Daniel and Jorge Explain the UniverseDaniel talks to Prof. Mark Colyvan, philosopher of mathematics, about whether math is something we invented or discovered.
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Terms apply. We know a lot of things about the universe. We know that everything around us is made of tiny little particles that obey strange quantum rules. We know that our planet moves through space curved by the mass of the Sun. We know that the Earth is four and a half billion years old, and that the universe is almost fourteen billion years old, and all of that knowledge has something in common. It's all expressed in terms of mathematics. Our quantum theories, our ideas about gravity, our understanding of the age of the Earth and the universe all depend deeply on math. And if we're going to dig deep into the foundations of reality and see if we understand what's there, shouldn't we do the same thing and ask ourselves some hard questions about the mathematics, asking what it is, why it works, and whether it's even necessary. Is a language of physics, then how certain are we that it reflects something true about the universe rather than something about our minds. Hi, I'm Daniel. I'm a particle physicist and a professor at UC Irvine, and I love speaking the language of math. Sometimes when I'm confused about how something works, it's the math that leads me through and shows me the answer. There's something wonderfully crisp about mathematics. I love how the patterns click together with an exactness and reliability. I love that it doesn't sag or break or wilt. That's six times nine is the same today as it is tomorrow and will be forever, and Welcome to the podcast Daniel and Jorge Explain the Universe, where we dig into the universe around us and try to find some answers. We have an unquenchable thirst to understand and an insatiable appetite for asking questions. My friend and co host jorgees on vacation, and so today we are going to ask some of the deepest of questions. Regular listeners know that we mostly talk about the physics of the universe, but that sometimes we dig a bit deeper and ask about the philosophy of it. We don't just want to know what the fundamental particles are, but we want to know why those particles. What does it mean that it's these particles? And also what is a particle? And why is the universe made out of them instead of something else. That's the philosophical side of physics. Today, we're going to follow our noses all the way down the philosophical rabbit hole and ask questions about what lies underneath all of that. If you dig far enough into physics, you always end up face to face with me. Our equations are written in math, our predictions and calculations are mathematical. So math provides the bricks for building our castle of physics. But that should intrigue us, that should inspire our curiosity. What are these bricks, the numbers and shapes and functions and sets that we use to build up our physics. Where do these mathematical bricks come from? Where do they live? Can we smash them together to learn about them? How do we know what rules they follow? Is this the only way to build physics? Or could we have done it without math? Would aliens use math in their theories? And if you're a regular listener, you'll hear me saying this all the time. It's amazing that math does describe the universe, that it works so well that we can devise these beautiful and simple mathematical stories about the universe in all different scenarios. Tiny particles seem to follow group theory, rushing rivers obey differential equations, and massive galaxies are bound together by geometry. What does it mean that works so well? Is it something about how our mind works? Or is it something deep and true about the universe itself? So today on the podcast, we'll be answering the question, is math the language of the universe and to help me sort through some of the slippery issues at the heart of this deep question is our guest, Professor Mark Coldivan. Mark is a professor of philosophy at the University of Sydney in beautiful Australia, where he thinks deeply about these questions all day long. He's also an accomplished writer, publishing The Indispensability of Mathematics and an Introduction to the Philosophy of Math, which I read recently cover to cover and found to be very compelling and accessible. The title makes it sound a little bit like a textbook from an introductory philosophy course, but it's very conversational and very easy to read. I learned a lot and it inspired me to invite Mark to join us on the podcast to chat about some of the questions at the heart of mathematics. So it's my pleasure then to welcome Professor Mark Colvin to the podcast. Mark, thanks very much for joining us, Thanks for having me, and I understand while I have never been to your part of the world, you have actually spent some time here in Irvine, Is that right.
That's right, Yeah, I had a visiting fellowship in Irvine back in two thousand and one.
So you can compare for us, then the glorious weather of Orange County with the weather of your local Sydney.
Nothing compares with the weather of Orange Counties. It's fabulous every day.
Correct answer, correct answer. All right, Now that we have your qualifications sorted out, tell me you're a philosopher of math. And I've never spoken to a philosopher of math before, So tell me what does a philosopher of math do all day? I mean, is it reading and writing and coffee and emails? What got you excited about philosophy of math? Well, I started out in mathematics. The usual story for a philosopher of mathematics. You start out in mathematics, you start getting interested in certain questions in mathematics that lead you to more philosophical pondering, and at some stage then you defect to the dark side and become a philosopher, which is what happened with me mathematics at undergraduate an honors level, and then that PhD switched to philosophy, mainly because those interesting questions about what counts as the right logic for mathematics and what is a proof in mathematics? And these are questions that mathematicians have a good handle on, but mainly by doing them. I mean, you trained in mathematics to do proofs by just doing proofs, and the question of what why is this a proof and that not a proof is mostly given to you by way of example, Right, there's a flaw in this proof, there's a gap.
In this proof, or this one is a good proof, and so on and so forth. But as a philosopher of mathematics, much more interested in a systematic answer to such questions. What is the correct logic? Is it classical logic, is it some other alternative logic that mathematicians are using, and so on and so forth. So these are sorts of questions that I was interested in, or became interested in by studying mathematics and found that the answers really weren't in the mathematics department. So I, you know, straight over to the philosophy department occasionally, and they didn't have the answers either, but at least they recognized these were interesting questions. So that was my particular path into the philosophy of mathematics at least.
And so, why do you think it is that mathematicians aren't that interested in like why proofs work or whether proofs should work. You know, well, why is it that it's the philosophy department to ask those kinds of questions? I mean, are there folks in the mathematics side of it that do that and just don't call it philosophy.
Yeah, I think so. I wouldn't say that mathematicians is not interested in this. I mean one of the things I think is interesting about philosophy of X, whatever X is. For me, it's philosophy of mathematics primarily. But if you're doing philosophy of something rather, then you need to engage with something or other. So if you're doing philosophy of quantum mechanics, you need to talk to folks doing quantum mechanics. If you're doing philosophy of biology, you need to speak to folks doing biology, and you learn a great deal about the other discipline as well. So for me, philosophy of mathematics was an excuse to kind of do a bit more mathematics, talk to mathematicians. Mathematicians some are interested in such questions, some are not. That's as you would expect. Some are interested in topology, some are not. So it's just a particular bunch of questions that some mathematicians are interested in, and as a philosopher of mathematics, it's good to talk to them about these things. You know, I'm interested in mathematical intuitions about such things, not just sitting back in the philosophical armchair, as it were, and coming up with my own theories of these things. Right.
Do you feel like there's something of an asymmetry there that maybe philosophers of mathematics are more interested in what mathematicians are thinking about than mathematicians are interested in what philosophers are thinking about. In the case of the physics department, for example, we have a lot of people over here who are doing physics, and a few of us are interested in what philosophers of physics are saying about what we're doing. But a lot of people seem to subscribe to you know, Fineman's approach philosophy of science is about as useful to scientists as ornithology as to birds. Right, do you have that same reaction from mathematicians. They're like, look, Proof's work, Why do we care why they work?
It depends again on the mathematicians, as you rightly point out, amongst physicists, you've got things like trying to sort out the interpretation of quantum mechanics. That's a deeply philosophical question that a lot of physicists are engaged with. You don't kind of just dismiss that, as you know, as philosophy, that's not physics. It's crucial to quantum mechanics to have an appropriate interpretation of what's going on there. So there's a place where some physicists, not all physicists are interested in the interpretation of quantum mechanics, but those who are recognize that as a philosophical problem and interested in well informed opinions from suitable philosophers. Not every philosopher has an opinion on that either, And so in mathematics, I would say most mathematicians are not particularly interested in the philosophy of mathematics, but there are some well.
In physics, it seems sort of natural to ask these questions. We discover the universe is this way, and then we can ask, like, what does that mean, or why is it this way in that some other way? In the case of mathematics, what are the sort of foundational questions here? What are the questions the philosophy of math is answering.
Basic questions about what the subject matter is. I think one of the interesting things about philosophy of mathematics is the problems start really early on. So if you say, you know, someone gives you a scientific discipline, biology, what is biology? It's not always easy to answer such questions. But you can say something helpful like it's the study of living organisms, or it's the study of evolution, and say what that is. What's physics, Well, it's a study of the fundamental particles and large scale structures, theories of space, time, and so forth. Mathematics is the study of doc doc dot fill in the dots, right, It's not easy to answer that question. Tempting to say that it's about the study of numbers, functions, sets, and the like, but that immediately raises a question of what are they?
Then?
These are not the sorts of things that one can gain access to. It's not like fundamental particles. You can build accelerators and you can smash things into one another, and you can find traces of fundamental particles, but numbers are not the sorts of things that even leave traces. So you know, if the mathematics is the study of numbers, then how is it that mathematicians gain access to this mathematical realm. There's some of the questions you get right from the get go.
I think this is one of the most fast new questions, is this question of like what our numbers? Are they things in our minds or are they things in the universe? You know, what are the rules by which they operate? Are they rules that we invented the way we invented the rules of checkers, or are they rules that we've discovered that are true and deep in the universe. But it's amazing to me that we can do math without knowing these things, right, That we can calculate one plus one equals two, and one plus two equals three, and all sorts of complicated integrals in multiple dimensions without knowing the answers about what it is we are doing. How do we reconcile that, How do we understand why it's possible to do it without knowing what it is we are doing?
Right? That's the really I think the heart of philosophy of mathematics is to understand mathematical progress in light of these difficult questions. So it's not like you know, as a philosopher mathematics. I'm going to the mathematics department and telling them to hang on, stop, stop, until we sort these things out. Right, Mathematics is business as usual, with a few exceptions. There have been some interest in cases in the history of mathematics. So early in the twentieth century there was a movement called intuitionism or constructivism, and that came from within mathematics. So one of the great mathematicians of the twentieth century, Ali J. Brower, many theorems named after Brower, became concerned that if mathematics is a kind of construct, a mental construct, then you can't just assume that every mathematical proposition is either true or false until it's constructed. So if you think about fiction, for instance, in this way, what's true in a fiction is all that's said to be true in the fiction, plus a bunch of natural implications.
Right, So for example, shock Holms lives that twenty two and a half Baker Street or whatever that's true in that story, that's.
Right, and implied by that that he lived near other streets that are nearby London. The geography of London is supposed to be held fixed, so without even saying that that's a natural implication of that. So all of that can be taken to be true. But Sherlock Holmes walked down Good Street exactly fourteen times in his life. Neither true nor false, surely, right, there's not stated in any of the books. It's not stated that he didn't. So it seems like that's neither true nor false. That's very different to the actual world, the actual real world. Even if you don't know whether something is true or false, it's true or false nonetheless, or or at least that's a natural position. I don't know how many pairs of gray socks Napoleon owned, but there is a fact of the matter about that, and we'll probably never know that.
Right.
So Brow became concerned that if mathematics was a kind of construct like this, then you can't use certain proof methods, proof methods that require that the proposition in question is either true or false. So, in particular, reductive of proof methods which proceed by assuming the negation of the thing that you want to prove and then derive a contradict from that, and therefore include that the proposition unnegated is true. But if it's neither true nor false to start with, the neither negation nor the proposition itself are true or false. So Brow became concerned about particular proof methods and wanted to restrict mathematics to purely constructive methods, and this movement continues. There are still mathematicians who stand by this very very strongly and think that classical mathematics, which uses these kinds of non constructive methods, are problematic. Let's say that you want to reconstruct some of the important theorems of mathematics and provide alternative proofs that are constructive.
It might be that all of mathematics is just a big castle built on sand, and then the end none of it is real. Is that the idea?
Yeah, So the idea is if it is some sort of construction, mental construction, that doesn't mean that it's nothing. It's still kind of a you know, Shakespeare is a mental you know shakespeareks of Shakespeare. Mental constructions that great things, But you just need to be careful about the logic that you're using. So according to this line of thought, then some proofs are in fact not proofs at all.
I think this touches on a really interesting question here, which is like, why does it matter whether or not these things are real, or whether or not these things are just constructed? And I think it goes to the heart of sort of what we try to do, at least in physics, which is learned about the universe. Right, I am interested in physics because I want to know what's out there and what's real, Not because I want to build a complicated mathematical construct that I can use to play around with in my head and with my friends. Right, I want to know what's actually out there. So in the question of mathematics, that brings up this issue of like our numbers real, are these proofs real? Or are they just a game that we have invented? And to make it more concrete, I like to think about it in terms of like aliens. You know, if alien scientists showed up here, could we talk to them about the things in physics that we've discs. If the things we've discovered are real and part of the universe, then yes. If they're just like in our minds, then no, they would have different ways of thinking about it. So can we take that same sort of question and ask it about mathematics and ask like, well, would aliens have developed mathematics if it's part of the universe, or where they have some other way of putting together structured thought to figure out the universe. If in fact, mathematics is just part of our minds and the way we think, is that a reasonable way to think about the questions of the philosophy of math.
Yeah, I think that's a very good way of putting it. You might think, so, for instance, some things in mathematics are just artifacts of the way we are. So you might think using base ten, for instance, that's to do with numbers of fingers and so on and so forth. But something very natural about base two. And I believe you would know more about this than me. But I believe that there's thought that if you were going to contact extraterrestrials, then the initial segments of theNational expansion of pi in base two would be some that an intelligent life form would recognize. That's assuming that sometimes there's something really objective about pie. It's hard to imagine that that's just the kind of construct of ours. You know, you think that surely pie turns up in the most unexpected places. It's not just about the ratio of the circumference to the diameter of a circle. That's the initial definition, but you know, as you know, it turns up in just about everything.
Right everywhere in complex analysis, in geometry.
Yeah so yeah, So the thought that that's something that would be recognized by another intelligent life form seems reasonable. But that pushes you to this sort of objective point about mathematics, that it seems to be something that's objectively true, not just mental construction. You wouldn't expect another an alien life form to you know, recognize facts about Sherlock Holmes for instance.
Hey, you know, the Detective novel could be a universal construct that could be exists in intelligent beings everywhere. You know, I imagine sitting across the table from alien mathematicians and introducing them to ours, and I can imagine that it might be that the kind of elaborate constructs that we've built calculus and geometry, they might have very different ways of doing these kind of things. I mean, even in the history of our mathematics, our path to these sorts of things have been varied and could have gone differently. But it feels like maybe at the heart of it, there could be something in common that if you drill down to the core of mathematics, the fundamental ideas on which everything else is built. Maybe we could compare those with alien mathematicians. How well have we done in terms of like examining the foundations of our own mathematics, of understanding what our castle is built on? You know, what are the basic rules of mathematics?
Great question? I mean when we think about the foundations of mathematics, a lot of it is this program of trying to construct other bits of mathematics from some other mathematics. So set theory or category if you see theory fer but set theory, let's stick for that for the moment. There's these beautiful constructions in set theory where you can construct the natural numbers out of sets, and then you can construct ordered pairs of natural numbers out of sets, and then you can get functions and so on and so forth. So you can get great deal of mathematics built just out of set theory.
Can you explain that to me? Like, how do you get natural numbers out of sets? What does that mean?
So you just have a series of sets, So you start with the empty set, right, So the empty set is the set that has nothing in it. You identify that with zero. Just call that zero. It's not zero, that's an empty set. But let's just, you know, humor me call that zero. Then you have the set that contains the empty set that has one member. You know, so it's the set that has the empty set inside it. So it has one member. Let's call that one.
This is just arbitrary. We're just making this up as we go.
Yeah, And then you collect all of the sets from the previous stages and collect them together. So the next stage you take the empty set plus the set that contains the empty set that has two members in it. Suggestive name for that one two. This is a construction due to the mathematician John von Neumann, and they're called the von Neumann ordinals. So you can construct the natural numbers in this way. You can then define you know, I won't go into details, but you can define addition and so and forth in this set theoretic way.
And what does that accomplished for you? Now instead of having zero, one, and two, you have these weird sets. Why is that better or more foundational or what have you learned from doing that?
That's the kind of really interesting question here. In one sense, it's now no longer sort of transparent. Right, we're familiar with the natural numbers and sort of building it out of these these sets. They get really is you can imagine, they get really ugly really quickly. So once you start talking about numbers like seventeen, it's hard to write it down on the one page what that is. But no one's suggesting that you need to use these things instead of natural numbers. But it's an interesting exercise that you can construct the natural numbers out of sets. So you might think, in a way, sets are all you need. Sets are really like the fundamental particles of physics. Right, No one says there are no tables and chairs and there are no people. Well, some people say such scenes. But just because we can show that people are made out of fundamental particles, no one says, stop doing biology or sociology or psychology. Hand it all over particle physicists, because it's all particles. It's an interesting discovery that we're all made up out of these fundamental particles. So in that sort of vein, you might think it's interesting that you can reconstruct almost all of mathematics out of set theory like this, not suggesting you do it that way, but it's an interesting construction, and that maybe that the fundamental mathematical particles, as it were, are sets.
I see. So it's like reductionism to say what it's our fundamental and what bits emerge? If we can figure out which bits are fundamental, and then we can ask questions just about those and try to get some insight into like the actual nature of mathematics. So does that work? I mean, can you say I'm going to start with sets and from that build everything geometry and integrals and differential equations. Can you base all of mathematics on these weird sets?
Yes, I mean, with certain caveats. There are a couple of little areas of mathematics that don't succumb to this primarily category theory. But set that aside. All of the mathematics that most of us know and love you can build out of sets in this kind of way, and that's, you know, again, that's just an interesting fact about mathematics. It demonstrates firstly the power of set theory, but really it's such a versatile tool set theory. Secondly, you know, it does lend support to this idea that sets are the equivalent of the fundamental particle and mathematics, and again business as usual for topology and all the other areas of mathematics. Not suggesting they quit and go and do set theory instead, but rather it's an interesting fact that their area can be reduced in this admittedly cumbersome way, just as reducing a table or a chair to fundamental particles. Try and do that in particle physics. Give the full description of what a table is in particle right, If it's possible at all, it's going to be incredibly cumbersome and not terribly useful for furniture removalists and other people working with furniture.
All right, well, I have a lot more questions about the foundations of mathematics, but first let's take a quick break. With big wireless providers, what you see is never what you get. Somewhere between the store and your first month's bill, the price you thought to repaying magically skyrockets. With mint Mobile, you'll never have to worry about gotcha's ever again. When mint Mobile says fifteen dollars a month for a three month plan, they really need it. I've used mint Mobile and the call quality is always so crisp and so clear. I can recommend it to you. So say bye bye to your overpriced wireless plans, jaw dropping monthly bills and unexpected overages. You can use your own phone with any mint Mobile plan and bring your phone number along with your existing contacts. So dit your overpriced wireless with Mint Mobiles deal and get three months a premium wireless service for fifteen bucks a month. To get this new customer offer and your new three month premium wireless plan for just fifteen bucks a month, go to mintmobile dot com slash universe. That's mintmobile dot com slash universe. Cut your wireless bill to fifteen bucks a month. At mintmobile dot com slash Universe, forty five dollars upfront payment required equivalent to fifteen dollars per month new customers on first three month plan only speeds slower about forty gigabytes On unlimited plan. Additional taxi speed and restrictions apply. See mint mobile for details.
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Yes, So I think there are realists in mathematics, and those people will say, maybe there's just sets, but the sets are at least real. You've got to sort of think that the fundamental furniture of the universe includes, you know, all of the things that you particle physicists tell us about plus sets, and the anti realists about mathematics say, no, the sets are some sort of construction and their role is to build this edifice of mathematics upon sets. But that doesn't tell us anything about the nature of sets, whether they're real or not. The action, I think in modern debates in philosophy mathematics turns two applications of mathematics pretty quickly. Then you can think about it in parallel with physics. Why is it that we believe in certain bits of physics and not others, So take bits of physics that are more speculative at the moment. I take string theory to be such an area. Some people believe in strings, some people don't, and it's yet to be settled. Happy to take your advice on this, but that's my understanding of the current state of play there. And why is it that people are not concerned about other particles like electrons, Well, because electrons do a lot of work in your theories, and it's hard to imagine any of our current physical theories functioning with our electrons or something very much like them, Whereas there are alternatives for some of the more speculative parts of physics. Now the specultive parts very often get settled down the track somewhere. But that's how we go about deciding whether things are real. In physics, it seems right, is it indispensable to sort of greater physical theory? So people have turned to mathematics with that same kind of view and thought, okay, for mathematics, what would it take for mathematics to real? Well, perhaps if it's indispensable to our best science, that's a clue that is real. It's not just some construct of the human mind, for instance. So this line of thought often called an indispensability argument. If mathematics is indispensable to science, then the bits of mathematics that are in fact indispensable should have the same status as the science itself.
It's certainly a common feeling among physicists that mathematics is somehow the language of the universe itself, because we can so effectively describe these rules of physics in terms of this language. You know, of Stephen Weinberg said, it's positively spooky how the physicist finds a mathematician has been there before him or her. And you know, there are often these cases, especially in particle physics, where we struggle to understand something, then we discover there's some bit of mathematics like group theory invented just because of the curiosity of mathematicians playing games basically in their mind, turns out to be applicable to the world in a gorgeous way that clicks into place and suddenly gives us insight. And those moments mean that they're not religious moments or spiritual moments, but there are moments where you feel like you've gained some deep insight into the way the universe works. And it doesn't feel like here's a useful description of the universe. It feels like you're revealing the inner mechanisms of the universe itself. But how can we know the difference right? How can we tell whether these things are real, not just the physical particles we're talking about, but the mathematics that describes them. How do we distinguish between whether real or not before we meet alien scientists? So I'm very interested in this argument you suggest about the indispensability of mathematics. And I read a book recently called Science Without Numbers Much, which I'm sure you're familiar with, by Hartree Field because he suggests that math is very, very useful, you know, like the way making it to do list is a good way to organize your day, but you could probably get through your day without it, but that it's not actually I mean, he says in this book, and I'll quot him because I find this outrageous. I am denying that numbers or any similar entities exist. What a statement to me? Can you help us wrap our minds around this opposite argument, the one that suggests that we don't actually need math, that it's useful but not indispensable. How do we make sense of that?
Yeah? I mean, let me say from the get go, I just think that's a fantastic books. I disagree with heartree Field on these issues, but it is one of the absolute gems in philosophy of mathematics.
That book.
It's you know, outrageous or audacious, incredible project. I mean so, in response to this indispensability argument that mathematics, you know, you can think of this argument in the following form. We ought to believe in all of them, only the entities that are indispensable to our best science. So should we believe in electrons? Well, just go and see are they indispensable? Could you do science without electrons? No, you can't. Could you do science without point Well, it would be difficult, but you could recast all talk of point masses a little bit more carefully.
Right, Could you do science without coffee? No? Therefore coffee is real.
Like I think it was the old joke about the mathematician being a machine that turns caffeine into theorems.
There you go, that you've solved the question. There's the philosophy of math. What is mathematics? Exactly? It's turning coffee into papers. Anyway, you were telling us about the argument about indispensability, right.
The indispensability argument says, you know, is something indispensable, that's really all you need to do, so look to your science. You don't need to philosophize too much about this in a way. Just philosophizing is done by recognizing that things that are indispensable to your best scientific theories, that's what you ought to be committed to.
All right, But let's explore that a little bit more deeply. Actually, before we get back to hoship Field this indispensability argument, because there's some wrinkles there that I don't really understand. I read like Putnam and Quine arguing that because our best theories are mathematical and those theories are confirmed, that's sort of like also confirms the mathematics as you go along with it, Like, if you have electrons in your theory and your theory works, then you believe in electrons. Well, if your theory also has math as part of it, you're adding numbers, then you that sort of like comes along with the proof. But you know, that makes me wonder about things like infinity. You know, we can do experiments in the universe to explore particles, but as far as we know, there's a certain number of particles in the universe. It's like ten to the eighty or something, depending on how many you count. Does that mean that only numbers up to ten to the eighty are real and indispensable, and numbers bigger than that, like infinity, are not real or just parts of our mind?
You need to be careful where I mean, that's not the only application of infinity, right, So, if you think space time is continuous as it is treated in general relativity, at least quantum mechanics, it's an argument that it's treated discreetly there, but at least in general relativity, space time is treated continuously. So how many space time points are there? Right? Continuum?
Many?
You know, not just the basic infinity there that's in mathematical terms two to the aile of zero. That's the infinity of the continuum. So it's not just a number of particles. But are you going to need infinity in other places? Try and do probability theory without continuous distributions, right, So infinity props up in all sorts of places in science, not just counting things.
So that argument allows us to believe in numbers and also believe in numbers like up to infinity that are real. But that's only if we actually need them in our science. If we could do the science without the numbers, then we wouldn't be able to necessarily argue that the numbers are also real. So help us understand artrey fields argument that we don't need numbers to do science, right.
So that's the starting point, is this argument that you know science, you committed to everything that's indispensable to best science. Taken for granted that mathematics is indispensable for science. The action was really kind of on their first premise, so do you really want to believe in everything? In your best science, we've have frictionist planes, what about inertial rest frames and so on and so forth. But hartrey Field came along and said, maybe you could do science without numbers. Maybe you could just be realist about space time itself instead of having The basic idea is, instead of sort of treating space time in this mathematical way, you just deal with space time itself. So you're realist about the space time as an entity as it were, rather than treating space time as this mathematical structure that has metrics and coordinate systems and so on and so forth.
Okay, so heartree Field hasn't smoked so many binin appeals that he doesn't believe in the universe. He says, space is real. Time is real, but a mathematical description of that is not necessarily real. Is that where we are?
That's right. But in order to say that you can't just I mean, you can say it, but for anyone to believe what you're saying, you've got to deliver the goods. You've got to show how how you can do something like Newtonian mechanics is his case study. Show how you can do at least the differential fragment of Newtonian mechanics without talking about anything mathematical. So just looking at relational properties of space time points, and that's the basic trick. And it's surprisingly how far you can go with that.
How is that possible at all? I mean, if I think about Newtonian mechanics, the first thing that comes to my mind is F equals gmm over r squared. It's about relative distances, it's about masses, it's about forces. If you throw that out, what do you have left?
Well, the thing is you don't throw it out. You reconstruct it in a much more direct or indirect depending on how you're looking at it way. So instead of thinking about, for instance, a point in space having a gravitational potential and the gravitational potential function, then is this map from the space time to real numbers? Right, that's the standard presentation. You wouldn't have talked in those terms, of course, but you know now we think of it as a space time you clearly and manifold and you map from that to real numbers, and that's just your gravitational potential function. Artry Field shows that you can do that directly. Just think about compare space time points with respect to their gravitational potential, not having a gravitational potential function that sits on top of that. And what he shows is by doing it this way, you can recover the standard presentation. So you can actually prove these results that show that you get everything back that you would have had in the standard presentation. So again, the take home message from Field is not you should be doing it this way rather than the way everyone's done it. It's just a bit like the story with sets, right. The fact that you can do it gives you evidence that the mathematics is not indispensable. It's just a nice, quick, and much more l a way of doing it, But it's not indispensable, and you can recover everything. You can prove that you can recover everything that you get instand it in your tony mechanics this way.
Now I understand that it's a useful way to answer the question do we need math? By proving that you could do without it if you had to, it doesn't mean that you should do without it. Right, It's like asking a question of could you live without jelly beans? You could go without them for a year, and you could prove that you don't need to eat them. It doesn't mean that nobody should eat jelly beans. But I'm still not one hundred percent convinced. I mean, your description here of his formulation of gravity includes things like comparing potentials, and to me, potentials are numbers and comparing is a relationship. Are you seeing those things are not mathematical or they're just not numbers.
Yeah, you've got to be realist about the points themselves. And that's one of the criticisms of Field, is that you've got to be realist about space time points and that those things have properties. They have primitive properties like that gravitational potential, electromagnetic potential and st on. So they have those properties. So rather than those being a mathematical function that lives on top of that, it's just these are primitive properties of the space time points. And so a lot of people who are concerned about believing in mathematical objects, because you know, after all, that's kind of spooky, believing in the space time points is also rather spooky, Right. It's not enough for feel to just believe in the Manifold has actually got to believe that individual points have these properties. But whichever way you go on this, And as I said, I disagree with him about the upshot of all this, but the exercise itself is just incredible. You know, before he did this, no one would have thought that you could even get started there. So you've got a lot of criticism. People were saying, oh, well, what about Hamiltonian formulations of classical theories, what about chronum mechanics where the underlying space is infinite dimensional Hilbert spaces, and so on and so forth, And these are all fair and interesting criticisms. But before he started, no one thought you could do the differential fragment of Newtony mechanics either, you know. So for a great deal of time, a great deal of the debate was about how far can you go with this Field style program, because if you can go take it further, then that's going to suggest that mathematics isn't indispensable after all.
So we have, on one hand, folks arguing that mathematics is beautiful and elegant and unreasonably effective and beyond that, actually indispensable to understand in the universe, and Hurtsie Field, and some folks suggesting that maybe it's just useful but not actually necessary. So where do you come down on that. You're a philosopher of mathematics, You've thought deeply about these things. Do you think that numbers are real? Are they just part of our minds? Or are they something we found in the universe.
I'm a realist about mathematics, so I come down on the former. So I think that mathematics is in fact indispensable to have best scientific theories. And that's why I take mathematical ends, at least some mathematics. I think there can still be speculative parts of mathematics that we don't have reason to believe yet.
And so what convinces you.
Just as they are, respectively parts of physics right that we don't believe yet. So, you know, you might think some of the higher reaches of set theory, they are it's hard to go into the details here now, but there are bizarre higher reaches of set theory that don't look like they have any direct applications anywhere yet. You know, maybe if they do, then then you'd be realist about those. But it's not a blanket argument that because mathematics is indispensable, believe in all of it. Right, it's got to be the bits that get applied. And that's one of the criticisms of this line of thought. You don't get realism about mathematics. You get realism about calculus, you get realism about algebraic topology, you get realism about differential geometry. You know, you're going to get realist about the bits that get used and not all of it. But that's for me, that's as it should be. Just because physics is, you know, in the business of describing the universe doesn't mean we should believe all the physics. We should believe the bits of physics that is really indispensive to our understanding of the universe. And as you know, there are going to be speculative parts of physics, not just things like string theory, but non physical models. So for instance, massless universes. People study things like a universe with no mass. Can you have curvature in the universe with no mass? For instance, Not that we live in such a units. We know we don't in one of those universes, but the question is that will give us some understanding about our own universe if we can study these non existent universes. So same with mathematics, they're going to be speculative parts of mathematics like that that even someone like me is not going to be a realist about. But I am not convinced at the end of the day by heartre Field's project, despite the fact that I think it's a fascinating and beautiful technical exercise.
And so I understand that being convinced about the realism of sets doesn't mean that all of mathematics is real. But what is it that convinces you about the realism of sets? So what is the argument that persuades you? Is it the usefulness of mathematics? Is it, you know, seeing something beautiful in nature and seeing you know, mathematics in it, the Fibonacci sequence or you know, the golden ratio. What is it that convinces you that mathematics is real and not just a human construct.
Well, it comes back to this indispensability that we just I can't see how we could do science without mathematics. And moreover, it's not just that it's this language of science, as you often hear, I'm not quite sure what that really even means. I mean, it's not like, you know, once upon a time all academic work had to be carried out into Latin, and Latin was the language of science. It's not that people are talking about it like there's something much deeper about mathematics than Latin. No one for a moment really thought that you couldn't do science in English or whatever it had to be in Latin. That when people say mathematics is the language of science, they mean something much deeper than that. I take it. And one of the ways I think that mathematics is indispensable is in offering up explanations. So this is very controversial, great deal of debate in philosophy mathematics about this at the moment, about whether you can get explanations in physics. Say that mathematical in character, so the mathematics is not merely just this language, but is providing explanations for what's going on in the physical world.
And as I.
Said, very controversial. I do believe that I do think that mathematics is offering explanations, and that's a really important way in which you can be indispensable. If somethings play an explanatory role in your best scientific theory, then it really does look like you should be a realist about it. If you're going to be realist about anything. I mean, there are anti realists about a great deal of science as well, but I'm setting that aside. If you're going to be realist about science, then it's very odd to say that the reason for such and such an event occurring was because of some entity. But there's no such entity. You haven't explained anything if you say that. So if mathematics can play this on a b explanist free role, then that looks like really good grounds for thinking that it's indispensable to scientific explanation.
It is really compelling to me that mathematics can describe not just the fundamental bits of the universe and the fundamental elements of mathematics, but also that it seems like we can find these fairly simple mathematical stories to describe emergent things. You can imagine living in a universe where the basic thing are strings, and there's math about strings, And that doesn't mean that even if you understood how strings work, that you could use them to predict the path of a hurricane. Right, It's incredibly complicated to go from fundamental bits, even from water drops, to a hurricane, not to mention from strings. Right, So strings don't really like provide any explanation of the path of a hurricane, even if you knew what all the strings were doing. Right, But you can zoom out and find some higher level laws, you know, you find fluid mechanics, and you find gravitational rotational theorems about how galaxies move. We can seem to do physics at these higher levels even without understanding of the little bits underneath. And the same way it seems like we can find mathematics that describes the universe even if it's not necessarily connected to those fundamental little bits of the universe. Why do you think that is? Why do you think it is that mathematical descriptions of the universe emerge at all these different levels, even when they're not necessarily connected to each other or easily built from one to the other.
Again, one of the great puzzles in philosophy of mathematics, it's this often called the unreasonable effectiveness problem. How is it mathematics just turns up in all of these places, not just fundamental pit physics, but in chemistry and biology and psychology. At which level you you know, if you think of science as these levels, from the fundamental to the more complex, at every level, you've got relevant mathematics that appear is there, And again I wish I had a good answer to that. One suggestion is that when you know the old, the old sort of adage that if hammer is your only tool, then the whole world looks like a nail. Right. We've got differential equations, damn it, we're going to use them everywhere, right, But that just doesn't wash with me. It's not just that we're forcing everything to be thought of in this framework of a particular bit of mathematics like differential equation. And to be fair, there were times I think where physics was a bit like that. Everything had to be well behaved differential equations, linear first order, and you're try and do as much as you can with those because they were well understood. But I just don't think that's how we work now. I mean, there's so many different branches of mathematics that are turning up in all different places. As you mentioned earlier, group theory. A number of places where you need group theory grows by the day. It doesn't seem to be simply we have these tools and we're going to use it, damn it. It's more like these are the very tools that we would need to do any such science. And again, what does that tell us about the mathematics, Well, like it's intricately connected to the physical world and this kind way that you just can't understand the physical world without having the relevant mathematics under control. You know, as I said, that's controversial. Let's try and flag the things that are more controversial. But since we're talking philosophy here, we can just have a general disclaimer all of this is controversy.
Well, we like to get into the weeds on this show, and so I have a lot more deep questions about a philosophy of math. But we need to take another quick break. When you pop a piece of cheese into your mouth or enjoy a rich spoonful of Greek yogurt, you're probably not thinking about the environmental impact of each and every bite. But the people in the dairy industry are US. Dairy has set themselves some ambitious sustainability goals, including being greenhouse gas neutral by twenty to fifty. That's why they're working hard every day to find new ways to reduce waste, conserve natural resources, and drive down greenhouse gas emissions. Take water, for example, most dairy farmers reuse water up to four times The same water cools the milk, cleans equipment, washes the barn, and irrigates the crops. How is US Dairy tackling greenhouse gases? Many farms use anaerobic digestors that turn the methane from maneuver into renewable energy that can power farms, towns, and electric cars. So the next time you grab a slice of pizza or lick an ice cream cone, know that dairy farmers and processors around the country are using the latest practices and innovations to provide the nutrient dense dairy products we love with less of an impact. Visit usdairy dot com slash sustainability to learn more.
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All right, so we're back and we're having a lot of fun talking to Professor Mark Coldevin about whether mathematics is inherent in the universe and is it real? And I was wondering when you were talking earlier, since philosophy of mathematics asks like what is mathematics and is it real? Is there a branch a philosophy called philosophy of philosophy that asks like whether philosophy is real and what our philosopher is doing?
Anyway, Hi, there is, there is Very recently people have been working on philosophy or philosophy. I think in a way, philosophers have been doing this for a long time. It just didn't come up with the phrase, but not so much of the idea of whether philosophy is real, because you're not interested whether physics is real. You're interested in whether the things that physics posits are real. Right, And so in so far as philosophy is positing entities, are those things real? You could ask that sort of question. But my understanding, at least the philosophy of philosophy is more a kind of a systematic study of methodology, right, which is kind of how I think of philosophy of science is in many ways looking at science and trying to discern useful things to say about methodology, And so philosophy of philosophy is much more about methodological questions about philosophy, questions like does philosophy make progress? So philosophy cops criticism because the questions we're interested in, the questions we're talking about here now are numbers real? That goes back to at least back to Plato, right, And have we made much progress since then? Well, you know, I'd like to think we've made some, But certainly if you look at progress physics has made since such times to now, physics has done much better.
It firm a ground then, right, that's true. Maybe you guys just need more coffee, although you could also say that physics is just an outgrowth of philosophy. When a question becomes experimental, it determs its own science, and philosophy sort of loses control of it. But speaking of concrete questions, I want to come back to the framing we had earlier about aliens. Do you think that if aliens arrived that we could use mathematics as a sort of basis for building a mental connection with them, of understanding whether or not we're thinking in a similar way as them. Would you send mathematicians or philosophers of math out to meet the aliens? First? Thing?
I do think that it would be a good place to start, would be bits of mathematics do you think likely to be universal? I must say I haven't given a lot of thought to who I would send first to meet the aliens.
You don't realize that you're near the top of the list that.
Should concern me for all sorts of reasons. But yeah, I do think that looking for bits of mathematics that you think would be common. You know, again, you wouldn't want decimal expansion of pie base ten, but bansion of pie based two. That's something that you might think would be recognizable. Fundamental theorem suitably couched because you know that the notation you use is perhaps arbitrary in various ways. But the fundamental theorem fundamental theorem of calculus, for instance, you'd think that any reasonably advanced life forms who are capable of traveling to Earth from great distance would have come across the fundamental theorems of calculus. So how do you express those in a way that's not merely notational depended too much on the notation. You can't express it in English, obviously, but the standard notation using integral signs and so forth. I think that's kind of accidental. How do you get that idea across? That does seem like a good place to start. You'd think that an intelligent advanced race would know the fundamental theorem of calculus, but how would they write it and how should you convey it to them?
Well, we had gnome Times on the podcast a few weeks ago, and we asked him this question, and he said, I'll quote, there's a good chance that arithmetic is universal. It's a fair guess that at least the arithmetic would be close enough to be absolute, so that anything we might call intelligence, that we would recognize as intelligence would at least sit on that. And I suppose that he's making the argument that you're making that mathematics is probably fundamental. And in addition, he's drilling down and he saying, let's not start with something complicated, Let's go down to the basics, like you were saying earlier. Set theory, let's find the fundamental elements and see if we can begin from that. Do you think that program is likely to be successful if aliens arrive.
Yeah, yeah, I think that's a very good suggestion. You know, again back to the something like pi, not just any old numbers, because you might think that nothing special about one, two, three, four in particular, but it's really crucial to number theory, be a concept of prime number, for instance. So you might think certain numbers jump out at you, like prime numbers, pi, e some of these numbers in particular, and so if you can get a way of expressing those numbers, but fundamental parts of arithmetic. But the notion of primeness. Again, had to imagine an intelligent advanced race not having that concept. Again, just how do you convey it? But I do like the suggestion.
Yeah, well, I'd love to examine the sort of counter idea, like to think about what an intelligent race might have to have in their minds in order to not arrive at arithmetic. You know, I can imagine sitting across from their mathematicians and drawing a symbol for one and pointing at one thing and an apple, and bringing another apple and then writing the symbol for two or something, and you know, then you have one plus one equals to this kind of stuff. And you must have thought about this more deeply than I have. What assumptions are there inherent in that? You know, are we assuming that this concept of like abstraction to say like, oh, these two apples they have the similar property. They're both apples. Obviously they're not the same apple. They're different ones darker on you know, bright or whatever. But if the aliens are like, no, that's one apple and that's one apple, we don't know what you mean with this whole two business. That's nonsense. Aren't there fundamental assumptions we're making there? Even with one plus one equals two?
Yeah, I think so. I think it's exactly right. It's got to be counting the right sorts of things, right, So one cloud and another cloud is one big cloud. Right, So you don't think one cloud plus one cloud equals one. That's a falsification of basic arithmetic. You think, No, you're counting the wrong kinds of things there, One plus one is two. Given that you're counting the right kinds of things, discrete things that have a certain kind of property, you can still count an apple and an orange and get two. But then you've got to have this overarching concept of pieces of fruit, or things before me or something. But you've got to be some overarching concept there. So you're absolutely right, there are some preconditions for even understanding basic arithmetic.
I was reading about how Japanese people count and discovering that Japanese counting words are actually quite different from English counting words, that if you put a set of things in front of them, they tend to group them by sheep. So those are not just two things. These are two flat things and there's two tall things over there. It strikes me that if even across human cultures, we count and abstract things in different ways, it might be that aliens have no idea what we're talking about when we're demonstrating our basic arithmetic to them.
There are cases in English as well where you're not interested in the number of things for various reasons. She's just interested in existence. So we say it's raining, meaning that there are drops of rainfalling, and that's all we care about. You just want to know whether I need my umbrellas today or not. So you can imagine someone putting apples in front of them. They're saying it's appling, right. I have the concept of multiple apples, but I don't need two, three, four. Who cares how many apples? There's zero apples and there's appling again, and we do that. There are lots of instances of that. Raining is the obvious example, but there are other cases as well, where you're interested in zero or existence. You know of something, and you don't particularly worry about counting, even though you could. There are you know, rain drops are discrete. You could worry about how many rain drops there are. And it's not purely because it's a difficult exercise to count them. It's more just no, I'm not really interested in how many there are. I'm just interested is it raining or not? So again, you know, you could have this idea of zero things or many you know, zero, one and many, right, you.
Could imagine an intelligent civilization getting by with a very different kind of sense of what many is as many more than a thousand, or you know, is it just a few. I find that people have a different sense for like effective infinity. You know, what is a lot and what needs to be counted individually. Anyway, these are really fascinating questions, and I really thank you for answering them and for exploring these questions with me. I think. My last question to you is do we expect any sort of breakthroughs in philosophy of mathematics. I mean, you said we've been struggling with a question of our numbers and circles real since Plato. Do you think we're going to figure that out, that we'll ever have a point where we're like, yeah, we proved that, now we can move on to something else. Or is philosophy of math basically going to go forever until we meet the aliens?
I would like to think that it would be solved, but I don't think that's necessary for it to be a worthwhile exercise. So why I think that is we learn a lot along the way. Sometimes asking the right questions is more interesting than finding answers to them. And I don't think there's anything special about philosophy. I think you would find that in physics as well, not being able to answer some questions, but those questions giving rise to while the questions and new areas is what motivates us and what keeps the disciplines rolling. It sounds to an outside of that might sound rather closed shop as it you're only interested in these little questions. We're interested in getting the exercise rolling. But I do think that we've learned a lot about the relationship between mathematics and the physical realm, a lot of understanding about foundations of mathematics. We don't have the answer to what the foundations of mathematics are, but we have some interesting insights from set theory and the like. So are we likely to solve the problems in my lifetime? I don't think so. I hope not. I'd be out of a job, you know, But I don't think it's going to happen. But I don't think that that means that it's all a waste of time. I think we get in many interesting insights. In particular, the insights that motivated me early on in my career were this connection the why is it the mathematics is applicable? I read this paper, famous paper by Eugene Wigner called the Unreasonable Effective just Mathematics and Natural Sciences, and I read that as an undergraduate and it just captivated by that paper. And did he answer the questions? No? Has anyone answered those questions? No? But fascinating stuff to think about. And I think we have a much better understanding of the relationship between applied mathematics and physics. Now as a result of the asking me sort of other questions about his mathematics.
Real, Absolutely, no, I think it's very useful and also in a way maybe you haven't even anticipated that you have spent your life preparing, I think, to meet the aliens, and when they ask me who we should send, you know, in our first contingent to chat with our alien technological friends, I'm going to nominate you.
Good. As long as they're friendly, you know, well, we'll find out right, we'll send the military. Make sure they're friendly first.
All right, sounds good. Well, thank you very much for joining us and for talking to us about these crazy ideas. I hope that one day we do figure out our numbers, real why math works, and if in fact math is just a game we invented in our minds or the fundamental code of the universe itself. Thanks very much for joining us.
Today, my pleasure, Thanks for having me, Thanks.
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