Peter Woit is a theoretical physicist, mathematician, critic of string theory, and author of the popular science blog Not Even Wrong. Please support this podcast by checking out our sponsors:
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OUTLINE:
0:00 - Introduction
0:23 - Physics vs mathematics
14:52 - Beauty of mathematics
36:43 - String theory
1:05:16 - Theory of everything
1:25:24 - Twistor theory and spinors
1:41:51 - Nobel Prize likelihood for theory of everything
1:45:37 - Simulating physics
1:49:08 - Sci-Fi, aliens and space
1:58:20 - Responsibility of scientists

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The following is a conversation with Peter White, a theoretical physicist at Columbia, outspoken critic of string theory, and the author of the popular physics and mathematics blog called Not Even Wrong.

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And now, here's my conversation with Peter White.

You're both a physicist and a mathematician.

me ask, what is the difference between physics and mathematics?

Well, there's kind of a conventional understanding of the subject that they're 2 quite different things.

So that mathematics is about making rigorous statements about these abstract things, things of mathematics and proving them rigorously.

And physics is about doing experiments and testing various models and that.

But I think the more interesting thing is that there's a wide variety of what people do as mathematics, what they do as physics, and there's a significant overlap.

And that I think is actually a very, very interesting area.

And if you go back kind of far enough in history and look at figures like Newton or something, I mean, at that point you can't really tell if it was Newton, a physicist or a mathematician.

The mathematicians will tell you he was a mathematician, the physicists will tell you he was a physicist.

But Anyway, there was kind of no such distinction then that's more of a modern thing.

But anyway, I think these days there's a very interesting space in between the 2.

So in the story of the 20th century and the early 21st century, what is the overlap between mathematics and physics, would you say?

Well, I think it's actually become very, very complicated.

I think it's really interesting to see a lot of what my colleagues in the math department are doing.

Most of what they're doing, they're doing all sorts of different things, but most of them have some kind of overlap with physics or other.

So, I mean, I'm personally interested in 1 specific, 1 particular aspect of this overlap, which I think has a lot to do with the most fundamental ideas about physics and about mathematics.

But there's just, you kind of see this, this really, really everywhere at this point.

Which particular overlap are you looking at group theory?

Yeah, so the, at least what the way it seems to me that if you look at physics and look at the our most successful laws of fundamental physics, they're really, they have a certain kind of mathematical structure, it's based upon certain kind of mathematical objects and geometry connections and curvature, the spinners, the Dirac equation.

And that this very deep mathematics provides kind of a unifying set of ways of thinking that allow you to make a unified theory of physics.

But the interesting thing is that if you go to mathematics and look at what's been going on in mathematics the last 50, 100 years, and even especially recently, there's a similarly some kind of unifying ideas which bring together different areas of mathematics and which have been especially powerful in number theory recently.

And there's a book for instance by Edward Frankel about love and math.

But it is a nice audio book that I listened to while running an exceptionally long distance, like across the San Francisco Bridge.

And there's something magic about the way he writes about it but some of the group theory in there is a little bit difficult.

Yeah, that's the problem with any of these things, to kind of really say what's going on and make it accessible is very hard.

He, in this book and elsewhere, I think, takes the attitude that kinds of mathematics he's interested in and that he's talking about are provide kind of a grand unified theory of mathematics.

They bring together geometry and number theory and representation theory, a lot of different ideas in a really unexpected way.

But I think to me, the most fascinating thing is if you look at the kind of grand unified theory of mathematics he's talking about, and you look at the physicist's kind of ideas about unification, it's more or less the same mathematical objects are appearing in both.

So it's this, I think there's a really, we're seeing a really strong indication that, you know, the deepest ideas that we're discovering about physics and some of the deepest ideas that mathematicians are learning about are really, are intimately connected.

Is there something, like if I was 5 years old and you were trying to explain this to me, is there ways to try to sneak up to what this unified world of mathematics looks like.

You said number theory, you said geometry, words like topology.

What does this universe begin to look like?

Are these, what should we imagine in our mind?

And we're trying to say something about it.

Like what are we supposed to imagine in our minds?

What's a good thing to try to, for people that don't know any of these tools, except maybe some basic calculus and geometry from high school that they should keep in their minds as to the unified world of mathematics that also allows us to explore the unified world of physics.

I mean, what I find kind of remarkable about this is the way in which we've discovered these ideas, but they're actually quite alien to our everyday understanding.

We grow up in this 3 spatial dimensional world and we have intimate understanding of certain kinds of geometry and certain kinds of things.

But these things that we've discovered in both math and physics are that they're not at all close, have any obvious connection to kind of human everyday experience.

And I can say that some of my initial fascination with this when I was young and starting to learn about it was actually exactly this kind of arcane nature of these things.

It was a little bit like being told, well, there are these kind of semi-mystical experience that you can acquire by a long study and whatever, except that it was actually true and there's actually evidence that this actually works.

So, I'm a little bit wary of trying to give people that kind of thing because I think it's mostly misleading.

But 1 thing to say is that geometry is a large part of it.

And maybe 1 interesting thing to say that's about more recent, some of the most recent ideas is that when we think about the geometry of our space and time, it's kind of 3 spatial and 1 time dimension.

It's a physics is in some sense about something as kind of 4 dimensional in a way.

And a really interesting thing about some of the recent developments in number theory have been to realize that these ideas that we were looking at, naturally fit into a context where your theory is kind of 4 dimensional.

So, I mean, geometry is a big part of this and we know a lot and feel a lot about, you know, 2123 dimensional geometry.

So wait a minute, so we can at least rely on the 4 dimensions of space and time and say they can get pretty far by working in that, in those 4 dimensions.

I thought you were gonna scare me that we're gonna have to go to many, many, many, many

My point of view, which goes against a lot of these ideas about unification is that no, this is really, everything we know about really is about 4 dimensions and that you can actually understand a lot of these structures that we've been seeing in fundamental physics and in number theory, just in terms of 4 dimensions, that it's in some sense I would claim has been a really, has been kind of a mistake that physicists have made in for decades and decades to try to, to try to go to higher dimensions, to try to formulate a theory in higher dimensions, and then you're stuck with the problem of how do you get rid of all these extra dimensions that you've created, and because we only ever see anything in 4 dimensions.

That kind of thing leads us astray, you think?

So creating all these extra dimensions just to give yourself extra degrees of freedom.

I mean, isn't that the process of mathematics is to create all these trajectories for yourself, but eventually you have to end up at a final place, but it's okay to sort of create abstract objects on your path to proving something.

Yeah, certainly, and from a mathematician's point of view, mathematicians are also very different than physicists in that we like to develop very general theories.

We like to, if we have an idea, we want to see what's the greatest generality in which you can talk about it.

So from the point of view of most of the ways geometry is formulated by mathematicians, It really doesn't matter, it works in any dimension.

There's no particular, for most of geometry, there's no particular special thing about 4.

But anyway, but what physicists have been trying to do over the years is try to understand these fundamental theories in a geometrical way.

And it's very tempting to kind of just start bringing in extra dimensions and using them to explain the structure.

But typically this attempt kind of founders because you just don't know, you end up not being able to explain why we only see 4.

It is nice in the space of physics that, like if you look at Fermat's last theorem, it's much easier to prove that there's no solution for N equals 3 than it is for the general case.

And so I guess that's the nice benefit of being a physicist, is you don't have to worry about the general case, because we live in a universe with n equals 4 in this case.

Yeah, physicists are very interested in saying something about specific examples.

And I find that interesting, even when I'm trying to do things in mathematics, and I'm trying even teaching courses and to mathematics students, I find that I'm teaching them in a different way than most mathematicians, because I'm very often very focused on examples, on what's kind of the crucial example that shows how this powerful new mathematical technique, how it works and why you would want to do it.

And I'm less interested in kind of, you know, proving a precise theorem about exactly when it's gonna work and when it's not gonna work.

Do you usually think about really simple examples, like both for teaching and when you try to solve a difficult problem?

Are you, do you construct like the simplest possible examples that captures the fundamentals of the problem and try to solve it?

That's often a really fruitful way to, if you've got some idea to just kind of try to boil it down to what's the simplest situation in which this kind of thing is gonna happen and then try to really understand that and understand that.

And that is almost always a really good way to get insight into it.

Do you work with paper and pen, or, like, for example, for me, coming from the programming side, if I look at a model, if I look at some kind of mathematical object, I like to mess around with it sort of numerically.

I just visualize different parts of it, visualize however I can.

So most of the work is like with neural networks, for example, is you try to play with the simplest possible example and just to build up intuition by, you know, any kind of object has a bunch of variables in it.

And you start to mess around with them in different ways and visualize in different ways to start to build intuition.

Or do you go the Einstein route and just imagine like everything inside your mind and sort of build like thought experiments and then work purely on paper and pen.

Well, the problem with this kind of stuff I'm interested in is you rarely can kind of, it's rarely something that is really kind of, or even the simplest example you can kind of see what's going on by looking at something happening in 3 dimensions.

There's generally the structures involved are either they're more abstract or if you try to kind of embed them in some kind of space and where you could manipulate them in some kind of geometrical way, it's gonna be a much higher dimensional space.

So even simple examples, the embedding them into 3 dimensional space, you're losing a lot.

Yeah, but to capture what you're trying to understand about them, you have to go to 4 or more dimensions.

So it starts to get to be hard to, and you can train yourself to try it as much as to kind of think about things in your mind.

And I often use pad and paper, and I'm often, in my office I often use the blackboard.

And you are kind of drawing things, but they're really kind of more abstract representations of how things are supposed to fit together.

And they're not really, unfortunately, not just kind of really living in 3 dimensions where you can understand.

Are we supposed to be sad or excited by the fact that our human minds can't fully comprehend the kind of mathematics you're talking about?

It makes me, it makes it seem like there's a giant mystery out there that we'll never truly get to experience directly.

I mean, or I would put it a different way that, you know, most questions that people have about this kind of thing, you know, you couldn't, you can give them a really, a true answer and really understand it, but the problem is, is 1 more of, of time.

It's like, yes, you know, I could explain to you how this works, but you'd have to be willing to sit down with me and work at this repeatedly for hours and days and weeks.

And it's just gonna take that long for your mind to really wrap itself around what's going on.

So that does make things inaccessible, which is sad.

But again, it's just kind of part of life that we all have a limited amount of time and we have to decide what we're gonna spend our time doing.

Speaking of a limited amount of time, we only have a few hours, maybe a few days together here on this podcast.

Let me ask you the question of amongst many of the ideas that you work on in mathematics and physics, what's the most beautiful idea?

Or 1 of the most beautiful ideas, maybe a surprising idea?

And once again, unfortunately the way life works, we only have a limited time together to try to convey such an idea.

Okay, well, actually, let me just tell you something, which I'm tempted to kind of start trying to explain what I think is this most powerful idea that brings together math and physics, ideas about groups and representations and how it fits to quantum mechanics.

But in some sense, I wrote a whole textbook about that and I don't think we really have time to get very far into it.

Well, can I actually, on a small tangent, you did write a paper towards the Grant Unified Theory of Mathematics and Physics?

Well, I think we've kind of gone over that.

I think that the key idea is what we were talking about earlier that just kind of a claim that if you look and see what's the, have been successful ideas in unification in physics and over the last 50 years or so and what has been happening in mathematics and the kind of thing that Frenkel's book is about, that these are very much the same kind of mathematics.

And so it's kind of an argument that there really is, you shouldn't be looking to unify just math or just fundamental physics, but taking inspiration for looking for new ideas in fundamental physics that they are gonna be in the same direction of getting deeper into mathematics and looking for more inspiration in mathematics from these successful ideas about fundamental physics.

Could you put words to sort of the disciplines we're trying to unify?

Are we literally talking about all the major fields of mathematics?

So it's like number theory, geometry, so like differential geometry, topology.

so the, I mean, 1 name for this, that this is acquired in mathematics is the so-called Langlands program.

It's that Robert Langlands kind of realized that a lot of what people were doing and that was starting to be really successful in number theory in the 60s.

And so that this actually was, anyway, that this could be thought of in terms of these ideas about symmetry in groups and representations, and in a way that was also close to some ideas about geometry.

And then more later on in the 80s and 90s, there was something called geometric Langlands that people realized that you could take what people have been doing in number theory in Langlands and just forget about the number theory and ask what is this telling you about geometry?

And you get a whole some new insights into certain kinds of geometry that way.

So it's anyway, that's kind of the name for this area is Langlands and geometric Langlands.

And just recently in the last few months, there's been There's kind of really major paper that appeared by Peter Schultze and Laurent Farg, where they made some serious advance and try to understand a very much kind of a local problem of what happens in number theory near a certain prime number.

And they turned this into a problem of exactly the kind that geometric Langlands people had been doing, these kind of pure geometry problem.

And they found by generalizing the mathematics, they could actually reformulate it in that way and it worked perfectly well.

1 of the things that makes me sad is, you know, I'm a pretty knowledgeable person and then, what is it, at least I'm in the neighborhood of theoretical computer science, right?

And so many people talk about, like Langlands, for example, is 1 of the most brilliant people in mathematics and just really admire his work.

And I can't, it's like almost I can't hear the music that he composed and it makes me sad.

Yeah, well I mean, I think that unfortunately it's not just you, it's I think even most mathematicians have no, really don't actually understand what this is about.

I mean, the group of people who really understand all these ideas.

And so, for instance, this paper of Schultz and Farag that I was talking about, the number of people who really actually understand how that works is, anyway, very, very small.

And so, I think even you find, if you talk to mathematicians and physicists, even they will often feel that there's this really interesting sounding stuff going on, which I should be able to understand.

It's kind of in my own field I have a PhD in, but it still seems pretty clearly far beyond me right now.

Well, if we can step back to the question of beauty, is there an idea that maybe is a little bit smaller that you find beautiful in the space of mathematics or physics?

There's an idea that, you know, I kind of went, got a physics PhD and spent a lot of time learning about mathematics.

And I guess it was embarrassing that I hadn't really actually understood this very simple idea until, and kind of learned it when I actually started teaching math classes, which is maybe there's a simple way to explain the fundamental way in which algebra and geometry are connected.

So, you normally think of geometry as about these spaces and these points, and you think of algebra as this very abstract thing about these abstract objects that satisfy certain kinds of relations.

You can multiply them and add them and do stuff, but it's completely abstract.

But the kind of really fundamental idea is that unifies algebra and geometry is to think whenever anybody gives you what you call an algebra, some abstract thing of things that you can multiply and add, that you should ask yourself, is that algebra the space of functions on some geometry?

So 1 of the most surprising examples of this, for instance, is a standard kind of thing that seems to have nothing to do with geometry is the integers.

So then there you can multiply them and add them.

It's an algebra, but it has seems to have nothing to do with geometry, but what you can, it turns out, but if you ask yourself this question and ask, you know, is, are integers, can you think, if somebody gives you an integer, can you think of it as a function on some space, on some geometry?

And the space is the space of prime numbers.

And so what you do is you just, if somebody gives you an integer, you can make a function on the prime numbers by just, you know, at each prime number taking that, that integer modulo that prime.

So if, as you say, I don't know, if you get given 10, you know, 10, and you ask what is its value at 2?

So it's 0 at 2, it's 1 at 3, and you can kind of keep going.

And so this is really kind of a truly fundamental idea.

It's at the basis of what's called algebraic geometry, and it just links these 2 parts of mathematics that look completely different.

And it's just an incredibly powerful idea, and so much of mathematics emerges from this kind of simple relation.

So you're talking about mapping from 1 discrete space to another.

For a second I thought perhaps mapping like a continuous space to a discrete space, like functions over a continuous space.

Well, I mean, you can take, if somebody gives you a space, you can ask, you can say, well, let's, and this is also, this is part of the same idea.

The part of the same idea is that if you try and do geometry and somebody tells you, here's a space, that what you should do is you should wait, so say, wait a minute, maybe I should be trying to solve this using algebra.

And so if I do that, the way to start is, you give me the space, I start to think about the functions on the space, okay?

So for each point in the space, I associate a number.

I can take different kinds of functions and different kinds of values, but basically functions on a space.

So what this insight is telling you is that if you're a geometer, often the way to work is to change your problem into algebra by changing your space.

Stop thinking about your space and the points in it and think about the functions on it.

And if you're an algebraist and you've got these abstract algebraic gadgets that you're multiplying and adding, say, wait a minute, are those gadgets, can I think of them in some way as a function on a space?

And what kind of functions would they be?

And that going back and forth really brings these 2 completely different looking areas of mathematics together.

Do you have particular examples where it allowed to prove some difficult things by jumping from 1 to the other?

Is that something that's a part of modern mathematics where such jumps are made?

A lot, much of modern number theory is kind of based on this idea.

But, and when you start doing this, you start to realize that you need, what simple things, simple things on 1 side of the algebra, start to require you to think about the other side about geometry in a new way.

You have to kind of get a more sophisticated idea about geometry.

Or if you start thinking about the functions on a space, you may need a more sophisticated kind of algebra.

But in some sense, I mean, much or most of modern number theory is based upon this move to geometry.

And there's also a lot of geometry and topology is also based upon.

If you wanna understand the topology of something, you look at the functions, you do deramocomology and you get the topology.

Well, let me ask you then the ridiculous question.

Can you formalize the definition of the word beautiful?

I think there are many different things you can find beautiful for different reasons.

I mean, I think in this context, the notion of beauty, I think really is just kind of an idea is beautiful if it's packages a huge amount of kind of power and information into something very simple.

So in some sense, you can almost kind of try and measure it in the sense of what are the implications of this idea, what non-trivial things does it tell you versus how simply can you express the idea.

So the level of compression, what is it, correlates with beauty?

And so you can start to tell that an idea is becoming uglier and uglier as you start kind of having to, you know, it doesn't quite do what you want, so you throw in something else to the idea and you keep doing that until you get what you want.

But that's how you know you're doing something uglier and uglier, when you have to kind of keep adding in more into what was originally a fairly simple idea and making it more and more complicated to get what you want.

Okay, so let's put some philosophical words on the table and try to make some sense of them.

1 word is beauty, another 1 is simplicity, as you mentioned, another 1 is truth.

So do you have a sense, if I give you 2 theories, 1 is simpler, 1 is more complicated.

Do you have a sense of which 1 is more likely to be true to capture deeply the fabric of reality?

Yeah, I think all of our evidence, what we see in the history of the subject is the simpler 1, though often it's a surprise.

It's simpler in a surprising way, but yeah, that we just don't, we just, Anyway, the kind of best theories we've been coming up with are ultimately, when properly understood, relatively simple and much, much simpler than you would expect them to be.

Do you have a good explanation why that is?

Is it just because humans want it to be that way?

Are we just like ultra biased and we just kinda convince ourselves that simple is better because we find simplicity beautiful?

Or is there something about our actual universe that at the core is simple?

My own belief is that there is something about a universe that's simple and that I was trying to say that there is some kind of fundamental thing about math, physics, and physics and all this picture which is in some sense simple.

It's true that, it's of course true that our minds are very limited and can certainly do certain things and not others.

So it's in principle possible that there's some great insight, there are a lot of insights into the way the world works, which just aren't accessible to us because that's not the way our minds work, we don't.

And that what we're seeing, this kind of simplicity, is just because that's all we ever have any hope of seeing.

So there's a brilliant physicist by the name of Sabine Hasenfelder, who both agrees and disagrees with you, or I suppose agrees that the final answer will be simple.

But simplicity and beauty leads us astray in the local pockets of scientific progress.

Yes, I found it was really fascinating reading her book.

And anyway, I was finding disagreeing with a lot, but then at the end when she says yes, when we find there, when we actually figure this out, it will be simple.

But does beauty lead us astray, which is the core thesis of her work in that book.

I actually, I guess I do disagree with her on that so much.

I don't think, and especially, and I actually fairly strongly disagree with her about sometimes the way she'll refer to math.

And so the problem is, you know, physicists and people in general just refer to it as math and they're often meaning not what I would call math, which is the interesting ideas of math, but just some complicated calculation.

And so I guess my feeling about it is more that it's very, the problem with talking about simplicity and using simplicity as a guide is that it's very, it's very easy to fool yourself.

And it's very easy to decide to fall in love with an idea, you have an idea, you think, oh, this is great, and you fall in love with it.

And like any kind of love affair, it's very easy to believe that the object of your affections is much more beautiful than the others might think, and that they really are.

So if you say, I'm just gonna pursue ideas about beauty and mathematics and this, it's extremely easy to just fool yourself, I think.

And I think that's a lot of what the story she was thinking of about where people have gone astray that I think it's, I would argue that it's more people.

It's not that there was some simple, powerful, wonderful idea which they'd found and it turned out not to be useful, but it was more that they kind of fooled themselves that this was actually a better idea than it really was, and that it was simpler and more beautiful than it really was, is a lot of the story.

I see, so it's not that the simplicity or the beauty leads us astray, is that just people are people and they fall in love with whatever idea they have.

And then they weave narratives around that idea, or they present it in such a way that emphasizes the simplicity and the beauty.

But I mean, the thing about physics that you have is that you, what really can tell, if you can do an experiment and check and see if nature is really doing what your idea expects that you do in principle have a way of really of testing it.

And it's certainly true that if you thought you had a simple idea and that doesn't work and you got into an experiment and what actually does work is some more, maybe some more complicated version of it that can certainly happen and that can be true.

I think her emphasis is more that I don't really disagree with is that people should be concentrating on when they're trying to develop better theories, on more on self-consistency, not so much on beauty, but not, is this idea beautiful, but is there something about the theory which is not quite consistent, and use that as a guide, that there's something wrong there which needs fixing.

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