures and make things concrete,
I have a much better chance to understand what's going on
than when I'm just trying to reason about things
in a very abstract way.
- Yeah, but there may be a way to map the program
to your conscious experience.
So for example, when you play a video game,
you do a first person shooter,
you walk around inside this entity,
it's a very different thing than watching this entity.
So if you can somehow connect more and more,
connect this full conscious experience
to the subset of the cellular automata.
- Yeah, it's something like that.
But the difference in the first person shooter thing
is there's still your brain and your memory
is still remembering, you still have, it's hard to,
I mean, again, what one's gonna get,
one is not going to actually be able
to be the cellular automaton.
One's gonna be able to watch
what the cellular automaton does.
But this is the frustrating thing
that I'm trying to understand,
you know, how to think about being it, so to speak.
- Okay, so like in virtual reality,
there's a concept of immersion,
like with anything, with video game, with books,
there's a concept of immersion.
It feels like over time,
if the virtual reality experience is well done,
and maybe in the future it'll be extremely well done,
the immersion leads you to feel like,
you mentioned memories,
you forget that you even ever existed
outside that experience.
- Yeah. - It's so immersive.
I mean, you could argue sort of mathematically
that you can never truly become immersed,
but maybe you can.
I mean, why can't you merge with the cellular automaton?
- Yeah, right.
- I mean, aren't you just part of the same fabric?
Why can't you just like--
- Well, that's a good question.
I mean, so let's imagine the following scenario.
Let's imagine-- - Then can you return?
(laughing)
- What's that?
§10Transcript · 1:14:05–1:21:58
- But then can you return back?
- Well, yeah, right.
I mean, it's like, let's imagine you've uploaded,
you know, your brain is scanned,
you've got every synapse, you know, mapped out.
You upload everything about you,
the brain simulator, you upload the brain simulator,
and the brain simulator is basically, you know,
some glorified cellular automaton.
And then you say, well, now we've got an answer
to what does it feel like to be a cellular automaton?
It feels just like it felt to be ordinary you,
because they're both computational systems,
and they're both, you know, operating in the same way.
So in a sense, but I think there's somehow more to it,
because in that sense,
when you're just making a brain simulator,
it's just, you know, we're just saying
there's another version of our consciousness.
The question that we're asking is,
if we tease away from our consciousness
and get to something that is different,
how do we make a bridge
to understanding what's going on there?
And, you know, there's a way of thinking about this.
Okay, so this is coming on to sort of questions
about the existence of the universe and so on.
But one of the things is there's this notion
that we have of ruleal space.
So we have this idea of this physical space,
which is, you know, something you can move around in
that's associated with the actual,
the extent of the spatial hypergraph.
Then there's what we call branchial space,
the space of quantum branches.
So in this thing we call the multi-way graph
of all of this sort of branching histories,
there's this idea of a kind of space
where instead of moving around in physical space,
you're moving from history to history, so to speak,
from one possible history to another possible history.
And that's kind of a different kind of space
that is the space in which quantum mechanics plays out.
Quantum mechanics, like for example,
or something like, I think we're slowly understanding
things like destructive interference in quantum mechanics.
That what's happening is branchial space
is associated with phase in quantum mechanics.
And what's happening is the two photons
that are supposed to be interfering
and destructively interfering
are winding up at different ends of branchial space.
And so us, as these poor observers that are trying to,
that have branching brains,
that are trying to conflate together
these different threads of history,
and say, we've really got a consistent story
that we're telling here.
We're really knitting together these threads of history.
By the time the two photons
wound up at opposite ends of branchial space,
we just can't knit them together to tell a consistent story.
So for us, that's sort of the analog
of destructive interference.
- Got it, and then there's rule-ial space too,
which is the space of rules.
- Yes, well, that's another level up.
So there's the question,
actually, I do want to mention one thing,
'cause it's something I've realized in recent times,
and I think it's really, really kind of cool,
which is about time dilation in relativity.
And it kind of helps to understand,
it's something that kind of helps
in understanding what's going on.
So according to relativity,
if you have a clock, it's ticking at a certain rate,
you send it in a spacecraft
that's going at some significant fraction
of the speed of light,
to you as an observer at rest,
that clock that's in the spacecraft
will seem to be ticking much more slowly.
And so in other words, it's kind of like
the twin who goes off to Alpha Centauri
and goes very fast will age much less
than the twin who's on Earth
that is just hanging out where they're hanging out.
Okay, why does that happen?
Okay, so it has to do with what motion is.
So in our models of physics, what is motion?
Well, when you move from somewhere to somewhere,
it's you having to sort of recreate yourself
at a different place in space.
When you exist at a particular place
and you just evolve with time,
you're again, you're updating yourself,
you're following these rules to update what happens.
Well, so the question is,
when you have a certain amount of computation in you,
so to speak, when there's a certain amount,
you know, you're computing,
the universe is computing at a certain rate,
you can either use that computation
to work out sitting still where you are,
what's going to happen successively in time,
or you can use that computation
to recreate yourself as you move around the universe.
And so time dilation ends up being,
it's really cool actually that this is explainable
in a way that isn't just,
imagine the mathematics of relativity.
But that time dilation is a story of the fact that
as you kind of are recreating yourself as you move,
you are using up some of your computation.
And so you don't have as much computation left over
to actually work out what happens progressively with time.
So that means that time is running more slowly for you
because it is, you're using up your computation,
your clock can't tick as quickly
because every tick of the clock
is using up some computation,
but you already use that computation up
on moving at half the speed of light or something.
And so that's why time dilation happens.
And so you can start,
so it's kind of interesting
that one can sort of get an intuition
about something like that,
because it has seemed like just a mathematical fact
about the mathematics of special relativity and so on.
- Well, for me, it's a little bit confusing
what the U in that picture is,
'cause you're using up computation.
- Okay, so we're simply saying the entity
is updating itself according to
the way that the universe updates itself.
And the question is, those updates,
let's imagine the U is a clock, okay?
And the clock is, there's all these little updates,
the hypergraph and a sequence of updates
cause the pendulum to swing back the other way
and then swing back and forth, okay?
And all of those updates are contributing
to the motion of the pendulum going back and forth
or the little oscillator moving, whatever it is, okay?
But then the alternative is that sort of situation one,
where the thing is at rest,
situation two, where it's kind of moving,
what's happening is it is having to recreate itself
at every moment.
The thing is going to have to do the computations
to be able to sort of recreate itself
at a different position in space.
And that's kind of the intuition behind,
so it's either going to spend its computation
recreating itself at a different position in space,
or it's gonna spend its computation
sort of doing the updating of the ticking of the clock,
so to speak.
- So the more updating it's doing,
the less the ticking of the clock update is doing.
- That's right.
The more it's having to update because of motion,
the less it can update the clock.
So that's, I mean, obviously there's a sort
of mathematical version of it that relates
to how it actually works in relativity,
but that's kind of, to me, that was sort of exciting to me
that it's possible to have a really mechanically
explainable story there that isn't,
and similarly in quantum mechanics,
this notion of branching brains,
perceiving branching universes,
to me, that's getting towards a sort
of mechanically explainable version
of what happens in quantum mechanics,
even though it's a little bit mind bending
to see these things about under what circumstances
can you successfully knit together
those different threads of history,
and when do things sort of escape,
and those kinds of things.
But the thing about this physical space,
in physical space, the main sort of big theory
is general relativity, the theory of gravity,
and that tells you how things move in physical space.
In branchial space, the big theory
is the Feynman path integral, which it turns out,
tells you essentially how things move
in the space of quantum phases.
So it's kind of like motion in branchial space.
And it's kind of a fun thing to start thinking about
what, oh, you know, all these things
that we know in physical space,
like event horizons and black holes and so on,
what are the analogous things in branchial space?
For example, the speed of light,
what's the analog of the speed of light in branchial space?
It's the maximum speed of quantum entanglement.
§11Transcript · 1:22:01–1:30:12
So the speed of light is a flashbulb goes off here,
what's the maximum rate at which the effect
of that flashbulb is detectable moving away in space?
So similarly, in branchial space, something happens,
and the question is, how far in this branchial space,
in the space of quantum states,
how far away can that get within a certain period of time?
And so there's this notion of a maximum entanglement speed,
and that might be observable.
That's the thing we've been sort of poking at,
is might there be a way to observe it
even in some atomic physics kind of situation?
'Cause one of the things that's weird in quantum mechanics
is when we study quantum mechanics,
we mostly study it in terms of small numbers of particles.
You know, this electron does this,
this thing on an ion trap does that, and so on.
But when we deal with large numbers of particles,
kind of all bets are off.
It's kind of too complicated to deal with quantum mechanics.
And so what ends up happening is,
so this question about maximum entanglement speed
and things like that may actually play in one of these,
in the sort of story of many-body quantum mechanics,
and even have some suspicions about things
that might happen even in,
one of the things I realized I'd never understood,
and it's kind of embarrassing,
but I think I now understand a little better,
is when you have chemistry and you have quantum mechanics,
it's like, well, there's two carbon atoms.
There's this molecule, and we do a reaction,
and we draw a diagram, and we say,
this carbon atom ends up in this place.
And it's like, but wait a minute.
In quantum mechanics, nothing ends up in a definite place.
There's always just some wave function for this to happen.
How can it be the case that we can draw these reasonable,
it just ended up in this place?
And you have to kind of say,
well, the environment of the molecule
effectively made a bunch of measurements on the molecule
to keep it kind of classical.
And that's a story that has to do with this whole thing
about measurements have to do with this idea of,
can we conclude that something definite happened?
Because in quantum mechanics,
the intrinsic quantum mechanics,
the mathematics of quantum mechanics is all about,
they're just these amplitudes
for different things to happen.
Then there's this thing of, and then we make a measurement,
and we conclude that something definite happened.
And that has to do with this thing, I think,
about sort of moving, about knitting together
these different threads of history and saying,
this is now something where we can definitively say
something definite happened.
In the traditional theory of quantum mechanics,
it's just like,
after you've done all this amplitude computation,
then this big hammer comes down and you do a measurement,
and it's all over.
And that's been very confusing,
for example, in quantum computing,
it's been a very confusing thing because when you say,
in quantum computing, the basic idea is you're gonna use
all these separate threads of computation, so to speak,
to do all the different parts of,
try these different factors for an integer
or something like this.
And it looks like you can do a lot
because you've got all these different threads going on.
But then you have to say, well, at the end of it,
you've got all these threads,
and every thread came up with a definite answer,
but we got to conflate those together
to figure out a definite thing that we humans
can take away from it, a definite,
so the computer actually produced this output.
- So having this branchial space
and this hypergraph model of physics,
do you think it's possible to then make predictions
that are definite about many body
quantum mechanical systems?
Is that the hope?
- I think it's likely, yes.
But I don't, you know, this is,
every one of these things,
when you go from the underlying theory,
which is complicated enough, and it's,
I mean, the theory at some level is beautifully simple,
but as soon as you start actually trying to,
it's this whole question about how do you bridge it
to things that we humans can talk about,
it gets really complicated.
And this thing about actually getting it
to a definite prediction about, you know,
definite thing you can say about chemistry
or something like this, you know,
that's just a lot of work.
So I'll give you an example.
There's a thing called the quantum Zeno effect.
So the idea is, you know, quantum stuff happens,
but then if you make a measurement,
you're kind of freezing time in quantum mechanics.
And so it looks like there's a possibility
that with sort of the relationship
between the quantum Zeno effect
and the way that many body quantum mechanics works
and so on, maybe just conceivably,
it may be possible to actually figure out a way
to measure the maximum entanglement speed.
And the reason we can potentially do that
is because the systems we deal with
in terms of atoms and things, they're pretty big.
You know, a mole of atoms is, you know,
it's a lot of atoms and, you know,
but it isn't a very, you know,
it's something where to get, you know,
when we're dealing with how can you see 10
to the minus 100, so to speak?
Well, by the time you've got, you know,
10 to the 30th atoms, you're not, you know,
you're within a little bit closer striking distance of that.
It's not like, oh, we've just got, you know,
two atoms and we're trying to see down
to 10 to the minus 100 meters or whatever.
So I don't know how it will work,
but this is a potential direction.
And if you can tell, by the way,
if we could measure the maximum entanglement speed,
we would know the elementary length.
These are all related.
So if we got that one number, we just need one number.
If we can get that one number, we can, you know,
the theory has no parameters anymore.
And, you know, there are other places.
Well, there's another hope for doing that is in cosmology.
In this model, one of the features is
the universe is not fixed dimensional.
I mean, we think we live in three-dimensional space,
but this hypergraph doesn't have any particular dimension.
It can emerge as something which on an approximation,
it's as if, you know, you say,
what's the volume of a sphere in the hypergraph
where a sphere is defined as how many nodes do you get to
when you go a distance R away from a given point?
And you can say, well, if I get to about R cubed nodes,
when I go a distance R away in the hypergraph,
then I'm living roughly in three-dimensional space.
But you might also get to R to the point, you know, 2.92,
you know, for some value of R in, you know,
as R increases, that might be the sort of fit
to what happens.
And so one of the things we suspect is that
the very early universe was essentially
infinite dimensional, and that as the universe expanded,
it became lower dimensional.
And so one of the things that is another little sort of point
where we think there might be a way
to actually measure some things is dimension fluctuations
in the early universe.
That is, is there a leftover dimension fluctuation of,
at the time of the cosmic microwave background,
100,000 years or something after the beginning
of the universe, is it still the case that there were pieces
of the universe that didn't have dimension three,
that had dimension 3.01 or something?
And can we tell that?
- Is that possible to observe the fluctuations
in dimensions?
I don't even know what that entails.
- Okay, so the question, which should be an elementary
exercise in electrodynamics, except it isn't,
is understanding what happens to a photon
when it propagates through 3.01 dimensional space.
So for example, the inverse square law is a consequence
of the surface area of a sphere is proportional
to R squared.
But if you're not in three dimensional space,
the surface area of a sphere is not proportional
to R squared, it's R to the whatever, 2.01 or something.
And so that means that I think when you try and do optics,
a common principle in optics is Huygens principle,
which basically says that every piece of a wave front
of light is a source of new spherical waves.
And those spherical waves, if they're different dimensional
spherical waves, will have other characteristics.
And so there will be bizarre optical phenomena,
which we haven't figured out yet.
- So you're looking for some weird photon trajectories
that designate that it's 3.01 dimensional space?
- Yeah, yeah, that would be an example of, I mean,
you know, there are only a certain number of things
we can measure about photons.
You know, we can measure their polarization,
we can measure their frequency,
we can measure their direction, those kinds of things.
And you know, how that all works out,
§12Transcript · 1:30:15–1:38:19
and you know, in the current models of physics,
you know, it's been hard to explain how the universe
manages to be as uniform as it is.
And that's led to this inflation idea that,
to the great annoyance of my then collaborator,
we figured out in like 1979, we had this realization
that you could get something like this,
but it seemed implausible that that's the way
the universe worked, so we put in a footnote.
And that was, so that's a, but in any case,
I've never really completely believed it,
but that's an idea for how to sort of puff out the universe
faster than the speed of light,
early moments of the universe,
that that's the sort of the inflation idea.
And that you can somehow explain how the universe manages
to be as uniform as it is.
In our model, this turns out to be much more natural,
because the universe just starts very connected.
The hypergraph is not such that the ball that you grow
starting from a single point has volume R cubed,
it might have volume R to the 500 or R to the infinity.
And so that means that you sort of naturally get
this much higher degree of connectivity and uniformity
in the universe.
And then the question is,
this is sort of the mathematical physics challenge,
is in the standard theory of the universe,
there's the Friedman-Robertson-Walker universe,
which is the kind of standard model where the universe
is isotropic and homogeneous,
and you can then work out the equations of general relativity
and you can figure out how the universe expands.
We would like to do the same kind of thing,
including dimension change.
This is just difficult mathematical physics.
I mean, the reason it's difficult
is the sort of fundamental reason it's difficult.
When people invented calculus 300 years ago,
calculus was a story of understanding change
and change as a function of a variable.
And so people study univariate calculus,
they study multivariate calculus,
it's one variable, it's two variables, three variables,
but whoever studied 2.5 variable calculus,
turns out nobody.
Turns out that, but what we need to have
to understand these fractional dimensional spaces,
which don't work like,
well, they're spaces where the effective dimension
is not an integer.
- So you can't apply the tools of calculus
and naturally and easily to fractional dimensions.
- No.
- So somebody has to figure out how to do that.
- Yeah, yeah, we're trying to figure this out.
I mean, it's very interesting.
I mean, it's very connected to very frontier issues
in mathematics.
It's very beautiful, but-
- So is it possible, is it possible,
we're dealing with a scale that's so, so much smaller
than our human scale.
Is it possible to make predictions versus explanations?
Do you have a hope that with this hypergraph model,
you'd be able to make predictions?
- Yeah.
- That then could be validated with a physics experiment,
predictions that couldn't have been done
or weren't done otherwise?
- Yeah, yeah, yeah.
I mean, you know, I think-
- In which domain do you think that prediction will-
- Okay, so there are gonna be cosmology ones
to do with dimension fluctuations in the universe.
That's a very bizarre effect.
Nobody, you know, dimension fluctuation is just something,
nobody ever looked for that.
If anybody sees dimension fluctuation, that's a huge flag,
that's something like our model is going on.
If, and how one detects that, you know,
that's a problem of kind of, you know,
that's a problem of traditional physics in a sense
of what's the best way to actually figure that out.
And for example, that's one,
there are all kinds of things one can imagine.
I mean, there are things that in black hole mergers,
it's possible that there will be effects
of maximum entanglement speed in large black hole mergers.
That's another possible thing.
- And all of that is detected through like what?
Do you have a hope for LIGO type of situation?
Like that's gravitational waves?
- Yeah, or alternatively, I mean, I think it's, you know,
look, figuring out experiments
is like figuring out technology inventions.
That is, you know, you've got a set of raw materials,
you've got an underlying model,
and now you've got to be very clever
to figure out, you know, what is that thing I can measure
that just somehow, you know,
leverages into the right place.
And we've spent less effort on that than I would have liked,
because one of the reasons is that I think that the,
you know, the physicists who've been working on our models,
and we've now, lots of physicists actually,
it's very, very nice.
It's kind of, you know, it's one of these cases
where I'm almost, I'm really kind of pleasantly surprised
that the sort of absorption of the things we've done
has been quite rapid and quite sort of,
you know, very positive.
- So it's a Cambrian explosion of physicists too,
not just ideas.
- Yes, I mean, you know, a lot of what's happened
that's really interesting, and again,
not what I expected is, there are a lot of areas
of sort of very elaborate,
sophisticated mathematical physics,
whether that's causal set theory,
whether it's higher category theory,
whether it's categorical quantum mechanics,
all sorts of elaborate names for these things,
spin networks, perhaps, you know,
causal dynamical triangulations,
all kinds of names of these fields.
And these fields have a bunch
of good mathematical physicists in them
who've been working for decades in these particular areas.
And the question is,
but they've been building these mathematical structures,
and the mathematical structures are interesting,
but they don't typically sit on anything.
They're just mathematical structures.
And I think what's happened is our models provide
kind of a machine code that lives underneath those models.
So a typical example, this is due to Jonathan Gorod,
who's one of the key people
who's been working on our project.
This is in, okay, so I'll give you an example,
just to give a sense of how these things connect.
This is in causal set theory.
So the idea of causal set theory is there are,
in space-time, we imagine that there's space and time,
it's a three plus one dimensional, you know, setup.
We imagine that there are just events that happen
at different times and places in space and time.
And the idea of causal set theory is the only thing
you say about the universe is there are a bunch of events
that happen sort of randomly
at different places in space and time.
And then the whole sort of theory of physics
has to be to do with this graph of causal relationships
between these randomly thrown down events.
So they've always been confused by the fact that
to get even Lorentz invariance,
even relativistic invariance,
you need a very special way to throw down those events.
And they've had no natural way
to understand how that would happen.
So what Jonathan figured out is that,
in fact, from our models,
they, instead of just generating events at random,
our models necessarily generate events in some pattern
in space-time effectively,
that then leads to Lorentz invariance
and relativistic invariance and all those kinds of things.
So it's a place where all the mathematics
that's been done on,
well, we just have a random collection of events.
Now, what consequences does that have
in terms of causal set theory and so on?
That can all be kind of wheeled in
now that we have some different underlying foundational idea
for what the particular distribution of events is
as opposed to just where we throw down random events.
And so that's a typical sort of example
of what we're seeing in all these different areas
of kind of how you can take really interesting things
that have been done in mathematical physics
and connect them.
And it's really kind of beautiful
because the sort of the abstract models we have
just seem to plug into all these different,
very interesting, very elegant, abstract ideas,
but we're now giving sort of a reason
for that to be the way,
for a reason for one to care.
I mean, it's like saying,
you can think about computation abstractly.
You can think about, I don't know,
combinators or something as abstract computational things.
And you can sort of do all kinds of study of them,
but it's like, why do we care?
Well, okay, Turing machines are a good start
'cause you can kind of see
they're sort of mechanically doing things.
But when we actually start thinking about computers
computing things, we have a really good reason to care.
And this is sort of what we're providing, I think,
§13Transcript · 1:38:21–1:46:12
is a reason to care about a lot of these areas
of mathematical physics.
So that's been very nice.
- So I'm not sure we've ever got to the question
of why does the universe exist at all.
Let's talk about that.
So it's not the simplest question in the world.
So it takes a few steps to get to it.
- And it's nevertheless even surprising
that you can even begin to answer this question,
as you were saying.
- I'm very surprised.
So the next thing to perhaps understand
is this idea of ruleal space.
So we've got kind of physical space,
we've got branchial space,
the space of possible quantum histories.
And now we've got another level of kind of abstraction,
which is ruleal space.
And here's where that comes from.
So you say, okay, you say,
we've got this model for the universe,
we've got a particular rule,
and we run this rule and we get the universe.
Okay, so that's interesting.
Why that rule?
Why not another rule?
And so that confused me for a long time.
And I realized, well, actually,
what if the thing could be using all possible rules?
What if at every step,
in addition to saying apply a particular rule
at all places in this hypergraph,
one could say, just take all possible rules
and apply all possible rules
at all possible places in this hypergraph.
Okay, and then you make this ruleal multi-way graph,
which both is all possible histories
for a particular rule and all possible rules.
So the next thing you'd say is,
how can you get anything reasonable out of it?
How can anything real come out of
the set of all possible rules applied
in all possible ways?
Okay, this is a subtle thing,
so which I haven't fully untangled.
There is this object,
which is the result of running all possible rules
in all possible ways.
And you might say, if you're running all possible rules,
why can't everything possible happen?
Well, the answer is because when you,
there's sort of this entanglement that occurs.
So let's say that you have
a lot of different possible initial conditions,
a lot of different possible states.
Then you're applying these different rules.
Well, some of those rules can end up with the same state.
So it isn't the case that you can just get
from anywhere to anywhere.
There's this whole entangled structure
of what can lead to what,
and there's a definite structure that's produced.
I think I'm gonna call that definite structure
the Rulliad,
the limit of kind of all possible rules
being applied in all possible ways.
- And you're saying that structure is finite,
so that somehow connects to maybe a similar kind of thing
as like causal invariance.
- Well, it happens that the Rulliad
necessarily has causal invariance.
That's a feature of,
that's just a mathematical consequence
of essentially using all possible rules
plus universal computation gives you the fact
that for many diverging paths,
you can always, the paths will always converge.
- But does that say that the Rulli,
does that necessarily infer that the Rulliad is a finite?
- In the end, it's not necessarily finite.
I mean, it's a,
just like the history of the universe may not be finite.
The history of the universe, time may keep going forever.
You can keep running the computations of the Rulliad
and you'll keep spewing out more and more and more structure.
It's like time doesn't have to end.
It's that, but the issue is there are three limits
that happen in this Rulliad object.
One is how long you run the computation for.
Another is how many different rules you're applying.
And another is how many different states you start from.
And the mixture of those three limits,
I mean, this is just mathematically a horrendous object.
And what's interesting about this object is
the one thing that does seem to be the case
about this object is it connects with ideas
in higher category theory.
And in particular, it connects to some of the 20th century's
most abstract mathematics done by this chap Grothendieck.
Grothendieck had a thing called the infinity groupoid,
which is closely related to this Rulliad object.
Although the details of the relationship,
I don't fully understand yet.
But I think that what's interesting is this thing
that is sort of this very limiting object.
So, okay, so a way to think about this,
that again, will take us into another direction,
which is the equivalence between physics and mathematics.
The way that, well, let's see, maybe this is,
just to give a sense of this kind of groupoid
and things like that.
You can think about, in mathematics,
you can think you have certain axioms,
they're kind of like atoms, and you,
well, actually, let's say,
let's talk about mathematics for a second.
So what is mathematics?
What is it made of, so to speak?
Mathematics, there's a bunch of statements,
like for addition, X plus Y is equal to Y plus X.
That's a statement of mathematics.
Another statement would be, you know,
X squared minus one is equal to X plus one X minus one.
There are infinite number of these possible statements
of mathematics.
- So it's not, I mean, it's not just, I guess, a statement,
but with X plus Y, it's a rule that you can,
it's a, I mean, you think of it as a rule.
- It's a, it is a rule.
It's also just a thing that is true in mathematics.
- Right, it's a statement of truth, okay.
- Right, and what you can imagine is,
you imagine just laying out this giant kind of ocean
of all statements, well, actually, you first start,
okay, this is where,
this, we're segueing into a different thing.
Let me not go in this direction for a second.
- Let's not go to meta-mathematics just yet.
- Yeah, we'll maybe get to meta-mathematics,
but it's, so let me not,
let me explain the groupoid and things later.
- Yes.
- But, so let's come back to the universe,
always a good place to be in, so to speak.
- Yeah, so what does the universe have to do
with the Rulliad, the Rullio space,
and how that's possibly connected
to why the thing exists at all,
and why there's just one of them?
- Yes, okay, so here's the point.
So the thing that had confused me for a long time was,
let's say we get the rule for the universe.
We hold it in our hand.
We say, "This is our universe."
Then the immediate question is,
"Well, why isn't it another one?"
And that's kind of the,
the sort of the lesson of Copernicus is,
we're not very special.
So how come we got universe number 312
and not universe quadrillion, quadrillion, quadrillion?
And I think the resolution of that is the realization
that the universe is running all possible rules.
So then you say, "Well, how on earth do we perceive
"the universe to be running according to a particular rule?
"How do we perceive definite things
"happening in the universe?"
Well, it's the same story.
It's the observer.
There is a reference frame that we are picking
in this Rullio space,
and that that is what determines
our perception of the universe.
With our particular sensory information and so on,
we are parsing the universe in this particular way.
So here's the way to think about it.
In physical space,
we live in a particular place in the universe.
And we could live on Alpha Centauri,
but we don't, we live here.
And similarly, in Rullio space,
we could live in many different places in Rullio space,
but we happen to live here.
And what does it mean to live here?
It means we have certain sensory input.
We have certain ways to parse the universe.
Those are our interpretation of the universe.
What would it mean to travel in Rullio space?
What it basically means is that we are successively
interpreting the universe in different ways.
So in other words,
to be at a different point in Rullio space
is to have a different, in a sense,
a different interpretation of what's going on
in the universe.
And we can imagine even things like an analog
of the speed of light as the maximum speed
of translation in Rullio space and so on.
- So wait, what's the interpretation?
§14Transcript · 1:46:15–1:54:20
So Rullio space, and we, I'm confused by the we
and the interpretation and the universe.
I thought moving about in Rullio space
changes the way the universe is.
- The way we would perceive it.
The way that--
- So it ultimately has to do with the perception.
So it doesn't, Rullio space is not somehow changing,
like branching into another universe or something like that.
- No, I mean, the point is, the whole point of this
is the Rulliard is sort of the encapsulated version
of everything that is the universe
running according to all possible rules.
- We think of our universe,
the observable universe as its thing.
So we're a little bit loose with the word universe then
because wouldn't the Rulliard potentially encapsulate
a very large number, like combinatorially large,
maybe infinite set of what we human physicists
think of as universes?
- That's an interesting parsing of the word universe, right?
Because what we're saying is,
just as we're at a particular place in physical space,
we're at a particular place in Rullio space,
at that particular place in Rullio space,
our experience of the universe is this.
Just as if we lived at the center of the galaxy,
our universe, our experience of the universe
will be different from the one it is
given where we actually live.
And so what we're saying is,
what you might say,
I mean, in a sense,
this Rulliard is sort of a super universe, so to speak,
but it's all entangled together.
It's not like you can separate out.
You can say, let me,
it's like when we take a reference, okay.
It's like our experience of the universe
is based on where we are in the universe.
We could imagine moving to somewhere else in the universe,
but it's still the same universe.
- So there's not like universes
existing in parallel.
- No, no, because,
because, and the whole point is that
if we were able to change our interpretation
of what's going on,
we could perceive a different reference frame
in this Rulliard.
- Yeah, but that's not,
that's not, that's just, yeah,
that's the same Rulliard,
that's the same universe.
You're just moving about.
These are just coordinates in the universe.
- So the way that's,
the reason that's interesting is,
imagine the extraterrestrial intelligence,
so the alien intelligence, we should say.
The alien intelligence might live on Alpha Centauri,
but it might also live at a different place in Rullio space.
- It can live right here on earth.
It just has a different reference frame
that includes a very different perception of the universe.
And then because that Rullio space is very large,
I mean,
- Do we get to communicate with them?
Right, that's.
- Yeah, but it's also,
well, one thing is how different
the perception of the universe could be.
- I think it could be bizarrely,
unimaginably, completely different.
And I mean, one thing to realize is,
even in kind of things I don't understand well,
you know, I know about the kind of Western tradition
of understanding, you know, science
and all that kind of thing.
And, you know, you talk to people who say,
well, I, you know, I'm really into some, you know,
Eastern tradition of this, that, and the other.
And it's really obvious to me how things work.
I don't understand it at all, but, you know,
it is not obvious, I think, with this kind of realization
that there's these very different ways
to interpret what's going on in the universe.
That kind of gives me at least,
it doesn't help me to understand
that different interpretation,
but it gives me at least more respect for the possibility
that there will be other interpretations.
- Yeah, it humbles you to the possibility that like,
what is it, reincarnation,
or all these like eternal recurrence,
when you share like just these ideas, yeah.
- Well, you know, the thing that I've realized
about a bunch of those things is that, you know,
I've been sort of doing my little survey
of the history of philosophy, just trying to understand,
you know, what can I actually say now
about some of these things?
And you realize that some of these concepts,
like the immortal soul concept, which, you know,
I remember when I was a kid and, you know,
it was kind of a lots of religion bashing type stuff
of people saying, you know, well, we know about physics,
tell us how much does a soul weigh?
And people are like, well, how can it be a thing
if it doesn't weigh anything?
Well, now we understand, you know,
there is this notion of what's in brains
that isn't the matter of brains
and it's something computational.
And there is a sense, and in fact, it is correct,
that it is in some sense immortal
because this pattern of computation is something abstract
that is not specific to the particular material of a brain.
Now, we don't know how to extract it, you know,
in our traditional scientific approach,
but it's still something where it isn't a crazy thing
to say there is something, it doesn't weigh anything.
That's a kind of a silly question.
How much does it weigh?
Well, actually, maybe it isn't such a silly question
in our model of physics
because the actual computational activity
has a consequence for gravity and things,
but that's a very subtle--
- You can start talking about mass and energy and so on.
There could be a, what would you call it, a solitron.
- Yes, yes, yes.
- A particle that somehow contains soleness.
- Yeah, right.
Well, that's what, by the way, that's what Leibniz said.
And, you know, one thing, I've never understood this.
You know, Leibniz had this idea of monads and monodology
and he had this idea that what exists in the universe
is this big collection of monads
and that the only thing that one knows about the monads
is sort of how they relate to each other,
which sounds awfully like hypergraphs, right?
But Leibniz had really lost me at the following thing.
He said, "Each of these monads has a soul
"and each of them has a consciousness."
And it's like, okay, I'm out of here.
I don't understand this at all.
I don't know what's going on.
But I realized recently that in his day,
the concept that a thing could do something,
could spontaneously do something,
that was his only way of describing that.
And so what I would now say is,
well, there's this abstract rule that runs
to Leibniz that would have been, you know,
in 1690 or whatever, that would have been kind of,
well, it has a soul, it has a consciousness.
And so, you know, in a sense, it's like one of these,
there's no new idea under the sun, so to speak.
That's a sort of a version of the same kinds of ideas,
but couched in terms that are sort of bizarrely different
from the ones that we would use today.
- Would you be able to maybe play devil's advocate
on your conception of consciousness,
that like the two characteristics of it,
that it's constrained and there's a single thread of time,
is it possible that Leibniz was onto something
that the basic atom, the screwed atom of space
has a consciousness?
So these are just words, right?
But like, is there some sense where consciousness
is much more fundamental than you're making it seem?
- I don't know.
I mean, you know, I think--
- Can you construct a world
in which it is much more fundamental?
- I think that, okay, so the question would be,
is there a way to think about kind of,
if we sort of parse the universe down
at the level of atoms of space or something,
could we say, well, so that's really a question
of a different point of view,
a different place in real space.
We're asking, you're asking the question,
could there be a civilization that exists?
Could there be sort of conscious entities
that exist at the level of atoms of space?
And what would that be like?
And I think that comes back to this question of,
can we, you know, what's it like
to be a cellular automaton type thing?
I mean, it's, you know, I'm not yet there.
I don't know.
I mean, I think that this is a,
and I don't even know yet quite how to think about this
in the sense that I was considering, you know,
I never write fiction, but I haven't written it
since I was like 10 years old.
And my fiction, I made one attempt,
which I sent to some science fiction writer friends of mine,
§15Transcript · 1:54:23–2:02:27
and they told me it was terrible.
So, but--
- This is a long time ago?
- No, it was recently.
- Recently, and they said it was terrible.
That'd be interesting to see you write a short story
based on what sounds like it's already inspiring
short stories by, or stories--
- Yeah, right. - By science fiction writers.
- But I think the interesting thing for me is,
you know, in the what does it,
what is it like to be a whatever?
- Yeah. - How do you describe that?
I mean, it's like, that's not a thing
that you describe in mathematics,
that what is it like to be such and such?
- Well, see, to me, when you say,
what is it like to be something,
presumes that you're talking about a singular entity.
So-- - Yeah.
- Like, there's some kind of feeling of the entity,
the stuff that's inside of it,
and the stuff that's outside of it.
And then that's when consciousness starts making sense.
But then, it seems like that could be generalizable.
If you take some subset of a cellular automata,
you could start talking about what does that subset--
- Maybe. - Feel.
But then you can, I think you could just take
arbitrary numbers of subsets.
Like, to me, like, you and I,
individually are consciousnesses,
but you could also say the two of us together
is a singular consciousness.
- Maybe, maybe.
I'm not so sure about that.
I think that the single thread of time thing
may be pretty important.
And that as soon as you start saying,
there are two different threads of time,
there are two different experiences,
and then we have to say, how do they relate?
How are they sort of entangled with each other?
I mean, that may be a different story
of a thing that isn't much like, you know,
what are the ants?
You know, what's it like to be an ant?
You know, where there's a sort of more collective view
of the world, so to speak.
I don't know.
I think that, I mean, this is, you know,
I don't really have a good, I mean,
you know, my best thought is, you know,
can we turn it into a human story?
It's like the question of, you know,
when we try and understand physics,
can we turn that into something
which is sort of a human understandable narrative?
And now what's it like to be a such and such?
You know, maybe the only medium in which we can describe
that is something like fiction,
where it's kind of like you're telling, you know,
the life story in that setting.
But this is beyond what I've yet understood how to do.
- Yeah, but it does seem so, like with human consciousness,
you know, we're made up of cells,
and like there's a bunch of systems that are networked
that work together that at the human level
feel like a singular consciousness when you take.
- Yes.
- And so maybe like an ant colony is just too low level.
Sorry, an ant is too low level.
Maybe you have to look at the ant colony.
- Yeah, I agree.
- There's some level at which it's a conscious being.
And then if you go to the planetary scale,
then maybe that's going too far.
So there's a nice sweet spot for consciousness.
- No, I agree.
I think the difficulty is that, you know, okay,
so in sort of people who talk about consciousness,
one of the terrible things I've realized,
'cause I've now interacted with some of this community,
so to speak, some interesting people
who do that kind of thinking,
but one of the things I was saying
to one of the leading people in that area,
I was saying that it must be kind of frustrating
because it's kind of like a poetry story.
That is, many people are writing poems,
but few people are reading them.
- Yes.
- So there are always these different,
everybody has their own theory of consciousness,
and they are very non-interdiscussable.
And by the way, I mean, my own approach
to sort of the question of consciousness,
as far as I'm concerned,
I'm an applied consciousness operative, so to speak,
because I don't really, in a sense,
the thing I'm trying to get out of it
is how does it help me to understand
what's a possible theory of physics?
And how does it help me to say,
how do I go from this incoherent collection
of things happening in the universe
to our definite perception and definite laws and so on,
and sort of an applied version of consciousness?
And I think the reason it sort of segues
to a different kind of topic,
but the reason that one of the things
I'm particularly interested in
is kind of what's the analog of consciousness
in systems very different from brains?
And so why does that matter?
Well, this whole description of this kind of,
well, actually, you know what?
We haven't talked about why the universe exists,
so let's get to why the universe exists,
and then we can talk about, perhaps a little bit about
what these models of physics kind of show you
about other kinds of things
like molecular computing and so on.
- Yes, that's good. - Okay,
why does the universe exist?
Okay, so we finally sort of more or less set the stage.
We've got this idea of this Rouliad,
of this object that is made
from following all possible rules,
the fact that it's sort of not just this incoherent mess,
it's got all this entangled structure in it, and so on.
Okay, so what is this Rouliad?
Well, it is the working out of all possible formal systems.
So the sort of the question of why does the universe exist,
its core question, which we kind of started with,
is you've got two plus two equals four,
you've got some other abstract result,
but that's not actualized, it's just an abstract thing.
And when we say we've got a model for the universe,
okay, it's this rule, you run it,
and it'll make the universe, but it's like,
but where's it actually running?
What is it actually doing, right?
What, is it actual,
or is it merely a formal description of something?
Okay, so the thing to realize with this,
the thing about the Rouliad is it's an inevitable,
it is the entangled running of all possible rules.
So you don't get to say, it's not like you're saying,
which Rouliad are you picking?
Because it's all possible formal rules.
It's not like it's just, you know,
well, actually it's only footnote,
the only footnote, it's an important footnote,
is it's all possible computational rules,
not hyper computational rules.
That is, it's running all the rules
that would be accessible to a Turing machine,
but it is not running all the rules
that will be accessible to a thing
that can solve problems in finite time
that would take a Turing machine infinite time to solve.
So you can, even Alan Turing knew this,
that you could make oracles for Turing machines,
where you say a Turing machine can't solve
a halting problem for Turing machines.
It can't know what will happen in any Turing machine
after an infinite time, in any finite time,
but you could invent a box, just make a black box.
You say, I'm gonna sell you an oracle
that will just tell you, you know, press this button,
it'll tell you what the Turing machine
will do after an infinite time.
You can imagine such a box.
You can't necessarily build one in the physical universe,
but you can imagine such a box.
And so we could say, well, in addition to,
so in this Rouliad, we're imagining
that there is a computational, that at the end,
it's running rules that are computational.
It doesn't have a bunch of oracle black boxes in it.
You say, well, why not?
Well, it turns out if there are oracle black boxes,
the Rouliad that is, you can make a sort of super Rouliad
that contains those oracle black boxes,
but it has a cosmological event horizon
relative to the first one, they can't communicate.
In other words, you can end up with,
what you end up happening, what ends up happening
is it's like in the physical universe,
we in this causal graph that represents
the causal relationships of different things,
you can have an event horizon,
where the causal graph is disconnected,
where the effect here, an event happening here
does not affect an event happening here
because there's a disconnection in the causal graph.
And that's what happens in an event horizon.
And so what will happen between this kind of
the ordinary Rouliad and the hyper Rouliad
§16Transcript · 2:02:30–2:12:14
is there is an event horizon and we in our Rouliad
will just never know that there is,
that they're just separate things, they're not connected.
- Maybe I'm not understanding,
but just because we can't observe it,
why does that mean it doesn't exist?
- It might exist, but it's not clear what it,
so what, so to speak, whether it exists.
What we're trying to understand
is why does our universe exist?
We're not trying to ask the question,
what, you know, it's, let me say another thing,
let me make a meta comment, okay?
Which is that I have not thought through
this hyper Rouliad business properly.
So I can't, the--
- The hyper Rouliad is referring to a Rouliad
in which hyper computation is possible.
- That's correct, yes.
- So like what, that footnote,
the footnote to the footnote is,
we're not sure why this is important.
- Yeah, that's right.
So let's ignore that, okay?
It's already abstract enough, okay?
So, okay, so the one question is,
we have to say, if we're saying,
why does the universe exist?
One question is, why is it this universe
and not another universe, okay?
So the important point about this Rouliad idea
is that it's, in the Rouliad are all possible formal systems.
So there's no choice being made.
There's no like, oh, we pick this particular universe
and not that one.
That's the first thing.
The second thing is that we have to ask the question.
So you say, why does two plus two equals four exist?
That's not really a, that is a thing
that necessarily is that way,
just on the basis of the meaning of the terms,
two and plus and equals and so on, right?
So the thing is that this Rouliad object
is in a sense a necessary object.
It is just a thing that is the consequence
of working out the consequence
of the formal definition of things.
You don't, it is not a thing where you're saying,
and this is picked as the particular thing.
This is just something which necessarily is that thing
because of the definition
of what it means to have computation.
- So the Rouliad, it's a formal system.
- Yes.
- But does it exist?
- Ah, well, where are we in this whole thing?
We are part of this Rouliad.
And so our, so there is no sense to say,
does two plus two equals four exist?
Well, that's, in some sense, it necessarily exists.
It's a necessary object.
It's not a thing that where you can ask.
Usually in philosophy, there's a sort of distinction made
between necessary truths, contingent truths,
analytic propositions, synthetic propositions,
there are a variety of different versions of this.
There are things which are necessarily true
just based on the definition of terms.
And there are things which happen to be true in our universe.
- But we don't exist in Roulial space.
That's one of the coordinates that define our existence.
- Well, okay, so yes, yes.
But this Rouliad is the set
of all possible Roulial coordinates.
So what we're saying is it contains that.
So what we're saying is we exist as, okay,
so our perception of what's going on
is we're at a particular place in this Rouliad,
and we are concluding certain things
about how the universe works based on that.
But the question is, do we understand,
is there something where we say,
so why does it work that way?
Well, the answer is, I think it has to work that way
because this Rouliad is a necessary object
in the sense that it is a purely formal object,
just like two plus two equals four.
It's not an object that was made of something.
It's an object that is just an expression
of the necessary collection of formal relations that exist.
And so then the issue is, can we, in our experience of that,
is it, can we have tables and chairs, so to speak,
in that just by virtue of our experience
of that necessary thing?
And what people have generally thought,
and honestly, I don't know of a lot of discussion of this
why does the universe exist question.
It's been a very, I've been surprised, actually,
at how little, I mean, I think it's one of these things
that's really kind of far out there.
But the thing that is, the surprise here
is that all possible formal rules,
when you run them together, and that's the critical thing,
when you run them together,
they produce this kind of entangled structure
that has a definite structure.
It's not just a random arbitrary thing.
It's a thing with definite structure.
And that structure is the thing,
when we are embedded in that structure,
when an entity embedded in that structure
perceives something which is,
then we can interpret as physics and things like this.
So in other words, we don't have to ask the question,
why does it exist?
It necessarily exists.
- I'm missing this part.
Why does it necessarily exist?
- Okay, let me try.
- So like you need to have it if you want to formalize
the relation between entities,
but why do you need to have relations?
- Okay, okay.
So let's say you say, well-
- It's like, why does math have to exist?
- Okay, fair question.
Yeah, okay, fair question.
Let's see.
I think the thing to think about is
the existence of mathematics is something where,
given a definition of terms,
what follows from that definition inevitably follows.
So now you can say, why define any terms?
But in a sense, the, well, that's okay.
So the definition of terms, I mean,
I think the way to think about this, let me see.
- So like concrete terms.
- Well, not very concrete.
I mean, they're just things like, you know,
logical or.
- Right, but that's a thing.
That's a powerful thing.
- Well, it's a, yes, okay.
But it's a, the point is that it is not a thing of a,
you know, people imagine there is, I don't know,
the, you know, an elephant or something,
or the, you know, elephants are presumably
not necessary objects.
They are, they happen to exist as a result of
kind of biological evolution and whatever else.
But the thing is that in some sense,
that there is, it is a different kind of thing
to say does plus exist.
It is not like an elephant.
- So a plus is, seems more fundamental,
more basic than an elephant, yes.
But you can imagine a world without plus
or anything like it.
Like why do formal things that are discrete,
that can be used to reason have to exist?
- Well, okay, so why, okay, so then the question is,
but the whole point is computation,
we can certainly imagine computation.
That is, we can certainly say there is a formal system
that we can construct abstractly in our minds
that is computation.
And that's the, you know, we can imagine it, right?
Now the question is, is it is that formal system,
once we exist as observers embedded in that formal system,
that's enough to have something which is like our universe.
And so then what you're kind of asking is perhaps,
is why, I mean, the point is we definitely can imagine it.
There's nothing that says that,
we're not saying that there's,
it's sort of inevitable that that is a thing
that we can imagine.
We don't have to ask, does it exist?
We're just, it is definitely something we can imagine.
Now that's, then we have this thing
that is a formally constructible thing that we can imagine.
And now we have to ask the question,
what, you know, given that formally constructible thing,
what is, what consequences does that,
if we were to perceive that formally,
if we were embedded in that formally constructible thing,
what would we perceive about the world?
And we would say, we perceive that the world exists
because we are seeing all of this mechanism
of all these things happening.
And, but that's something that is just a feature of,
it's something where we are--
- See another way of asking this,
that I'm trying to get at,
I understand why it feels like this Rouliad
is necessary,
but maybe it's just me being human,
but it feels like then you should be able to,
not us, but somehow step outside of the Rouliad.
Like what's outside the Rouliad?
- Well, the Rouliad is all formal systems.
§17Transcript · 2:12:16–2:20:50
So there's nothing because--
- But that's what a human would say.
- I know that's what a human would say
'cause we're used to the idea that there's,
but the whole point is that
by the time it's all possible formal systems,
it's like, it is all things you can imagine, but--
- Not all computations you can imagine,
but like we don't--
- Well, so the issue is can we encode?
Okay, so that's a fair question.
Is it possible to encode all,
I mean, once we,
is there something that isn't
what we can represent formally?
- Right.
- That is, is there something that,
and that's I think related to the hyper-Rouliad footnote,
so to speak, of which I'm afraid that the,
one of the things sort of interesting about this is,
there has been some discussion of this in theology
and things like that,
but which I don't necessarily understand all of,
but the key sort of new input is this idea
that all possible formal systems,
it's like, if you make a world,
people say, well, you make a world
with a particular, in a particular way,
with particular rules, but no, you don't do that.
You can make a world that deals with all possible rules
and then merely by virtue of living in a particular place
in that world, so to speak,
we have the perception we have of what the world is like.
Now, I have to say,
it's sort of interesting 'cause I've,
I wrote this piece about this
and I, this philosophy stuff is not super easy
and I've, as I'm talking to you about it
and I actually haven't,
people have been interested in lots of different things
we've been doing, but this, why does the universe exist,
has been, I would say, one of the ones
that you would think people would be most interested in,
but actually, I think they're just like,
oh, that's just something complicated that,
so I haven't explained it as much
as I've explained a bunch of other things
and I have to say, I think I may be missing
a couple of pieces of that argument that would be,
so it's kind of like--
- Well, you're conscious being is computationally bounded,
so you're missing-- - Indeed.
- Having written quite a few articles yourself,
you're now missing some of the pieces.
- Yes, well, it's always-- - That's the limitation
of being human.
- Right, one of the consequences of this,
why the universe exists thing and this kind of concept
of Rouillades and places in there representing
our perception of the universe and so on,
one of the weird consequences is,
if the universe exists, mathematics must also exist.
And that's a weird thing because mathematics,
people have been very confused, including me,
have been very confused about the question
of kind of what is the foundation of mathematics?
What is, what kind of a thing is mathematics?
Is mathematics something where we just write down axioms
like Euclid did for geometry and we just build the structure
and we could have written down different axioms
and we'd have a different structure?
Or is it something that has a more fundamental
sort of truth to it?
And I have to say, it's one of these cases where
I've long believed that mathematics has a great deal
of arbitrariness to it, that there are particular axioms
that kind of got written down by the Babylonians
and that's what we've ended up with the mathematics
that we have.
And I have to say, actually, my wife has been telling me
for 25 years, she's a mathematician,
she's been telling me, "You're wrong about the foundations
"of mathematics."
And I'm like, "No, no, no, I know what I'm talking about."
And finally, she's much more right than I've been.
So it's one of the--
- So I mean, her sense and your sense, are we just,
so this is to the question of mathematics,
are we just kind of on a trajectory through
ruleal space except in mathematics,
is there a trajectory, a certain kind of--
- I think that's partly the idea.
So I think that the notion is this.
So 100 years ago, a little bit more than 100 years ago,
well, people have been doing mathematics for ages,
but then in the late 1800s, people decided to try
and formalize mathematics and say,
mathematics is, we're gonna break it down,
we're gonna make it like logic, we're gonna make it
out of sort of fundamental primitives.
And that was people like Frege and Piano and Hilbert
and so on, and they kind of got this idea of,
let's do kind of Euclid, but even better,
let's just make everything just in terms of this sort of
symbolic axioms, and then build up mathematics from that.
And that, they thought at the time,
as soon as they get these symbolic axioms,
that they made the same mistake,
the kind of computational irreducibility mistake.
They thought as soon as we've written down the axioms,
then it'll just, we'll just have a machine,
kind of a super Mathematica, so to speak,
that can just grind out all true theorems of mathematics.
That got exploded by Godel's theorem,
which is basically the story
of computational irreducibility.
It's that even though you know those underlying rules,
you can't deduce all the consequences in any finite way.
And so that was, but now the question is,
okay, so they broke mathematics down into these axioms,
and they say, now you build up from that.
So what I'm increasingly coming to realize is,
that's similar to saying, let's take a gas
and break it down into molecules.
There's gas laws that are the large scale structure
and so on that we humans are familiar with,
and then there's the underlying molecular dynamics.
And I think that the axiomatic level of mathematics,
which we can access with automated theorem proving
and proof assistance and these kinds of things,
that's the molecular dynamics of mathematics.
And occasionally we see through to that molecular dynamics.
We see undecidability, we see other things like this.
One of the things I've always found very mysterious
is that Godel's theorem shows that there are sort of things
which cannot be finitely proved in mathematics.
There are proofs of arbitrary length,
infinite length proofs that you might need.
But in practical mathematics,
mathematicians don't typically run into this.
They just happily go along doing their mathematics.
And I think what's actually happening is that
what they're doing is they're looking at this,
they are essentially observers in meta-mathematical space,
and they are picking a reference frame
in meta-mathematical space,
and they are computationally bounded observers
of meta-mathematical space,
which is causing them to deduce
that the laws of meta-mathematics
and the laws of mathematics,
like the laws of fluid mechanics,
are much more understandable
than this underlying molecular dynamics.
And so what gets really bizarre is thinking about
kind of the analogy between meta-mathematics,
this idea of you exist in this kind of,
in this sort of space of possible,
in this kind of mathematical space
where the individual kind of points
in the mathematical space are statements in mathematics,
and they're connected by proofs,
where one statement, you know,
you take a couple of different statements,
you can use those to prove some other statement,
and you've got this whole network of proofs.
That's the kind of causal network of mathematics,
of what can prove what, and so on.
And you can say at any moment
in the history of a mathematician,
of a single mathematical consciousness,
you are in a single kind of slice
of this kind of meta-mathematical space.
You know a certain set of mathematical statements.
You can then deduce with proofs,
you can deduce other ones, and so on.
You're kind of gradually moving
through meta-mathematical space.
And so it's kind of the view is
that the reason that mathematicians perceive mathematics
to have the sort of integrity
and lack of kind of undecidability and so on that they do,
is because they, like we as observers
of the physical universe,
we have these limitations associated
with computational boundedness, single thread of time,
consciousness limitations, basically,
that the same thing is true of mathematicians
perceiving sort of meta-mathematical space.
And so what's happening is that when you look at,
if you look at one of these formalized mathematics systems,
something like Pythagoras' theorem,
it'll be, it'll take, oh, I don't know, what is it?
Maybe 10,000 individual little steps
to prove Pythagoras' theorem.
And one of the bizarre things,
that's sort of an empirical fact
that I'm trying to understand a little bit better,
if you look at different proof,
if you look at different formalized mathematics systems,
§18Transcript · 2:20:52–2:28:26
they actually have different axioms underneath,
but they can all prove Pythagoras' theorem.
And so, in other words,
it's a little bit like what happens with gases.
We can have air molecules, we can have water molecules,
but they still have fluid dynamics.
Both of them have fluid dynamics.
And so similarly, at the level that mathematics,
that mathematicians care about mathematics,
it's way above the molecular dynamics, so to speak.
And there are all kinds of weird things,
like for example, one thing I was realizing recently
is that the quantum theory of mathematics,
that's a very bizarre idea.
But basically, when you prove what is, you know,
a proof is you've got one statement in mathematics,
you go through other statements,
you eventually get to a statement
you're trying to prove, for example,
that's a path in metamathematical space.
And that's a single path, a single proof is a single path.
But you can imagine there are other proofs
of the same result.
There are a bundle of proofs.
There's this whole set of possible proofs.
- You could think of it as branching,
similar to the quantum mechanics model
that you were talking about.
- Exactly, and so-
- And then there's some invariance that you can formalize
in the same way that you can for the quantum mechanical.
- Right, so the question is in proof space,
you know, as you start thinking about multiple proofs,
are there analogs of, for example,
destructive interference of multiple proofs?
So here's a bizarre idea, it's just a couple of days old,
so not yet fully formed.
But as you try and do that,
when you have two different proofs,
it's like two photons going in different directions.
You have two proofs which at an intermediate stage
are incompatible.
And that's kind of like destructive interference.
- Is it possible for this to instruct the engineering
of automated proof systems?
- Absolutely, I mean, as a practical matter,
I mean, you know, this whole question,
in fact, Jonathan Gorad has a nice heuristic
for automated theorem provers
that's based on our physics project
that is looking for essentially using kind of,
using energy, in our models,
energy is kind of level of activity in this hypergraph.
And so there's sort of a heuristic
for automated theorem proving
about how do you pick which path to go down
that is based on essentially physics.
And I mean, the thing that gets interesting about this
is the way that one can sort of have the interplay
between like, for example, a black hole.
What is a black hole in mathematics?
So the answer is, what is a black hole in physics?
A black hole in physics is where,
in the simplest form of black hole, time ends.
That is all, you know, everything is crunched down
to the space-time singularity
and everything just ends up at that singularity.
So in our models, and that's a little hard to understand
in general relativity with continuous mathematics
and what does singularity look like.
In our models, it's something very pragmatic.
It's just, you're applying these rules,
time is moving forward,
and then there comes a moment where the rules,
no rules apply, so time stops.
It's kind of like the universe dies.
There's, you know, nothing happens in the universe anymore.
Well, in mathematics, that's a decidable theory.
That's a theory, so theories which have undecidability,
which are things like arithmetic, set theory,
all the serious models, theories in mathematics,
they all have the feature that there are proofs
of arbitrary long length.
In something like Boolean algebra,
which is a decidable theory,
there are, you know, any question in Boolean algebra,
you can just go crunch, crunch, crunch,
and in a known number of steps, you can answer it.
You know, satisfiability, you know, might be hard,
but it's still a bounded number of steps
to answer any satisfiability problem.
And so that's the notion of a black hole in physics
where time stops.
That's analogous to in mathematics
where there aren't infinite length proofs.
Where when in physics, you know,
you can wander around the universe forever
if you don't run into a black hole.
If you run into a black hole and time stops, you're done.
And it's the same thing in mathematics
between decidable theories and undecidable theories.
So it's an example, and I think we're sort of,
the attempt to understand, so another question is kind of,
what is the generativity of metamathematics?
What is the bulk theory of metamathematics?
So in the literature of mathematics,
there are about 3 million theorems
that people have published.
And those represent, it's kind of on this,
it's like on the earth, we would be, you know,
we've put cities in particular places on the earth,
but yet there is ultimately, you know,
we know the earth is roughly spherical
and there's an underlying space.
And we could just talk about, you know,
the world of space in terms of where our cities
happen to be, but there's actually an underlying space.
And so the question is, what's that for metamathematics?
And as we kind of explore what is, for example,
for mathematics, which is always likes taking
sort of abstract limits.
So an obvious abstract limit for mathematics to take
is the limit of the future of mathematics.
That is, what will be, you know,
the ultimate structure of mathematics?
And one of the things that's an empirical observation
about mathematics that's quite interesting
is that a lot of theories in one area of mathematics,
algebraic geometry or something, might have,
they play into another area of mathematics.
That same kind of a fundamental construct
seem to occur in very different areas of mathematics.
And that's structurally captured a bit
with category theory and things like that.
But I think that there's probably an understanding
of this metamathematical space that will explain
why different areas of mathematics
ultimately sort of map into the same thing.
And I mean, you know, my little challenge to myself
is what's time dilation in metamathematics?
In other words, as you basically,
as you move around in this mathematical space
of possible statements, you know,
what's, how does that moving around?
It's basically what's happening is that
as you move around in the space of mathematical statements,
it's like you're changing from algebra to geometry
to whatever else.
And you're trying to prove the same theorem.
But as you try, if you keep on moving
to these different places,
it's slower to prove that theorem
'cause you keep on having to translate what you're doing
back to where you started from.
And that's kind of the beginnings of the analog
of time dilation in metamathematics.
- Plus there's probably fractional dimensions
in this space as well.
- Oh, this space is a very messy space.
This space is much messier than physical space.
I mean, even in the models of physics,
physical space is very tame
compared to branchial space and ruleal space.
I mean, the mathematical structure, you know,
branchial space is probably more like Hilbert space,
but it's a rather complicated Hilbert space.
And ruleal space is more like this weird
infinity groupoid story of growth and decay.
And I can explain that a little bit
because in metamathematical space,
a path in metamathematical space
is a path between two statements
is a way to get by proofs,
is a way to find a proof
that goes from one statement to another.
And so one of the things you can do,
you can think about is you've got,
between statements, you've got proofs.
They are paths between statements.
Okay, so now you can go to the next level
and you can ask,
what about a mapping from one proof to another?
And so that's in category theory,
that's kind of a higher category,
a notion of higher categories
where you're mapping not just between objects,
but you're mapping between the mappings
between objects and so on.
And so you can keep doing that.
You can keep saying higher order proofs.
I want mappings between proofs between proofs and so on.
And that limiting structure,
oh, by the way,
one thing that's very interesting is
imagine in proof space,
you've got these two proofs.
And the question is,
what is the topology of proof space?
In other words, if you take these two paths,
can you continuously deform them into each other?
§19Transcript · 2:28:29–2:35:43
Or is there some big hole in the middle
that prevents you from continuously deforming them
one into the other?
It's kind of like,
when you think about some,
I don't know, some puzzle, for example,
you're moving pieces around on some puzzle,
and you can think about the space
of possible states of the puzzle,
and you can make this graph that shows
from one state of the puzzle
to another state of the puzzle and so on.
And sometimes you can easily get
from one state to any other state,
but sometimes there'll be a hole in that space.
And there'll be,
you always have to go around the circuitous route
to get from here to there.
There won't be any direct way.
That's kind of a question
of whether there's sort of an obstruction in the space.
And so the question is, in proof space,
what is the, what are,
what does it mean if there's an obstruction in proof space?
- Yeah, I don't even know what an obstruction means
in proof space,
'cause for it to be an obstruction,
it should be reachable some other way
from some other place, right?
So this is like an unreachable part of the graph.
- No, it's not just an unreachable part.
It's a part where there are paths that go one way,
there are paths that go the other way.
And this question of homotopy and mathematics
is this question, can you continuously deform,
from one path to another path,
or do you have to go in a jump, so to speak?
So if you're going around a sphere, for example,
if you're going around, I don't know,
a cylinder or something, you can wind around one way
and you can, there's no,
there are paths where you can easily deform one path
into another, because it's just sort of sitting
on the same side of the cylinder.
But when you've got something
that winds all the way around a cylinder,
you can't continuously deform that down to a point
because it's stuck, wrapped around a cylinder.
- My intuition about proof spaces,
you should be able to deform it.
I mean that, 'cause then otherwise
it doesn't even make sense.
'Cause if the topology matters
of the way you move about the space,
then I don't even know what that means.
- Well, what it would mean is that you would have
one way of doing a proof of something over here in algebra
and another way of doing a proof of something
over here in geometry,
and there would not be an intermediate way
to map between those proofs.
- But how would that be possible
if they start at the same place and end at the same place?
- Well, it's the same thing as we've got points on a,
if we've got paths on a cylinder.
- Now I understand how it works in physical space,
but it just doesn't,
it feels like proof space shouldn't have that.
Okay, I mean--
- I'm not sure, I don't know.
We'll know very soon,
'cause we get to do some experiments.
This is the great thing about this stuff,
is that in fact, in the next few days,
I hope to do a bunch of experiments on this.
- So you're playing with like proofs in this kind of space?
- Yes, yes.
I mean, so this is toy theories,
and we've got good,
so this kind of segues to perhaps another thing,
which is this whole idea of multi-computation.
So this is another kind of bigger idea that,
so, okay, this has to do with
how do you make models of things?
And it's going to, it's some,
so I've sort of claimed that there've been sort of four epochs
in the history of making models of things.
And this multi-computation thing is the fourth,
is a new epoch.
- What are the first three?
- The first one is back in antiquity,
ancient Greek times,
people were like, what's the universe made of?
Oh, it's made of, you know, everything is water, Thales,
you know, or everything is made of atoms.
It's sort of what are things made of?
Or, you know, there are these crystal spheres
that represent where the planets are, and so on.
It's like a structural idea
of how the universe is constructed.
There's no real notion of dynamics,
it's just what is the universe, how is the universe made?
Then we get to the 1600s,
and we get to the sort of revolution of mathematics
being introduced into physics.
And then we have this kind of idea
of you write down some equation,
the what happens in the universe
is the solving of that equation,
time enters, but it's usually just a parameter.
We just can, you know, sort of slide it back and forth
and say, here's where it is.
Okay, then we come to this kind of computational idea
that I kind of started really pushing in the early 1980s.
As a result, you know, the things we were talking about
before about complexity, that was my motivation,
but the bigger story was the story
of kind of computational models of things.
And the big difference there from the mathematical models
is in mathematical models, there's an equation,
you solve it, you got kind of slide time
to the place where you want it.
In computational models, you give the rule
and then you just say, go run the rule.
And time is not something you get to slide,
time is something where it just, you run the rule,
time goes in steps, and that's how you work out
how the system behaves.
You don't, time is not just a parameter,
time is something that is about the running of these rules.
And so there's this computational irreducibility,
you can't jump ahead in time,
but there's still, important thing is
there's still one thread of time.
It's still the case, you know, the cellular automaton state,
then it has the next state and the next state and so on.
The thing that is kind of, we've sort of tipped off
by quantum mechanics in a sense,
although it actually feeds back even into relativity
and things like that,
that there are these multiple threads of time.
And so in this multi-computation paradigm,
the kind of idea is, instead of there being
this single thread of time,
there are these kind of distributed asynchronous threads
of time that are happening.
And the thing that's sort of different there
is if you want to know what happened,
if you say what happened in the system,
in the case of the computational paradigm,
you just say, well, after a thousand steps,
we got this result, right?
But in the multi-computational paradigm,
after a thousand steps,
not even clear what a thousand steps means,
because you've got all these different threads of time,
but there is no state.
There's all these different possible,
you know, there's all these different parts.
And so the only way you can know what happened
is to have some kind of observer who is saying,
here's how to parse the results of what was going on.
- Right, but that observer is embedded
and they don't have a complete picture, so.
- In the case of physics, that's right, yes.
And then, but that's, but so the idea is
that in this multi-computation setup,
that it's this idea of these multiple threads of time
and models that are based on that.
And this is similar to what people think about
in non-deterministic computation.
So you have a Turing machine,
usually it has a definite state,
it follows another state, it follows another state.
But typically what people have done
when they've thought about these kinds of things
is they've said, well, there are all these possible paths,
a non-deterministic Turing machine
can follow all these possible paths,
but we just want one of them.
We just want the one that's the winner,
that factors the number or whatever else.
And similarly, you know, it's the same story
in logic programming and so on,
where we say, we've got this goal,
find us a path to that goal.
I just want one path, then I'm happy.
Or theorem proving, same story.
I just want one proof and then I'm happy.
What's happening in multi-computation in physics
is we actually care about many paths.
And well, there is a case, for example,
probabilistic programming is a version of multi-computation
in which you're looking at all the paths,
you're just asking for probabilities of things.
But in a sense in physics,
we're taking different kinds of samplings.
For example, in quantum mechanics,
we're taking a different kind of sampling
of all these multiple paths.
§20Transcript · 2:35:46–2:43:35
But the thing that is notable
is that when you're an observer embedded in this thing,
et cetera, et cetera, et cetera,
with various other sort of footnotes and so on,
it is inevitable that the thing that you parse
out of the system looks like general relativity
and quantum mechanics.
In other words, that just by the very structure
of this multi-computational setup,
it inevitably is the case
that you have certain emergent laws.
Now, why is this perhaps not surprising?
In thermodynamics and statistical mechanics,
there are sort of inevitable emergent laws
of sort of gas dynamics that are independent
of the details of the molecular dynamics,
sort of the same kind of thing.
But I think what happens is,
what's a sort of a funny thing
that I've just been understanding very recently
is when I kind of introduced this whole
sort of computational paradigm complexity-ish thing
back in the '80s, it was kind of like a big downer
because it's like there's a lot of stuff you can't say
about what systems will do.
And then what I realized is,
and then you might say, now we've got multi-computation,
it's even worse.
You know, it isn't just one thread of time
that we can't explain, it's all these threads of time
we can't explain anything.
But the following thing happens.
Because there is all this irreducibility
and any detailed thing you might want to answer,
it's very hard to answer.
But when you have an observer
who has certain characteristics
like computational boundedness,
sequentiality of time and so on,
that observer only samples certain aspects
of this incredible complexity going on
in this multi-computational system.
And that observer is sensitive only to
some underlying core structure
of this multi-computational system.
There is all this irreducible computation going on,
all these details, but to that kind of observer,
what's important is only the core structure
of multi-computation, which means that observer
observes comparatively simple laws.
And I think it is inevitable that that observer
observes laws which are mathematically structured
like general relativity and quantum mechanics,
which by the way are the same law in our model of physics.
- So that's an explanation why there's simple laws
that explain a lot for this observer.
- Potentially, yes.
But the place where this gets really interesting
is there are all these fields of science
where people have kind of gotten stuck,
where they say we'd really love to have
a physics-like theory of economics.
We'd really love to have a physics-like law in linguistics.
- We gotta talk about molecular biology here.
- Okay.
- So where does multi-computation come in for biology?
Economics is super interesting too, but biology--
- Okay, let's talk about that.
So let's talk about chemistry for a second, okay?
So I mean, I have to say, you know,
this is, it's such a weird business for me
because, you know, there are these kind of
paradigmatic ideas and then the actual applications.
And it's like I've always said,
I know nothing about chemistry.
I learned all the chemistry I know, you know,
the night before some exam when I was 14 years old.
But I've actually learned a bunch more chemistry.
And in Wolfram language these days,
we have really pretty nice symbolic
representation of chemistry.
And in understanding the design of that,
I've actually, I think, learned a certain amount of chemistry
that if you quizzed me on sort of basic high school
chemistry, I would probably totally fail.
But, okay, so what is chemistry?
I mean, chemistry is sort of a story of, you know,
chemical reactions are like you've got
this particular chemical, it's represented
as some graph of, you know, these are,
these are this configuration of molecules
with these bonds and so on.
And a chemical reaction happens,
you've got these sort of two graphs,
they interact in some way, you've got another graph
or multiple other graphs out.
So that's kind of the, the sort of the abstract view
of what's happening in chemistry.
And so when you do a chemical synthesis, for example,
you are given certain sort of these are possible reactions
that can happen, and you're asked, can you piece together
this sequence of such reactions, a sequence of such sort of
axiomatic reactions, usually called name reactions
in chemistry, can you piece together a sequence
of these reactions so that you get out at the end
this great molecule you were trying to synthesize?
And so that's a story very much like theorem proving.
And people have done, actually, they started in the 1960s,
looking at kind of the theorem proving approach to that,
although it didn't really, it didn't, it didn't,
was sort of done too early, I think.
But anyway, so that's kind of the view is that chemistry,
chemical reactions are the story of all these different
sort of paths of possible things that go on.
Okay, let's go to an even lower level.
Let's say, instead of asking about which species
of molecules we're talking about,
let's look at individual molecules.
And let's say we're looking at individual molecules,
and they are having chemical reactions,
and we're building up this big graph
of all these reactions that are happening, okay?
So then we've got this big graph, and by the way,
that big graph is incredibly similar
to these hypergraph rewriting things.
In fact, in the underlying theory of multi-computation,
there are these things we call token event graphs,
which are basically, you've broken your state into tokens,
like in the case of a hypergraph,
you've broken it into hyperedges,
and each event is just consuming some number of tokens
and producing some number of tokens.
- Yeah, but then you have to,
there's a lot of work to be done on update rules.
- In terms of what they actually are for chemistry?
- Yeah, what they are for our observed chemistry.
- Yes, indeed, yes, indeed.
We've been working on that, actually,
because we have this beautiful system in Wolfram language
for representing chemistry symbolically.
So we actually have, this is an ongoing thing
to actually figure out what they are
for some practical cases.
- Does that require human injection,
or can it be automatically discovered, these update rules?
- Well, if we could do quantum chemistry better,
we could probably discover them automatically,
but I think in reality right now,
it's like there are these particular reactions.
And really, to understand what's going on,
we're probably gonna pick a particular subtype of chemistry,
and just, because let me explain where this is going.
The place that, here's where this is going.
So, you've got this whole network of all these molecules
having all these reactions and so on.
And this is some whole multi-computational story,
because each sort of chemical reaction event
is its own separate event.
We're saying they all happen asynchronously.
We're not describing in what order they happen.
Maybe that order is governed
by some quantum mechanics thing, doesn't really matter.
We're just saying they happen in some order.
And then we ask, what is the,
how do we think about the system?
Well, this thing is some kind
of big multi-computational system.
The question is, what is the chemical observer?
And one possible chemical observer is,
all you care about is,
did you make that particular drug molecule?
You're just asking for the one path.
Another thing you might care about is,
I want to know the concentration of each species.
I want to know, at every stage,
I'm going to solve the differential equations
that represent the concentrations,
and I want to know what those all are.
But there's more, because when,
and it's kind of like,
you're going below in statistical mechanics,
there's kind of all these molecules bouncing around,
and you might say, we're just going to ignore,
we're just going to look at the aggregate densities
of certain kinds of molecules,
but you can look at a lower level.
You can look at this whole graph of possible interactions.
And so, the kind of the idea would be,
what, is the only chemical observer,
one who just cares about overall concentrations,
or can there be a chemical observer
who cares about this network of what happened?
And so, that, the question then is,
so let me give an analogy.
So this is where I think this is potentially
very relevant to molecular biology and molecular computing.
When we think about a computation,
usually we say it's input, it's output.
We, you know, or chemistry, we say, there's this input,
we're going to make this molecule as the output.
§21Transcript · 2:43:38–2:51:29
But what if what we actually encode,
what if our computation,
what if the thing we care about
is some part of this dynamic network?
What if it isn't just the input and the output
that we care about?
What if there's some dynamics of the network
that we care about?
Now, imagine you're a chemical observer.
What is a chemical observer?
Well, in molecular biology,
there are all kinds of weird sorts of observers.
There are membranes that exist
that have, you know, different kinds of molecules
that can bind to them, things like this.
It's not obvious that the, from a human scale,
we just measure the concentration of something,
is the relevant story.
We can imagine that, for example,
when we look at this whole network of possible reactions,
we can imagine, you know, at a physical level,
we can imagine, well, what was the actual
momentum direction of that molecule?
What was it, which we don't pay any attention to
when we're just talking about chemical concentrations.
What was the orientation of that molecule?
These kinds of things.
And so here's the place where I have a little suspicion.
Okay?
So one of the questions in biology
is what matters in biology?
And that is, you know, we have all these chemical reactions.
We have all these molecular processes going on
in biological systems.
What matters?
And, you know, one of the things is
to be able to tell what matters,
well, so a big story of the what matters question
was what happened in genetics in 1953,
when DNA, when it was figured out how DNA worked.
Because before that time, you know,
genetics had been all these different effects
and complicated things.
And then it was realized, ah, there's something new.
A molecule can store information,
which wasn't obvious before that time.
A single molecule can store information.
So there's a place where there can be something important
that's happening in molecular biology,
and it's just in the sequence
that's storing information in a molecule.
So the possibility now is, imagine this dynamic network,
this, you know, causal graphs and multi-way causal graphs
and so on, that represent all of these different reactions
between molecules.
What if there is some aspect of that
that is storing information
that's relevant for molecular biology?
- And the dynamic aspect of that.
- Yes, that's right.
- So it's similar to how the structure
of a DNA molecule stores information.
It could be the dynamics of the system
somehow stores information.
And this kind of process might allow you
to give predictions of what that would be.
- Well, yes, but also imagine that you're trying to do,
for example, imagine you're trying
to do molecular computation, okay?
You might think the way we're gonna do
molecular computation is, we're just gonna run the thing,
we're gonna see what came out,
we're gonna see what molecule came out.
This is saying, that's not the only thing you can do.
There is a different kind of chemical observer
that you can imagine constructing,
which is somehow sensitive to this dynamic network.
Exactly how that works, how we make that measurement,
I don't know, but I have a few ideas,
but that's what's important, so to speak.
And that means, and by the way,
you can do the same thing even for Turing machines.
You can say, if you have a multi-way Turing machine,
you can say, how do you compute
with a multi-way Turing machine?
You can't say, well, we've got this input and this output,
because the thing has all these threads of time,
and it's got lots of outputs.
And so then you say, well, what does it even mean
to be a universal multi-way Turing machine?
I don't fully know the answer to that.
- That's an interesting idea, it would freak Turing out,
for sure, 'cause then the dynamics,
the trajectory of the computation matters.
- Yes, yes, but the thing is,
so this is, again, a story of what's the observer,
so to speak, in chemistry, what's the observer there?
Now, to give an example of where that might matter,
a very sort of present-day example is in immunology,
where we have whatever it is,
10 billion different kinds of antibodies
that are all these different shapes and so on.
We have a trillion different kinds of T-cell receptors
that we produce, and the traditional theory of immunology
is this clonal selection theory
where we are constantly producing,
randomly producing all these different antibodies,
and as soon as one of them binds to an antigen,
then that one gets amplified,
and we produce more of that antibody and so on.
Back in the 1960s, an immunologist called Nils Joerner,
who was the guy who invented monoclonal antibodies,
various other things, kind of had this network theory
of the immune system, where it would be like,
well, we produce antibodies,
but then we produce antibodies to the antibodies,
anti-antibodies, and we produce anti-anti-antibodies,
and we get this whole dynamic network of interactions
between different immune system cells,
and that was kind of a qualitative theory at that time,
and I've been a little disappointed
'cause I've been studying immunology a bit recently,
and I knew something about this 35 years ago or something,
and I knew, I'd read a bunch of the books,
and I'd talked to a bunch of the people and so on,
and it was like an emerging theoretical immunology world,
and then I look at the books now,
and they're very thick because they've got,
there's just a ton known about immunology,
and all these different pathways,
all these different details and so on,
but the theoretical sections seem to have shrunk,
and so it's, so the question is, what,
for example, immune memory,
where does the immune memory reside?
Is it actually some cell sitting in our bone marrow
that is living for the whole of our lives
that's gonna spring into action
as soon as we're shown the same antigen,
or is it something different from that?
Is it something more dynamic?
Is it something more like some network of interactions
between these different kinds of immune system cells
and so on, and it's known
that there are plenty of interactions between T cells,
and there's plenty of dynamics,
but what the consequence of that dynamics is is not clear,
and to have a qualitative theory for that
doesn't seem to exist.
In fact, I was just been trying to study this,
so I'm quite incomplete in my study of these things,
but I was a little bit taken aback
'cause I've been looking at these things,
and it's like, and then they get to the end
where they have the most advanced theory that they've got,
and it turns out it's the cellular automaton theory.
It's like, okay, well, at least I understand that theory,
but I think that the possibility is that
this is a place where if you want to know,
explain roughly how the immune system works,
it ends up being this dynamic network,
and then the immune consciousness, so to speak,
the observer, ends up being something that in the end,
it's kind of like does the human get sick or whatever,
but it's something which is a complicated story
that relates to this whole sort of dynamic network
and so on, and I think that's another place
where this kind of notion of,
where I think multi-computation has the possibility.
See, one of the things, okay,
you can end up with something where,
yes, there is a general relativity in there,
but it turns out, but it may turn out
that the observer who sees general relativity
in the immune system is an observer that's irrelevant
to what we care about about the immune system.
I mean, it could be, yes, there is some effect
where there's some time dilation of T cells
interacting with whatever, but it's like,
that's an effect that's just irrelevant,
and the thing we actually care about is things about
what happens when you have a vaccine
that goes into some place in shape space,
and how does that affect other places in shape space,
and how does that spread?
What's the analog of the speed of light in shape space,
for example?
That's an important issue
if you have one of these dynamic theories.
It's like you poke into shape space by having,
let's say, a vaccine or something
that has a particular configuration in shape space.
How quickly, as this dynamic network spreads out,
how quickly do you get sort of other antibodies
in different places in shape space, things like that.
- When you say shape space,
you mean the shape of the molecules?
And then, so this is like,
could be deeply connected to the protein
and multi-protein folding, all of that kind of stuff.
§22Transcript · 2:51:32–2:59:29
- To be able to say something interesting
about the dance of proteins.
- Right, exactly.
- That actually has an impact on
helping develop drugs, for example,
or has an impact on virology, immunology,
helping to deal with viruses.
- Well, I think the big thing is,
when we think about molecular biology,
what is the qualitative way to think about it?
In other words, is it chemical reaction networks?
Is it genetics, DNA, big news,
it's kind of, there's a digital aspect to the whole thing.
What is the qualitative way to think about
how things work in biology?
When we think about, I don't know,
some phenomenon like aging or something,
which is a big, complicated phenomenon,
which just seems to have all these different tentacles,
is it really the case that that can be thought about
in some, without DNA,
when people were describing genetic phenomena,
there were dominant, recessive, this, that, and the other,
it got very, very complicated,
and then people realized, oh, it's just,
and what is a gene, and so on, and so on, and so on,
then people realize it's just base pairs,
and there's this digital representation.
And so the question is,
what is the overarching representation
that we can now start to think about using in microbiology?
I don't know how this will work out,
and this is, again, one of these things where,
and the place where that gets important is,
maybe molecular biology is doing molecular computing
by using some dynamic process
that is something where it is very happily saying,
oh, I just got a result.
It's in the dynamic structure of this network.
Now I'm gonna go and do some other thing
based on that result, for example,
but we're like, oh, I don't know what's going on.
It's just, we just measured the levels of these chemicals,
and we couldn't conclude anything,
but it just, we're looking at the wrong thing.
And so that's kind of the potential there,
and it's, I mean, these things are,
I don't know, it's, for me, it's like,
I've not really, that was not a view.
I mean, I've thought about molecular computing
for ages and ages and ages,
and I've always imagined that the big story
is kind of the original promise of nanotechnology
of like, can we make a molecular scale constructor
that will just build a molecule in any shape?
But I don't think, I'm now increasingly concluding
that's not the big point.
The big point is something more dynamic.
That will be an interesting end point
for any of these things,
but that's perhaps not the thing,
'cause the one example we have in molecular computing
that's really working is us biological organisms.
And maybe the thing that's important there
is not this, what chemicals do you make, so to speak,
but more this kind of dynamic process.
- Dynamic process, and then you can have a good model
like the hypergraph to then explore,
what, like simulate, again, make predictions,
and if they-- - I think just have a way
to reason about biology.
I mean, it's hard.
First of all, biology doesn't have theories like physics.
Physics is a much more successful
sort of global theory kind of area.
Biology, what are the global theories of biology?
Pretty much Darwinian evolution.
That's the only global theory of biology.
Any other theory is just a, well, the kidneys work this way,
this thing works this way, and so on.
There isn't, I suppose another global theory
is digital information in DNA.
That's another sort of global fact about biology.
- But the difficult thing to do is to match
something you have a model of in the hypergraph
to the actual, like how do you discover the theory?
How do you discover the theory?
Okay, you have something that looks nice and makes sense,
but like you have to match it to validation.
- Oh, sure, right. - And that's tricky
because you're walking around in the dark.
- Not entirely.
I mean, so, you know, for example,
in what we've been trying to think about
is take actual chemical reactions, okay?
So, you know, one of my goals is
can I compute the primes with molecules?
Okay, that's, if I can do that,
then I kind of, maybe I can compute things.
And, you know, there's this nice automated lab,
these guys, these Emerald Cloud Lab people
have built with Wolfram Language and so on.
That's an actual physical lab,
and you send it a piece of Wolfram Language code,
and it goes and, you know,
actually does physical experiments.
And so one of my goals,
because I'm not a test tube kind of guy,
I'm more of a software kind of person,
is can I make something where, you know,
in this automated lab, we can actually get it
so that there's some gel that we made,
and, you know, the positions of the stripes are the primes.
- Is the primes, I love it, yeah.
- I mean, that would be an example of where,
and that would be with a particular, you know,
framework for actually doing the molecular computing,
you know, with particular kinds of molecules.
And there's a lot of kind of ambient technological mess,
so to speak, associated with, oh, is it carbon?
Is it this, is it that?
You know, is it important that there's a bromine atom here?
Et cetera, et cetera, et cetera.
This is all chemistry that I don't know much about.
And, you know, that's a sort of, you know,
unfortunately, that's down at the level, you know,
I would like to be at the software level,
not at the level of the transistors, so to speak.
But in chemistry, you know, there's a certain amount
we have to do, I think, at the level of transistors
before we get up to being able to do it,
although, you know, automated labs
certainly help in setting that up.
- I talked to a guy named Charles Hoskinson.
He mentioned that he's collaborating with you.
He's an interesting guy.
He sends me papers on, speaking of automated theorem
proving a lot.
He's exceptionally well read on that area as well.
So what's the nature of your collaboration with him?
He's the creator of Cardano.
What's the nature of the collaboration between Cardano
and the whole space of blockchain and Wolfram,
Wolfram Alpha, Wolfram Blockchain, all that kind of stuff?
- Well, okay, we're segueing to a slightly different world.
But so, although not completely unconnected.
- Right, the whole thing is somehow connected.
- I know, I mean, you know, the strange thing in my life
is I've sort of alternated between doing basic science
and doing technology about five times in my life so far.
And the thing that's just crazy about it is, you know,
every time I do one of these alternations,
I think there's not gonna be a way back to the other thing.
And like I thought for this physics project,
I thought, you know,
we're doing fundamental theory of physics,
maybe it'll have an application in 200 years.
But now I've realized,
actually this multi-computation idea
is applicable here and now.
It's, and in fact, it's also giving us this way.
I'll just mention one other thing
and then we can talk about blockchain.
The question of,
actually that relates to several different things,
but one of the things about, okay,
so our Wolfram Language,
which is our attempt to kind of represent everything
in the world computationally.
And it's the thing I kind of started building 40 years ago
in the form of actual Wolfram Language 35 years ago.
It's kind of this idea of,
can we express things about the world
in computational terms?
And, you know, we've come a long way
in being able to do that.
Wolfram Alpha is kind of the consumer version of that
where you're just using natural languages as input.
And it turns it into our symbolic language.
And that's, you know, the symbolic language,
Wolfram Language is what people use
and have been using for the last 33 years.
Actually, Mathematica, which is its first instantiation,
will be one third of a century old in October.
And that, it's kind of interesting.
- What do you mean one third of a century?
Does that mean 33 or 30?
What are we--
- 33 and a third.
- 33 and a third.
(laughing)
- I've never heard of anyone celebrating that anniversary,
but I like it.
- I know, a third of a century though.
It's like, you don't get many slices of a century
that are interesting.
But, you know, I think that the thing
that's really striking about that is that means,
you know, including the whole sort of technology stack
I built around that's about 40 years old,
§23Transcript · 2:59:32–3:07:21
and that means it's a significant fraction
of the total age of the computer industry.
And it's, I mean, I think it's cool
that we can still run, you know,
Mathematica version one programs today and so on,
and we've sort of maintained compatibility.
And we've been just building this big tower
all those years of just more and more
and more computational capabilities.
It's sort of interesting.
We just made this picture of kind of the different
kind of threads of computational content,
of, you know, mathematical content,
and, you know, all sorts of things with, you know,
data and graphs and whatever else.
And what you see in this picture
is about the first 10 years,
it's kind of like it's just a few threads.
And then about maybe 15, 20 years ago,
it kind of explodes in this whole collection
of different threads of all these different capabilities
that are now part of Wolfram Language
and representing different things in the world.
But the thing that was super lucky in some sense
is it's all based on one idea.
It's all based on the idea of symbolic expressions
and transformation rules for symbolic expressions,
which was kind of what I originally
put into this SMP system back in 1979
that was a predecessor of the whole Wolfram Language stack.
So that idea was an idea that I got
from sort of trying to understand
mathematical logic and so on.
It was my attempt to kind of make a general
human comprehensible model of computation
of just everything is a symbolic expression
and all you do is transform symbolic expressions.
And, you know, in retrospect,
I was very lucky that I understood
as little as I understood then,
because had I understood more,
I would have been completely freaked out
about all the different ways
that that kind of model can fail.
Because what do you do when you have a symbolic expression,
you make transformations for symbolic expressions?
Well, for example, one question is there may be
many transformations that could be made
in a very multi-computational kind of way.
But what we're doing is picking,
we're using the first transformation that applies.
And we keep doing that until we reach a fixed point.
And that's the result.
And that's kind of a very,
it's kind of a way of sort of sliding around
the edge of multi-computation.
And back when I was working on SMP and things,
I actually thought about these questions
about how, you know, how,
what determines this kind of evaluation path.
So for example, you know, you work out Fibonacci,
you know, Fibonacci is a recursive thing.
F of N is F of N minus one plus F of N minus two.
And you get this whole tree of recursion, right?
And there's the question of how do you evaluate
that tree of recursion?
Do you do it sort of depth first,
where you go all the way down one side?
Do you do it breadth first,
where you're kind of collecting the terms together,
where you know that, you know,
F of eight plus F of seven,
F of seven plus F of six,
you can collect the F of sevens and so on.
These are, you know, I didn't realize it at the time.
It's kind of funny.
I was working on gauge field theories back in 1979.
And I was also working on the evaluation model in SMP
and they're the same problem.
But it took me 40 more years to realize that.
And this question about how you do
this sort of evaluation front,
that's a question of reference frames.
It's a question of kind of the story of,
I mean, that is basically this question of
in what order is the universe evaluated?
And that, and so what you realize is
there's this whole sort of world
of different kinds of computation that you can do
sort of multi computationally.
And that's an interesting thing.
It has a lot of implications
for distributed computing and so on.
It also has a potential implication for blockchain,
which we haven't fully worked out,
which is, and this is not what we're doing with Cardano,
but this is a different thing.
This is something where one of the questions is
when you have, in a sense,
blockchain is a deeply sequentialized story of time.
Because in blockchain,
there's just one copy of the ledger.
And you're saying, this is what happened.
You know, time has progressed in this way.
And there are little things around the edge
as you try and reach consensus and so on.
And, you know, actually we just recently,
we've had this little conference we organized
about the theory of distributed consensus.
'Cause I realized that a bunch of interesting things
that some of our science can tell one about that.
But that's a different, let's not go down that path.
- Yeah, but distributed consensus,
that still has a sequential, there's like one-
- There's still sequentiality.
So distributed consensus-
- Don't tell me you're thinking through like
how to apply multi computation to blockchain.
- Yes, and so that becomes a story of, you know,
instead of transactions all having to settle in one ledger,
it's like a story of all these different ledgers
and they all have to have some ultimate consistency,
which is what causal invariance would give one,
but it can take a while.
And the, it can take a while
is kind of like quantum mechanics.
So it's kind of what's happening is
there are these different paths of history
that correspond to, you know,
in one path of history, you got paid this amount.
In another path of history, you got paid this amount.
In the end, the universe will always become consistent.
Now, the way it works is,
okay, it's a little bit more complicated than that.
What happens is the way space is knitted together
in our theory of physics is through all these events.
And the idea is that the way that economic space
is knitted together is there these autonomous events
that essentially knit together economic space.
So there are all these threads of transactions
that are happening.
And the question is, can they be made consistent?
Are there, is there something forcing them
to be sort of a consistent fabric of economic reality?
And sort of what this has led me to
is trying to realize how does economics fundamentally work?
And, you know, what is economics?
And, you know, what are the atoms of economics,
so to speak?
And so what I've kind of realized is that,
that sort of the, perhaps,
I don't even know if this is right yet.
There's sort of events in economics of transactions.
There are states of agents
that are kind of the atoms of economics.
And then transactions are kind of agents
transact in some,
transact in some way, and that's an event.
And then the question is,
how do you knit together sort of economic space from that?
What is there in economic space?
Well, all these transactions,
there's a whole complicated collection
of possible transactions.
But one thing that's true about economics
is we tend to have the notion
of a definite value for things.
We could imagine that, you know,
you buy a cookie from somebody
and they want to get a movie ticket.
And there is some way that AI bots could make some path
from the cookie to the movie ticket
by all these different intermediate transactions.
But in fact, we have an approximation to that,
which is we say they each have a dollar value.
And we have this kind of numeraire concept
of there's just a way of kind of,
of taking this whole complicated space of transactions
and parsing it in something
which is a kind of a simplified thing
that is kind of like a parsing of physical space.
And so my guess is that the, yet again, I mean, it's crazy
that all these things are so connected.
This is another multi-computation story,
another story of where what's happening
is that the economic consciousness,
the economic observer is not going to deal
with all of those different microscopic transactions.
They're just going to parse the whole thing by saying,
there's this value, it's a number.
And that's their understanding of their summary
of this economic network.
And there will be all kinds of things,
like there are all kinds of arbitrage opportunities,
which are kind of like the quantum effects
in this whole thing.
And that's, you know, and places where there's sort
of different paths that can be followed and so on.
And so the question is,
can one make a sort of global theory of economics?
§24Transcript · 3:07:24–3:15:35
And then my test case is, again,
what is time dilation in economics?
And I know for, you know,
if you imagine a very agricultural economics
where people are growing lettuces in fields
and things like this, and you ask questions about,
well, if you're transporting lettuces to different places,
you know, what is the value of the lettuces
after you have to transport them
versus if you're just sitting in one place and selling them?
And you can kind of get a little bit of an analogy there,
but I think there's a better and more complete analogy.
And that's the question of,
is there a theory like general relativity
that is a global theory of economics?
And is it about something we care about?
It could be that there is a global theory,
but it's about a feature of economic reality
that isn't important to us.
Now, another part of the story is,
can one use those ideas to make essentially
a distributed blockchain,
a distributed generalization of blockchain,
kind of the quantum analog of money, so to speak,
where you have, for example,
you can have uncertainty relations where you're saying,
you know, well, if I insist on knowing my bank account
right now, there'll be some uncertainty.
If I'm prepared to wait a while,
then it'll be much more certain.
And so there's, you know, is there a way of using,
and so we've made a bunch of prototypes of this,
which I'm not yet happy with,
because what I realized is
to really understand these prototypes,
I actually have to have a foundational theory of economics.
And so that's kind of a, you know,
it may be that we could deploy one of these prototypes
as a practical system,
but I think it's really gonna be much better
if we actually have an understanding
of how this plugs into kind of the economic system.
- And that means like a fundamental theory
of transactions between entities.
- Well, and how- - That's what you mean
by economics.
- Yes, I think so.
But I mean, you know, how there emerge
sort of laws of economics,
I don't even know, and I've been asking friends of mine
who are economists and things, what is economics?
You know, is it an axiomatic theory?
Is it a theory that is a kind of
a qualitative description theory?
Is it, you know, what kind of a theory is it?
Is it a theory, you know, what kind of thinking?
It's like in biology, in evolutionary biology, for example,
there's a certain pattern of thinking
that goes on in evolutionary biology,
where if you're a good evolutionary biologist,
somebody says, "That creature has a weird horn."
And they'll say, "Well, that's because this and this
"and this and this selection of this kind and that kind,
"and that's the story."
And it's not a mathematical story,
it's a story of a different type of thinking
about these things.
By the way, evolutionary biology is yet another place
where it looks like this multi-computational idea
can be applied.
And that's where maybe speciation is related
to things like event horizons,
and there's a whole other kind of world of that.
But- - It seems like this kind of model
can be applicable to so many aspects,
like at the different levels of understanding of our reality.
So it could be the biology, the chemistry,
at the physics level, the economics,
and you could potentially...
The thing is, it's like, okay, sure,
at all these levels, it might rhyme,
it might make sense as a model.
The question is, can you make useful predictions
at one of these levels? - That's right.
That's right, and that's really a question of...
It's a weird situation, because it's a situation
where the model probably has definite consequences.
The question is, are they consequences we care about?
And that's some...
In the case of, in the economic case,
so one thing is this idea of using physics-like notions
to construct a distributed analog of blockchain.
Okay, the much more pragmatic thing
is a different direction,
and it has to do with this computational language
that we built to describe the world
that knows about different kinds of cookies
and knows about different cities
and knows about how to compute all these kinds of things.
One of the things that is of interest is,
if you want to run the world,
you need, with contracts and laws and rules and so on,
there are rules at a human level,
and there are kind of things like...
And so this gets one into the idea
of computational contracts.
Right now, when we write a contract,
it's a piece of legalese, it's just written in English,
and it's not something that's automatically
analyzable, executable, whatever else.
It's just English.
Back in Gottfried Leibniz, back in 1680 or whatever,
was like, "I'm gonna figure out how to use logic
"to decide legal cases," and so on,
and he had kind of this idea of,
"Let's make a computational language for human law.
"Forget about modeling nature,
"forgot about natural laws, what about human law?
"Can we make kind of a computational
"representation of that?"
Well, I think finally we're close to being able to do that,
and one of the projects that I hope to get to
as soon as there's a little bit of slowing down
of some of this Cambrian explosion that's happening
is a project I've been meaning to really do for a long time,
which is what I'm calling a symbolic discourse language.
It's just finishing the job of being able
to represent everything, like the conversation we're having
in computational terms, and one of the use cases for that
is computational contracts.
Another use case is something like the Constitution
that says what the AIs, what we want the AIs to do.
- But this is useful, so you're saying,
so these are like, you're saying computational contracts,
but smart contracts, this is what's in the domain
of cryptocurrencies known as smart contracts,
and so the language you've developed,
this symbolic or seek to further develop
symbolic discourse language enables you
to write a contract.
- Write a contract that richly represents
some aspect of the world, yeah.
- But so, I mean, smart contracts tend to be right now
mostly about things happening on the blockchain.
Sometimes they have oracles, and in fact,
Wolfram Alpha API is the main thing people use
to get information about the real world, so to speak,
within smart contracts.
- So Wolfram Alpha, as it stands, is a really good oracle
for whoever wants to use it.
That's perhaps where the relationship with Cardano is.
- Yeah, well, that's how we started getting involved
with blockchains is we realized people were using,
you know, Wolfram Alpha as the oracle
for smart contracts, so to speak,
and so that got us interested in blockchains in general,
and what was ended up happening is Wolfram Language is,
with its symbolic representation of things,
is really very good at representing things like blockchains,
and so I think we now have,
and we don't really know all the comparisons,
but we now have a really nice environment
within Wolfram Language for dealing with the sort of,
you know, for representing what happens in blockchains,
for analyzing what happens in blockchains.
We have a whole effort in blockchain analytics,
and, you know, we've sort of published some samples
of how that works, but it's, you know,
because our technology stack, Wolfram Language
and Mathematica, are very widely used
in the quant finance world, there's a sort of immediate
sort of co-evolution there of sort of
the quant finance kind of thing, and blockchain analytics,
and that's, so it's kind of the representation
of blockchain in computational language,
then ultimately, it's kind of like,
how do you run the world with code?
That is, how do you write sort of all these things
which are right now regulations, and laws,
and contracts, and things in computational language,
and kind of the ultimate vision is that sort of
something happens in the world,
and then there's this giant domino effect
of all these computational contracts
that trigger based on the thing that happened,
and there's a whole story to that,
and of course, you know, I like to always pay attention
to the latest things that are going on,
and I really, I kind of like blockchain
because it's another rethinking of kind of computation.
It's kind of like, you know, cloud computing
was a little bit of that, of sort of persistent
kind of computational resources and so on,
and, you know, this multi-computation
is a big rethinking of kind of what it means to compute.
Blockchain is another bit of rethinking
of what it means to compute.
The idea that you lodge kind of these autonomous lumps
of computation out there in the blockchain world,
and one of the things that just sort of,
§25Transcript · 3:15:38–3:23:44
for fun, so to speak, is we've been doing a bit of stuff
with NFTs, and we just did some NFTs on Cardano,
and we'll be doing some more,
and, you know, we did some cellular automaton NFTs
on Cardano, which people seem to like quite a bit,
and, you know, one of the things I've realized about NFTs
is that there's kind of this notion,
and we're really working on this, you know,
I like recording stuff, you know,
one of the things that's come out of kind of my science,
I suppose, is this history matters type story
of, you know, it's not just the current stage,
it's the history that matters,
and I've kind of, I don't think this is,
I should be realizing, maybe it's not coincidental
that I'm sort of the human who's recorded
more about themselves than anybody else,
and then I end up with these science results
that say history matters, which was not,
those things, I didn't think those were connected, but--
- They're at least correlated, yes.
- Yeah, right, so, you know, this question
about sort of recording what has happened,
and having sort of a permanent record of things,
one of the things that's kind of interesting there
is, you know, you put up a website,
and it's got a bunch of stuff on it,
but, you know, you have to keep paying the hosting fees,
or the thing's gonna go away,
but one of the things about blockchain
that's kind of interesting is,
if you put something on a blockchain,
and you pay, you know, your commission to get that thing,
you know, put on, you know, mine, put on the blockchain,
then, in a sense, everybody who comes after you is,
you know, they are motivated to keep your thing alive,
'cause that's what keeps the consistency of the blockchain.
So, in a sense, with sort of the NFT world,
it's kind of like, if you want to have something permanent,
well, at least for the life of the blockchain,
but even if the blockchain goes out of circulation,
so to speak, there's gonna be enough value
in that whole collection of transactions
that people are gonna archive the thing,
but that means that, you know, pay once,
and you're kind of, you're lodged in the blockchain forever,
and so we've been kind of playing around
with sort of a hobby thing of mine,
of thinking about sort of the NFTs,
and how you, and sort of the consumer idea
of kind of the, it's the anti, you know,
it's the opposite of the Snapchat view of the world.
- Sure, there's a permanence to it
that's heavily incentivized,
and thereby you can have a permanence of history.
- Right, and that's kind of the, now, you know,
so that's one of the things we've been doing with Cardano,
and it's kind of fun.
I think that, I mean, this whole question about,
you know, you mentioned automated theorem proving,
and blockchains, and so on,
and as I've thought about this kind of physics-inspired
distributed blockchain,
obviously there, the sort of the proof that it works,
that there are no double spends, there's no whatever else,
that proof becomes a very formal kind of,
almost a matter of physics, so to speak.
And, you know, it's been an interesting thing
for the practical blockchains
to do kind of actual automated theorem proving,
and I don't think anybody's really managed it
in an interesting case yet.
It's a thing that people, you know, aspire to,
but I think it's a challenging thing,
'cause basically the point is,
one of the things about proving correctness of something
is, well, you know, people say, "I've got this program,
"and I'm going to prove it's correct."
And it's like, what does that mean?
You have to say what correct means.
I mean, it's kind of like,
then you have to have another language,
and people were very confused back in past decades of,
you know, "Oh, we're going to prove the correctness
"by representing the program in another language,
"which we also don't know whether it's correct."
And, you know, often by correctness,
we just mean it can't crash, or it can't scribble on memory,
but the thing is that there's this complicated trade-off,
because as soon as you're really using computation,
you have computational irreducibility,
you have undecidability.
If you want to use computation seriously,
you have to kind of let go of the idea
that you're going to be able to box it in,
and say, "We're going to have just this happen,
"and not anything else."
I mean, this is an old fact.
I mean, Godel's theorem tries to say, you know,
Peano arithmetic, the axioms of arithmetic,
can you box in the integers,
and say these axioms give just the integers,
and nothing but the integers.
Godel's theorem showed that wasn't the case.
There's a, you know, you can have all these wild,
weird things that obey the Peano axioms,
but aren't integers.
And there's this kind of infinite hierarchy
of additional axioms you would have to add,
and it's kind of the same thing.
You don't get to, you know, if you want to say,
"I want to know what happens," you're boxing yourself in,
and there's a limit to what can happen, so to speak.
So it's a complicated trade-off,
and it's a big trade-off for AI, so to speak.
It's kind of like, do you want to let computation
actually do what it can do, or do you want to say,
"No, it's very, very boxed in to the point
"where we can understand every step?"
And that's kind of a thing that becomes difficult to do.
But that's some, I mean, in general,
I would say one of the things that's kind of complicated
in my sort of life and the whole sort of story
of computational language and all this technology
and science and so on, I mean, I kind of,
in the flow of one's life, it's sort of interesting
to see how these things play out,
because I, you know, I've kind of concluded
that I'm in the business of making kind of artifacts
from the future, which means, you know,
there are things that I've done, I don't know,
this physics project, I don't know whether anybody
would have gotten to it for 50 years.
You know, the fact that Mathematica
is a third of a century old, and I know
that a bunch of the core ideas are not well-absorbed.
I mean, that is, people finally got this idea
that I thought was a triviality of notebooks.
That was 25 years.
And, you know, some of these core ideas
about symbolic computation and so on are not absorbed.
I mean, people use them every day in Wolfram Language
and, you know, do all kinds of cool things with them,
but if you say, what is the fundamental
intellectual point here, it's not well-absorbed.
And it's something where you kind of realize
that you're sort of building things,
and I kind of made this thing about, you know,
we're building artifacts from the future, so to speak,
and I mentioned that at our, we have a conference,
it's coming up actually in a couple of weeks,
our annual technology conference,
where we talk about all the things we're doing.
And, you know, so I was talking about it last year
about, you know, we're making artifacts from the future,
and I was kind of like, I had some version of that
that was kind of a dark and frustrated thing of like,
you know, I'm building things which nobody's gonna care
about until long after I'm dead, so to speak.
But then I realized, you know, people were sort of telling
me afterwards, you know, that's exactly how, you know,
we're using Wolfram Language in some particular setting
in some computational X field or some organization
or whatever, and it's like people are saying,
oh, you know, what did you manage to do?
You know, well, we know that in principle
it will be possible to do that,
but we didn't know that was possible now,
and it's kind of like that's sort of the business we're in,
and in a sense, with some of these ideas in science,
you know, I feel a little bit the same way,
that there are some of these things where, you know,
some things, like for example, this whole idea,
well, so to relate to another sort of piece of history
and the future, one of, you know, I mentioned,
we mentioned at the beginning kind of complexity
as this thing that I was interested in back 40 years ago
and so on, where does complexity come from?
Well, I think we kind of nailed that.
The answer is in the computational universe,
even simple programs make it, and that's kind of the secret
that nature has that allows you to make it.
So that's kind of the, that's that part,
but the bigger picture there was this idea
of this kind of computational paradigm,
the idea that you could go beyond mathematical equations,
which have been sort of the primary modeling medium
for 300 years, and so it was like, look,
it is inexorably the case that people will use programs
rather than just equations, and you know,
I was saying that in the 1980s, and people were,
you know, I published my big book, "New Kind of Science,"
that'll be 20 years ago next year, so in 2002,
and people were saying, oh no, this can't possibly be true,
you know, we know for 300 years,
we've been doing all this stuff, right?
To be fair, I now realize, on a little bit more analysis
of what people actually said, in pretty much every field
other than physics, people said, oh, these are new models,
that's pretty interesting, and physics, people were like,
§26Transcript · 3:23:47–3:31:31
we've got our physics models, we're very happy with them.
- Yeah, in physics, there's more resistance
because of the attachment and the power of the equations.
The idea that programs might be the right way
to approach this field was, there's some resistance,
and like you're saying, it takes time.
For somebody who likes the idea of time dilation
and all these applications,
I thought you would understand this.
- Yeah, right, but you know,
and computational irreducibility as well.
- Yes, exactly.
- But I mean, it is really interesting that just 20 years,
the span of 20 years, it's gone from, you know,
pitchforks and horror to, yeah, we get it,
and you know, it's helped that we've, you know,
in our current effort in fundamental physics,
we've gotten a lot further,
and we've managed to put a lot of puzzle pieces together
that make sense.
But the thing that I've been thinking about recently
is this field of complexity.
So I've kind of was a sort of a field builder
back in the 1980s.
I was kind of like, okay, you know, can we, you know,
I'd understood this point
that there was this sort of fundamental phenomenon
of complexity, and it showed up in lots of places.
And I was like,
this is an interesting sort of field of science.
And I was recently was reminded,
I was at this, the very first sort of get together
of what became the Santa Fe Institute.
I was like, in fact, there's even an audio recording
of me sort of saying, people have been talking about,
oh, what should this, you know, outfit do?
And I was saying, well, there is this thing
that I've been thinking about.
It's this kind of idea of complexity.
And it's kind of like, and that's what that ended up.
- And you planted the seed of complexity at Santa Fe.
That's beautiful.
That's a beautiful vision.
- But, I mean, so that, but what's happened then
is this idea of complexity and, you know,
and I started the first research center
at University of Illinois for doing that
in the first journal, complex systems and so on.
And it's kind of an interesting thing in my life,
at least that it's kind of like you plant the seed,
you have this idea, it's a kind of a science idea.
You have this idea of sort of focusing
on the phenomenon of complexity.
The deeper idea was this computational paradigm.
But the nominal idea is this kind of idea of complexity.
Okay, then you roll time forward 30 years or whatever,
35 years, whatever it is, and you say, what happened?
Okay, well, now there are a thousand
complexity institutes around the world.
I think more or less, we've been trying to count them.
And, you know, there are 40 complexity journals, I think.
And so it's kind of like what actually happened
in this field, right?
And I look at a lot of what happened
and I'm like, you know, I have to admit
to some eye rolling, so to speak,
because it's kind of like, what's actually going on?
Well, what people definitely got
was this idea of computational models.
And then they got, but they thought,
one of the kind of cognitive mistakes I think is,
they say, we've got a computational model
and we're looking at a system that's complex
and our computational model gives complexity.
By golly, that must mean it's right.
And unfortunately, because complexity
is a generic phenomenon and computational irreducibility
is a generic phenomenon that actually tells you nothing.
And so then the question is, well, what can you do?
You know, there's a lot of things
that have been sort of done under this banner of complexity.
I think it's been very successful
in providing sort of an interdisciplinary way
of connecting different fields together.
- Which is powerful in itself.
- Right, I mean, it's a very useful--
- Biology and economics and yeah, it is.
- Right, it's a good organizing principle.
But in the end, a lot of that is around
this sort of computational paradigm, computational modeling.
That's the raw material that powers
that kind of correspondence, I think.
But the question is sort of what is the,
you know, I was just thinking recently,
you know, we've been, I mean, the other,
we've been, for years people have told me,
you should start some Wolfram Institute
that does basic science.
You know, all I have is a company that builds software
and we, you know, we have a little piece
that does basic science as kind of a hobby.
People are saying, you should start
this Wolfram Institute thing.
And I've been, you know, 'cause I've known
about lots of institutes and I've seen kind of
that flow of money and kind of, you know,
what happens in different situations and so on.
So I've been kind of reluctant, but I have realized
that, you know, what we've done with our company
over the last 35 years, you know,
we built a very good machine for doing R&D
and, you know, innovating and creating things.
And I just applied that machine to the physics project.
That's how we did the physics project
in a fairly short amount of time with a, you know,
efficient machine with, you know,
various people involved and so on.
And so, you know, it works for basic science
and it's like, we can do more of this.
And so now-- - In biology and chemistry,
so it's become an institute.
- Yes, well, it needs to become an institute.
- An official institute. - Right, right.
But the thing that, so I was thinking about,
okay, so what do we do with complexity?
You know, what, there are all these people who've,
you know, what should happen to that field?
And what I realized is there's kind of this area
of foundations of complexity that's about these questions
about simple programs, what they do,
that's far away from a bunch of the detailed applications
that people, well, it's not far away.
It's the under, you know,
the bedrock underneath those applications.
And so I realized recently,
this is my recent kind of little innovation of a sort,
a post that I'll do very soon,
about kind of, you know, the foundations of complexity,
what really are they?
I think they're really two ideas, two conceptual ideas
that I hadn't really enunciated, I think, before.
One is what I call metamodeling, the other is ruleology.
So what is metamodeling?
So metamodeling is, you've got this complicated model,
and it's a model of, you know, hedgehogs interacting
with this, interacting with that.
And the question is, what's really underneath that?
What is it?
You know, is it a Turing machine?
Is it a cellular automaton?
You know, what is the underlying stuff
underneath that model?
And so there's this kind of meta-science question
of, given these models, what is the core model?
And I realized, I mean, to me,
that's sort of an obvious question,
but then I realized I've been doing language design
for 40 years, and language design is exactly that question.
You know, underneath all of this detailed stuff people do,
what are the underlying primitives?
And that's a question people haven't tended to ask
about models.
They say, "Well, we've got this nice model
for this and that and the other.
What's really underneath it?
And what, you know, because once you have the thing
that's underneath it, well, for example,
this multi-computation idea
is an ultimate metamodeling idea,
because it's saying underneath all these fields
is one kind of paradigmatic structure.
And, you know, you can imagine the same kind of thing
in much more sort of, much sort of shallower levels
in different kinds of modeling.
So the first activity is this kind of metamodeling,
the kind of the models about models, so to speak.
You know, what is the, what's, you know,
drilling down into models?
That's one thing.
The other thing is this thing that I think
we're going to call ruleology,
which is kind of the, okay, you've got these simple rules.
You've got cellular automata, you've got Turing machines,
you've got substitution systems,
you've got register machines,
you've got all these different things.
What do they actually do in the wild?
And this is an area that I've spent a lot of time,
you know, working on.
It's a lot of stuff in my new kind of science book
is about this.
You know, this new book I wrote about combinators
is full of stuff like this.
And this journal, "Complex Systems,"
has lots of papers about these kinds of things.
But there isn't really a home
for people who do ruleology or whatever.
- As you call the basic science of rules.
- Yes, yes, right.
So it's like you've got some, what is it?
§27Transcript · 3:31:34–3:38:36
Is it mathematics?
No, it isn't really like mathematics.
In fact, from my now understanding of metamathematics,
I understand that it's the molecular dynamics level.
It's not the level that mathematicians
have traditionally cared about.
It's not computer science,
because computer science is about writing programs
that do things, you know, that were for a purpose,
not programs in the wild, so to speak.
It's not physics.
It doesn't have anything to do with, you know,
it may be underneath some physics,
but it's not physics as such.
So it just hasn't had a home.
And if you look at, you know,
but what's great about it is it's a surviving field,
so to speak.
It's something where, you know,
one of the things I find sort of inspiring
about mathematics, for example,
is you look at mathematics that was done,
you know, in ancient Greece, ancient, you know,
Babylon, Egypt, and so on.
It's still here today.
You know, you find an icosahedron
that somebody made in ancient Egypt.
You look at it, oh, that's a very modern thing.
It's an icosahedron.
You know, it's a timeless kind of activity.
And this idea of studying simple rules and what they do,
it's a timeless activity.
And I can see that over the last 40 years or so,
as, you know, even with cellular automata,
it's kind of like, you know,
you can sort of catalog what are the different
cellular automata used for,
and, you know, like the simplest rules,
like one, you might even know this one, rule 184.
It's rule 184 is a minimal model for road traffic flow.
And, you know, it's also a minimal model
for various other things.
But it's kind of fun that you can literally say,
you know, rule 90 is a minimal model
for this and this and this.
Rule four is a minimal model for this.
And it's kind of remarkable that you can just,
by in this raw level of this kind of study of rules,
they then branch, they're the raw material
that you can use to make models of different things.
So it's a very pure basic science,
but it's one that, you know, people have explored it,
but they've been kind of a little bit in the wilderness.
And I think, you know, one of the things
that I would like to do finally is, you know,
I always thought that sort of this notion of pure
and chaos pure and chaos being the acronym for my book,
"New Kind of Science" was something
that people should be doing.
And, you know, we tried to figure out
what's the right institutional structure to do this stuff.
You know, we dealt with a bunch of universities.
Oh, you know, can we do this here?
Well, what department would it be in?
Well, it isn't in a department.
It's its own new kind of thing.
That's why the book was called "A New Kind of Science."
It's kind of the, because that's an increasingly
good description of what it is, so to speak.
We're actually, we were thinking about kind
of the ruleological society, because we're realizing
that it's kind of, it's, you know,
there's a, it's very interesting.
I mean, I've never really done something like this before,
because there's this whole group of researchers
who've been doing things that are really,
in some cases, very elegant, very surviving, very solid.
You know, here's this thing that happens
in this very abstract system.
But it's like, what is that part of?
You know, it doesn't have a home.
And I think that's something, you know,
I kind of fault myself for not having been more,
you know, when complexity was developing in the '80s,
I didn't understand the danger of applications.
That is, it's really cool that you can apply this
to economics, and you can apply it to evolutionary biology,
and this and that and the other.
But what happens with applications is everything
gets sucked into the applications.
And the pure stuff, it's like the pure mathematics,
there isn't any pure mathematics, so to speak.
It's all just applications of mathematics.
And I failed to kind of make sure
that this kind of pure area
was kind of maintained and developed.
And I think now, you know,
one of the things I want to try to do,
and you know, it's a funny thing,
because I'm used to, look, I've been a tech CEO
for more than half my life now.
So, you know, this is what I know how to do.
And, you know, I can make stuff happen
and get projects to happen,
even as it turns out basic science projects
in that kind of setting,
and how that translates into kind of, you know,
there are a lot of people working on, for example,
our physics project sort of distributed
through the academic world, and that's working just great.
But the question is, you know,
can we have a sort of accelerator mechanism
that makes use of kind of what we've learned
in sort of R&D innovation?
And, you know, but on the other hand,
it's a funny thing because, you know,
in a company, in the end, the thing is, you know,
it's a company, it makes products,
it sells things to people.
And, you know, when you're doing basic research,
one of the challenges is there isn't that same kind
of sort of mechanism.
And so it's, you know, how do you drive the thing
in a kind of, in a lead kind of way
so that it really can make forward progress
rather than, you know, what can often happen in,
you know, in academic settings where it's like,
well, there are all these flowers blooming,
but it's not clear that they're, you know, that it's-
- You have to have a central mission and a drive,
just like you do with a company
that's delivering a big overarching product.
And that's- - Right.
But the challenge is, you know, when you have a,
the economics of the world are such that, you know,
when you're delivering a product and people say,
"Wow, that's useful, we'll buy it."
And then that kind of feeds back and, okay,
then you can pay the people who build it to eat,
you know, so they can eat and so on.
And with basic science, the payoff is very much less visible
and, you know, with this physics project, as I say,
the big surprise has been that, I mean, you know,
for example, well, the big surprise with the physics project
is that it looks like it has near-term applications.
And I was like, I'm guessing this is 200 years away.
It's, I was kind of using the analogy of, you know,
Newton starting a satellite launch company,
which would have been kind of wrong time.
And, you know, but it turns out that's not the case,
but we can't guarantee that.
And if you say we're signing up to do basic research,
basic rheology, let's say, and, you know,
and we can't, we don't know the horizon, you know,
it's an unknown horizon.
It's kind of like an undecidable kind of proposition
of when is this proof going to end, so to speak?
When is it going to be something that gets applied?
- Well, I hope this is,
this becomes a vibrant new field of rheology.
I love it.
Like I told you many, many times,
it's one of the most amazing ideas
that has been brought to this world.
So I hope you get a bunch of people to explore this world.
Thank you once again
for spending your really valuable time with me today.
- Fun stuff.
Thank you.
- Thanks for listening to this conversation
with Stephen Wolfram.
To support this podcast,
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And now let me leave you with some words
from Richard Feynman.
"Nature uses only the longest threads
"to weave her patterns.
"So each small piece of her fabric
"reveals the organization of the entire tapestry."
Thank you for listening and hope to see you next time.
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