00:00In a sense, we know that black holes or in
the Big Bang or something, that's probably

00:03an abstraction that loses usefulness and eventually
will be superseded by something more foundational.

00:08Our universe seems to be neither maximally
simple, nor is it kind of maximally complicated.

00:13There's some regularity, but it's not completely
logically trivial. It's not like every little

00:17particle follows its own set of laws, but
it's also not like we can just reduce everything

00:21to one logical tautology.
Jonathan Garrard is a researcher in mathematical

00:27physics at Princeton University and in my
opinion is the sharpness and the brains behind

00:32the rigor at the Wolfram's Physics Project.
Today's conversation is quite detailed as

00:37we go into the meticulous technicalities,
as if this were a conversation between two

00:41friends behind closed doors.
In this discussion, we elucidate the core

00:45principles and claims of the Wolfram's Physics
Project. We distinguish them from the surrounding

00:50hype. Specifically, we explore potential connections
between category theory and quantum gravity.

00:55We also delve into refining truth and representations,
the pros and the perils of peer review. And

01:01furthermore, we highlight the differences
between Jonathan and Stephen Wolfram, particularly

01:07in the context of computational and consciousness
related aspects.

01:11You should also know that there are three
interviews with Stephen Wolfram on this channel.

01:15Each is linked in the description. In it,
we detail the Wolfram's Physics Project with

01:19Stephen Wolfram himself and why he thinks
it's a potential candidate for a theory of

01:23everything.
My name is Curt Jaimungal. For those of you

01:26who are unfamiliar, this is a channel called
Theories of Everything where we explore theories

01:31of everything in the physics sense, using
my background in mathematical physics from

01:35the University of Toronto, but as well as
explore other large grand questions. What

01:40is consciousness? Where does it come from?
What is reality? What defines truth? What

01:44is free will? And do we have it?
Of course, increasingly, we've been exploring

01:48artificial intelligence and its potential
relationship to the fundamental laws.

01:52Also, the string theory video that Jonathan
mentions is called the Iceberg of String Theory

01:58and I recommend you check it out. It took
approximately two months of writing, four

02:02months of editing with four editors, four
rewrites, 14 shoots, and there are seven layers.

02:08It's the most effort that's gone into any
single theories of everything video. It's

02:12a rabbit hole of the math of string theory
geared toward the graduate level. There's

02:16nothing else like it.
If that sounds interesting to you, then check

02:20out the channel or hit subscribe to get notified.
Enjoy this episode with Jonathan Girard.

02:26So Jonathan, what is the Wolfram's Physics
Project and what's your role in it?

02:31That's a really good question, Curt. So I
guess, I don't know, there are various people

02:37involved and I think you'll get slightly different
answers or perhaps very different answers

02:40depending on who you ask. I'm someone who,
you know, I think when we first launched the

02:46physics project back in April 2020, we kind
of, we lent hard on this billing of it's a

02:51project to find the fundamental theory of
physics. That was not really how I viewed

02:57it at the time and it's become even less how
I view it over time.

03:00Interesting.
And so, you know, I'm just, I'm saying this

03:05as a kind of prelude to clarify that what
you're about to hear is my own perspective

03:08on it and it will probably differ quite a
lot from the perspective given by some other

03:13members of the project. So essentially, my
view is that the Wolfram Physics Project is

03:18an attempt to answer a kind of counterfactual
history question.

03:23So back in the 17th century, Newton, Leibniz,
a little bit earlier people like Descartes,

03:32Galileo, you know, they kind of set the stage
for modern theoretical kind of mathematical

03:37physics and more broadly for our kind of modern
conception of how the exact sciences work.

03:42And you know, so essentially the idea was,
you know, rather than just describing phenomena

03:46in these kind of philosophical terms, you
could actually construct kind of robust quantitative

03:51models of what, you know, what natural systems
do. And that was enabled by a particular piece

03:57of mathematical technology or a particular
piece of, I guess, cognitive technology, which

04:00was calculus, which later became, you know,
mathematical analysis and the basis of differential

04:04geometry and all the kind of machinery of
modern mathematical physics. So, you know,

04:08Newton, Leibniz, you know, building off earlier
work by people like Archimedes and so on kind

04:12of, you know, they built up this formalism
of calculus that sort of enabled modern physics.

04:18And arguably, that choice of formalism, that
choice to base physical models on, you know,

04:25analytic calculus-based mathematical formalisms
has had an impact on our physical intuition,

04:31right? So, you know, it involves thinking
about things in terms of smooth analytic functions,

04:36in terms of continuously varying kind of gradients
of quantities. You know, it necessitates us

04:40formalizing notions like space and time in
terms of, you know, smooth manifolds or real

04:46numbers. It involves, you know, thinking about
things like energy and momenta as being continuously

04:50varying quantities. And those are, of course,
extremely good idealizations of what's really

04:56happening. But I think there's always a danger
whenever you have a model like that, that

04:59you start to kind of believe in the ontological
validity of the model. And so, for a lot of

05:03physicists, I feel like, you know, it's kind
of seeped in and percolated our intuition

05:08to the extent that we actually think that
space is a, you know, smooth Riemannian manifold.

05:13We think that energy is a kind of real valued
function, rather than these just being idealizations

05:17of some potentially quite different, you know,
underlying reality. Okay, now fast

05:22forward about 300 years, and you have people
like Alan Turing and Alonzo Church and Curt

05:27GÃ¶del in the early 20th century building
up the beginnings of what became theoretical

05:30computer science, right? As an offshoot of
mathematical logic. There were people interested

05:35in the question of, you know, what is mathematics?
What is mathematical proof? You know, what

05:40are mathematical theorems? And that kind of
necessitated them building this really quite

05:44different mathematical formalism, which initially
had different manifestations. It had, you

05:50know, Turing machines, lambda calculus, you
know, general recursive functions, etc., which

05:53then gradually got unified thanks to things
like the Church-Turing thesis. But so now,

05:59so in a way, again, at least the way I like
to think about it is, you know, the sort of

06:03stuff that Newton and Leibniz and people were
doing in the 1600s, that gave, you know, with

06:08analysis, that gave you a systematic way of
understanding sort of an exploring continuous

06:13mathematical structures. What Turing and Church
and GÃ¶del and people did in the early 20th

06:17century with computability theory gave one
a systematic way of understanding discrete

06:21mathematical structures. You know, the kinds
of things that could be represented by simple

06:26computations and simple programs. Now, by
that point, as I say, you know, calculus,

06:31the sort
of calculus-based approaches had had a 300-year

06:33head start in terms of the exact sciences.
And it took a little while before people started

06:38thinking, hmm, actually, you know, maybe we
could use these formalisms from computability

06:41theory to construct models of natural phenomena,
to construct, you know, scientific models

06:45and models for things like fundamental physics.
But of course, that necessitates being a quite

06:50radical departure in how we think about physical
laws, right? That, you know, suddenly you

06:54have to deviate from thinking about space
as some smooth continuous structure and start

06:58thinking about it in terms of some discrete
combinatorial structure, like a kind of network

07:01or a graph. It necessitates you moving away
from thinking about dynamics in terms of continuous

07:07partial differential equations and thinking
about it in terms of kind of discrete time-step

07:10updates, like, say, the kinds that you can
represent using network rewriting rules.

07:15And so, you know, a lot of physicists who
are kind of trained in the traditional mathematical

07:20formalism find this quite counterintuitive
because, as I say, you know, those ideas from

07:26mathematical analysis have seeped so far into
our intuition that we think that's actually

07:30how the universe works, rather than just thinking
of it as being a model.

07:33And so, the way, the slightly poetic way that
I like to think about what the physics project

07:37is doing is we're trying to address this kind
of counterfactual history question of

07:41what would have happened if, you know, Turing
was born 300 years before Newton, not the

07:46other way around. In other words, if we had,
if discrete mathematical approaches based

07:50on computability theory had a 300-year head
start in the foundations of natural science

07:54over continuous, you know, mathematical structures
based on analysis. That's my kind of zoomed

08:00out picture of what it is that we're trying
to do.

08:02So, okay, that's, there's a lot more that
can be said about that, of course, and I'm

08:07sure we'll discuss more of it later. But that's
at least my kind of, that's my big picture

08:13summary of what I think the physics project
is about. It's about trying to reconstruct

08:16the foundations of physics, not in terms of,
you know, Lorentzian manifolds and continuous

08:20base times, but in terms of things like graphs,
hypergraphs, hypergraphic writing, causal

08:25networks, and, you know, the kinds of discrete
structures that can be represented in a very

08:29explicit, computable way. There are some nice
connections there, by the way, to things

08:32like the constructivist foundations of mathematics
that arose in the 20th century as well. And

08:37again, we'll likely talk about that later,
too.

08:41In terms of my own role within it, so, you
know, Stephen Wolfram, who I know has appeared

08:46on TOA a number of times and has been kind
of the, by far the single most energetic evangelist

08:54of these ideas for a very long time. He wrote
back in 2002, this book, A New Kind of Science,

09:01in which he first postulated these, you know,
the beginnings of these ideas about, you know,

09:05maybe it's useful to think of fundamental
physics in terms of network automata and things

09:08like that. And, you know, had some initial
hints towards, okay, here's how we might be

09:13able to get general relativity, you know,
beginnings of quantum mechanics, those kinds

09:16of things out of those systems. But then,
you know, those ideas basically lay dormant

09:22for a long time. I mean, NKS had, you know,
had this kind of maelstrom of attention for

09:27a couple of years, and then mostly, at least
physicists mostly ignored it, is kind of at

09:31least my impression. Where, you know, I, as
a teenager, you know, I read NKS and I, like

09:39many people, found certain aspects of the
way the book is written a little bit off-putting.

09:43But I thought that there were many, many core
ideas in it that were really, really quite

09:47foundationally important.
And one of them was this idea about fundamental

09:50physics.
And so, for a while, I kind of advocated like,

09:54we should be trying to build physics on these
kind of computable models, if only just to

09:58see what happens, right?
Just to see where that leads us.

10:03And so I started to do some initial kind of
work in those directions, nothing particularly

10:08profound.
But also, I would repeatedly badger Stephen,

10:12you know, maybe every year or so and say,
we should, you know, we should go and like,

10:16actually try and do a more serious investigation
of these things.

10:19And then finally...
Sorry, just a moment.

10:21You said that you would be working on these
prior to going to the Wolfram School, the

10:24summer school?
Yes, yeah, exactly.

10:27So I went to the Wolfram Summer School in
2017, as a consequence of my interest in these

10:32models.
So I'd already been doing a little bit of

10:35my own kind of work on this, the stuff, trying
in large part to, in a sense, to rediscover

10:40what Stephen had already done, right?
He had these big claims in NKS about being

10:45able to derive, you know, Einstein equations
and things from these graph rewriting models.

10:50But the details were never included in the
book.

10:52And I tried to ask Stephen about them, and
he kind of said, oh, I can't really remember

10:55how I did that now.
And so I spent quite a lot of time trying

10:58to kind of reconstruct that.
And that eventually ended up, you know, that

11:02was the thing that resulted in me, you know,
ending the summer school and then being kind

11:06of pulled into Stephen's orbit.
And is it your understanding that Stephen

11:12actually did have a proof?
He just wasn't able to recall it like Fermat,

11:16or it just was too small of a space to publish?
Or that he thinks he was able to prove it,

11:21but the tools weren't available at the time?
And you think back, like, maybe he had a sketch,

11:24but it wasn't, well, it's Leonardo's sketch
versus the Mona Lisa.

11:32Right, right.
I think the Leonardo sketch versus the Mona

11:34Lisa analogy is probably the right one.
So my suspicion, based on what I know of the

11:41history of that book, and also based on what
I know of Stephen's personality, is that Stephen

11:45had proved it to his own satisfaction, probably
not to the satisfaction of anyone else, right?

11:49So I think, you know, many of us are like
this, right?

11:52Like if you encounter some problem, or, you
know, some phenomenon you don't really understand,

11:59and you go away and you try and understand
how it works, or you try and prove some of

12:02the results about it, and eventually you convince
yourself that it can be done, or

12:06that you convince yourself that there is an
explanation, and you don't necessarily tie

12:09together all the details to the point where
you could actually publish it and make it

12:11understandable to other people.
But kind of to your own intellectual satisfaction,

12:15it's like, oh yeah, now I'm at least convinced
that that can work.

12:19My impression is that that's basically, that's
essentially where the kind of physics project

12:24formalism ended up in 2002, that Stephen thought
about it for a while, had some research assistants

12:28look at it, and eventually they kind of convinced
themselves, yes, it would be possible to derive

12:32Einstein equations from these kinds of formalisms.
But I highly, from what I've seen of the material

12:37that was put together and so on, I don't think
anyone actually traced that proof, you know,

12:40with complete mathematical precision.
Eventually in 2019, Stephen, myself, and Max

12:46Piskunov, we decided, for various reasons,
that it was kind of the right time for us

12:52to do this project in a serious way.
Stephen had some new ideas about how we could

12:56simplify the formalism a little bit.
I'd made some recent progress in kind of understanding

13:01the mathematical underpinnings of it.
Max had just finished writing some really

13:05quite nicely optimized kind of low-level C++
code for enumerating these hypergraph systems

13:09really efficiently.
And so we decided like, okay, if we're not

13:13going to do it now, it's never going to happen.
And so that was then the beginnings of the

13:17physics project.
And so now I'm less, I guess, less actively

13:22involved in the project as a kind of branding
entity.

13:27But I'm still kind of actively working on
the formalism and still trying to push ahead

13:31in various mathematical directions, trying
to kind of concretify the foundations of what

13:35we're doing and make connections to existing
areas of mathematical physics.

13:40So I also noticed a similar problem as yourself
across society, so across history, that people

13:46entwine this prevalent application with some
ontological status.

13:50So what I mean by that is, you'll have a tool
which is ubiquitous and usefulness, and then

13:56you start to think that there's some reality
synonymous with that.

13:59So another example would be an ancient poet
who would see the power of poetry and think

14:06that what lies at the fundament is narrative
pieces.

14:10Or a mystic who sees consciousness everywhere
almost by definition and then believes consciousness

14:14must lie at the root of reality.
And some people, Max Tegmark would be an example

14:20of this, find that math is so powerful it
must be what reality is.

14:25So it's also not clear to me whether computation
is another such fashionable instance of a

14:30tool being so powerful that we mistake its
effectiveness with its substantiveness.

14:36And I understand that Stephen may think differently,
I understand that you may think differently,

14:39so please explain.
That's a fantastic point, and I suspect, at

14:45least from what you've said, I think our
views may be quite similar on this, that I'm

14:49reminded of this meme that circulated on Twitter
a little while ago about people saying, immediately

14:55after the invention of writing systems and
narrative structure, everyone goes, ah, yes,

15:00the cosmos must be a book.
And then immediately after the invention of

15:02mathematics, ah, yes, the cosmos must be made
of mathematics.

15:06And then immediately after the invention of
the computer, ah, yes, the cosmos must be

15:09a computer.
So I think that it's a folly that we've fallen

15:14into throughout all of human history.
And so, yeah, my feeling about this is always

15:19that, you know, we build models using the
kind of ambient technology of our time.

15:27And when I say technology, I don't just mean,
you know, nuts and bolts technology, I also

15:30mean kind of thinking technology, right?
So you know, there are kind of ambient ideas

15:36and processes that we have access to, and
we use those as a kind of raw substrate for

15:41making models of the world.
So you know, it's unsurprising that when people

15:45like Descartes and Newton built models
of the cosmos, you know, of the solar system

15:49and so on, they described them in terms of
clockwork by analogies to clockwork mechanisms,

15:53right?
And you know, Descartes even sort of more

15:56or less directly wrote that he thought that,
you know, the solar system was a piece of

15:59clockwork.
Whether he actually thought that in an ontological

16:02sense or whether it was just a kind of poetic
metaphor, I don't completely know.

16:05But you know, it's sort of obvious that that
would happen, right?

16:08Because you know, the 15th century, 16th century,
that was sort of the height of clockwork technology

16:12in ambient society.
And so you know, we live right now in arguably

16:16the zenith of kind of computational technology.
And so again, it's completely unsurprising

16:21that we build models of the cosmos based largely
on computation, or based largely or partly

16:25on computational ideas.
Yeah, I agree.

16:28I think it would be a folly, and I think you're
right.

16:31This is maybe one area where perhaps Stephen
and I differ slightly in our kind of philosophical

16:36conception.
I personally feel like it's folly to say,

16:38oh, therefore, you know, the universe must
be a computer, right?

16:42Or that, you know, that, yeah, my feeling
about it is, the strongest we can say is that,

16:50you know, modeling the universe as a Turing
machine is a useful scientific model.

16:54And it's a useful thinking tool by which to
reason through kind of various problems.

16:59I think it's, yeah, I would be uncomfortable
endowing it with any greater ontological significance

17:06than that.
That being said, of course, you know, there

17:08are also lots of examples where people have
made the opposite problem, right, where, you

17:11know, made the opposite mistake, I mean.
So, you know, the classic example is people,

17:16you know, say Hendrik Lorentz, right, who
invented, basically invented the whole formalism

17:19of special relativity.
But he said, oh, no, no, this is just a mathematical

17:23trick, right?
You know, he discovered the right form of

17:26time dilation and length contraction.
But he said, this is just some coordinate

17:29change, it doesn't have any physical effect,
it's just a formalism.

17:31And then really, the contribution of Einstein
was to say, no, it's not just a formalism,

17:35this is an actual physical effect, and here's
how we might be able to measure it.

17:38And so, yeah, I'm just trying to indicate
that you have to thread a delicate needle

17:47So you mentioned Turing, and there's another
approach called constructor theory, which

17:53generalizes Turing machines, or universal
Turing machines, to universal constructors,

17:58so-called universal constructors.
So I'd like you to explain what those are

18:01to the degree that you have studied it, and
then its relationship to what you work on

18:05at the Wolfram Physics Project.
And by the way, string theory, loop quantum

18:09gravity, they have these succinct names, but
WPP doesn't have a graspable, apprehensible

18:15name, at least not to me, to be able to echo
that.

18:19So is there one that you all use internally
to refer to it?

18:22Okay, so on that, yeah, I agree.
I'm not a fan of the naming of the Wolfram

18:29Physics Project, or indeed even the Wolfram
model, which is a slightly more succinct version.

18:35In a lot of what I've written, I use the term
hypergraph dynamics, or sometimes hypergraphic

18:40writing dynamics, because I think that's a
more descriptive title for what it really

18:47is.
But no, I agree.

18:48I think as a branding exercise, there's still
more work that needs to be done.

18:51So for the sake of us speaking more quickly,
we'll say the HD model.

18:55So in this HD model, what is its relationship
to, what was it, category?

19:00No, it wasn't category.
It was-

19:02Constructor theory.
Constructor, right.

19:03So the HD model's relationship to constructor
theory.

19:07Although that's an interesting Freudian slip,
because I think basically the relationship

19:10is category theory, right?
So yeah, okay, so I mean, with the proviso

19:15that, you know, again, I know that you've
had Chiara Maletto on TOE before, right?

19:20So I'm certainly not an expert on constructor
theory.

19:23I've read some of Chiara's and David Deutsch's.
papers on these topics, but so as you say,

19:29I can give an explanation to the extent that
I understand it. So with, you know, as I understand

19:35it, the central idea with constructor theory
is rather than describing physical laws in

19:40terms of kind of, you know, equations of motion,
right? So in the traditional conception of

19:44physics, we would say, you know, you've got
some initial state of a system, you have some

19:47equations of motion that describe the dynamics
of how it evolves, and then you, you know,

19:51it evolves down to some final stage. The idea
with constructor theory is you say, rather

19:54than formulating stuff in terms of equations
of motion, you formulate things in terms of

19:58what classes of transformations are and are
not permitted. So, and I think one of the

20:06classic examples that I think Deutsch uses
in one of his early papers, and I know that

20:10Chiara's done additional work on, is the second
law of thermodynamics, and indeed the first

20:13law of thermodynamics, right? So thermodynamic
laws are not really expressible in terms of

20:18equations of motion, or at least not in a
very direct way. They're really saying quite

20:22global statements about what classes of physical
transformations are and are not possible,

20:26right? They're saying you cannot build a perpetual
motion machine of the first kind or the second

20:31kind or whatever, right? That there is no
valid procedure that takes you from this class

20:36of initial states to this class of final states
that, you know, reduce global entropy or that,

20:40you know, create free energy or whatever,
right? And that's a really quite different

20:43way of conceptualizing the laws of physics.
So constructor theory, as I understand it,

20:47is a way of applying that to physics as a
whole, to saying we formalize physical laws

20:53not in terms of initial states and equations
of motion, but in terms of initial substrates,

20:59final substrates, and constructors, which
are these general processes that I guess one

21:04can think of as being like generalizations
of catalysts. It's really a kind of grand

21:09generalization of the theory of catalysis
in chemistry, right? You know, you're describing

21:15things in terms of, you know, this enables
this process to happen, which allows this

21:19class of transformations between these classes
of substrates or something. Now, you brought

21:25up, inadvertently, you brought up this question
of category theory or this concept of category

21:29theory. And I have to be a little bit careful
with what I say here because I know that the

21:33few people I know who work in constructor
theory say that what they're doing is not

21:37really category theory. But I would argue
it has some quite, in terms of the philosophical

21:42conception of it, it has some quite remarkable
similarities. So to pivot momentarily to talk

21:51about the duality between set theory and category
theory as foundations for mathematics. So

21:58since the late 19th century, early 20th century,
it's been the kind of vogue to build mathematical

22:04foundations based on set theory, based on
things like Zemillo-Fraenkel set theory or

22:08Hilbert-Bernays-GÃ¶del set theory and other
things, where the idea, you know, your fundamental

22:13object is a set, some collection of stuff,
which then, you know, you can apply various

22:18operations to. And the idea is you build mathematical
structures out of sets. Now, set theory is

22:27a model of mathematics that depends very heavily
on internal structure, right? So for instance,

22:33in the standard axioms of set theory, you
have things like the axiom of extensionality

22:37that essentially says two sets are equivalent
or two sets are identical if they have the

22:41same elements. So it involves you identifying
sets based on looking inside them and seeing

22:47what's inside. But there's another way that
you can think about mathematical structure,

22:51which is you say, I'm not going to allow myself
to look inside this object. I'm just going

22:56to treat it as some atomic thing. And instead,
I'm going to give it an identity based on

23:01how it relates to all other objects of the
same type. So what transformations can I,

23:06so, you know, to give a concrete example,
right, suppose I've got some topological space.

23:12So one, the kind of set theoretic view is,
okay, that topological space is a set of points.

23:18It's a collection of points that have a topology
defined on them. The kind of more category

23:22theoretic view would be to say, actually,
that topological space is defined as the collection

23:28of continuous transformations that can be
applied to it. So that space can be continuously

23:33deformed into some class of other spaces.
And that class of other spaces that it can

23:37be deformed into is what identifies the space
you started from. And so that's a, so, and

23:42you can define that without ever having to
talk about points or, you know, what was inside

23:46it, right? In fact, there's this whole generalization
of topology called pointless topology or locale

23:51theory, which is all about doing topology
without an a priori notion of points. So in

23:57a way, it necessitates this conceptual shift
from an internal structure view to a kind

24:02of process theoretic view. And so that was
a viewpoint that was really advocated by the

24:08pioneers of Catsby theory, Samuel Eilenberg
and Saunders MacLean, and also some other

24:13people who were working in topology, like
Jean-Pierre Serres and Alexander Grotendieck

24:17and others. There was a kind of radically
different way to conceptualize the foundations

24:21of mathematics.
Sorry to interrupt. Just as a point for the

24:24audience, you mentioned the word duality between
sets and categories. Now, do you mean that

24:28in a literal sense or just morally there's
a duality? Because category theorists make

24:33a huge fuss that what they're dealing with
aren't always like small categories or sets,

24:39but, or can be thought of as sets, but not
categories as such.

24:44Right, right. Okay. And I shouldn't have said,
I mean, yes, no. The short answer is no, I

24:50don't mean duality in any formal sense. And
in particular, it's a dangerous word to use

24:55around category theorists because it means
something very precise. It means that dual

24:59concepts are ones that are equivalent up to
reversal of the direction of morphisms. I

25:04certainly don't mean that.
Right, right, right.

25:05No, I meant duality in the sense that, so
there is a precise sense in which set theory

25:12and category theory are equivalently valid
foundations for mathematics. And that precise

25:19sense is, and hopefully, I mean, we can go
deep in the weeds if you want. We'll see

25:26where the conversation goes. But the basic
idea is there's a branch of category theory

25:32called elementary topos theory, which is all
about using category theory as a foundation

25:36for logic and mathematics. And the idea there
is, so from a category theoretic perspective,

25:43sets are just, they just form one particular
category. There is a category called set,

25:48whose objects are sets and whose transformations,
whose morphisms are set-valued functions.

25:53And so then you might say, well, you know,
why is set so important? Like what's so great

25:57about set that we build all mathematics on
that? It's just one random category in this

26:01space of all possible categories.
So elementary topos theory is all about asking

26:04what are the essential properties of set that
make it a quote-unquote good place to do mathematics?

26:11And can we abstract those out and figure out
some much more general class of mathematical

26:15structures, some more general class of categories
internal to which we can build mathematical

26:21structures? And that gives us the idea of
an elementary topos. I'm saying elementary

26:25because there's a slightly different idea
called a Grothendieck topos that's related,

26:29but not quite equivalent and whatever. But
generally when logicians say topos, they mean

26:35elementary topos.
So yeah, there's a particular kind of category

26:39which has these technical conditions that
it

26:41has all finite limits and it possesses a sub-object
classifier or equivalently a power object.

26:45But basically what it means is that it has
the minimal algebraic structure that sets

26:50have,
that you can do analogs of things like set

26:52intersections, set unions, that you can take
power sets, you can do subsets. And it kind

26:58of detects for you a much larger class of
mathematical structures, these elementary

27:04topos, which have those same features.
And so then the argument goes, well, therefore

27:10you can build mathematics internal to any
of

27:14those topos. And the mathematical structures
that you get out are in some deep sense isomorphic

27:19to the ones that you would have got if you
built mathematics based on set.

27:23So that's the precise meaning of, I guess,
what I was saying. That in a sense,

27:28there are these set theoretic foundations,
there are these category theoretic foundations

27:32that
come from topos theory, and there is some

27:34deep sense in which it doesn't matter which
one you

27:37use. That somehow the theorems you prove are
equivalent up to some notion of isomorphism

27:43in the two cases.
Yes. And now the relationship between constructor

27:47theory and HD,
which is the hypergraph dynamics or Wolfram's

27:51physics project for people who are just tuning
in.

27:53Right, right. So yes, the excursion to talk
about category theory is in a sense,

28:01my reason for bringing that up is because
I think that same conceptual shift that I

28:04was
describing, where you go from thinking about

28:06internal structure to thinking about kind
of

28:08process theories, that's been applied to many
other areas. It's been applied, say,

28:11in quantum mechanics, right? So where there's,
in the traditional conception, you'd say quantum

28:16states are fundamental, and you have Hilbert
spaces that are spaces of quantum states,

28:20and then you have operators that transform
those Hilbert spaces, but they're somehow

28:24secondary.
Then there's this rather different, and that's

28:26the kind of von Neumann-Dirac picture.
Then there's this rather different formalization

28:31of the foundations of quantum mechanics that's
due to Samson Abramski and Bob Coker, which

28:34is categorical quantum mechanics,
where the idea is you say, actually, the spaces

28:38of states, those are secondary,
and what really matters are quantum processes.

28:42What matters are the transformations from
one

28:44space of states to another. You describe quantum
mechanics purely in terms of the

28:48algebra of those processes. There are many
other examples of that. Things like functional

28:56programming versus imperative programming,
or lambda calculus versus Turing machines,

28:59in a sense that these are all instances of
thinking about things in terms of processes

29:04and functions rather than in terms of states
and sets.

29:08I view constructor theory as being the kind
of processes and functions version of physics,

29:13whereas traditional mathematical physics is
the kind of sets and structures version of

29:16physics. In a sense, the hypergraph dynamics
view, slash Wolfram model view, however you

29:22want to describe it, is one that nicely synthesizes
both cases, because in the hypergraph dynamics

29:29case, you have both the internal structure,
that you have an actual hypergraph, and you

29:34can look inside it, and you can talk about
vertices and edges and so on.

29:40But you also have a kind of process algebra,
because you have this multi-way system where

29:45I apply lots of different transformations
to the hypergraph, and I don't just get a

29:49single transformation path. I get this whole
tree or directed acyclic graph of different

29:54transformation paths. In a sense, you can
imagine defining an algebra, and we've done

29:59this in other work, where you have a rule
for how you compose different edges in the

30:06multi-way system, both sequentially and in
parallel. You get this nice algebraic structure

30:10that happens to have a category theoretic
interpretation.

30:15In a way, the pure hypergraph view, that's
a kind of set theory structural view. The

30:20pure multi-way system view, that's a kind
of pure process theory category theoretic

30:26view. One of the really interesting ideas
that comes out of thinking about physics in

30:31those terms is that general relativity and
quantum mechanics emerge from those two limiting

30:36cases. In a sense, if you neglect all considerations
of the multi-way system, and you just care

30:42about the internal structure of the hypergraph
and the causal graph, and you define a kind

30:47of discrete differential geometric theory
over those, what you get in some appropriate

30:53limit is general relativity for some class
of cases. On the other hand, if you neglect

30:57all considerations of the internal structure
of the hypergraph, and you care only about

31:00the process algebra of the multi-way system,
what you get is categorical quantum mechanics.

31:04You get a symmetric monoidal category that
has the same algebraic structure as the category

31:08of finite dimensional Hilbert spaces in quantum
mechanics. In a sense, the traditional physics,

31:17which is very structural, gives you one limit,
gives you the general relativistic limit.

31:21The kind of more constructive theoretic view,
which is more process theoretic,

31:25more category oriented, gives you another
limit, gives you the quantum mechanics limit.

31:28JS Yeah, and do you need a dagger symmetric
monoidal category, or is the symmetric monoidal

31:33enough?
SIMON You do need it to be dagger

31:35symmetric. Yeah, no, that's a very important
point. I'm going to assume not all of your

31:42followers and listeners are card-carrying
category theorists. Just as a very quick summary

31:47of what
means by dagger symmetric. Actually, maybe

31:50we should say what we mean by symmetric monoidal.
If you have a category, if you just think

31:56of it as some collection of simple processes,
like in the multi-way system cases, just individual

32:00rewrites of a hypergraph,
then you can compose those things together

32:05sequentially. You can apply rewrite one,
then rewrite two, and you get some result.

32:09There's also a case where you can do that
in

32:11any category. There are also cases where you
can apply them in parallel. You can do rewrite

32:15one
and rewrite two simultaneously. In a multi-way

32:18system, that's always going to be possible.
Then you get what's called a monoidal category,

32:21or actually a symmetric monoidal category,
if it doesn't matter which way around you

32:24compose them. That kind of generalizes the
tensor product structure in quantum mechanics.

32:30Then you can also have what's called a dagger
structure. The dagger structure is the thing

32:35that generalizes the Hermitian adjoint operation
in quantum mechanics, the thing that gives

32:38you time reversal. In that case, then you
have some

32:42operation that you can take a rewrite and
you can reverse it. For hypergraph rewriting

32:47rules,
there's a guarantee that you can always do

32:49that. There's yet another level of structure,
which is what's called a compact closed structure,

32:55which means that you can essentially do the
analog of taking duals of spaces. For those

33:02people who know about quantum mechanics, that's
the

33:04generalization of exchanging brows for cats
and vice versa. Again, you can do that in

33:10the case
of hypergraphs. There's a natural duality

33:12operation because for any hypergraph, you
can

33:15construct a dual hypergraph whose vertex set
is the hyperedge set of the old hypergraph,

33:21and whose
hyperedge set is the incident structure of

33:24those hyperedges in the new case. That gives
you a

33:27duality that satisfies the axioms of compact
closure. In a sense, the key idea behind

33:34categorical quantum mechanics is that if you
have one of these dagger structures, you have

33:37a compact
closed structure, you have a symmetric monoidal

33:40structure, and they're all compatible, then
what

33:42you've got is, again, by analogy to topos
theory, some mathematical structure in which

33:49you can build
a theory that is isomorphic to quantum mechanics.

33:52That's what we have in the case of multi-way
systems.

33:56So when we spoke approximately three years
ago, I believe we had a Zoom meeting. It could

34:01have
been a phone call. I recall that you were

34:03saying that you were working, maybe the year
prior,

34:07on something where your operators, your measurements,
don't have to be self-adjoint.

34:13And the reason was, self-adjointness is there
because we want real eigenvalues, and that

34:18just
means for people who are listening, you want

34:19to measure something that's real, not imaginary.
What is an imaginary tick? It usually comes

34:25down to ticks or not ticks in the measurement
device.

34:29But then I recall you said that you were working
on constructing quantum mechanics with observables

34:34that weren't. So self-adjointness implies
real eigenvalues, but there were other ways

34:42of getting
real eigenvalues that aren't self-adjoint.

34:45I don't know if I misunderstood what you said,
or if I'm recapitulating incorrectly, but

34:49please spell out that research if this rings
a bell to

34:53you.
So your memory is far better than mine. That

34:56sounds like a very accurate summary of something
I would have said, but I actually have no

35:03memory of saying it. To be clear, that's by
no means my

35:10idea. There's a field called PT-symmetric
quantum mechanics, and sometimes known as

35:14non-Hermitian
quantum mechanics, which have various developers.

35:18Carl Bender is one of them. I think there's
a guy

35:21called Jonathan Brody, or Jorge Brody. I can't
remember.

35:26Carl Bender. So I just spoke to him about
a couple months ago, coincidentally.

35:29Oh, well, you should have asked him this question.
He's the expert, right?

35:34Yeah.
Yeah, so Bender and Brody and others. Jorge

35:37Brody. I don't know why there's another person.
Maybe Jonathan Keating is involved somehow.

35:42Sure.
But anyway, so it's been a little while since

35:45I thought about this, as you can tell. So
yes,

35:48there's a generalization of standard unitary
Hermitian quantum mechanics. So yeah, as Curt

35:55mentioned, in the standard mathematical formalism
of quantum mechanics, your measurements seem

36:00to
be Hermitian. So when you take the adjoint

36:04of the operator, you get the same result.
And your evolution operators are assumed to

36:09be unitary, so that when you take the adjoint,
you get the time reversal of the result. In

36:15a sense, that's the key difference between
evolution and measurement in standard formalism.

36:20And we know that, yeah, if your Hamiltonian
is

36:27Hermitian, the thing that appears in the Schrodinger
equation, if that's a Hermitian operator,

36:32then the solution to the Schrodinger equation
that gives you the unitary evolution,

36:34that gives you the evolution operator, sorry,
is guaranteed to be unitary.

36:38And also the eigenvalues of the measurement
operator, which is, as Curt said, are in a

36:42sense, those are your measurement outcomes.
Those are guaranteed to be real. That's a

36:47sufficient
condition, Hermeticity, but it's not a necessary

36:50one. So that you can have non-Hermitian measurement
operators that still give you real eigenvalues.

36:56And where you don't get a unitary evolution
operator in the algebraic sense, but you get

37:03what is often called, I think in these papers
it's referred to as kind of physical unitarity.

37:07So unitarity means a bunch of things, right?
So algebraically, as I say, it means that

37:11when you apply the adjoint operator,
you get the time reversal. And therefore,

37:17if you take a unitary evolution operator and
it's

37:19adjoint, you get the identity matrix or the
identity operator. So as soon as you have

37:23non-Hermitian Hamiltonians, that ceases to
be true. And also you end up with probabilities.

37:29So in the interpretation where your quantum
amplitudes are really kind of related to

37:36probabilities, right? Where you take the absolute
value of the amplitude squared,

37:39and that gives you the probability. Now, as
soon as you have non-unitary evolution operators,

37:44your probability amplitudes or your probabilities
are not guaranteed to sum to one.

37:48So that looks on the surface like it's completely
hopeless. But actually, as I say, you can

37:57still
get real measurement outcomes. The interpretation

37:59of the norm squareds of the amplitudes as
being

38:02probabilities, that's simply an interpretation.
It's not mandated by the formalism.

38:06And what people like Bender and Brody showed
was that you could still have a consistent

38:11theory
where you have parity time symmetry. So you

38:15still have a time symmetric theory of quantum
mechanics.

38:16It's still invariant under parity transformations.
And it's still possible, even when you apply

38:22one
of these non-unitary evolution operators to

38:26some initial state, it's still always possible
to

38:28reconstruct what the initial state was from
the final state. I mean, that's really what

38:32time
symmetry means. And so it was widely believed,

38:36I think, for a long time that if you didn't
have

38:37amplitudes whose normal squareds sum to one,
then you wouldn't be able to do that. And

38:42what
Bender and Brody showed was that no, you can.

38:44You just have to be
â€“ you still have restrictions, but they're

38:47just weaker than the restrictions we thought
existed.

38:50So I was probably bringing that up because
at the time â€“ well, okay, two reasons. One

38:55was it turns
out there are these nice connections, which

38:58I was a little bit obsessed with a few years
back,

39:02between PT symmetric quantum mechanics and
the Riemann hypothesis.

39:05So a colleague of mine, a former colleague
of mine from Wolfram Research, Paul Abbott,

39:10was the person who actually told me about
this. And so the idea there is, there's this

39:15thing
called the â€“ okay, let me get this right.

39:18So there's a thing called the Hilbert-Polya
conjecture,

39:21which is the conjecture that â€“ which I think
is reasonably well-known. At least some people,

39:28people in our kind of area have often heard
about. Yeah, which is the idea that somehow

39:34the
non-trivial zeros of the Riemann zeta function

39:37should be related to the eigenspectrum of
some

39:42manifestly self-adjoint operator. And so it's
somehow a connection between the analytic

39:48number
theory of the zeta function and the kind of

39:51foundation, the operator-theoretic foundations
of quantum mechanics. And then there's the

39:55thing called the Berry-Keating Hamiltonian.
So Mike Berry and Jonathan Keating constructed

40:00a case of what they conjectured to be a Hilbert-Polya
type Hamiltonian. So in other words, a Hamiltonian

40:07where if you could prove that
it was manifestly self-adjoint, it would be

40:11equivalent to proving the Riemann hypothesis.
The problem is that Hamiltonian is actually

40:16not â€“ it's not self-adjoint. It's not
Hermitian in the traditional sense. But it

40:21is Hermitian in this PT symmetric sense. It
is â€“ so

40:24it's not algebraically Hermitian. It's not
equal to its own adjoint. But it's still a

40:29valid
Hamiltonian for parity time symmetric quantum

40:33mechanics. And so by trying to think about
the Riemann hypothesis in terms of quantum

40:41formalism, you end up being kind of inevitably
drawn into thinking about non-Hermitian foundations

40:45and these kind of PT symmetric
formulations. So that's how I first learned

40:49about this. And I suspect I was talking about
it at the

40:50time partly because I was just interested
in that connection. It turns out that the

40:55spectrum of
these operators are related not just to the

40:58Riemann zeta function, but also to what's
called

41:00the Hovitz zeta function and several other
objects in analytic number theory. But also

41:05at the time,
this has turned out to be false, but at the

41:08time I thought that the version of quantum
mechanics

41:12that we would end up with from these multiway
systems would be a PT symmetric

41:16formalism for quantum mechanics, not standard
quantum mechanics. As it turns out, actually,

41:21there's a way you can do it where you get
standard quantum mechanics complete with proper

41:23Hermeticity
and Unitarity, so you don't really need to

41:26worry about that. But at the time, I was quite
nervous

41:28that we weren't going to get that, but we
were going to get some weird non-Hermitian

41:31version
of quantum mechanics, and we'd have to worry

41:33about that.
Jay, do you end up getting both or just one?

41:37So there is a construction where you can getâ€¦
I mean, what I want to stress is that there's

41:44no
canonicalâ€¦ If you're just given a multiway

41:47system and you're said,
derive quantum mechanics, right? There's no

41:49canonical way to do that.
The approach that we ended up taking was to

41:53show that, as I say, there's this algebraic
structure

41:55that has this dagger symmetric compact closed
monoidal category feature. Therefore, you

42:03can
get standard quantum mechanics because standard

42:05quantum mechanics is what's developed kind
of

42:06internal to that category. But in order to
do that, we had to make a whole bunch of really

42:13quite arbitrary choices. So I strongly suspect
that there are ways that you could define

42:19an
algebraic structure that is essentially a

42:21non-Hermitian PT symmetric formulation. I
just

42:24I don't personally know the way to do it.
So just as an aside, a pedagogical aside for

42:29the people who aren't mathematicians or physicists,
they hear terms like closed,

42:33compact, symmetric, monoidal, dagger, unitary,
adjoint, and they're wondering,

42:37why are we using these words to describe physical
processes? And the reason is because the

42:43mathematicians got there first. So physicists
are trying to describe something and then

42:48they see
that there's some tools invented by other

42:50people, goes by other names, and then they
end up applying

42:52in the physical situations. But when the physicist
gets there first, they're often intuitive

42:56names,
momentum, spin up, spin down. It's actually,

42:59it makes more sense. So just in case people
are

43:02wondering, this terminology is needlessly
complex. Well, it can be to the outsider,

43:06but as you become
familiar with them, you just realize historically,

43:09if you want to communicate to mathematicians
and

43:11vice versa, then just use whatever terms were
invented first. I would say there's the opposite

43:17problem as well, right? I mean, there are
cases where physicists discovered concepts

43:21first that
have been subsumed into mathematics, and the

43:23physical names don't really make any sense
in

43:25the mathematical context. There are things
like physicists, because of general relativity,

43:29were really the first people to seriously
think about and formalize notions like torsion

43:34in
differential manifolds and concepts like metric

43:37affine connections. So the standard connection
that you define on a manifold with torsion

43:44is the spin connection, so named because it
was

43:47originally used in these metric affine theories
where you have a spin tensor that describes

43:50the
spin of particles. So now there are these

43:54ideas in algebraic and differential geometry
called

43:56spin connections and spin holonomies, and
they have nothing to do with spin, nothing

44:00to do with
particle physics. But it's the relic of the

44:04physical origins of the subject. There are
several cases of that too. Yeah, I haven't

44:08announced this, and I'm not sure if I'll
end up doing this. I've been writing a script

44:12for myself on words that I dislike in physics
and math.

44:16Sometimes they'll say something like, what's
the callback? Wow, what is it called? The

44:22callback?
Callback-Leibler divergence.

44:24Callback-Leibler divergence. Okay, if you
just say that, it doesn't mean anything. You

44:28have to know
what it's defined as. So calling something

44:31the earth mover's distance is much more intuitive.
And then I have this whole list of words that

44:37I say, okay, it's so foolish to call it this.
Why don't you just call it by its descriptive

44:42name? And then I suggest some descriptive
names.

44:45And there's another class of foolish names
to myself. Torsion is one of them,

44:50but it's not because it's a bad name. It's
because it's used in different senses.

44:56On an elliptic curve, there's torsion, but
it has nothing to do with the torsion in differential

45:00geometry, which as far as I can tell, maybe
you can tell me the difference here. In cohomology,

45:05there's torsion where if you are using the
field of the integers and then you go to the

45:10reals,
if they're not equivalent, then you say it

45:13has torsion.
Yes, yes.

45:14But it's not the same as the differential
geometric torsion as far as I can tell.

45:19I think that's true. Yeah, so I think that
confusion has arisen because it's one of

45:24these examples of independent evolution. So
there was a notion of torsion that appeared

45:28in group
theory, but then because of that got subsumed

45:30into, as you say, things like homology theory
and cohomology theory. So in group theory,

45:36a group is defined as being torsion if it's
if it has only finite generators, generators

45:45of finite order. So the generators of a group,
the things that you multiply, you exponentiate

45:52to get all elements of the group. If the group
is generated only by generators of finite

45:58order, then you say it's a torsion group.
You can talk about torsion subgroups, or you

46:02could talk about the torsion part of a group.
And so yeah, it appears a lot in the theory

46:06of elliptic curves because there are things
like the

46:09Mordell-Weythe theorem that are talking about
when you take rational points on elliptic

46:14curves,
you can ask about how large is the torsion

46:16part, how large is the non-torsion part.
And there are things like Birch-Swinson-Dyer

46:19conjecture that are all about relating those
ideas. But then yeah, then quite independently,

46:24there was a notion of torsion that appeared
in

46:26differential geometry that, as you know, is
that it's just essentially it's a measure

46:29of,
you know, I have points x and y, how different

46:33is the distance from x to y and the difference
from y to x. And the name there comes from

46:38the fact that in the classical kind of Gaussian
theory

46:41of geometry of surfaces, it's the concept
that gives you the torsion of a curve, right?

46:46You know,
how much the curve is twisting. Yeah, as far

46:50as I know, those two names are unrelated.
And as you

46:53say, there are these awkward areas like homology
theory where it's partly about spaces and

46:59it's
partly about groups. And so it's kind of unclear

47:01which one you're talking about.
This is a great point to linger on here, particularly

47:04about torsion,
because I have a video that is controversially

47:08titled that gravity is not curvature.
For some context, here's the string theory

47:12iceberg video that's being referenced where
I

47:15talk about gravity is not curvature. The link
is in the description. If you listen to this

47:19podcast,
you'll hear me say often that it's not so

47:21clear gravity is merely the curvature of space-time.
Yes, you heard that right. You can formulate

47:25the exact predictions of general relativity,
but with a model of zero curvature with torsion,

47:31nonzero torsion, that's Einstein-Cartan. You
can

47:33also assume that there's no curvature and
there's no torsion, but there is something

47:37called
non-matricity. That's something called symmetric

47:39teleparallel gravity. Something else I like
to

47:41explore are higher spin gravitons. That is
controversially titled that gravity is not

47:47curvature. It's just the saying that there
are alternative formulations with torsion

47:51or
non-matricity. For people who don't know,

47:54general relativity is formulated as gravity
is curvature

47:57of space-time, but you can get equivalent
predictions if you don't think of curvature.

48:02You can think of zero curvature, but the presence
of so-called torsion, or zero curvature and

48:07zero
torsion, but the presence of so-called non-matricity.

48:10Okay, these are seen as equivalent formulations,
but I'm wondering if the Wolfram's physics

48:17project or the hypergraph dynamical approach
actually identifies one of them as being more

48:22canonical.
Unfortunately, I think at least based on stuff

48:30that I've done, I think the answer is no.
Also,

48:34I think it actually makes the problem worse.
If you're concerned by the fact that there's

48:41this apparent arbitrary freedom of do you
choose to fix the contortion tensor or the

48:45non-matricity tensor or the curvature tensor
or whatever, thinking about things in terms

48:49of hypergraphs, you actually get yet another
free parameter, which is dimension. In a hypergraph

48:58setting, again, I know you've had Stephen
on here before, and I know that he's covered

49:02a lot of these ideas, so I'll just very briefly
summarize. Hypergraphs have no a priori notion

49:08of dimension. They have no a priori notion
of curvature. You can calculate those things

49:12subject to certain assumptions where you say,
I'm going to look at, I take a node and I

49:17look at all nodes adjacent to it and all nodes
adjacent to those nodes and so on. I

49:20grow out some ball and I ask, what is the
scaling factor of that ball as a function

49:23of radius? I can take logarithmic differences.
That gives me the exponent. That exponent

49:28is like a Hausdorff dimension. Then I can
ask, what's the correction? Does that give

49:32me some leading order term in the expansion?
What are the correction terms? Those correction

49:36terms give me projections of the Riemann tensor.
That's just using the analogy to kind of classical

49:41differential geometry. But the point is that
to get the curvature terms, as we do in, say,

49:45the derivation of general relativity, you
have to assume that the hypergraph is kind

49:50of uniform dimensional, right? Even to be
able to take that Taylor expansion, you have

49:55to assume that the dimension is uniform. So
then an obvious question is, what happens

49:59if you relax that assumption? And the answer
is, well, you get a theory that is equivalent

50:05to general relativity in the kind of observational
sense, but now you can have fixed curvature,

50:12fixed contortion, fixed non-matricity, but
you just have variable dimension. The point

50:17is that in the expansion for that volume element,
the dimension gives you the kind of leading

50:24order exponential term. The Ricci scalar curvature
gives you a quadratic correction to that,

50:29and then you have higher order corrections.
Because of this very basic mathematical fact

50:36that if you're zoomed in really far, it's
very hard to distinguish an exponential curve

50:42from a quadratic curve, right? You kind of
have to zoom out and see it very globally

50:45before you can really tell the difference
between the two. And so in a sense, what that

50:49translates to is if you're looking only at
the microstructure of space-time, there's

50:52no way for you to systematically distinguish
between a small change in dimension and a

50:58very large change in curvature. So if you
had a region of space-time that was kind of

51:02rather than being four-dimensional, was 4.00
one-dimensional, but we were to kind of measure

51:08it as though it were four-dimensional, it
would manifest to us as a curvature change.

51:12It would be observationally indistinguishable
from a curvature change. So what I would say

51:18is that in the hypergraph dynamics view, yeah,
you again have this arbitrarity of you have

51:24to make choices of connections which fix torsion
and non-matricity and so on. But you have

51:28this additional problem that you also have
to make choices about trade-offs between curvature

51:31and dimension.
So let's go back to category theory for just

51:35a moment. When I was speaking to Wolfram about
that, Stephen Wolfram, he said that he's not

51:39a fan of category theory because he believes
it circumvents computational irreducibility.

51:45I said, why? He said, well, because you go
from A to B, yes, then you can go from B to

51:49C, but then you also have an arrow that goes
directly from A to C. But when I was thinking

51:54about it, that's only the case if you think
that each mapping takes a time step. But when

51:59I look at category theory, I don't see
it as any time step. At least I don't. I see

52:04it as just this timeless creation. So please
tell me your thoughts.

52:08Right. Okay. Well, so I'm in the fortunate
position of having written quite a long paper

52:15on exactly this problem. So there's a paper
that I wrote back in 2022 called A Functorial

52:20Perspective on Multicomputational Irreducibility,
which is all about exactly this idea. So as

52:29you say, category theory, as it's ordinarily
conceived, is just a kind of algebraic theory

52:34that has no notion of, there's nothing computational
about it, right? There's no notion of time

52:38step. There's no statement made about what's
the computational complexity of any given

52:42morphism.
But then an obvious question is, well, okay,

52:46is there a version of category theory which
does care about those things, a kind of resource

52:49limited version, or some version where individual
morphisms are kind of tagged with computational

52:54complexity information? And it turns out the
answer is yes. And it has some very nice connections

52:59to not just categorical quantum mechanics,
but also things like functorial quantum field

53:03theory. But also it gives you a new... I think
Stephen is wrong in that statement that it

53:12doesn't care about computational irreducibility,
because actually it gives you a very clean

53:16way of thinking about computational irreducibility.
So what I mean by that is, so computational

53:21irreducibility, this idea that there are some
computations that you kind of can't shortcut

53:25in some fundamental sense. As far as I know,
I was the first person to actually give a

53:29formal definition of that in a paper back
in 2018 or something.

53:33Sorry, a formal definition of computational
irreducibility?

53:37Of computational irreducibility. Nothing very
profound, but just essentially you say, I've

53:42got some Turing machine that maps me from
this state to that state. Does there exist

53:45a Turing machine of the same signature that
gets me to the same output state with fewer

53:49applications of the transition function? And
so I mean, I needed that for another result

53:55that I was proving. But having looked in the
literature, I'm not aware of anyone previously

54:00who'd formalized that definition.
Sorry, I don't mean to cut you off, so please

54:03just remember where you are. Because it's
my understanding that Wolfram said that rule

54:0730, something like that, maybe you would recall
it more vividly because it's in his book,

54:12rule 30 is computationally irreducible. I've
always wondered, how do you prove that? Now,

54:16I imagine that he proved it, or maybe it's
one of those Wolfram proofs, so proof to himself.

54:21But in order for him to prove it, even to
himself, he would have had to have a definition

54:25of it.
Right. Okay. So that's an important point.

54:31So rule 30 is not proved to be computationally
irreducible. And in fact, there's a prize.

54:36So if you go to, I think it's rule30prize.org.
I'm ostensibly on the prize committee. This

54:42is a prize that Wolfram put out back in 2018.
There's actually three prizes, none of which

54:48have been claimed. Each one is $10,000. And
one of which is prove that rule 30 is computationally

54:53irreducible. And so yeah, it's unproven. And
in fact, there's really only one, up to some

55:01notion of equivalence, there's really only
one of the elementary cellular automata in

55:05NKS that's been proven to be computationally
irreducible in any realistic sense. And that's

55:10rule 110. And that was proved by showing that
it's capable of doing universal computation,

55:15that it's a Turing-complete rule. And so intuitively,
you can kind of say, well, if it's Turing-complete,

55:23then questions about termination are going
to be undecidable, and therefore it has to

55:26be irreducible. But it's a kind of slightly
hand-wavy thing. But yeah, so in a way, it's

55:33an interesting question. Can you prove that
something is computationally irreducible without

55:38proving that it's universal? And of course,
as you say, for that, you would need a formal

55:42definition of irreducibility.
Okay. And now going back to your paper on

55:47functoriality and computational irreducibility,
you were able to formalize this.

55:52Yes. So sorry. Yes. So what I was saying was,
yes, so there was this existing formal definition

55:57of computational irreducibility. But I then
realized that if you think about it from a

56:02category theoretic standpoint, there's actually
a much nicer definition, a much less kind

56:04of ad hoc definition, which is as follows.
So imagine a version of category theory where

56:09your morphisms, as I say, are tagged with
computational complexity information. So each

56:12morphism has a little integer associated to
it. So you fix some model of computation,

56:17you fix Turing machines, and you say, each
morphism, I'm going to tag with an integer

56:21that tells me how many operations was needed
to compute this object from that object. In

56:26other words, how many applications of the
transition function of the Turing machine

56:30did I need to apply?
So now if I compose two of those morphisms

56:37together, I get some composite. And that composite
is also going to have some computational complexity

56:43information. And that computational complexity
information, it's going to satisfy some version

56:46of the triangle inequality, right? So if it
takes some number of steps to go from X to

56:50Y and some number of steps to go from Y to
Z, I can't go from X to Z in fewer computational

56:56steps that it would have taken to go from
X to Y or from Y to Z. So it's going to at

57:02least satisfy the axioms of something like
a metric space. There's some kind of triangle

57:06inequality there.
But then you could consider the case where

57:11the complexities are just additive, right?
Where to get from X to Z, it takes the same

57:16number of steps as it takes to go from X to
Y plus the number of steps it takes to go

57:19from Y to Z. And that's precisely the case
where the computation is irreducible, right?

57:23Because it's saying you can't shortcut the
process of going from X to Z. Which then means

57:27you could define the case of computational
reducibility as being the case where the algebra

57:34of complexities is sub-additive under the
operation of morphism composition.

57:39And there's a way that you can formulate this.
So you take your initial category, and you

57:44take a category whose objects are essentially
integers and discrete intervals between integers.

57:52And then you have a functor that maps each
object in one category to an object in another,

57:58each morphism in one to a morphism in another.
And then the composition operation in the

58:02second category is just discrete unions of
these intervals. And then you can ask essentially

58:08whether the cardinality of those intervals
is discretely additive or discretely sub-additive

58:12under morphism composition. And that gives
you a way of formalizing computational reducibility.

58:16And the really lovely thing about that is
that not only can you then measure irreducibility

58:21and reducibility in terms of defamation of
this functor, but it also generalizes to the

58:27case of multi-way systems. You can formalize
notions of multi-computational irreducibility

58:32by essentially just equipping these categories
with a monoidal structure, with a tensor product

58:35structure.
Aaron Powell So my understanding of computational

58:39irreducibility
is either that a system has it or it doesn't,

58:41but it sounds like you're able to formulate
an index so that this system is more irreducible

58:46than another. Like you can actually give a
degree to it.

58:49Tom Clougherty Exactly, exactly. So there's
a limit case

58:53where it's exactly additive, and anything
that's less than that, you know, where the

58:58complexities are exactly additive, that's
kind of maximally irreducible. But anything

59:01less than that is sort of partially reducible,
but not necessarily fully reducible.

59:06Aaron Powell Now, are there any interesting
cases of something

59:08that is completely reducible, like has zero
on the index of computational irreducibility?

59:13Is there anything interesting? Even trivial
is interesting, actually.

59:16Tom Clougherty Yes, I mean, well, okay, so
any computation

59:24that doesn't change your data structure, that's
just a repetition of the identity operation

59:30is going to have that property. I'm not sure
I can necessarily prove this. I don't think

59:35there are any examples other than that. I
think any example other than that must have

59:39at least some minimal amount of irreducibility.
But yes, I mean, this also gets into a bigger

59:50question that actually relates to some things
I'm working on at the moment, which is exactly

59:55how you equivalence objects in this kind of
perspective, right? Because even to say it's

01:00:01a trivial case, right, where I'm just applying
some identity operation, I'm getting the same

01:00:05object, you have to have some way of saying
that it is the same object. And that's actually,

01:00:11I mean, that sounds like a simple thing, but
it's actually quite a philosophically thorny

01:00:17issue, right? Because, you know, in a very
simple case, you could say, well, okay, so

01:00:21sorry, first thing to say is, everything we're
talking about at the moment, this is all internal

01:00:25to this category, which in the paper I call
comp, this category whose objects are in a

01:00:30sense elementary data structures, and whose
morphisms are the morphisms that freely generate

01:00:37this category are elementary computations.
And so the collection of all morphisms that

01:00:40you get from compositions are essentially
the class of all possible programs. So within

01:00:45this category, when two objects are equivalent,
and therefore when two programs are equivalent

01:00:50is a fairly non-trivial thing, right? So you
can imagine having a data structure where

01:00:54nothing substantively changes, but you've
just got like a time step or something that

01:00:58goes up every time you apply an operation.
So it just increments from one, two, three,

01:01:01four. So in that case, you're never going
to have equivalences. Every time you apply

01:01:04an operation, even if the operation morally
does nothing, it's going to be a different

01:01:09object. So even that would show up as being
somehow irreducible. But there are also less

01:01:14trivial cases of that, like with hypergraphs,
right? So with hypergraphs, you have to determine

01:01:19equivalence, you have to have some notion
of hypergraph isomorphism. And that's a complicated

01:01:24to define, let alone to formalize algorithmically.
And so you quickly realize that you can't

01:01:33really separate these notions of reducibility
and irreducibility from these notions of equivalencing.

01:01:39And somehow it's all deeply related to what
data structures do you kind of define as being

01:01:46equivalent or equivalent up to natural isomorphism
or whatever. And that's really quite a difficult

01:01:51problem that relates to definitions of things
like observers in these physical systems,

01:01:56right? If you have someone who is embedded
in one of these data structures, what do they

01:02:00see as equivalent, which might be very different
to what a kind of God's eye perspective views

01:02:04as being equivalent from the outside.
JSON So are there close timelike curves in

01:02:08Wolfram's
physics project? Sorry, HD project.

01:02:11SIMON No, we can say Wolfram physics. I mean,
that's

01:02:15how it's known, right? No, so yeah, that's
a really good question, right? Because in

01:02:20a way, it's very easy to say no, because we
can do that trick that I just described, where

01:02:27you just tag everything with a time step number.
And then of course, even if the hypergraph

01:02:31is the same, the time step is different. So
there's no equivalence thing. In the multiway

01:02:35system or the causal graph, you never see
a cycle. But that's somehow cheating, right?

01:02:39And when people ask about CTCs, what they
care about is not this very nerdy criterion

01:02:46of, oh, do you actually get exactly equivalent
data structures? What they care about isâ€¦

01:02:50JSON Nerdy criterions seems to define this
entire

01:02:53conversation up until this point.
SIMON Well, yes, I suppose. You know, you

01:02:59take two
people with math backgrounds and get them

01:03:01to discuss stuff.
JSON Yeah, exactly, exactly.

01:03:02SIMON That's going to happen, right? But yeah,
soâ€¦

01:03:05JSON But yeah, what they care about, people
who

01:03:07care about time travel.
What one cares about is, yeah, exactly, is

01:03:12time travel and causality violations and things
which don't necessarily care about your equivalency

01:03:18or care about them in a slightly different
way. Yeah, I mean, so my short answer is I

01:03:25don't know. Because I think we can'tâ€¦
My personal feeling is we are not yet at this

01:03:32level of maturity where we can even pose that
question precisely for the following reason,

01:03:37right? So even defining a notion of causality
is complicated. So in most of what we've done

01:03:46in that project, in derivations of things
like the Einstein equations and so on, we've

01:03:50used what on the surface appears like a very
natural definition of causality for hypergraph

01:03:54rewriting. So you have two rewrites. You know,
each one is going to ingest some number of

01:04:01hyperedges. It's going to output some other
number of hyperedges. Those hyperedges have

01:04:04some identifier. And then you can ask, okay,
did this future event ingest edges that were

01:04:09produced in the output of this past event?
And so if it did, then the future event couldn't

01:04:13have happened unless the past event had previously
happened. And so we say that they're causally

01:04:17related. So somehow, if the output set of
one has a non-trivial intersection with the

01:04:20input set of another, we say that they're
causally related. That seems like a perfectly

01:04:26sensible definition, except it requiresâ€¦
It has exactly the problem we've been discussing,

01:04:31right? It requires having an identifier for
each of the hyperedges. You need to be able

01:04:34to say this hyperedge that this event ingested
is the same as this hyperedge that the other

01:04:39event output. But if they're just hyperedges,
they're just structural data, there's no canonical

01:04:44choice of universal identifier, of UUID.
And so what that means is you can have these

01:04:51degenerate trivial cases where, for instance,
you have an event that ingests a hyperedge,

01:04:56changes its UUID, but doesn't actually change
anything structurally. The graph is still

01:05:00the same. Nothing has actually changed, interestingly.
But the identifier is different. But now,

01:05:05any event in the future that uses that edge
is going to register as being causally related

01:05:11to this other event that didn't do anything,
right? And so you have a bunch of these spurious

01:05:14causal relations. So it's clear that that
definition of causality isn't quite right.

01:05:19And so what's really needed is some definition
of causality that isn't subject to this problem,

01:05:24but it's very unclear what that is. And I've
worked a little bit on trying to formalize

01:05:28that by recursively identifying hyperedges
based on their complete causal history, right?

01:05:35So the identifiers are not chosen arbitrarily
as random integers or something. But instead,

01:05:40each hyperedge encodes, in a slightly blockchain-y
way, a directed acyclic graph representation

01:05:46of its complete causal history. And so then
two hyperedges are treated as the same if

01:05:49and only if they have the same history of
causal relationships in the rewriting system.

01:05:54And that's somewhat better, but again, is
quite complicated to reason about. And it's

01:05:59all deeply related to this question of what
data structures do you ultimately treat as

01:06:04being equivalent, which is really an observer-dependent
thing. It depends on the computational sophistication

01:06:09of the person or entity who is trying to decode
what the system is doing. It's not a kind

01:06:14of inherent property of the system itself.
So what do you make of observer theory, which

01:06:19is a recent large blog post by Stephen, and
a theory, well, an outlook. So what do you

01:06:27make of it?
Yeah, so observer theory really has, it's

01:06:31a rebranding of this thing that's been a feature
of the physics project since before we started

01:06:36it, right? So this idea that, yes, exactly,
that you cannot sort of consider a computational

01:06:44system independent of the observer that is
interpreting its results. And somehow, both

01:06:51the computational sophistication of the observer
and the computational sophistication of the

01:06:55system have to be factored into that description
somehow. So in a way, it's a very natural

01:07:01idea, which is really the prelude to the work
we did on quantum foundations and other things

01:07:07in the context of the physics project.
I like to think of it as a kind of natural

01:07:11extension of a bunch of stuff that happened
in 20th century physics, right? Because of

01:07:16course, this is not how these things were
viewed at the time, but both general relativity

01:07:22and quantum mechanics can in some sense be
thought of as being theories that you arrive

01:07:27at by being more realistic about what the
observer is capable of, right? So if you say,

01:07:35okay, a lot of traditional scientific models
made this assumption.

01:07:39That the observer was kind of infinitely far
removed from the system that they were observing.

01:07:43That they somehow, you know, they were these
kind of omnipotent entities.

01:07:45They didn't have any influence over the systems.
They weren't constrained by the same laws.

01:07:49But if you then say, okay, well maybe the
observer has some limitations.

01:07:52Maybe they can't travel faster than light,
right?

01:07:54What does that imply?
Well, in some, if you follow the right chain

01:07:57of logical deduction, what that implies is
general covariance and therefore general relativity.

01:08:00That as soon as you have observers who can't
travel faster than light, they don't necessarily

01:08:05agree on the ordering of space-like separated
events and suddenly you get general relativity.

01:08:08Equivalently, if you have observers who are
constrained by the same physical laws that

01:08:15of the systems that they're observing, then
what that means is, you know, if you try and

01:08:19measure some property of a system, what happens
when you measure it?

01:08:22Well, you have to have some interaction with
it.

01:08:24You have to kind of poke it somehow and, you
know, and the poke that you receive back is

01:08:28going to be equal in magnitude to the poke
that you gave to the system.

01:08:31And so anytime you try and measure some quantity,
there's a minimum amount that you have to

01:08:34disturb it.
And again, if you kind of follow that chain

01:08:37of reasoning to its logical conclusion, you
get at least the kind of Heisenberg picture

01:08:41of quantum mechanics.
So in a way, both general relativity and quantum

01:08:45mechanics are, as I say, you know, ways of
becoming more realistic about what observers

01:08:48are capable of and ways of coming to terms
with the fact that observers are constrained

01:08:55by the same physical laws as the systems that
they observe.

01:08:58So observer theory, which, I mean, I don't,
I don't think it's yet a theory, so I'm not

01:09:04sure it's, you know, I'm not, I'm sure I,
I, I, I'm hugely fond of the terminology,

01:09:09but I mean, it's, it's a, it's a, yeah, it's
a conceptual idea is really just the kind

01:09:16of computational instantiation of that.
And you know, so my feet, okay.

01:09:22You mentioned before this very interesting
thing about geometry that, that somehow, you

01:09:26know, you, you have this freedom of, do you
choose to vary curvature, do you choose to

01:09:30vary torsion, do you choose to vary non-matricity?
My feeling is that there's a similar free

01:09:35parameter that exists in our scientific models
with regards to the role of the observer.

01:09:39And this is again, maybe a point of philosophical
departure from, between me and Stephen is,

01:09:47so you have these kind of, you can imagine
these two extreme cases, right?

01:09:51You can imagine the case where all you care
about is the computation that the system is

01:09:54doing.
It's picking up some, some structure from,

01:09:57from, you know, from, from bottom up rules.
And so the observer, so to speak is just some

01:10:03trivial object that's seeing the data structure
and all of the kind of computational burden

01:10:07is being shouldered by the system itself.
And you know, that's kind of the, that's the

01:10:12way that the physics project is often presented,
right?

01:10:14You just have a hypergraph and it's doing
its thing and we kind of, we, we, we perform

01:10:18analyses on it.
That's one extreme.

01:10:20There's another extreme where you could say,
well, maybe the system itself is trivial.

01:10:23You know, the computation it's doing is, is,
is essentially trivial and all of the sophistication

01:10:28is all the kind of computational burden is
shouldered by the observer.

01:10:31So the case of that would be what Stephen
refers to as the Ruliad, which is really

01:10:34just this, what I was describing earlier,
this kind of category of, you know, all possible

01:10:39elementary data structures and all possible
computations.

01:10:43And so in that picture, I mean, that, that's
a kind of, that's an object that minimizes

01:10:48algorithmic complexity, right?
It minimizes Kolmogorov complexity, the, you

01:10:52know, the, the, the, the, the, the set of
all possible computations has the same algorithmic

01:10:57complexity as the set of no computations just
purely for information theoretic reasons.

01:11:02And so in that case, you know, the, the actual
computation that generates it is trivial.

01:11:06It's, you know, it's trivial to specify, but
in order to get a particular computational

01:11:12path or in order to restrict down to a particular
multi-way system, you have to have an observer,

01:11:17some generalized observer who is making equivalences
between different paths.

01:11:21And the sophistication of that observer can
be arbitrarily high.

01:11:25And so you have these two extreme cases, one,
one case where the observer is trivial, all

01:11:29the computation is being done by the system.
Another case where the system is trivial,

01:11:32all the computations being done by the observer.
And my argument is these two cases, I mean,

01:11:37there's no observational way of distinguishing
between them.

01:11:40And in fact, there's the whole interstitial
space in the, in the middle where you have

01:11:43some of the burden being shouldered by the
system, some of the burden being shouldered

01:11:45by the observer.
And these are not really things that we can

01:11:49observationally distinguish.
And so in a sense, it's a, it's a genuinely

01:11:51free parameter in how we construct our models.
And I would even go so far as to say that

01:11:56I think in some sense, this argument that
occurred in early European philosophy between

01:12:03the kind of empiricists and the rationalists,
right, between people like, you know, Locke

01:12:08and, and, and Hume on the kind of empiricist
side and people like, you know, Descartes

01:12:13and, and Bishop Berkeley and so on, and on
the, on the rationalist side, that's really

01:12:18the kind of, this is really the modern version
of that same argument, right?

01:12:20The empiricists saying, we need to get the
observer out of the picture as much as possible

01:12:25and just describe the systems.
The rationalists saying, no, no, you know,

01:12:28what matters is the internal representation
of the world.

01:12:30And, you know, the, the external reality is
somehow some secondary emergent phenomenon.

01:12:35That's exactly this, this case, right?
That, that, that, that, in a sense, the two

01:12:39extremes of, you know, maximal algorithmic
complexity of the observer versus maximal

01:12:43algorithmic complexity of the system.
I'm confused as to the difference between

01:12:48observation and perception, because Steven
would say that, look, because you're an observer

01:12:53of the kind that you are, you then derive
general relativity or have that as a property

01:12:58or quantum mechanics.
But then firstly, we all don't perceive the

01:13:02same.
And then we also don't perceive quantum mechanics

01:13:05nor general relativity.
In fact, in many ways, we perceive the earth

01:13:07as being flat and we don't perceive any of
the other colors outside of the spectrum of

01:13:12visible light.
So yeah, it's a painstaking process to then

01:13:15say, well, what are the laws of physics?
We have to somehow derive that, test that.

01:13:20And then the question is, well, does a cat
perceive the same laws?

01:13:24Well, a cat doesn't perceive, perceive, this
is what I mean.

01:13:28We don't perceive the same.
The cat doesn't perceive the same, but presumably

01:13:33it's on the same field.
We're playing on the same field.

01:13:36The cat is playing on the same field of general
relativity and quantum mechanics as we are.

01:13:41So sure, our perceptions are different, but
then would Wolfram say that our observations

01:13:46are the same?
Like delineate for me an observation and a

01:13:53perception.
Yeah, that's a really important distinction,

01:13:54right?
And it goes back to some really kind of foundational

01:13:58ideas in early philosophy of science and people
like Thomas Kuhn and others and Karl Popper,

01:14:05who stressed the idea of theory-ladenness
of observation.

01:14:08So I think in the way that you're using those
terms, I think it's an important distinction.

01:14:14The perceptions are kind of much closer to
just the qualia that we perceive, the qualia

01:14:18that we experience.
And the observations are some kind of interpretation

01:14:21that we give to them.
And so the important point, I think the point

01:14:25that people like Kuhn and Popper were making
with theory-ladenness is that, in a sense,

01:14:30we perceive nothing as it, quote, really is.
Anytime we make a scientific observation,

01:14:38we're not perceiving the phenomenon.
It's filtered through many, many layers of

01:14:41observation and interpretation and analysis.
So when we say that we have detected this

01:14:51particle in this particle accelerator, what
does that actually mean?

01:14:54Well, it means that there was some cluster
of photons in this detector that were produced

01:14:59by some Cherenkov radiation, which would then
stimulate some photovoltaic cells on the scintillator.

01:15:07There may be a hundred layers of models and
theories and additional bits of interpretation

01:15:16in between whatever was going on in that particle
accelerator and the bits of photosensitive

01:15:20cells that were stimulated in the scientists'
eyes as they looked at the screen and saw

01:15:25this thing.
And so if you actually try and trace out how

01:15:28many levels of abstraction are there between
the quote-unquote perceptions and the quote-unquote

01:15:33scientific observations, it's huge, right?
And it only takes one of those to be wrong

01:15:39or tweaked a little bit.
And suddenly, the model that you have of the

01:15:43world, which is still just as consistent with
your own perceptions, is completely different,

01:15:48right?
So yeah, I think that's an important thing

01:15:51to bear in mind.
It's a thing in a sense which annoys me a

01:15:56little bit with regards to some criticisms
of experimental validation, because I think

01:16:04people tend to get...
That's an area where people kind of get confused

01:16:07in terms of that distinction.
The people say...

01:16:10It annoys you just a bit?
Only a bit?

01:16:14Well, maybe I don't have to deal with it as
much as you do.

01:16:17Well, no, I don't deal with it.
I just mean, I'm curious if it annoys you

01:16:20more than that, or if you're just being polite.
Well, I mean, it maybe would annoy me if I

01:16:26was being confronted with it all the time.
But when you see people kind of saying that,

01:16:33oh, the multiverse is fundamentally unobservable,
that seems to me to make this exactly the

01:16:42mistake that you're characterizing, right?
It's not perceivable, sure.

01:16:47Most things that we care about in science
aren't perceivable, right?

01:16:49I think David Deutsch has this nice example
that no one has ever seen a dinosaur.

01:16:54No one ever will see a dinosaur.
We'll never get a dinosaur in a lab, right?

01:16:57If you restrict science to only be about things
that we can directly perceive, or test in

01:17:03the laboratory or something, then you can't
make statements about dinosaurs.

01:17:05You can make statements about the composition
and distribution of fossils, but that's not

01:17:09very interesting.
Or at least if you only care about the properties

01:17:12of certain rocks, you would be a geologist,
not a paleontologist.

01:17:15The point is that when we look at the composition
and distribution of fossils,

01:17:19that perceptual data is consistent with a
model of the world that logically implies

01:17:26the existence of dinosaurs. And that's really
what we mean when we say we have evidence

01:17:31of dinosaurs. To be clear, not that I'm particularly
defending the multiverse view or anything

01:17:35like that, but there's a really important
distinction between, yes, the multiverse is

01:17:39not perceivable, which is true, and it's not
possible on the basis of perceptions that

01:17:45we can have to validate a model of the world
that is logically consistent with the existence

01:17:51of a multiverse, which is a very different
statement, and a much more reasonable statement.

01:17:55And yet, in the popular discourse about these
things, those are things that often get confused.

01:17:59So yeah, it annoys me when I see it, and maybe
would annoy me more if I saw it more often.

01:18:06JSL
Speaking of points of annoyance, what are

01:18:08your thoughts on the state of publishing?
So what's your stance on peer review, and

01:18:16where academic publishing is headed, even
in its current state?

01:18:21PG
Yeah. So I had the slightly depressing experience

01:18:27recently, I'm not sure whether you've done
this, of going to Google Scholar and searching,

01:18:30you know, in inverted commas, as an AI language
model or, you know, some other similar thing,

01:18:35right? And just seeing the sheer volume of
papers that have passed so-called peer review

01:18:40in so-called prestigious journals, that are
just obviously, you know, not human written,

01:18:45with no indication of that fact. And there
are obviously plenty of examples, you know,

01:18:51the Sokol affair, and, you know, other things
where, you know, this process that, on the

01:18:58surface, sounds like a very reasonable idea,
this, you know, the idea that, you know, you

01:19:02claim some new result, you get people who
know the field to kind of say, yes, that's

01:19:05a reasonable result, or no, this is not quite
right. That's a perfectly reasonable model.

01:19:09It's just not what peer review actually is
in practice. And, yeah, it's important to

01:19:15remember, as well, that in a sense, the modern
system of scientific publishing, and indeed,

01:19:21the modern system of academia, was not really
designed, right? Like, no one sat down and

01:19:26said, this is how we should do science. It
just kind of happened, right? This model of

01:19:29scientific journals, and peer review, and
editors, and so on, you can trace that back

01:19:34to a direct extension of these early proto-journals,
like the Transactions of the Royal Society,

01:19:41which, if you go back and look at them, were
very different to modern scientific journals,

01:19:45right? It's always kind of entertaining when
you go and read, you know, submissions to

01:19:49the Transactions of the Royal Society that
were made by Robert Hooke, and Robert Boyle,

01:19:52and Isaac Newton, and so on, because they
basically read like blog posts. They're actually

01:19:57very, very informal. You know, you have these
guys that just go in and they say, oh, I did

01:20:02this, I did that, I mixed this chemical
with this, and I saw this thing, and then

01:20:05my cat knocked my experiment over, and whatever.
It's very conversational. It's very discursive.

01:20:14And yes, it was reviewed, but the review process
was much less formalized than it is. I'm not

01:20:20saying that something like that could work
today. I mean, science is much more sort of

01:20:23industrialized, and so on. You clearly need
a more systematic way of processing the volume

01:20:29of scientific literature that's being produced.
But still, it's pretty evident that there

01:20:35was never any person who said, this is a good
model for scientific research and dissemination.

01:20:39This is how it should be done. It naturally
evolved from a system that really wasn't set

01:20:43up to accommodate what it's become.
Another important thing to remember is that

01:20:50the notion of scientific publishing and the
notion of peer review served a pair of purposes,

01:20:58which in the modern world have essentially
become distinct. So it used to be that journal

01:21:02publishers served two roles. They were there
for quality control, because of peer review,

01:21:06and they were there for dissemination, because
they actually printed the physical manuscripts

01:21:09that got sent to libraries and things. In
the modern era, with things like archive,

01:21:13and sci-archive, and bio-archive, and generally
pre-print servers, and people able to host

01:21:18papers on their website, dissemination, which
was always the expensive part of journal publishing,

01:21:24we don't need that anymore. We've
got that covered.

01:21:27So peer review is for quality control?
Yes, exactly. The real role for journals now

01:21:34is quality control, in my opinion. The issue
with that is that's incredibly cheap, because

01:21:40I review papers as does every other academic,
and we do it for free. We do it because it's

01:21:45public service and whatever, and it's an important
thing to do. So we don't get paid. The people

01:21:51writing the papers don't get paid. The journals
shouldn't need to spend lots of money to print

01:21:56physical copies. So really, journal publication
should be not quite free, but basically incredibly

01:22:01cheap, and it's not. The reason is because
you have these journals who are essentially

01:22:06kind of holding on to this very outmoded model,
where they're pushing the dissemination part,

01:22:11I would argue, at the expense of the quality
control part. And so that's why I've been

01:22:15a great advocate. There are these new kinds
of journals that are coming out. There's one

01:22:20called Discrete Analysis and a few others
that are these so-called archive overlay journals,

01:22:26which I think are a fantastic idea. The idea
is we say the content itself is going to be

01:22:32hosted on the archive preprint server, so
we don't need to care about dissemination.

01:22:34So that's all incredibly cheap. We just literally
post a link to an archive paper. And so all

01:22:38we're going to do is worry about the quality
control. And then once you start to think

01:22:43about that, and once you're not bound to having
physical copies that have to go to printers

01:22:47and things, you can actually do peer review
in a very different and, I would argue, much

01:22:50more productive way. You can have open post-publication
peer review, where rather than pre-publication,

01:22:58the manuscript gets sent to some anonymous
reviewers and then they spend six months deliberating

01:23:01and they get the result back and no one ever
sees it. You can have something where someone

01:23:05posts a preprint on archive, it goes on an
open review site, and then anyone in that

01:23:10area, or anyone outside the area, can come
in and say, I don't understand this, or this

01:23:14doesn't make sense, or oh, this is a great
paper or whatever. And then you can kind of

01:23:17upvote, downvote, you can say, oh yeah, I
agree with your criticism, et cetera. And

01:23:20the whole thing can be open and de-anonymized.
And it would have to be anonymized by the

01:23:24person who's publishing, who's posting it
up there, because otherwise, if people see

01:23:28that Ed Witten posted something, more eyes
will go toward that. But you can also, if

01:23:33you're in the field, you can discern sometimes
who's publishing what.

01:23:36Yeah, absolutely. And certainly in math and
physics, and computer science, in places where,

01:23:44in those fields, it's been standard for many
decades now, for several decades, that everyone

01:23:48posts their work on archive. And they post
their work on archive typically before or

01:23:52possibly simultaneously with submitting their
work to a journal. So because of that, physics

01:24:00journals, journals like J-HEP or Classical
Quantum Gravity, et cetera, they don't even

01:24:03try and anonymize their manuscripts, because
they know if they anonymized it, you could

01:24:07just Google the first sentence and go find
the archive paper and see who posted it.

01:24:10So yes, I think double-blind peer review,
et cetera, made sense in a particular era

01:24:17to eliminate exactly the kinds of biases that
you're characterizing and other ones. But

01:24:23for math and physics, where the workflow is,
you put your paper on archive and then maybe

01:24:26a couple of weeks later you submit it to a
journal, it doesn't make sense at all. And

01:24:29so people don't even try.
So about the journal's inflated prices, outside

01:24:34of an oligarchy or collusion, what's keeping
it high?

01:24:42I mean, I'm, I'm reticent to claim that it's
a collusion. I mean, so, you know, a lot of

01:24:52it is just that a lot of it is tied into the
promotion structure in academia, right? So

01:24:58a lot of it is tied into, if you want to get
a permanent job in academia, if you want

01:25:03to advance up that, that ladder, you need
to get, you know, there's this general view

01:25:06that you need to get published in the fancy
journals. And then that means that the journals

01:25:10that are generally perceived by university
administrators as being the fancy ones know

01:25:14that they can charge essentially arbitrarily
high prices and people will pay them because

01:25:17they kind of, because you know, their livelihoods
depend on it, right?

01:25:22It's a really quite sordid situation when
you think about it.

01:25:26I saw a talk recently by someone who was going
into the academic world saying that

01:25:30some of the applications for professorship
or postdocship, that the second question after

01:25:35what is your name is how many citations do
you have? And then people try to game this

01:25:39because you can publish something that is
just worthy of publication and do that many

01:25:44times rather than produce something that you
feel like it's of high quality, but we'll

01:25:48get less citations than if you were to split
that up and then you just flood the market.

01:25:53Yeah, absolutely. And you know, there are
these metrics, there's author level metrics

01:25:58like the H index and so on, which measure,
you know, so H index equals N means that you

01:26:03have N papers that have been cited at least
N times. And that gets used actually quite

01:26:06frequently in hiring committees and tenure
committees and things like that. And yeah,

01:26:10it's incredibly easy to game, right? It's
this classic Goodhart's law example where,

01:26:14you know, as soon as you know that you're
being measured on that criterion, you can

01:26:17then say, oh, I'm going to just cite myself
in all, you know, every future paper I'm going

01:26:22to write, I'm going to cite myself in all
previous ones. And then I can very easily

01:26:25get some kind of N squared dependence on my
H index and then I can get my friends to cite

01:26:29me too. And I can, as you say, rather than,
you know, rather than investing a year to

01:26:34write this one really good polished definitive
paper on this subject, I'm going to write

01:26:3910 like salami sliced mini, you know, minimum
publishable unit things.

01:26:43Yeah, yeah, right. That's a great way of saying
it.

01:26:46Right. And yeah, and all of that happens,
right? And it requires, and I know I'm guilty

01:26:51of some of that too, you know, not because
I want to be, but because, you know, I need,

01:26:55you know, I, I live in the academic system
and that's kind of how one has to operate

01:26:58to a certain extent. If you're competing with
other people who are doing that, it's, it's

01:27:01awful. Right. And I don't, I don't want to
be in that situation. And you know, I, yeah,

01:27:06if obviously if given the choice, I always
try to be someone who, yeah, if I'm going

01:27:10to invest the time to write a paper on something,
I want to write in as much as possible, the

01:27:15definitive paper on that thing and have it
clean and polished and, and something that

01:27:18I'm proud of. But yeah, it's, I think it's
my impression at least is that it's becoming

01:27:23increasingly hard for that to be a viable
career strategy.

01:27:25Yeah. What's fortunate in your case is that
you were employed by Wolfram for some time.

01:27:30And so you were able to work on the ideas
that were interesting to you and not have

01:27:34to concern yourself. Maybe I'm incorrect,
but at least from my perspective, you didn't

01:27:37have to concern yourself with incremental
publications on ideas that aren't innovative

01:27:42in order for you to build the credit to your
name, but maybe I'm incorrect.

01:27:47Well, I mean, there was certainly an element
of that, right? So during the time I was employed

01:27:53at Wolfram, I, I, you know, I also was, I
mean, initially I was a graduate student or

01:27:58actually very early stages. I was an undergraduate,
then I was a graduate student, and then I

01:28:01was a kind of junior academic. So I still
had some academic position during that time.

01:28:07And for that reason, it wasn't something I
could completely ignore, right? I think because,

01:28:11you know, that would have been kind of irresponsible
from a career standpoint, but yes, in a way

01:28:14it did take the pressure off because it meant
that I, it meant that I had a kind of more

01:28:18or less guaranteed funding source for at least
part of my research. And I wasn't having to

01:28:23repeatedly kind of beg, you know, government
funding agencies for more money and things

01:28:27and show them long lists of papers. It was
also useful in a different way, which is that

01:28:32it meant that the stuff I was doing got much
more exposure than it would have done otherwise.

01:28:36I mean, like, you know, we wouldn't have met,
you know, if, if it hadn't been for Steven,

01:28:40right. And, and, and the, the kind of the
additional, both the additional cache and

01:28:44the additional flack that is associated with,
you know, with having his name attached to

01:28:49the project. And so, yeah, in a way it meant
that there was for, you know, for my level

01:28:54in the, in the academic hierarchy, my work
ended up being significantly overexposed and

01:28:59yeah, that was good in a way, it was bad in
another way. It, why would it be bad? Well,

01:29:05it meant that in a, okay. So, you know, so
one negative aspect of it, which has not been

01:29:12hugely problematic, but is, you know, Steven
has a certain reputation, right. And that

01:29:17reputation is positive in many ways and negative
in many other ways. And by, you know, if you

01:29:23are billed as, you know, you are the person
where you are the other person or one of the

01:29:27other people working on the Wolfram Physics
Project, you get, there's a, there's a sense

01:29:30in which you're elevated by association and
you get tainted by association. And people

01:29:34assume that, you know, yeah. People assume
that, that many of the negative characteristics

01:29:39associated with, you know, I don't know, not
giving appropriate credits to, to prior sources

01:29:44or, you know, having slightly inflated ego
issues, et cetera, right. Many of those things

01:29:49kind of get projected on you rightly or wrongly,
but yeah, by virtue of association.

01:29:53Yeah. Or that you're supporting that. So maybe
you don't have those qualities.

01:29:56Okay.
Right. Right. And, and it's, it's a, yeah,

01:29:59it's a difficult thing to, to throw. I mean,
in a way it helped because it meant that a

01:30:05lot of the criticism of the project got leveled
at

01:30:08Steven, not at the rest of us. Right. So in
a way it was useful, but yeah, but in other

01:30:13senses,
you know, it was a, yeah, it's a delicate

01:30:15balance.
So how do you see academics engagement with

01:30:17the ideas from the Wolfram Physics Project?
Yeah, it's been mixed, very mixed. So on the

01:30:25kind of traditional fundamental physics people,
it's mostly been, you know, ignored. Right.

01:30:32So like if you look at your average string
theorist,

01:30:35many of them will have, or you talk to them,
many of them will have heard of the project

01:30:38and will
say, oh, that's that weird kooky thing that

01:30:40that guy did. And we don't really know anything
about

01:30:42it. Right. That's at least that's the general
response that I've seen.

01:30:45They'll say they scrolled through the blog
posts, but then didn't find anything

01:30:48readily applicable to their field. And so
they're just waiting for it to produce results.

01:30:52That's
the general state. Right. Exactly. And then

01:30:55they'll say the necessary, well, I wish him
luck,

01:30:58but firstly, I don't think they actually mean
that. Secondly, if they do, they only mean

01:31:02that
because they're not competing for the same

01:31:04dollars. Yes. And I've certainly had conversations
with people who are not quite so polite, but

01:31:10yes. So there's that crowd. There are some
people in

01:31:16the quantum gravity community who have actually
taken some interest and have started, you

01:31:19know,
have cited our work and have used it and it's

01:31:22been incorporated in other things. So causal
set

01:31:23theory is one example of a, that's again,
a slightly unconventional branch to quantum

01:31:30gravity
that's really quite formalistically similar

01:31:33in a way. Causal sets are really just, you
know,

01:31:36they're partially ordered sets. They're really
the same as causal graphs in some sense. And

01:31:39so
there's a precise sense in which you can say

01:31:42that the, you know, that the hypergraphic
writing

01:31:44formalism is just giving you a dynamics for
causal set theory, which causal set theory

01:31:48does not
possess a priori because it's essentially

01:31:50a kinematic theory. And so in those communities,
it's been somewhat more receptive. There's

01:31:56been, again, there are in areas, this is essentially
unsurprising, right? So in areas where there

01:32:02is formalistic similarity, like say loop quantum
gravity, where there's some similarity in

01:32:06the setup of things like spin networks and
spin

01:32:08foams, there's been some interest in these
kinds of topological quantum field theory

01:32:12models or
topological quantum computing models, where

01:32:14again, there's this interest in this intersection
between, you know, combinatorial structure,

01:32:19topology, et cetera, and fundamental physics.
There's been some interest. An area where

01:32:23we've got a lot of interest is in applied
category

01:32:25theory. So, you know, people who, I would
say that's been, at least in terms of the

01:32:31stuff that
I've worked on, that's been by far our kind

01:32:34of most warm reception are people working
on

01:32:37categorical quantum mechanics and particularly
these kinds of diagrammatic graph rewriting

01:32:41approaches to quantum mechanics like ZX calculus
and so on. We've had some very,

01:32:45very productive interactions with that crowd.
And also with people not directly on the physics

01:32:50side,
but interested in the formalism for other

01:32:52reasons. So there are people like
the algebraic graph rewriting crowd, many

01:32:57of whom are in areas like Paris and Scotland,
who again, you know, have been very interested

01:33:02in what we've been doing. Not necessarily
for

01:33:04physics reasons, but because they're interested
in the algebraic structure of how we're setting

01:33:08things up or they're interested in how the
formalism can be applied to other things like

01:33:12chemical reaction networks or, you know, distributed
computing and that kind of stuff.

01:33:16Aaron Powell You're currently at Princeton,
correct?

01:33:18Tom Clougherty
Right.

01:33:19Aaron Powell Okay. So what do you do day to
day?

01:33:22Tom Clougherty
So mostly I work on computational physics.

01:33:26So I work on, you know, developing algorithms
and

01:33:32things for understanding physical phenomena
through computational means, which is more

01:33:36or
less a direct extension of, you know, of the

01:33:39stuff that I was doing at Wolfram Research.
But

01:33:41yeah, I'm in a sense, having been associated
with the physics project and with Wolfram

01:33:47Research for
some time, I now consider in part my role

01:33:51to be trying to get some of those ideas more
deeply

01:33:53embedded in sort of traditional scientific
and academic circles. And, you know, not so

01:33:59much tied
to, yeah, as you were putting it earlier,

01:34:02you know, Stephen's own personal research
dollars

01:34:04and that kind of thing.
How do you feel when the popular press almost

01:34:08invariably ascribes all,
if not the majority, of the credit of the

01:34:12Wolfram Physics Project to Wolfram himself?
Tom Clougherty

01:34:16Yeah, it's difficult, right? So as I say,
in a way, there is a positive aspect to that,

01:34:24which is that it means that, you know...
Aaron Powell

01:34:28You're shielded from direct criticism.
Tom Clougherty

01:34:31Right, right. Less likely to be blamed. But
no, I mean, yeah, it's emotionally difficult,

01:34:37right? I think, I don't know, maybe not for
everyone, but certainly for me, I find it

01:34:42quite
psychologically tough if, you know, if there's

01:34:46an idea that I've had that I'm reasonably
proud of,

01:34:48or a result that I've proved that I'm reasonably
proud of, etc., it's not the best feeling

01:34:52to see,
you know, headlines and Twitter threads and

01:34:55whatever, where it's all being ascribed to
one

01:34:57person. And in my small way, I try to push
back against that. But sorry, go on.

01:35:03Aaron Powell
I love Wolfram. I love Stephen. But so this

01:35:07goes without saying, he doesn't do
many favors in that regard. So when someone

01:35:14gives him the accolade, it's rare that I'll
see him say, oh, and by the way, that result

01:35:19was from Jonathan Garrard.
Tom Clougherty

01:35:22Right, right. And again, I guess we're all
guilty of that to a certain extent, right?

01:35:27I mean,
I'm acutely aware that in the course of this

01:35:30conversation, I haven't mentioned,
for instance, Manoganir Namaduri, who is the

01:35:32person who I kind of did a lot of this work
on categorical quantum mechanics with, right?

01:35:36And who deserves, again, a reasonable fraction
of the credit for that insight. So I'm guilty

01:35:41of this too, and I guess everyone is to an
extent.

01:35:47Stephen, maybe more than many people, but
it's a feature of this personality that

01:35:55I can't claim to have been ignorant of.
Aaron Powell

01:35:57Sure, sure. So he has another claim, which
is that he solved the second law of thermodynamics.

01:36:03And from my reading of it, I wasn't able to
see what the problem was with the second law

01:36:09and how it was solved, other than you say
you derive it from statistical mechanics,

01:36:15which was there before. I must be missing
something because I don't imagine Stephen

01:36:20would make that claim without there being
something more to it. So please enlighten

01:36:25me.
Tom Clougherty

01:36:26Yeah, okay. So I think, as with many of these
things, that series of three blog posts about

01:36:36the second law, I think there was interesting,
just like with NKS, right? I think there was

01:36:41a lot of interesting stuff there.
After they got figured out, it wasn't quite

01:36:45as grandiose as I think Stephen made it out
to be. But again, that's the responsibility

01:36:50of any scientist, right? It's to slightly
inflate the significance of what they're doing.

01:36:54So my reading of it is as follows. So there's
a kind of standard textbook, popular science

01:37:03type way that entropy increase gets explained,
which is you say, if you define entropy as

01:37:10being the number of microstates consistent
with a given macrostate or the logarithm of

01:37:15that, which is Boltzmann's equation, then
the fact that entropy has to increase is kind

01:37:19of obvious in some sense because the number
of ordered microstates or the number of microstates

01:37:27consistent with an ordered macrostate is always
going to be smaller than the number of microstates

01:37:31consistent with a disordered macrostate. And
so if you're just ergotically sampling in

01:37:38your space of states, you're going to tend
towards ones which are less orderly and not

01:37:42towards ones that are more orderly. And that
argument or that explanation seems convincing

01:37:48for a few seconds until you really start to
think about it and you realize that it can't

01:37:52possibly make sense. And one reason, I mean
a very foundational reason why it can't possibly

01:37:57make sense is because that explanation is
time symmetric. So if it's the case that you're

01:38:03ergotically sampling your space of
possible states, and yes, okay, the less ordered

01:38:08ones are always going to be more numerous
than the more ordered ones, then yes, it's

01:38:11true that evolving forwards in time, you're
going to tend towards the less ordered ones.

01:38:16But it's also true that if you're evolving
backwards in time, you would tend towards

01:38:19the less ordered ones. But of course, that's
not what we observe in thermodynamic systems.

01:38:23So that explanation can't be right, or at
the very least, that can't be the complete

01:38:27answer. And so I think the conceptual problem
is a real one. I think it is true that we

01:38:37really don't fully understand the second law
of thermodynamics from a statistical mechanical

01:38:40point of view. And as soon as you start trying
to apply it to more general kinds of systems,

01:38:45the problems become worse. I mean, there's
a famous example that was brought up by Penrose

01:38:51of what happens when you try and apply the
second law of thermodynamics to the early

01:38:56universe. And again, you seemingly get these
two contradictory answers. So as the universe

01:39:00evolves forwards, if we believe the second
law, as we get further and further away from

01:39:06the initial singularity, entropy should be
getting higher and higher. And yet, when you

01:39:12look back close to the initial singularity
and you look at the cosmic microwave background

01:39:15and so on, it looks very, very smooth. It
looks basically Maxwellian, like a Boltzmann

01:39:20distribution. It looks more or less like a
maximum entropy state. So we have this bizarre

01:39:26situation where as you move away from the
Big Bang, entropy gets higher. But as you

01:39:30go towards the Big Bang, entropy gets higher.
So something must be wrong. And Penrose has

01:39:35these arguments about conformal cyclic cosmology
and how the role of gravitational fields is

01:39:40essentially to decrease global entropy and
all that kind of stuff. But that's all, again,

01:39:44fairly speculative. And I would say at some
deep level, that's still a story we don't

01:39:47really understand. So that, I think, is the
problem that's being solved. And that series

01:39:55of blog posts proposes... And again, this
is not really that... I mean, even in NKS,

01:40:02there were indications of this idea. But yeah,
the basic idea is that you can explain the

01:40:09time asymmetry in terms of computational irreducibility.
Where you say, okay, so even if you have a

01:40:15system whose dynamics are exactly reversible,
in practice, because of computational irreducibility

01:40:21effects, the system can become pragmatically
arbitrarily hard to reverse. And that you

01:40:26can think about it essentially as being a
kind of cryptanalysis problem, right? So in

01:40:30a sense, the dynamics of a computationally
irreducible system are progressively encrypting

01:40:35certain microscopic details of the initial
condition. So that in practice, even if it

01:40:39is in principle possible to reverse from a
computability standpoint, if you try and think

01:40:43about the computational complexity of
that operation, it's equivalent to solving

01:40:46some arbitrarily difficult cryptanalysis problem
to work out, okay, where exactly was that

01:40:51gas molecule at time t equals zero? And that
goes some way towards explaining this time

01:40:55asymmetry problem. I don't think it's a complete
explanation. I think there's a yet deeper

01:41:01mystery there, but I do think it's an interesting
collection of ideas.

01:41:04Yeah, so that's observer-dependent. So it
would be difficult for you. Sorry, not difficult

01:41:09for anyone, but difficult for an observer.
But for the system itself, would there still

01:41:16be that issue of having to decrypt for the
system itself?

01:41:19Well, no, I would argue not. Because, yeah,
it's a very important point, right, that these

01:41:25notions are all observer-dependent. Because
in a sense, the Boltzmann equation requires

01:41:31the existence of a macro state, right? And
the macro state is an observer. It's a synthetic

01:41:38kind of observer theoretic idea, right? It's
like you've got a bunch of molecules bouncing

01:41:42around in a box, and so they have some micro
state details. But then you want to describe

01:41:48that box in terms of gas kinematics. You want
to describe it in terms of a density, and

01:41:51a pressure, and a temperature, and whatever.
So those give you your macro states. But the

01:41:56choice to aggregate this particular collection
of micro states and say, these are all consistent

01:42:01with an ideal gas, with this temperature,
and this adiabatic index, whatever, that's

01:42:06an observer-dependent thing.
And so, yeah, that's another point that, again,

01:42:10I don't think is completely original, but
I think has not been adequately stressed until

01:42:15these blog posts, which is that different
definitions of an observer will yield different

01:42:20definitions of entropy. Different choices
of coarse grainings yield different definitions

01:42:23of entropy. And therefore, in that sense,
it's kind of unsurprising that, as von Neumann

01:42:31and Claude Shannon and people kind of pointed
out, that the term entropy is so poorly understood,

01:42:36and that there are so many different definitions
of it. There's entropy in quantum mechanics,

01:42:38there's entropy in thermodynamics, there's
entropy in stat mech, there's entropy in information

01:42:43theory. They're all similar vibes, but they're
formally different. You can have situations

01:42:49where one entropy measure is increasing, one
entropy measure is decreasing, and that becomes

01:42:52much more easy to understand when you realise
that they are all measures of entropy relative

01:42:57to different formalisations of what it means
to be an observer.

01:43:01And yeah, so with regards to the decryption
thing, yes, I would say there's an aspect

01:43:08of it that is fundamental, that is purely
a feature of the system. Even if you don't

01:43:15have any model of the observer and you're
just looking directly at the data structures,

01:43:19you can have the situation where the forward
computation is much more easy or much more

01:43:23difficult than the reverse computation. And
obviously those kind of one-way functions,

01:43:26those get used in things like cryptography,
right? And the existence of those is quite

01:43:31well studied in cryptanalysis. So those certainly
exist, and those can give you some form of

01:43:36time asymmetry.
But arguably, the version of time asymmetry

01:43:39that's relevant for physics is the observer-dependent
one. It's the one where you say, actually,

01:43:45for this particular aggregation of microstates
and this particular interpretation of that

01:43:49aggregation as this macrostate, this is the
computational complexity of the reversal operation.

01:43:54And that is an observer-dependent
thing.

01:43:56You mentioned Penrose, and I want to get to
some of your arguments. I don't know if you

01:43:59still have them, but I recall from a few years
ago, you mentioned that you have some issues

01:44:04with Penrose's non-computational mind argument.
So I want to get to that, but I want to say

01:44:10something in defense of Stephen, that people
don't realize what it's like when you're not

01:44:14in academia to one, get your ideas taken seriously
by academia, and then also what it's like

01:44:19in terms of funding. So people will say that,
yeah, sure, Stephen is rodomontade or self-triumphant,

01:44:26but you have to be that to the public because
that's your funding source. Whereas for the

01:44:31academics, they are that to the grant agencies,
to the people they're asking for money, you

01:44:35have to big yourself up. It's just that you
don't get to see that.

01:44:38Yeah, I know. I absolutely agree here.
Great, great. Now for Penrose, please outline

01:44:44what are your issues with, I think it's the
Penrose-Lucas argument. Although I don't know

01:44:47if Penrose and Lucas, I know Lucas had
an argument and it's called the Penrose-Lucas

01:44:52argument. I don't know their historical relationship.
Right, right. And yeah, there's an original

01:44:58argument that's purely using kind of mathematical
logic and Turing machines and things. And

01:45:02then there's the Penrose-Hameroff mechanism,
which is the proposed biochemical mechanism

01:45:06by which there exists this non-computability
in the brain. Yeah, I mean, so, okay, there's

01:45:13an, okay, how to phrase this. There's an element
of this, which I'm quite sympathetic to, which

01:45:19goes back actually to one of the very first
things we discussed, right? Which is the distinction

01:45:23between what is model versus what is reality.
Turing machines are a model. And so if you

01:45:30say, well, the mind is not a Turing machine.
I mean, if that's your only statement, then

01:45:36I agree, right? But then nothing, you know,
like the universe isn't a Turing machine in

01:45:39that sense, right? The question is, is
it useful to model the mind as a Turing machine,

01:45:43or is it useful to model the universe as a
Turing machine? And there, I think the answer

01:45:46is emphatically yes. And, you know, okay,
are you going to be able to model everything?

01:45:51Well, not necessarily. So again, to that extent,
I do have some sympathy with the Penrose-Lucas

01:45:57argument. I'm open to the possibility that
there may be aspects of cognition that are

01:46:04not amenable to analysis in terms of Turing
machines and lambda calculus and that kind

01:46:07of thing. I just don't think that the particular
examples that Penrose gives, for instance,

01:46:14in his book, Emperor's New Mind, are especially
convincing examples, right? I mean, so he

01:46:18has this argument that, you know, mathematics,
the process of apprehending mathematical truth...

01:46:25...must be, you know, a non-computable process,
because we know from GÃ¶del's theorems that,

01:46:30you know, for any given formal system, if
it's consistent, then there must be statements

01:46:36that are independent of that system, where
both the statement and its negation are consistent

01:46:41with the underlying axioms. But we, you know,
so GÃ¶del's original argument proved that

01:46:48for Peano arithmetic, for the standard axiom
system for arithmetic, and later on it was

01:46:52worked for any axiom system that's at least
as strong as Peano arithmetic.

01:46:57And so Penrose's argument, I mean, I'm caricaturing
a bit and it's a little unfair,

01:47:01but, you know, the basic argument is, well,
we can obviously see that arithmetic is consistent.

01:47:07So when we construct this GÃ¶del sentence
that says this statement is unprovable,

01:47:11we can see that it has to be true. And yet,
you know, within the formal axioms of arithmetic,

01:47:17as they are computable, it cannot be decided
in finite time that that statement is true.

01:47:22And, okay, so most of that is correct. But
the part where you say, well, we as human

01:47:29observers
can clearly see that that statement is true,

01:47:31well, that presupposes that we are able to,
you know, we are able to know the consistency

01:47:36of integer arithmetic, which we have strong
reason

01:47:38to believe is consistent. But GÃ¶del's second
incompleteness theorem says that, well, we

01:47:43can't
know that formally either. So in a sense,

01:47:46he's presupposing the conclusion. He's already
presupposing that we can know the truth value

01:47:52of an independent proposition, namely the
consistency

01:47:54of Peano arithmetic, in order to prove that
we can know the truth value of another independent

01:47:59proposition, namely this GÃ¶del sentence.
And so for me, it just feels extremely circular.

01:48:03So it doesn't...
C.S.: Sorry, can he not use, like, what if

01:48:06he didn't say that it's irrefutable,
rather that probably, so far, it seems like

01:48:12Peano arithmetic is consistent.
And if it was to explode, it'd be so odd that

01:48:17it hasn't exploded already.
And we've explored it quite extensively. Every

01:48:22day, we increase our credence in the
consistency of it. Can he not use an argument

01:48:26like that?
HB. He absolutely could, and that would be

01:48:29correct. But then the problem with that is,
there's nothing in that argument that a computer

01:48:34could not replicate, right?
A machine could also make that same argument.

01:48:38You could also write a computer program
that says, okay, I'm going to test loads of

01:48:41propositions in Peano arithmetic and
see whether I find an inconsistency. And the

01:48:46more propositions I test,
the less likely it is that Peano arithmetic

01:48:50is inconsistent. So I can construct â€“ this
is the machine speaking here â€“ I can construct

01:48:54some kind of Bayesian argument that says,
you know, I'm this level of confident that

01:48:58this proposition is true.
So yes, human beings can do that kind of Bayesian

01:49:02reasoning, but then so can a machine.
And so the crux of the Penrose argument, or

01:49:08the Penrose-Lucas argument, is that
there is this additional non-computable step

01:49:14where the human somehow knows â€“ not assumes,
but just knows â€“ that Peano arithmetic is

01:49:19consistent, and from that deduces that T has
to be true. And I don't see how you can justify

01:49:24that without essentially presupposing the
conclusion.

01:49:26CW. So what's the difference between intuitionist
logic and constructivist logic?

01:49:31Ah, okay, that's a fantastic question. And
it cycles back to the stuff we were talking

01:49:36about
at the beginning with regards to constructivist

01:49:38foundations for physics, right? So I would
say

01:49:42constructivism is really a kind of broad â€“ okay,
the simple answer is intuitionistic logic

01:49:46is a
special case of constructivist logic. So constructivism

01:49:50is a broad philosophical movement
where the idea is â€“ so okay, for people

01:49:55who don't know the history of this â€“ so
in the aftermath of GÃ¶del's incompleteness

01:49:59theorems, and Tarski's undefinability theorem,
and Turing's proof of the undecidability of

01:50:03the halting problem, and all these
limitative results in mathematical logic that

01:50:06happened in the early 20th century,
people started saying, okay, well, how can

01:50:10we trust that anything is true in mathematics,
right? So if we always have to make some unprovable

01:50:15assumption about the consistency
of our axiom system, how can we ever be confident

01:50:19of anything beyond just the kind of heuristic
argument that we made before? And so then

01:50:24various people, especially a guy called Brouwer,
and later in his later years, David Hilbert,

01:50:30coddled on to the idea that, okay, what you
could

01:50:32do is you could say, well, if we strengthen
our criterion for mathematical proof, if we

01:50:40say that
when you reason about a mathematical object,

01:50:43it's not enough just to reason about it abstractly.
You actually have to give an algorithm a finite,

01:50:48deterministic procedure that constructs that
object before your statements can even make

01:50:54sense. That's a much stronger condition,
and it immediately rules out certain forms

01:50:58of mathematical proof. So for instance,
a proof by contradiction, it would not be

01:51:01allowed in such a paradigm because if you
prove a

01:51:05statement, okay, so obviously, suppose I want
to convince you that this equation has a solution.

01:51:13So one way I could convince you is to make
a proof by contradiction. I could say,

01:51:16assume it doesn't have a solution, and then
derive some piece of nonsense.

01:51:18Yes, yes, yes.
So then my assumption had to be wrong.

01:51:21Yes. You can prove existence without construction.
Right, right. But that only works if I assume

01:51:27that the axiom system I was using to prove
that

01:51:29is consistent, and that the inference rules
I was using to derive that contradiction were

01:51:34actually
sound. If they weren't, if it was an inconsistent

01:51:36axiom system or the inference rules were not
sound,

01:51:38then I could derive a contradiction even from
a statement that was true, and so it would

01:51:43be
invalid. And of course, we know from GÃ¶del's

01:51:47theorems and from Turing's work that we cannot,
for any non-trivial formal system, know conclusively

01:51:52that the system is consistent
or that the inference rules are sound. Whereas

01:51:57instead, if I try and convince you by saying,
look, here's a program, here's an actual algorithm

01:52:01that constructs a solution for you,
and you can just go and check whether it solves

01:52:05the equation, somehow that's much more convincing.
You don't have to assume anything except that

01:52:10maybe the validity of the model of computation
can check that too, right? So you're placing

01:52:16a much lower epistemological burden on the
underlying axioms of mathematics. You can

01:52:24use those to guide you in how you search for
things,

01:52:26but ultimately, the ultimate criterion, the
ultimate test for truth is, can you define

01:52:33a deterministic algorithm that actually witnesses
the structure that you're talking about?

01:52:38And so this was intended to be a kind of almost
a get-out clause from these limitative results

01:52:43to
say, this is a way that we can kind of bypass

01:52:45many of these, not all of them, of course,
but many of these issues. Now, it's a very,

01:52:50very significant limitation because it immediately
means that there are very large classes of

01:52:54mathematical structures that you just can't
talk about at all, the structures where you

01:52:58can't avoid undecidability and independence.
But rather astonishingly, there are large

01:53:04parts of mathematics, including areas like
analysis,

01:53:07which you maybe wouldn't have thought would
be amenable to constructivism, where many

01:53:10of the
most interesting results, the Heine-Borel

01:53:13theorem or whatever, you can actually prove
using purely

01:53:16constructivist means. So that's really what
constructivism is about. Then intuitionism,

01:53:21which is a particular flavor of constructivism
that's due to Brouwer. So once you've decided

01:53:28that you want to work in constructivist mathematical
foundations, then you still have

01:53:32the problem of, okay, what are my underlying
rules going to be? How do I actually impose

01:53:37those constraints in a systematic way? And
so intuitionism is just one approach to doing

01:53:41that,
where you say, okay, I want to outlaw non-constructive

01:53:46proofs like proof by
contradiction. How do I do that? So one thing

01:53:52that should be outlawed is any use of double
negation. So the axiom of double negation,

01:53:57that not not x is equivalent to x. I shouldn't
be

01:53:59able to do that because that allows me to
do non-constructive proofs. And it turns out

01:54:03that
if you're going to outlaw that, you also need

01:54:04to outlaw what's called the law of excluded
middle,

01:54:06a statement that a or not a is true for any
proposition a.

01:54:09Sorry, you need to outlaw it or it's equivalent
to outlawing that?

01:54:15It's equivalent to it. So one necessitates
the other. And then in the kind of logical

01:54:23foundations, that's what you need to do. And
then that implies certain things like, say,

01:54:26the axiom of choice in set theory. The statement
that if you have some collection of non-empty

01:54:33sets
and you assemble a new set by choosing one

01:54:35element from each element of that collection,
that that set is necessarily non-empty. Something

01:54:40which is very intuitively obvious for finite
collections, but very not intuitively obvious

01:54:45for finite and countable collections, but
not

01:54:47intuitively obvious for uncountable collections
of sets.

01:54:50Is that the root of the word intuitionism?
Is it actually meant to say that this is more

01:54:54intuitively the case?
It's more... So my understanding is that it's

01:55:01more that these were meant to be the minimum
rules that somehow... Yeah, I mean, in a way,

01:55:08yes. These were meant to be kind of the minimum
conditions that matched human mathematical

01:55:13intuition.
Yeah, I don't know. I know there's a whole

01:55:16history of, like I mentioned, I want to do
a

01:55:18whole video on gripes with names. So it could
be something philosophical about Kant and

01:55:24intuition.
I have no clue. But do intuitionists not have

01:55:28a concept of infinity? Because you mentioned
Heine-Borel, and so it's not embedded in the

01:55:33infinitesimals?
Right, right.

01:55:35If you're saying you can do analysis, I don't
understand how that can be done.

01:55:40Yeah, okay. This is a really important point.
So I mentioned that intuitionism is just one

01:55:45flavor of constructivism, and there are many
others. And there are ones that are

01:55:49more or less strict. There's a stricter version
of constructivism called finitism,

01:55:56which is exactly that, where you say,
not only am I going to be constructivist,

01:56:02but my algorithms have to terminate in finite
time.

01:56:05So if you're an intuitionist,
and you don't subscribe to the kind of finitism

01:56:11idea, you might say, well, I can write down
an

01:56:13algorithm that solves this. There is a deterministic
procedure, but it may not necessarily terminate

01:56:18in
finite time. So an example of that would be

01:56:23the integers, right? So with the integers,
I can write down an algorithm which provably

01:56:28constructs the complete set of integers.
That algorithm doesn't terminate. If I were

01:56:32to run it on a finite machine, it wouldn't
terminate.

01:56:35But any given integer can eventually be derived
by just repeatedly applying that procedure.

01:56:41So there is actually a way, subject to this
kind of weaker version of intuitionism,

01:56:46there is a way that you can reason about infinite
mathematical structures. But if you then say,

01:56:51oh no, I'm not going to allow myself to do
that. I want all the deterministic procedures

01:56:56that I
write down have to be constrained so that

01:56:59they always terminate in finite time. Then
you become

01:57:02a finitist. And then there's variants of that,
like ultra-finitism, which I think is quite

01:57:06fun,
where one effectively believes that there

01:57:10is a largest number and that that number is
decreasing

01:57:14over time because of essentially physical
constraints. Yeah, I like it. I don't believe

01:57:19in it, but I like it. Well, again, it's this
question of what do you mean by belief, right?

01:57:26I mean, if mathematics is intended to be a
kind of toolset for modelling certain processes

01:57:32of
thought, then there are certain kinds of problems

01:57:37where I think it's useful to take a
finitist or ultra-finitist mindset. Yeah,

01:57:41I agree. If you're a mathematical Platonist,
which I'm not,

01:57:45then you might say, okay, well, I believe
that the mathematical universe is much larger

01:57:48than
in some ontological sense than the universe

01:57:51that's conceived by ultra-finitists. But
you know, I at least am a pragmatist, and

01:57:56I say, well, you know, I'm going to use whatever
version of mathematics I think makes sense

01:57:59for this particular problem.
So what do you believe to be the primary issue

01:58:04between combining, well, the primary
difficulty? What do you believe to be the

01:58:08primary difficulty between combining general
relativity and quantum mechanics? Right, so

01:58:17that's been formulated in many ways.
So having just sort of slightly slated Penrose

01:58:22for his consciousness views, let me
try and right that wrong a little bit by saying

01:58:29I think Penrose has a really, really nice
argument

01:58:31for why, even just at a conceptual level,
quantum mechanics and general relativity are

01:58:37incompatible,
which is the following. That if you take two

01:58:41of the most foundational principles,
which in a sense delineate how quantum mechanics

01:58:51is different from classical mechanics and
how

01:58:53general relativity is different from classical
mechanics, those would be the superposition

01:58:57principle in quantum mechanics. The principle
that if you have a system that can be in this

01:59:01eigenstate or this eigenstate, it can also
be in some complex linear combination of them.

01:59:04And on the side of the Einstein equations
of general relativity, it's the principle

01:59:09of
equivalence, right? It's the principle that

01:59:10accelerating reference frames and gravitational
reference frames are really the same, or to

01:59:14translate that into slightly more mathematical
terms, that anything that appears on the left-hand

01:59:19side of the field equations in the Einstein
tensor

01:59:21you can move as a negative contribution to
the right-hand side in the stress energy tensor.

01:59:26So Penrose has this really nice argument for
why those two principles are logically

01:59:31inconsistent. And the argument goes like this.
So suppose that you've got something like

01:59:38a
Schrodinger cat-type experiment, where you've

01:59:40got, I don't know, you have like a robotic
arm

01:59:42that contains a mass at the end that's producing
a gravitational field. And it's connected

01:59:46up to,
I don't know, radioactive nucleus that has

01:59:49some probability of decaying.
So that arm can be in one of two positions.

01:59:52It can be position A, position B. And the
position

01:59:56that it's in depends on the quantum state
of that nucleus. So now, just naively, what

02:00:00you appear to
have done is created a superposition of two

02:00:02different gravitational field configurations.
Okay. So if you do that, you can write down

02:00:08the wave function that corresponds to that
superposition

02:00:10and everything looks just fine. So far, there's
no problem. But then if you believe the equivalence

02:00:16principle, then you should get the same wave
function if you then do the same calculation

02:00:21in an accelerating frame. So if you take that
whole desktop apparatus, and rather than doing

02:00:24it here on the Earth, you do it in a spaceship
that's accelerating at 9.81 meters per second

02:00:29square, and you have exactly the same experimental
setup with the same robotic arm, you should

02:00:34get
the same wave function. But if you calculate

02:00:37it, which again is just a standard calculation
in

02:00:39relativistic quantum mechanics, you get almost
the same answer. The two wave functions differ

02:00:44by a phase factor, which normally wouldn't
be too much of a problem. Normally, if they

02:00:48differ by a
phase factor, you say that they're somehow

02:00:50the same quantum system. But the phase factor
depends

02:00:53on time to the power four. And because of
some slightly technical reasons that have

02:01:00to do with
the fact that quadratics have two solutions,

02:01:03if you have a phase factor that depends on
time to

02:01:04the power four, that's telling you that the
wave function you've written down corresponds

02:01:07to a
superposition of two different vacuum states.

02:01:11And one of the core axioms of quantum mechanics
is

02:01:13that you can't superpose two different vacuum
states for the very simple reason that the

02:01:17vacuum
state is the kind of zero point from which

02:01:19you measure energies using your Hamiltonian.
So if

02:01:22you have a superposition of two different
vacuum states, there's no longer a uniquely

02:01:26defined
Hamiltonian. There's no longer a uniquely

02:01:28defined energy because there's no rule for
how you

02:01:31superpose those vacua. So it is inherently
illegal in quantum mechanics to produce those

02:01:36superpositions. So somehow by just assuming
that you could superpose gravitational fields,

02:01:40you've been able to use the equivalence principle
to violate the superposition principle or

02:01:45equivalently vice versa. There's a more mathematical
way of seeing the same thing,

02:01:50which is to say that at a very basic level,
quantum mechanics is linear and has to be

02:01:55linear
by the Schrodinger equation. The Schrodinger

02:01:58equation has to be linear because of the
superposition principle. So if I have two

02:02:01solutions to the Schrodinger equation, then
a

02:02:04complex linear combination of those states
with appropriate normalization has to also

02:02:08be a valid
solution to the Schrodinger equation. General

02:02:11relativity is non-linear and has to be non-linear
because in a sense, if you take the Einstein

02:02:18field equations and you linearize them, you
linearize the gravitational interaction, then

02:02:22what you get is a version of general relativity
that

02:02:25doesn't possess gravitational self-energy.
So in other words, the reason why general

02:02:30relativity
is a non-linear theory is because in Newtonian

02:02:33gravity, if I have a mass, that mass produces
a

02:02:36gravitational potential, but the gravitational
potential doesn't produce a gravitational

02:02:41potential. But in general relativity, because
of the mass-energy equivalence, I have a mass

02:02:45that
produces a gravitational potential, but that

02:02:46gravitational potential has some energy associated
to it. So it also produces a gravitational

02:02:51field, and that produces another gravitational
field,

02:02:52and so on. So there's actually a whole infinite
set of these smaller gravitational fields

02:02:57that
are being produced. So this is often summarized

02:02:59by the slogan that gravity gravitates.
And that appears as a non-linear contribution

02:03:06to the Einstein field equations,
these off-diagonal terms that appear in the

02:03:09Einstein tensor. And so it has to be non-linear
because if you were to take two solutions

02:03:15to the Einstein equations, two metrics, and
just

02:03:16try and add them together, you quite clearly
wouldn't get a third solution to the Einstein

02:03:21equations in general. Because what you've
done is you've added the gravitational potentials,

02:03:24which is really what the metric tensors are
indicating, but you haven't incorporated all

02:03:29these additional non-linear contributions
induced by the sum of the gravitational potentials

02:03:34themselves. So the basic problem is that you
can't superpose gravitational fields,

02:03:41and that's really what the Penrose argument
is indicating. That if I try and take two

02:03:44metric
tensors and just add them in a way that's

02:03:46consistent with the Schrodinger equation,
I'll violate the Einstein field equations.

02:03:49And if I try and take two solutions to the
Einstein field equations and combine them

02:03:53in a non-linear way that's compatible with
general

02:03:55relativity, I'll violate the linearity of
the Schrodinger equation. And at some level,

02:04:00that's the basic problem. The problem is that
the linearity of Schrodinger versus the non-linearity

02:04:04of Einstein means that superpositions of gravitational
fields cannot be described

02:04:08without violating at least one of those two
formalisms.

02:04:12Does the conceptual difficulty still persist
in quantizing linearized general relativity?

02:04:19So my understanding is that you can certainly
get further with quantizing linearized.

02:04:26So if you just linearize your gravitational
interaction, you can not only evolve quantum

02:04:33fields on top of a curved space-time described
in terms of linearized gravity, which you

02:04:38can
do for Einstein gravity, but you can also

02:04:40describe the back reaction of the quantum
fields onto

02:04:45the metric tensor. I actually don't know how
much further than that you can go.

02:04:49But what I do know is that it's definitely
a lot easier. You can make much more rapid

02:04:53progress
with quantizing gravity if you assume linearizations

02:04:56than if you don't. I think there
are still some problems that persist, but

02:04:59I think they're nowhere near as difficult.
So how is it that higher category theory overcomes

02:05:04this?
That's a great question. The basic answer

02:05:11is I don't know, but there's a very tempting
kind of hypothesis. I mentioned towards the

02:05:18beginning that there are these category theoretic
models for quantum mechanics, and also I think

02:05:22I even mentioned briefly that there are these
models for quantum field theory as well. The

02:05:26way that that works is, so we talked at the
start

02:05:28about these dagger-symmetric compact closed
monoidal categories, which are the basic

02:05:34mathematical setup for categorical quantum
mechanics. The problem with that, though,

02:05:38is that every time you apply one of these
morphisms, every time you apply one of these

02:05:41time evolution operators, you are essentially
picking out a preferred direction of time,

02:05:46right? You are assuming you've got
you know, if you imagine each of your quantum

02:05:50states, each of your spaces of states is a
space

02:05:52of states on a particular space like hypersurface.
Once you construct a unitary evolution operator

02:05:57that's a solution to the Schrodinger equation,
you are selecting a preferred direction of

02:06:01time,
which is of course not relativistic, that's

02:06:04not covariant. So to go from the non-relativistic
version of quantum mechanics to a version

02:06:09that's compatible at least with Lorentz symmetry,
you need to have some systematic way of transforming

02:06:14one time direction to another.
Well, if you think about it in the category

02:06:18theoretic perspective, through the category
theoretic lens, there's a systematic way to

02:06:23do that, which is through higher categories.
So if you consider categories which have,

02:06:28you know, objects and morphisms, you can also
consider

02:06:30two categories that have two morphisms between
those morphisms that allow you to transform

02:06:34morphisms to each other, not just objects
to each other. And so if you take the two

02:06:40category
version of the one category picture of categorical

02:06:43quantum mechanics,
you can allow the two categories to correspond

02:06:47to gauge transformations between your evolution
operators. So you're transforming the direction

02:06:51of time in a way that's consistent with, say,
with the generators of the Lorentz group.

02:06:56And so what you get in some appropriate special
case

02:06:59is what's called a functorial quantum field
theory. So Baez and Dolan constructed this

02:07:05axiomatization of functorial and particularly
topological quantum field theories based on

02:07:10what's called the Atiyah-Segal axiomatization
that use these two categories and indeed even

02:07:15higher categories as a way of formalizing
this notion of gauge transformations, of being

02:07:18able
to transform between time directions. Okay,

02:07:22so that's a nice piece of mathematics.
And in my opinion, is one of the more promising

02:07:29avenues towards constructing a kind of
mathematically rigorous foundation for quantum

02:07:32field theory. What does it have to do with
quantum gravity? Well, this is where it necessarily

02:07:38becomes very speculative.
But so there's an idea that goes back to Alexander

02:07:42Grothendieck, who I mentioned,
this amazing algebraic geometer from the early

02:07:4720th century who really developed a whole
bunch

02:07:49of these ideas in higher category theory while
he was sort of living as basically a hermit

02:07:54in
the Pyrenees, I think. But so Grothendieck

02:07:59made this hypothesis that's now called Grothendieck's
hypothesis or the homotopy hypothesis, which

02:08:04goes as follows. Okay, let me motivate it
like this.

02:08:06So if I have a topological space, it has some
collection of points and it has paths that

02:08:13connect those points. But I can also have
paths that connect the paths and those are

02:08:19called
homotopies, right? So I can continuously deform

02:08:21one path into another and I can use that information
to tell me stuff about the topology of the

02:08:26space. So you can use the homotopy information
to tell

02:08:28you about the homology, right? You can find
that if you're in a donut, you can see that

02:08:33there's a
hole there because if you have a loop, a path

02:08:35that loops around that hole, you can't
continuously contract it to a point without

02:08:40encountering some discontinuity. So those
homotopies you can formalize as kind of higher

02:08:47order paths between paths. So in the language
of category theory, you could say my initial

02:08:52topological space is a one category that has
points and paths between the objects and morphisms.

02:08:58The first homotopy type is the two
categories I construct from that, where the

02:09:02two morphisms are the homotopies between those
paths.

02:09:04But then I can also consider homotopies between
homotopies and so on. So I can construct this

02:09:08whole hierarchy of higher categories and higher
homotopy types. Then that terminates at this

02:09:14infinity category level, which is that the
hierarchy has some natural endpoint.

02:09:22And somehow we know that from various results
in higher category theory that

02:09:30all the information that you care about up
to weak homotopy equivalents, about not just

02:09:34the
space you started from, but all of the intermediate

02:09:36spaces that were in that hierarchy, all of
that

02:09:39information is somehow contained in the algebraic
structure of that infinity category. So the

02:09:43infinity category determines up to weak homotopy
equivalents everything that comes in the hierarchy

02:09:48below it. And that's why kind of infinity
category theory is so different to even just

02:09:52normal
finite higher category theory. Infinity categories

02:09:54somehow contain far more information. There's
actually a specific type of infinity category

02:09:59called an infinity groupoid because the paths
are invertible. And Grotendieck was really

02:10:06one of the first people who encouraged topologists
to stop thinking about fundamental groups

02:10:11and start thinking about fundamental groupoids
without needing to define distinguished base

02:10:16points and things like that.
But the homotopy hypothesis is this really

02:10:20deep statement that kind of goes in the other
direction.

02:10:23So we know that starting from a space and
doing this hierarchical construction, you

02:10:29build up to
this infinity category that tells you up to

02:10:32weak homotopy equivalents, all the topological
information about that space and all of its

02:10:35homotopy types. Grotendieck then said, well,
maybe that's really the definition of a topological

02:10:42space, that infinity categories are
just spaces. Infinity groupoids are spaces,

02:10:47or at least they define the structure of a
space and all

02:10:50of its homotopy types up to weak homotopy
equivalents. So it's kind of a converse

02:10:53direction of that statement. And that's the
homotopy hypothesis. It's not proven. It's

02:10:58not
even precisely formulated, but it's a very

02:11:00interesting idea that I think is largely
believed to be correct. It aligns well with

02:11:05our intuitions for how algebraic topology
should work.

02:11:08So therefore, attempting speculation about
the relationship between that and physicsâ€¦

02:11:14So going
back to the quantum field theory picture for

02:11:17a moment. So suppose you don't just stop at
two

02:11:19categories, or indeed three categories, but
you keep going, right? You keep adding these

02:11:23higher
gauge transformations. So not just gauge transformations

02:11:26that deform time direction
to time direction, but higher gauge transformations

02:11:30that deform gauge transformation to gauge
transformation. You build up a higher homotopy

02:11:34type that way. What happens when you get to
the

02:11:36infinity category limit? Well, so what you
end up with is something that has the structure

02:11:41of a
topological space. So starting from something

02:11:43that's completely non-spatial, you've ended
up

02:11:45with a topological space. And so in the spirit
of these kind of emergent space-time views,

02:11:53you know, like ER equals EPR and so on, one
hypothesis that's quite tempting to make is

02:11:58maybe that infinity category defines the structure
of our space-time, right? The topology and

02:12:04geometry of space-time emerges in that infinity
category limit that I take by just adding

02:12:08higher
and higher gauge transformations starting

02:12:10from categorical quantum mechanics. And so
if that's

02:12:14true, which again, to be clear, we have no
idea whether that's true or not, right? But

02:12:19if that
were true, then the coherence conditions,

02:12:21the conditions that define how the infinity
category

02:12:24relates to all of the lower categories in
that hierarchy, those coherence conditions

02:12:29would
essentially be an algebraic parameterization

02:12:31for possible quantum gravity models.
And so if that ended up being correct, that

02:12:36would be a really nice way to kind of
conceptualize and formalize the essential

02:12:40problem of quantum gravity, that we're really
trying to

02:12:42nail down the coherence conditions that relate
that infinity category to all the higher categories

02:12:48in
that hierarchy. Now what would it be like

02:12:50to study the topology? So there's something
called

02:12:53stone duality, I'm sure you're aware of, which
relates topology to syntax. So I've never

02:13:00heard
of someone studying stone duality at the infinity

02:13:03categorical level, at the topology that's
induced

02:13:05from that category. What does that look like?
Yeah, that's a really interesting question.

02:13:11So yes, the way that stone duality works isâ€¦
Again, as with many of these things,

02:13:20there's a nice categorical interpretation
in terms of Boolean topos and things. But

02:13:24the basic idea is that if you have a Boolean
algebra, a kind of minimal algebraic axiomatization

02:13:31for logic, there's a way that you can formalize
that in terms of this mathematical structure

02:13:35of
a lattice, specifically an orthomodular lattice,

02:13:38I think. I may be getting that wrong. I think
it's

02:13:41an orthomodular lattice. But so in which essentially
every point in that lattice is a proposition,

02:13:47and then you have these meet operations and
these join operations that become equivalent

02:13:50to your and and or operations in logic. And
the reason that's significant is because those

02:13:56same
class of lattices also appear in topology

02:13:59because there are specific spaces called stone
spaces that

02:14:03are essentially theâ€¦ So okay, sorry, let
me say that less confusingly. So if you take

02:14:09a topological
space and you look at it, it's open. Doesn't

02:14:12like topological spaces. No. Okay, let's try
that

02:14:15again. Okay. That's being kept in. We'll take
that part in. So wait, wait, is it angry at

02:14:24you?
No, it was angry at someone. There's a gate

02:14:27just outside, which sometimes opens and closes.
And

02:14:31this is my fiance's Dachshund, who is very,
very territorial. And he was up until now

02:14:36sleeping
very soundly and has just woken up. And so

02:14:39we may get some interruptions.
Well, congratulations on the engagement.

02:14:43Thank you. Thank you. Yes. Anyway, so what
was I saying? Yes. Okay. So if you take a

02:14:50topological
space, then you can look at its open set structure.

02:14:54So if you take the collection of all open
sets,

02:14:56you can look at, in particular, you can look
at the open set containment structure. You

02:15:00can look
at which open sets are included in which others.

02:15:04And when you do that, you again get the structure
and orthomodular lattice, because the lattice

02:15:08operations are essentially defined by the
inclusion relations between the open sets.

02:15:13And so there's this duality between topological
spaces

02:15:15and this class of lattices. So you could ask,
what are the particular topological spaces

02:15:21that
you get if you look for topological spaces

02:15:24whose open set lattices are the lattices that
you get

02:15:27from looking at Boolean algebras? And those
are the stone spaces.

02:15:30So they are the kind of topological spatial
interpretation of logic in some sense. And

02:15:36in a way, you could say topos theory is really
about trying to generalize that idea, right?

02:15:40That's another way to think about it. So every
elementary topos has an internal logic. And

02:15:48also every elementary topos has some kind
of spatial interpretation, because the axioms

02:15:53of elementary topos theory, this finite limit
axiom and this existence of power objects

02:15:58or subobject classifiers is really some generalization
of the axioms of point set topology, right?

02:16:04Because they're the topos theoretic analog
of saying that your open sets have to be closed

02:16:10and the collection of open sets has to be
closed under arbitrary unions and finite intersections

02:16:15and so on.
So topos have spatial interpretations, and

02:16:19they also have an internal logic. So there's
a particular kind of topos called a Boolean

02:16:24topos whose internal logic is Boolean algebra
and whose spatial interpretation is therefore

02:16:28a stone space. But actually, you can do the
same construction for any elementary topos

02:16:34that you like. And so then really what you're
asking is, okay, when you go to higher topos

02:16:38theory, if we take the higher category, which
turns out that infinity category that you

02:16:43get from the Grothendieck construction admits
a topos structure. So then you could ask,

02:16:47what is the internal logic to that? And what
is its relationship to its spatiality? And

02:16:53what you end up with is the spatial structure
of an infinity homotopy type in homotopy type

02:16:58theory. So in homotopy type theory, this
is another kind of logic interpretation of

02:17:05higher categories, where, my apologies, crying
somewhat. Hang on, wait. Okay, there we go.

02:17:13I'm slightly more restricted in my emotions
now. But if you imagine taking a proof system

02:17:20and you say, okay, so now I'm going to interpret
every proposition in that proof system as

02:17:25being a point in some space, and every proof
as being a path, right? So a proof just connects

02:17:29two propositions together. So I can prove
one proposition for another, or I could prove

02:17:34that two propositions are equivalent. I can
also prove that two proofs are equivalent,

02:17:37right? I can take two paths and I can continuously
deform them. But that proof exists in the

02:17:42next homotopy type, right? Because that's
interpreted topologically as a homotopy between

02:17:46those parts. And so you can do exactly the
same construction. And so in the infinity

02:17:51category limit, what you get is a logic which
allows not just for proofs of propositions,

02:17:57but proofs of equivalence between proofs,
and proofs of equivalence between those proofs,

02:18:01and so on, right? So that's the internal logic
of one of those higher topos. It's a logic

02:18:07that allows for proofs of equivalence between
proofs up to arbitrarily high order.

02:18:11JS So in theories of truth, there's one called
Tarski's theory of truth, where your truth

02:18:18can only speak about the level that's beneath
it. And then, right, and this is one of the

02:18:22ways of getting around the liar's paradox,
is that you say, well, it's truth level one,

02:18:27and then you're speaking about a truth level
two or falsity level two, etc. And then the

02:18:31criticism is, well, what happens Tarski when
you go all the way up to infinity? And I don't

02:18:36think he had an answer. But it's sounding
like there can be a metaphor here for some

02:18:41answer.
PW Yes, I mean, potentially. It's not something

02:18:47I've thought about a huge amount, but it's
certainly the case that in these kind of higher

02:18:51order logic constructions, there are things
that happen at the infinity level that don't

02:18:56happen at any finite level. And it's conceivable
that, yes, you might be able to do a kind

02:19:01of Tarski thing of evading the liar, or you
may be able to do some kind of Quine's paradox.

02:19:05I think the same thing happens with Quine's
paradox, right? You try and construct liar

02:19:15paradox type scenarios without self-reference,
where you say, you know, the next sentence

02:19:19is false, the previous sentence is true or
something. But then the logical structure

02:19:24of those things changes. As soon as you go
from having a finite cycle of those things

02:19:28to having an infinite cycle, the logical structure
changes. And I think the same is true of things

02:19:33like the Tarski theory of truth. And yeah,
it may be that there's some nice interpretation

02:19:37of that in terms of what happens as you build
up to these progressively higher-order toposses

02:19:43in homotopy type theory. I don't know. But
it's an interesting speculation.

02:19:47JS What would be your preferred interpretation
of truth?

02:19:52PW So from a logic standpoint, I'm quite taken
with the definition of semantic truth that

02:20:00exists in things like Tarski's undefinability
theorem, which is the idea that you say a

02:20:04proposition is true if you can incorporate
it into your formal system without changing

02:20:08its consistency properties, right? So if you
have formal system S and your proposition

02:20:14T, T is true if and only if S plus T is, you
know, if and only if con S plus T is the same

02:20:21as con S. And that's a fairly neat idea that
I think, I mean, it's used a lot in logic

02:20:27and it's quite useful for formalizing certain
concepts of mathematical truth, and particularly

02:20:30for distinguishing these kind of concepts
of completeness versus soundness versus decidability,

02:20:36which often get confused. Those become a lot
easier to understand, in my experience, if

02:20:40you start to think of truth in those terms.
JS Yeah, great. John, that's a formal definition

02:20:44of truth that works for formal statements,
but what about colloquial informal ones?

02:20:48PW No, no, no, I agree. It's extremely formal.
But I was actually about to say that I think

02:20:53it also aligns quite well with some basic
intuition we have for how truth works when

02:20:58we reason about things informally, right?
So if, you know, we have some model of the

02:21:02world, right? And that's like our formal system
or some informal system, right? And if you

02:21:07take on board some new piece of information,
generally speaking, the way that humans seem

02:21:12to work is if we can incorporate that new
piece of information without fundamentally

02:21:16changing the consistency properties of our
model of the world, we are much more likely

02:21:20to believe that statement is true than if
it necessitates some radical reimagining of,

02:21:24you know, of the consistency properties
of our internal representation. And so I think

02:21:30informally, there's a version of that same
definition of truth that has a bit of slack,

02:21:36right? Where you say, okay, a proposition
could be provisionally true, but how likely

02:21:41I am to accept it as true depends on how radically
I have to reformulate, you know, my foundations

02:21:47of reality in order to incorporate it in a
consistent way.

02:21:50I see. Well, John, I don't know what subject
we haven't touched on. This is a fascinating

02:21:57conversation. Thank you, man.
No, this was fantastic. As you say, I'm really,

02:22:02you know, it's been a long time coming, but
I'm really glad we had this opportunity to

02:22:06chat. And, yeah, I really look forward to
staying in touch. I've become, I have to confess,

02:22:11when you first reached out, I hadn't heard
of you, but in part because you reached out

02:22:16and in part because, you know, of the explosion
of your channel, I've been following a lot

02:22:20of what you've been doing subsequently. And
I think, no, I think TOE is a really fantastic

02:22:24resource. And the, yeah, your particular niche
is one that definitely, that desperately needs

02:22:31to be filled. And I think you're doing a fantastic
job of filling it.

02:22:33What would you say that niche is? And I ask
just because it's always interesting for me

02:22:37to hear, well, I have an idea as to what TOE
is or what TOE is doing, what theories of

02:22:41everything the project is. It doesn't always
correspond with what other people think of

02:22:46it.
Right. So the reason I really like your channel

02:22:51and the reason I like witnessing these conversations
and to some limited extent participating in

02:22:56them as well is the following reason. It feels
to me like you've got these two extremes out

02:23:00there, right? There are these really quite
vacuous kind of popular science, popularization

02:23:07or philosophy, popularization, YouTube channels
and documentary series and things where you

02:23:11often have a host who, you know, goes very
far to kind of play up the fact that they're

02:23:17ignorant of what's being discussed and they
don't really have any strong opinions. And

02:23:21it's just, you know, they go and ask some
brain boxes for what they think and it all

02:23:25gets assembled in some nice documentary package.
That's kind of one extreme. Then you have

02:23:29the other extreme of, you know, you take some
physicist, some philosopher who's been working

02:23:34on their own pet theory for 30 years and they
go make some, you know, some, you know, long

02:23:39YouTube video about it, just advocating that
and shouting down all the competition and

02:23:43being very kind of bigoted and dogmatic or
whatever.

02:23:46And it feels like what you are managing to
do by, you know, because you are an extremely

02:23:52intelligent and well-read person with a background
in math and physics and who has

02:23:55very wide interests outside of that and who,
you know, more so than any other YouTuber

02:24:01I've
encountered actually makes an effort to really

02:24:04understand, you know, the stuff that they're
talking about and the stuff that their guests

02:24:07are talking about. You know, that's even just
in itself, that would be incredibly valuable.

02:24:12But then what I think that allows you to do
is

02:24:19to do something that's somehow a really nice
synthesis of the best aspects of those two

02:24:23approaches whilst avoiding their more unpleasant
aspects, which is to be the kind of interested,

02:24:29educated, motivated interlocutor who is, you
know, not completely inert, like in the

02:24:35kind of the sort of popular science documentary
case, but also not, you know, dogmatically

02:24:41pushing
and saying, ah, you know, you're completely

02:24:43wrong. You need to be thinking about the quantum
gravity

02:24:45or something, but just saying, oh, but how
does this connect to that? Or is it possible

02:24:50you could
think of things in this, you know, being that

02:24:53kind of Socratic dialogue partner in a way
that I think

02:24:57you are almost uniquely placed because of
your skill set and your personality to, you

02:25:01know,
that's a role you're almost uniquely placed

02:25:03to play in that space. I've never really seen
that

02:25:05work in any context outside of your channel.
And I think that's something really quite

02:25:11special.
Well, man, that's the hugest compliment and

02:25:13I appreciate that. Thank you so much. I think
you've captured, well, I don't know if I'm

02:25:17the bigot in that, but I'll interpret that
as me not being a bigot just to sleep at night.

02:25:24No, no, no, exactly. I mean, I think you handle
the balance really well as someone

02:25:28who clearly has ideas and has opinions and
has views as, you know, as you have every

02:25:33right to as someone who's thought about this
as much as anyone else, right? But you're

02:25:37not, you're not trying to shout down opposition.
You're not trying to force some view down

02:25:41someone's throat. You are, as far as I can
tell, you are actually, you know, in completely

02:25:49good faith, just trying to explore with genuine
intellectual curiosity, the space of ideas

02:25:55and, you know, and present new perspectives
and point in directions that people may not

02:26:00have previously thought of in a way that I
think a lot of people say that they're trying

02:26:04to do. But I've very rarely seen anyone actually,
you know, and people might be able to simulate

02:26:10that for a while, but after a while, you know,
the mask kind of slips and you see, oh, really

02:26:14they're kind of pushing this viewpoint or
whatever.

02:26:16So part of that is that I don't have that
incentive structure of having to produce and

02:26:22get citations in order for me to live. Because
if I was, then I would have to specialize

02:26:27much earlier and I wouldn't be able to survey
as much before I specialize. So currently

02:26:32I'm still in the surveying mode. I'm like
a gannet before I go down and eat. So I'm

02:26:37lucky in that regard. And man, like, holy
moly, super cool. So I have many questions

02:26:42from the audience, by the way.
I mean, just, just informally on the following

02:26:46up on that. I mean, I think the, in many ways,
I think the String Theory landscape video

02:26:49is the, is the perfect embodiment of that,
of that sort of side of you, right? It's the

02:26:55fact that I don't know any other person really
who could have done something like that because

02:27:00it requires both, you know, you're not, you
know, you come across quite critical of String

02:27:06Theory, right? So no, no, no String Theorist
would have made that video, but also no one

02:27:11whose paycheck depends on them investigating
loop quantum gravity would have invested the

02:27:16time to understand String Theory at the level
that you had to understand it in order to

02:27:19make the video. And so it's like, I don't
know who else would have filled that, that

02:27:22niche, right?
Yeah, that was a fun project. I find it's

02:27:26just, it's so terribly in vogue to say I dislike
String Theory, but then simultaneously to

02:27:32feel like you're voicing a controversial opinion.
And I wanted to understand String Theory before

02:27:37I said, and I, by the way, I love String Theory.
I think it may be describing elements of reality

02:27:43correctly. And that may be why it has, I misspoke
by the way, when I said in the video that

02:27:49it has no predictions, it had mathematical
predictions. Maybe it still does. And this

02:27:53is something Richard Borcherds emailed me
because he said, that's something I would

02:27:56correct in the video. It has mathematical
predictions. It doesn't have physical ones.

02:28:00But anyhow, I think that's why it may prove
so fruitful mathematically.

02:28:07And it also, I mean, like parts of it have
physical predictions that are, but they just

02:28:13happen to not strictly depend on the String
Theoretic interpretation, right? So there

02:28:17are condensed matter predictions of ADS-CFT
that have been quite experimentally, you know,

02:28:21validated, right? It's just that ADS-CFT came
from String Theory, but it doesn't strictly

02:28:26depend on String Theory.
Oh, right. Exactly. Exactly. Okay. So one

02:28:29of the questions from the audience is, has
John ever done psychedelics?

02:28:33Yes. So I have tried psychedelics and actually
I consider it, I don't want to come across

02:28:40as too much of a kind of drug pusher, but
I consider it's one of the most important

02:28:45things I've ever done. I don't do it regularly
because I'm afraid of the effect that it has

02:28:51on the brain and things like that. So I had
a list of things I wanted to try and I tried

02:28:56each of them once and I'm very glad that I
did. And the main takeaway was, you know,

02:29:02the stuff we were talking about before about,
you know, there's kind of, there's the computation

02:29:08that a system is doing and there's the computation
that the observer is doing and, you know,

02:29:12so, you know, really what you've got is that,
you know, you've got these two computations

02:29:15and you've got a third computation that is
sort of the encoding function, the thing that

02:29:19maps a concrete state of the system to an
abstract state in the internal representation

02:29:24of the observer. And really all three of those
things are kind of free parameters.

02:29:28And, you know, I'd been thinking about that
kind of stuff since I, you know, not in those

02:29:34terms precisely, but in some form for a long
time, you know, from when I was a teenager

02:29:38onwards and kind of in this very kind of nerdy
intellectual way thinking about, oh, yes,

02:29:44you
know, surely if my model of reality changes

02:29:47even slightly, then, you know, the interpretations
of

02:29:50the perceptions and qualia that I experienced
is going to be radically different. But it

02:29:55doesn't
matter how much you intellectualize that idea.

02:29:58It's very, very different if you just like
subjectively experience it, right? And that's

02:30:02in a sense, driving home the fact that if
you make

02:30:05what is, in the grand scheme of things, an
absolutely trivial modification to your brain

02:30:11chemistry, your modes of decomposing and understanding
the world completely just dissolve,

02:30:18as happens with things like LSD. Actually
experiencing that from a first-hand perspective

02:30:23is really, really important. It kind of convinced
me. I don't want to, again, I don't want to

02:30:27seem
too... Okay, it would be too strong to say

02:30:31it ultimately convinced me of the validity
of that

02:30:33way of thinking about things, but it definitely
is something that occurs to me when I'm worried

02:30:40that I'm overplaying this observer-dependence-of-phenomena
line. I kind of think,

02:30:45well, no, actually, if you modify even just
very slightly neurotransmitter balances in

02:30:50the brain,
the internal perception of reality changes,

02:30:53you know, kind of really, really radically.
JS Yes. Okay, well, here's a physics question.

02:31:00What would happen if an object wider than
a wormhole throat flies into the wormhole?

02:31:05Does the wormhole widen? Does the object cork
the wormhole? Does it deform the object?

02:31:11If it deforms it, how? What about if the object
flies at an even faster speed, so 0.9 speed

02:31:16of light?
Okay, interesting question. So, I mean, wormholes

02:31:22obviously are not known to be physical. They
are

02:31:24valid solutions to the Einstein equations.
Einstein rows and bridges and extended Schwarzschild

02:31:30solutions are valid solutions, but the Einstein
equations are incredibly permissive, and they

02:31:34permit many, many more solutions than things
that we believe to be physical. So if you

02:31:39just take
the Einstein field equations on face valueâ€¦

02:31:42Okay, one thing to remember is that when an
object is

02:31:46falling into the wormhole, it's not like it
has to fit into the throat, so to speak, right?

02:31:53If you imagine the topology of what's going
on, you've got this two-sheet sort of hyperboloid,

02:31:58and the wormhole throat that's connecting
them, but any object you throw in is localized

02:32:02to one
of the sheets. So it's traveling on that sheet

02:32:05and follows the world lines on that sheet.
It's not

02:32:09like it's some plug that's trying to go through
the throat, through the space in the middle.

02:32:15So it may well be that the world linesâ€¦
I mean, this will happen due to tidal deformations,

02:32:19that the object will be stretched in the radial
direction and compressed in the angular directions

02:32:25as it gets pulled in, just due to gravitational
tidal effects. But the fact that the object

02:32:31is
quote-unquote bigger than the wormhole throat

02:32:34doesn't matter. From its perception, its world
lines are traveling on some smooth region

02:32:41of space. It never encounters any kind of
discontinuity,

02:32:44anything that has to sort of fit through,
so to speak.

02:32:46Okay. Would you kindly ask him, how would
he tie science and spirituality together?

02:32:59I think one always has to be a bit careful
with that, right? In the sense that I don't

02:33:05want to
take either of the two extreme positions of

02:33:07saying, oh, science validates the existence
of an immortal soul or something, which I

02:33:13don't believe. But nor do I want to say,
oh, science invalidates whatever, the numinous

02:33:19dimension. I think they're largely agnostic
to one another. Okay, so actually it comes

02:33:27back to the stuff we were talking about at
the beginning,

02:33:29in a way, about the language that we use and
the models that we use for constructing reality,

02:33:36right? Do you actually believe that the universe
is a computer? Do you actually believe that

02:33:42the
solar system is made of clockwork or something?

02:33:44And again, the answer is no, right? My view
is

02:33:47that these are just models we use based on
the ambient technology of our time.

02:33:52And I kind of have a similar feeling about
a lot of theology and a lot of spirituality,

02:33:58right? If you go and read writings by people
like John Duns Scotus or medieval scholastic

02:34:05theologians, the questions they're grappling
with are really the same questions that I'm

02:34:10interested in. Okay, to take a concrete example,
right? So I realize I'm talking about religion

02:34:16here, not necessarily spirituality, but I'll
tie it together in a sec. So you could ask

02:34:22the
question, so our universe, right? It seems

02:34:26to be neither completely trivial, right? It's
neither

02:34:28kind of maximally simple, nor is it kind of
maximally complicated, right? So there's some

02:34:34regularity, but it's not completely logically
trivial. You know, it's not like every little

02:34:38particle follows its own set of laws, but
it's also not like we can just reduce everything

02:34:42to
one, as far as you can tell, we can just reduce

02:34:44everything to one logical tautology. So as
far

02:34:50as I can tell, the first people to really
discuss that question

02:34:53in a systematic way, at least from European
theology and philosophy, I'm less, I'm more

02:34:58ignorant of other traditions, were the scholastic
theologians, were people like Duns Scotus,

02:35:04who asked, you know, why did God create a
world which is neither maximally simple nor

02:35:09maximally complex, effectively? And Duns Scotus'
answer is a perfectly reasonable answer, right?

02:35:14Which is because God created the world that
way because that world is the most interesting.

02:35:20If I were to formulate that question in modern
terminology, I would formulate it in terms

02:35:25of
Kolmogorov complexity, right? I would say,

02:35:28why is the algorithmic complexity of the universe
neither zero nor infinity? Why is it some

02:35:33finite value? And the answer, as far as you
can tell,

02:35:36is essentially because of information theory.
Because we learned from Shannon that the kind

02:35:41of the most interesting or the highest information
density, you know, the most interesting signal

02:35:46is one that is neither completely noisy, maximum
information, nor completely simple,

02:35:51but somewhere in the middle. So really, Duns
Scotus hit upon a really foundational idea

02:35:56in
modern algorithmic information theory. He

02:35:59didn't formulate it in those terms because,
you know,

02:36:02he didn't know what Kolmogorov complexity
was. He had no way of, you know, that ambient

02:36:06thinking
technology didn't exist. So he formulated

02:36:08the answer in terms of the ambient thinking
technology

02:36:11of the time, which was God and the Bible and,
you know, all that kind of stuff.

02:36:15And so, I don't want to be someone who sits
here and says, oh, look at those people. They

02:36:19were
talking about, you know, God and whatever,

02:36:21and weren't they so ignorant? Because I don't
want

02:36:24people to look at, you know, not that I think
they're wrong. But I don't want people to

02:36:27look
at my work in a thousand years and say, oh,

02:36:29look, he thought the universe was a computer,
how silly he was, right? I don't think the

02:36:33universe is a computer. I think it's a useful
model just as they thought God was a useful

02:36:35model, which it was and maybe to an extent
still is.

02:36:41So that's kind of my general view about sort
of theology and spirituality is that

02:36:46I think there are, you know, there are some
classes of questions where it's useful to

02:36:48think about things in terms of Turing machines
or, you know, fiber bundles or whatever it

02:36:53is.
And there are some classes of questions where

02:36:54it is useful to couch them in terms of the
soul or,

02:36:57you know, an immortal spirit or God or whatever.
And you can do those things without believing

02:37:01in the ontological reality of any of them,
as indeed I don't. But that doesn't make them

02:37:07Now, can you actually distinguish those two
if you're a pragmatist? Because it's my understanding

02:37:12if you're like William James, the utility
of it is tied to the truth of it.

02:37:16Yeah, I mean, that's, it's a tricky one. That's
something I, okay, being completely honest,

02:37:22I don't know. It's something I've gone back
and forth on over the years, right? Because

02:37:25in a way,
so yes, you might say, okay, do I believe

02:37:28in God or do I believe in the soul in some
ontological

02:37:32sense? And the answer is no. But if that's
your definition of exist, or that's your definition

02:37:38of belief, then I also don't believe in electrons,
right? I don't believe in space-time.

02:37:42You know, I think all of these things are
just models, right? Like, do I think that,

02:37:46you know, space-time is a useful mathematical
abstraction? But in a sense, we know that,

02:37:50you know, in black holes or in the Big Bang
or something, that's probably an abstraction

02:37:54that
loses usefulness and eventually will be superseded

02:37:58by something more foundational.
So do I believe in space-time in an ontological

02:38:01sense? No. Do I believe in particles in an
ontological sense? No. So whereas you might

02:38:07say, okay, well, therefore, that means probably
that

02:38:11my definition of the word exist is not very
useful, right? I should loosen that definition

02:38:14a
bit and be a bit more permissive. So then

02:38:17you might take the William James view of,
okay, well,

02:38:20you could say, I believe that space-time exists
in as much as I think it's a useful model

02:38:26for a
large class of natural phenomena. Again, it's

02:38:30a bit like the dinosaur thing we were talking
about earlier. You could say, well, I don't

02:38:33believe that space-time doesn't exist in an
ontological sense, but it's kind of consistent

02:38:37with a model of reality that does have good
experimental validation or observational validation.

02:38:43But then, if that's your criterion,
then I kind of have to admit that, okay, well,

02:38:47in that sense, maybe I do believe in a soul,
because there areâ€¦ So for instance, I don't

02:38:56believe that there's any hard-line distinction
between the computations that are going on

02:39:03inside the brain and the computations that
are going on

02:39:05inside lumps of rock or something. Really,
the distinction is, it comes back to the point

02:39:11you
were making earlier about what laws of physics

02:39:14would a cat formulate? So in a sense, okay,
maybe they exist in the same objective reality,

02:39:19whatever that means.
But whatever their internal model of the world