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Is math the language of the Universe? · Daniel and Kelly’s Extraordinary Universe

• 0:08
  Speaker 1: We know a lot of things about the universe. We
• 0:11
  Speaker 1: know that everything around us is made of tiny little
• 0:15
  Speaker 1: particles that obey strange quantum rules. We know that our
• 0:18
  Speaker 1: planet moves through space curred by the mass of the Sun.
• 0:22
  Speaker 1: We know that the earth is four and a half
• 0:24
  Speaker 1: billion years old and that the universe is almost fourteen
• 0:27
  Speaker 1: billion years old, and all of that knowledge has something
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  Speaker 1: in common. It's all expressed in terms of mathematics. Our
• 0:35
  Speaker 1: quantum theories, our ideas about gravity, our understanding of the
• 0:40
  Speaker 1: age of the earth and the universe all depend deeply
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  Speaker 1: on math. And if we're going to dig deep into
• 0:46
  Speaker 1: the foundations of reality and see if we understand what's there,
• 0:50
  Speaker 1: shouldn't we do the same thing and ask ourselves some
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  Speaker 1: hard questions about the mathematics, asking what it is, why
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  Speaker 1: it works and whether it's even necessary? If math is
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  Speaker 1: a language of physics, then how certain are we that
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  Speaker 1: it reflects something true about the universe rather than something
• 1:09
  Speaker 1: about our minds? Hi, I'm Daniel. I'm a particle physicist
• 1:28
  Speaker 1: and a professor at UC Irvine, and I love speaking
• 1:31
  Speaker 1: the language of math. Sometimes, when I'm confused about how
• 1:35
  Speaker 1: something works, it's the math that leads me through and
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  Speaker 1: shows me the answer. There's something wonderfully crisp about mathematics.
• 1:43
  Speaker 1: I love how the Patterns Click together with an exactness
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  Speaker 1: and reliability. I love that it doesn't zag or break
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  Speaker 1: or wilt, that six times nine is the same today
• 1:53
  Speaker 1: as it is tomorrow and will be forever and welcome
• 1:57
  Speaker 1: to the podcast Daniel and Jorge Explode in the universe,
• 2:00
  Speaker 1: where we dig deep into the universe around us and
• 2:03
  Speaker 1: try to find some answers. We have an unquenchable thirst
• 2:08
  Speaker 1: to understand and an insatiable appetite for asking questions. My
• 2:12
  Speaker 1: friend and Co host, Jorge is on vacation and so
• 2:15
  Speaker 1: today we are going to ask some of the deepest
• 2:18
  Speaker 1: of questions. Regular listeners know that we mostly talk about
• 2:22
  Speaker 1: the physics of the universe, but that sometimes we dig
• 2:25
  Speaker 1: a bit deeper and ask about the philosophy of it.
• 2:28
  Speaker 1: We don't just want to know what the fundamental particles are,
• 2:32
  Speaker 1: but we want to know why those particles, what does
• 2:35
  Speaker 1: it mean that it's these particles, and also what is
• 2:38
  Speaker 1: a particle and why is the universe made out of
• 2:41
  Speaker 1: them instead of something else? That's the philosophical side of physics.
• 2:47
  Speaker 1: Today we're gonna follow our noses all the way down
• 2:51
  Speaker 1: the philosophical rabbit hole and ask questions about what lies
• 2:55
  Speaker 1: underneath all of that. If you dig far enough into physics,
• 2:59
  Speaker 1: you always end up face to face with math. Our
• 3:02
  Speaker 1: equations are written in Math, our predictions and calculations are mathematical.
• 3:07
  Speaker 1: So math provides the bricks for building our castle of physics.
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  Speaker 1: But that should intrigue us, that should inspire our curiosity.
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  Speaker 1: What are these bricks, the numbers and shapes and functions
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  Speaker 1: and sets that we use to build up our physics?
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  Speaker 1: Where do these mathematical bricks come from? Where do they live?
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  Speaker 1: Can we smash them together to learn about them? How
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  Speaker 1: do we know what rules they follow? Is this the
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  Speaker 1: only way to build physics, or could we have done
• 3:35
  Speaker 1: it without math? Would aliens use math in their theories?
• 3:40
  Speaker 1: And if you're a regular listener, you'll hear me saying
• 3:43
  Speaker 1: this all the time. It's amazing that math does describe
• 3:47
  Speaker 1: the universe, that it works so well, that we can
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  Speaker 1: devise these beautiful and simple mathematical stories about the universe
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  Speaker 1: and all different scenarios. Tiny particles seem to follow group theory,
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  Speaker 1: rushing rivers obeyed differential equations and massive galaxies are bound
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  Speaker 1: together by geometry. What does it mean that it works
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  Speaker 1: so well? Is it something about how our mind works,
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  Speaker 1: or is it something deep and true about the universe itself?
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  Speaker 1: So today on the podcast will be answering the question.
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  Speaker 1: Is Math the language of the universe? And to help
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  Speaker 1: me soart through some of the slippery issues at the
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  Speaker 1: heart of this deep question, is our guest, professor, Mark Coldy.
• 4:32
  Speaker 1: Van Mark is a professor of philosophy at the University
• 4:35
  Speaker 1: of Sydney in Beautiful Australia, where he thinks deeply about
• 4:39
  Speaker 1: these questions all day long. He's also an accomplished writer,
• 4:42
  Speaker 1: publishing the indispensability of mathematics and an introduction to the
• 4:46
  Speaker 1: philosophy of math, which I read recently cover to cover
• 4:50
  Speaker 1: and found to be very compelling and accessible. The title
• 4:54
  Speaker 1: makes it sound a little bit like a textbook from
• 4:56
  Speaker 1: an introductory philosophy course, but it's very conversation national and
• 5:00
  Speaker 1: very easy to read. I learned a lot and it
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  Speaker 1: inspired me to invite mark to join us on the
• 5:05
  Speaker 1: podcast to chat about some of the questions at the
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  Speaker 1: heart of mathematics. So it's my pleasure, then, to welcome
• 5:14
  Speaker 1: Professor Mark Colevin to the podcast. Mark, thanks very much
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  Speaker 1: for joining us, thanks for having me, and I understand
• 5:19
  Speaker 1: while I have never been to your part of the world,
• 5:22
  Speaker 1: you have actually spent some time here in Irvine. Is
• 5:24
  Speaker 1: that right? That's right. Yeah, I had a visiting fellowship
• 5:27
  Speaker 1: in Irvine back in two thousand and one. So you
• 5:31
  Speaker 1: can compare for us then, the glorious weather of Orange
• 5:33
  Speaker 1: County with the weather of your local Sydney. Nothing compares
• 5:37
  Speaker 1: with the weather of Orange County. Is it's faculous every day?
• 5:41
  Speaker 1: Correct answer. Correct answer. All right, now that we have
• 5:44
  Speaker 1: your qualifications sorted out. Hell me, you're a philosopher of
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  Speaker 1: math and I've never spoken to a philosopher of math before.
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  Speaker 1: So tell me. What does a philosopher of math do
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  Speaker 1: all day? I mean, is it reading and writing and
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  Speaker 1: coffee and emails? What got you excited about philosophy of math? Well,
• 5:58
  Speaker 1: I started out in mathematics. The usual story for a
• 6:01
  Speaker 1: philosopher of mathematics. You start out in mathematics, you start
• 6:05
  Speaker 1: getting interested in certain questions in mathematics. That leads you
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  Speaker 1: to more philosophical pondering and at some stage then you
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  Speaker 1: defect to the dark side and become a philosopher, which
• 6:18
  Speaker 1: is what happened with me mathematics, that undergraduate an honors
• 6:23
  Speaker 1: level and then that PhD, switched to philosophy, mainly because
• 6:28
  Speaker 1: I was interesting questions about what counts as the right
• 6:32
  Speaker 1: logic for mathematics and what is a proof in mathematics,
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  Speaker 1: and these are questions that mathematicians have a good handle on,
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  Speaker 1: but mainly by doing them, I mean you're trained in
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  Speaker 1: mathematics to do proofs, by just doing proofs, and the
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  Speaker 1: question of what, why is this a proof and that
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  Speaker 1: not a proof, is mostly given to you by way
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  Speaker 1: of example. Right, there's a flaw in this proof, there's
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  Speaker 1: a gap in this proof, or this one is a
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  Speaker 1: good proof, and so on and so for but as
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  Speaker 1: a philosopher of mathematics you're much more interested in a
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  Speaker 1: systematic answer to such such questions. What is the correct logic?
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  Speaker 1: Is that classical logic? Is it something other alternative logic
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  Speaker 1: mathematicians are using, and so on and so forth. So
• 7:13
  Speaker 1: these are sorts of questions that I was interested in,
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  Speaker 1: or became interested in by studying mathematics and found that
• 7:18
  Speaker 1: the answers really weren't in the mathematics department so I
• 7:23
  Speaker 1: straight over to the philosophy department occasionally and they didn't
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  Speaker 1: have the answers either, but at least they recognized that
• 7:29
  Speaker 1: these were interesting questions. So that was my particular path
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  Speaker 1: into the philosophy of mathematics at least. And so why
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  Speaker 1: do you think it is that mathematicians aren't that interested
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  Speaker 1: in like why proofs work or whether proofs should work?
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  Speaker 1: You know. Well, why is it that it's the philosophy
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  Speaker 1: department to ask those kinds of questions? I mean, are
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  Speaker 1: there folks in the mathematics side of it that do
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  Speaker 1: that and just don't call it philosophy? Yeah, I think so.
• 7:54
  Speaker 1: I wouldn't say that mathematicians not interested in this. I
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  Speaker 1: mean one of the things that I think is interesting
• 8:00
  Speaker 1: about philosophy of x, whatever X is. For me it's
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  Speaker 1: philosophy of mathematics primarily. But if you're doing philosophy of
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  Speaker 1: something rather, then you need to engage with there's something
• 8:12
  Speaker 1: or other. So if you're doing philosophy of quantum mechanics,
• 8:14
  Speaker 1: you've need to talk to folks doing quantum mechanics. If
• 8:17
  Speaker 1: you're doing philosophy of biology, you need to speak to
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  Speaker 1: folks doing biology and you learn a great deal about
• 8:23
  Speaker 1: the other discipline as well. So for me philosophy of
• 8:26
  Speaker 1: mathematics was an excuse to kind of do a bit
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  Speaker 1: more mathematics, talk to mathematicians. Mathematicians, some are interested in
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  Speaker 1: such questions, some are not. That's as you would expect.
• 8:36
  Speaker 1: Some are interested in Topology, some are not. So it's
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  Speaker 1: just a particular bunch of questions that some mathematicians are
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  Speaker 1: interested in and as a philosophy of mathematics it's good
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  Speaker 1: to talk to them about these things. You know, I'm
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  Speaker 1: interested in mathematical intuitions about such things, not just sitting
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  Speaker 1: back in the philosophical armchair, as it were, and coming
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  Speaker 1: up with my own theories of these things. Right. Do
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  Speaker 1: you feel like there's something of an asymmetry there, that
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  Speaker 1: maybe philosophers of mathematics are more interested in what mathematicians
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  Speaker 1: are thinking about than mathematicians are interested in what philosophers
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  Speaker 1: are thinking about? In the case of the physics department,
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  Speaker 1: for example, we have a lot of people over here
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  Speaker 1: who are doing physics and a few of us are
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  Speaker 1: interested in what philosophers of physics are saying about what
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  Speaker 1: we're doing, but a lot of people seem to subscribe
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  Speaker 1: to you know, fine, men's approach. Philosophy of science is
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  Speaker 1: about as useful to scientists as ornithology as the birds right.
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  Speaker 1: Do you have that same reaction from mathematicians? They're like look,
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  Speaker 1: proofs work. Why do we care why they work? It
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  Speaker 1: depends again on the mathematicians, as you rightly point out.
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  Speaker 1: Amongst physicists you've got things like trying to sort out
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  Speaker 1: the interpretation of quantum mechanics. That's a deeply philosophical question
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  Speaker 1: that a lot of physicists are engaged with. You know'
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  Speaker 1: you've got to just dismissed that. As you know, that's philosophy,
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  Speaker 1: that's not physics. It's crucial to quantum mechanics to have
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  Speaker 1: an appropriate interpretation of what's going on there. So there's
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  Speaker 1: a place where some physicists, not all physicists, are interested
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  Speaker 1: in the reputation of quantum mechanics, but those who are
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  Speaker 1: recognized that as a philosophical problem and interested in well
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  Speaker 1: informed opinions from suitable philosophers. Not every philosopher has an
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  Speaker 1: opinion on that either, and so in mathematics I would
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  Speaker 1: say most mathematicians are not particularly interested in the philosophy
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  Speaker 1: of mathematics. But there are some. Well, in physics it
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  Speaker 1: seems sort of natural to ask these questions. We discover
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  Speaker 1: the universe is this way and then we can ask
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  Speaker 1: like what does that mean, or why is it this way,
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  Speaker 1: in that some other way? In the case of mathematics,
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  Speaker 1: what are the sort of foundational questions here? What are
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  Speaker 1: the questions? The philosophy of math is answering basic questions
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  Speaker 1: about what the subject matter is. I think that one
• 10:41
  Speaker 1: of the interesting things about philosophy of mathematics is the
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  Speaker 1: problems start really early on. So if you say you
• 10:47
  Speaker 1: know someone who gives you a scientific discipline, biology, what
• 10:50
  Speaker 1: is biology? It's not always easy to answer such questions.
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  Speaker 1: But you can say something helpful like it's the study
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  Speaker 1: of living organisms or it's the study of evolution, and
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  Speaker 1: say what that is. What's physics? Well, it's the study
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  Speaker 1: of the fundamental particles and large scale structures, theories of
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  Speaker 1: spacetime and so forth. Mathematics is the study of dot
• 11:15
  Speaker 1: dot dot, filling the dots right. It's not easy to
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  Speaker 1: answer that question, attempting to say that it's about the
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  Speaker 1: study of numbers, functions, sets and the like, but that
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  Speaker 1: immediately raises a question of what are they, then these
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  Speaker 1: are not the sorts of things that one can gain
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  Speaker 1: access to. It's not like fundamental particles. You can build
• 11:36
  Speaker 1: accelerators and you can smash things into one another and
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  Speaker 1: you can find traces of fundamental particles, but numbers are
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  Speaker 1: not the sorts of things that even leave traces. So,
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  Speaker 1: you know, if the mathematics is the study of numbers,
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  Speaker 1: then how is it that mathematicians gain access to this
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  Speaker 1: mathematical realm? And there's some of the you know, the
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  Speaker 1: questions you get right from the get I think this
• 12:01
  Speaker 1: is one of the most fascinating questions. Is this question
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  Speaker 1: of like, what our numbers? Are they things in our
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  Speaker 1: minds or are they things in the universe? You know,
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  Speaker 1: what are the rules by which they operate? Are they
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  Speaker 1: rules that we invented the way we like invented the
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  Speaker 1: rules of checkers, or are they rules that we've discovered
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  Speaker 1: that are true and deep in the universe? But it's
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  Speaker 1: amazing to me that we can do math without knowing
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  Speaker 1: these things right, that we can calculate one plus one
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  Speaker 1: equals two and one plus two equals three and all
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  Speaker 1: sorts of complicated into rules in multiple dimensions without knowing
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  Speaker 1: the answers about what it is we are doing. How
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  Speaker 1: do we reconcile that? How do we understand why it's
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  Speaker 1: possible to do it without knowing what it is we
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  Speaker 1: are doing? Right? That's the really, I think the heart
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  Speaker 1: of philosophy of mathematics is to understand mathematical progress in
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  Speaker 1: light of these difficult questions. So it's not like you
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  Speaker 1: know as a philosopher mathematics. So I'm going to the
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  Speaker 1: mathematics department and telling them to hang on, stop, stop,
• 12:56
  Speaker 1: until we sort these things out. Right, mathematics is business
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  Speaker 1: as usually. With a few exceptions, there have been some
• 13:01
  Speaker 1: interesting cases in the history of mathematics. So early in
• 13:04
  Speaker 1: the twentieth century there was a movement called intuitionism or
• 13:10
  Speaker 1: constructive ism, and that came from within mathematics. So one
• 13:13
  Speaker 1: of the Great Mathematicians of the Twentieth Century, Alie J Brauer.
• 13:18
  Speaker 1: Many theorems named after Brauer, became concerned that if mathematics
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  Speaker 1: is a kind of construct, a mental construct, then you
• 13:26
  Speaker 1: can't just assume that every mathematical proposition is either true
• 13:30
  Speaker 1: or false until it's constructed. So if you think about
• 13:34
  Speaker 1: the fiction, for instance, in this way, what's true in
• 13:36
  Speaker 1: a fiction is all that's said to be true in
• 13:40
  Speaker 1: the fiction, plus a bunch of natural implications. Right. So,
• 13:43
  Speaker 1: for example, she collins lives that twenty two and a
• 13:46
  Speaker 1: half Baker Street or whatever. That's true in that story.
• 13:48
  Speaker 1: That's right. And implied by that that he lived near
• 13:54
  Speaker 1: other streets that are nearby London. The geography of London
• 13:57
  Speaker 1: is supposed to be held fixed. So without even saying that,
• 14:00
  Speaker 1: that's a natural implication of that. So all of that
• 14:03
  Speaker 1: can be taken be true. But Sherlock Holmes walked down
• 14:07
  Speaker 1: Good Street exactly fourteen times in his life. Neither true
• 14:11
  Speaker 1: nor false. Surely right? There's not stated in any of
• 14:13
  Speaker 1: the books. It's not stated that he didn't. So it
• 14:17
  Speaker 1: seems like that's neither true nor false. That's very different
• 14:20
  Speaker 1: to the actual world, the actual real world. Even if
• 14:23
  Speaker 1: you don't know whether something is true or false, it's
• 14:26
  Speaker 1: true or false nonetheless, or at least that's a natural position.
• 14:30
  Speaker 1: I don't know how many pairs of gray socks Napoleon owned,
• 14:34
  Speaker 1: but there is a fact of the matter about that
• 14:36
  Speaker 1: and we'll probably never know that. Right. So brow became
• 14:40
  Speaker 1: concerned that if mathematics was a kind of construct like this,
• 14:44
  Speaker 1: then you can't use certain proof methods, proof methods that
• 14:49
  Speaker 1: require that the proposition in question is either true or false,
• 14:52
  Speaker 1: in particular reductive of proof methods which proceed by assuming
• 14:57
  Speaker 1: the negation of the thing that you want to prove
• 15:00
  Speaker 1: and then derive a contradiction from that and therefore include
• 15:04
  Speaker 1: that the proposition unnegated, is true. But if it's neither
• 15:09
  Speaker 1: true nor false to start with, that neither negation nor
• 15:12
  Speaker 1: the proposition itself are true or false. So brow became
• 15:16
  Speaker 1: concerned about particular proof methods and wanted to restrict mathematics
• 15:20
  Speaker 1: too purely constructive methods, and this movement continues. There are
• 15:24
  Speaker 1: still mathematicians who stand by this very, very strongly and
• 15:29
  Speaker 1: think that classical mathematics, which uses these kinds of non
• 15:33
  Speaker 1: constructive methods, problematic. That's saying that you want to reconstruct
• 15:39
  Speaker 1: some of the important theorems of mathematics and provide alternative
• 15:43
  Speaker 1: proofs that are constructive. There may be that all of
• 15:46
  Speaker 1: mathematics is just a big castle built on sand and
• 15:48
  Speaker 1: then the end none of it is real. Is that
• 15:50
  Speaker 1: the idea? Yeah, so the idea is if it is
• 15:53
  Speaker 1: some sort of construction, mental construction. That doesn't mean that
• 15:56
  Speaker 1: it's nothing, it's still kind of you know, Shakespeare is
• 15:59
  Speaker 1: a meant. You know, shakes, works of Shakespeare Mental Constructions.
• 16:02
  Speaker 1: That great, great things, but you just need to be
• 16:06
  Speaker 1: careful about the logic that you're using. So, according to
• 16:09
  Speaker 1: this line of thought, then some proofs are in fact
• 16:13
  Speaker 1: not proofs at all. I think this touches on a
• 16:15
  Speaker 1: really interesting question here, which is like, why does it
• 16:18
  Speaker 1: matter whether or not these things are real or whether
• 16:22
  Speaker 1: not these things are just constructed? And I think it
• 16:24
  Speaker 1: goes to the heart of sort of what we're trying
• 16:26
  Speaker 1: to do, at least in physics, which is learned about
• 16:28
  Speaker 1: the universe. Right, I am interested in physics because I
• 16:31
  Speaker 1: want to know what's out there and what's real, not
• 16:33
  Speaker 1: because I want to build a complicated mathematical construct that
• 16:37
  Speaker 1: I can use to play around with in my head
• 16:39
  Speaker 1: and with my friends. Right, I want to know what's
• 16:41
  Speaker 1: actually out there. So, in the question of mathematics, that
• 16:44
  Speaker 1: brings up this issue like our numbers real, are these
• 16:47
  Speaker 1: proofs real, or are they just a game that we
• 16:50
  Speaker 1: have invented? And to make it more concrete, I like
• 16:52
  Speaker 1: to think about it in terms of like aliens. You know,
• 16:55
  Speaker 1: if alien scientists showed up here. Could we talk to
• 16:59
  Speaker 1: them about out the things in physics that we've discovered?
• 17:02
  Speaker 1: If the things we've discovered are real and part of
• 17:04
  Speaker 1: the universe, then yes, if they're just like in our minds,
• 17:07
  Speaker 1: then no, they would have different ways of thinking about it.
• 17:09
  Speaker 1: So can we take that same sort of question and
• 17:12
  Speaker 1: ask it about mathematics and ask like what would aliens
• 17:15
  Speaker 1: have developed mathematics if it's part of the universe, or
• 17:19
  Speaker 1: where they have some other way of like putting together
• 17:21
  Speaker 1: structured thought to figure out the universe? If, in fact,
• 17:24
  Speaker 1: mathematics is just part of our minds and the way
• 17:26
  Speaker 1: we think, is that a reasonable way to think about
• 17:28
  Speaker 1: the questions of the philosophy of math? Yeah, I think
• 17:30
  Speaker 1: that's a very good way of putting it. You might
• 17:32
  Speaker 1: think so. For instance, some things in mathematics are just
• 17:36
  Speaker 1: artifacts of the way we are. So you might think
• 17:39
  Speaker 1: using based ten, for instance, that's you know, to do
• 17:42
  Speaker 1: with numbers of fingers and so on and so forth.
• 17:45
  Speaker 1: But something very natural about base two. And I don't
• 17:48
  Speaker 1: believe you would know more about this than me, but
• 17:50
  Speaker 1: I believe that it's thought that if you were going
• 17:52
  Speaker 1: to contact extraterrestrials, then the initial segments of the international
• 17:57
  Speaker 1: expansion of Pie in base two would be something that
• 18:02
  Speaker 1: an intelligent life form would recognize. That's assuming that sometimes
• 18:05
  Speaker 1: there's something really objective about Pie. It's hard to imagine
• 18:09
  Speaker 1: that that's just the kind of construct of ours. You know,
• 18:12
  Speaker 1: you think that surely pie turns up in the most
• 18:15
  Speaker 1: unexpected places. It's not just about the ratio of the
• 18:19
  Speaker 1: circumference to the diameter of a circle. That's the initial definition,
• 18:24
  Speaker 1: but you know, as you know, it turns up in
• 18:26
  Speaker 1: it's just about everything right everywhere, in complex analysis, in geometry.
• 18:31
  Speaker 1: So yeah, so that the thought that that's something that
• 18:33
  Speaker 1: would be recognized by another intelligent life form seems reasonable.
• 18:38
  Speaker 1: But that pushes you to this sort of objective point
• 18:41
  Speaker 1: about mathematics, that it seems to be something that's objectively true,
• 18:45
  Speaker 1: not just mental construction. You wouldn't expect another an alien
• 18:50
  Speaker 1: life form too recognize facts about Sherlock Holmes, for instance.
• 18:55
  Speaker 1: You know, the detective novel could be a universal kind
• 19:00
  Speaker 1: throughout that could be exists and all intelligent beings everywhere.
• 19:03
  Speaker 1: You know, I imagine sitting across the table from alien
• 19:06
  Speaker 1: mathematicians and introducing them to ours, and I can imagine
• 19:09
  Speaker 1: that it might be that the kind of elaborate constructs
• 19:12
  Speaker 1: that we've built calculus and geometry, they might have very
• 19:15
  Speaker 1: different ways of doing these kind of things. I mean,
• 19:18
  Speaker 1: even in the history of our mathematics, our path to
• 19:20
  Speaker 1: these sorts of things have been varied and could have
• 19:23
  Speaker 1: gone differently. But it feels like maybe at the heart
• 19:25
  Speaker 1: of it there could be something in common that if
• 19:27
  Speaker 1: you drill down to the core of mathematics, of fundamental
• 19:30
  Speaker 1: ideas on which everything else is built, maybe we could
• 19:33
  Speaker 1: compare those with alien mathematicians. How well have we done
• 19:36
  Speaker 1: in terms of like examining the foundations of our own mathematics,
• 19:40
  Speaker 1: of understanding what our castle is built on? You know,
• 19:43
  Speaker 1: what are the basic rules of mathematics? Great Question. I mean,
• 19:48
  Speaker 1: when we think about the foundations of mathematics, a lot
• 19:51
  Speaker 1: of it is this program of trying to construct other
• 19:54
  Speaker 1: bits of mathematics from some other mathematics. So set theory
• 19:59
  Speaker 1: or category, if the theory for prefer, but set theoryt'st
• 20:03
  Speaker 1: stick for that for the moment. There's beautiful constructions in
• 20:07
  Speaker 1: set theory where you can construct the natural numbers out
• 20:10
  Speaker 1: of sets and then you can construct ordered pairs of
• 20:13
  Speaker 1: natural numbers out of sets and then you can get
• 20:16
  Speaker 1: functions and so on and so forth. So you can
• 20:19
  Speaker 1: get a great deal of mathematics built just out of
• 20:22
  Speaker 1: set theory. Can you explain that to me? Like, how
• 20:24
  Speaker 1: do you get natural numbers out of sets? What does
• 20:26
  Speaker 1: that mean? So you just have a series of sets.
• 20:29
  Speaker 1: So you start with the empty set, right. So the
• 20:31
  Speaker 1: empty set is the set that has nothing in it.
• 20:33
  Speaker 1: You identify that with zero. Just call that zero. It's
• 20:38
  Speaker 1: not zero. That's an empty set, but let's let's just
• 20:40
  Speaker 1: humor me call that zero. Then you have the set
• 20:44
  Speaker 1: that contains the empty set that has one member, you know.
• 20:47
  Speaker 1: So it's the set that has the empty set inside it.
• 20:50
  Speaker 1: So it has one member. Let's call that one. This
• 20:53
  Speaker 1: is just arbitrary. We're just making this up as we go.
• 20:56
  Speaker 1: And then you collect all of the sets from the
• 20:58
  Speaker 1: previous stages and collect them together. So the next stage
• 21:02
  Speaker 1: you take the empty set plus the set that contains
• 21:06
  Speaker 1: the empty set that has two members in it. Suggestive
• 21:10
  Speaker 1: name for that one too. This is a construction due
• 21:13
  Speaker 1: to the mathematicians John Von Neuman, and they're called the
• 21:17
  Speaker 1: von Neuemen ordinals. So you can construct the natural numbers.
• 21:20
  Speaker 1: In this way, you can then define, you know, I
• 21:23
  Speaker 1: won't go into details, but you can define edition and
• 21:25
  Speaker 1: so forth in this set theoretic way. And what does
• 21:28
  Speaker 1: that accomplished for you? Now, instead of having zero, one
• 21:31
  Speaker 1: and two, you have these weird sets. Why is that
• 21:34
  Speaker 1: better or more foundational, or what have you learned from
• 21:37
  Speaker 1: doing that? That's the kind of really interesting question here.
• 21:40
  Speaker 1: In one sense, it's now no longer sort of transparent, right,
• 21:43
  Speaker 1: we're familiar with the natural numbers and sort of building
• 21:46
  Speaker 1: it out of these these sets. They get really is
• 21:49
  Speaker 1: you can imagine. They get really ugly really quickly. So
• 21:52
  Speaker 1: once you start talking about numbers like seventeen, it's hard
• 21:55
  Speaker 1: to write it down on the one page what that is.
• 21:58
  Speaker 1: But no one suggests thing that you need to use
• 22:01
  Speaker 1: these things instead of natural numbers. But it's an interesting
• 22:06
  Speaker 1: exercise that you can construct the natural numbers out of sets.
• 22:10
  Speaker 1: So you might think in a way sets are all
• 22:13
  Speaker 1: you need. Sets really like the fundamental particles of physics. Right,
• 22:17
  Speaker 1: no one says there are no tables and chairs and
• 22:21
  Speaker 1: there are no people. Well, some people say such things,
• 22:23
  Speaker 1: but just because we can show that people are made
• 22:26
  Speaker 1: out of fundamental particles no one says stop doing biology
• 22:31
  Speaker 1: or sociology or psychology handed all over particle physicists, because
• 22:35
  Speaker 1: it's all particles. It's an interesting discovery that we're all
• 22:38
  Speaker 1: made up out of these fundamental particles. So in that
• 22:42
  Speaker 1: sort of vein you might think it's interesting that you
• 22:44
  Speaker 1: can reconstruct almost all of mathematics out of set theory
• 22:49
  Speaker 1: like this. Not suggesting you do it that way, but
• 22:52
  Speaker 1: it's an interesting construction. And that may be that the
• 22:54
  Speaker 1: fundamental mathematical particles, as it were, our sets, I see.
• 22:59
  Speaker 1: So it's like reduct Shinism to say what bits are
• 23:02
  Speaker 1: fundamental and what bits emerge. If we can figure out
• 23:06
  Speaker 1: which bits are fundamental and then we can ask questions
• 23:08
  Speaker 1: just about those and try to get some insight into,
• 23:10
  Speaker 1: like the actual nature of mathematics. So does that work?
• 23:15
  Speaker 1: I mean, can you say I'm going to start with
• 23:17
  Speaker 1: sets and from that build everything geometry and into roles
• 23:21
  Speaker 1: and differential equations? Can you base all of mathematics on
• 23:25
  Speaker 1: these weird sets? Yes, I mean with certain caveats. Are are
• 23:29
  Speaker 1: are a couple of little areas of mathematics that don't
• 23:32
  Speaker 1: succumb to this primarily category theory. But but set that
• 23:36
  Speaker 1: aside all of the mathematics that most of us know
• 23:39
  Speaker 1: and love. You can build out of sets in this
• 23:42
  Speaker 1: kind of way, and that's, you know, again, that's just
• 23:45
  Speaker 1: an interesting fact about mathematics. It demonstrates, firstly, the power
• 23:50
  Speaker 1: of set theory, but really it's such a versatile tool
• 23:55
  Speaker 1: set theory. Secondly, you know, it does lend support to
• 23:58
  Speaker 1: this idea that sets are the equivalent of the fundamental
• 24:01
  Speaker 1: particles and mathematics. And again, business is usual for topology
• 24:05
  Speaker 1: and all the other areas of mathematics. Not suggesting they
• 24:08
  Speaker 1: quit and go and do set theory instead, but rather
• 24:11
  Speaker 1: it's an interesting fact that their area can be reduced
• 24:14
  Speaker 1: in this, you know, admittedly cumbersome way, just as reducing
• 24:19
  Speaker 1: a table or a chair to fundamental particles. Try and
• 24:23
  Speaker 1: do that in particle physics, give the full description of
• 24:26
  Speaker 1: what a table is in particles. Right, if it's possible
• 24:29
  Speaker 1: at all, it's going to be incredibly cumbersome and not
• 24:32
  Speaker 1: terribly useful to you furniture removalists and other people working
• 24:36
  Speaker 1: with furniture. All right, well, I have a lot more
• 24:38
  Speaker 1: questions about the foundations of mathematics, but first let's take
• 24:42
  Speaker 1: a quick break. Okay, we are back and we're talking
• 24:57
  Speaker 1: to professor Mark Collivan about the fundamental part of goals
• 25:00
  Speaker 1: of mathematics and he is suggesting that if aliens arrive
• 25:05
  Speaker 1: and we are sitting across the table from their mathematicians,
• 25:07
  Speaker 1: that we might be able to talk to them about
• 25:09
  Speaker 1: the foundations of mathematics, which may be built out of sets.
• 25:14
  Speaker 1: We understand now that these sets follow some of the
• 25:16
  Speaker 1: rules that we identify with, for example basic arithmetic, and
• 25:19
  Speaker 1: that from that you can build everything else. So then,
• 25:23
  Speaker 1: what does it mean that the fundamental units of math
• 25:26
  Speaker 1: are sets? Does that mean that sets are real in
• 25:28
  Speaker 1: some way, or does it just mean that if sets
• 25:31
  Speaker 1: are real, then everything else is real? Or if those
• 25:34
  Speaker 1: rules about sets are real, they're from the universe, then
• 25:38
  Speaker 1: we can rely on everything else being true? Is that
• 25:41
  Speaker 1: the situation? Yes, so I think there are realists in
• 25:44
  Speaker 1: mathematics and those people will say maybe there's just sets,
• 25:49
  Speaker 1: but the sets are at least real. You've got to
• 25:51
  Speaker 1: sort of think that the fundamental furniture of the universe includes,
• 25:55
  Speaker 1: you know, all of the things that your particle for
• 25:57
  Speaker 1: physicists tell us about, plus sets. And the anti realists
• 26:02
  Speaker 1: about mathematics say no, the sets are some sort of
• 26:05
  Speaker 1: construction and their their role is to build this edifice
• 26:11
  Speaker 1: of mathematics upon sets. But that doesn't tell us anything
• 26:15
  Speaker 1: about the nature of sets, whether they're real or not.
• 26:19
  Speaker 1: The action, I think, in modern debates in philosophy mathematics
• 26:22
  Speaker 1: turns to applications of mathematics pretty quickly. Then you can
• 26:26
  Speaker 1: think about it in parallel with physics. Why is it
• 26:29
  Speaker 1: that we believe in certain bits of physics and not others?
• 26:33
  Speaker 1: So take bits of physics that are more speculative at
• 26:37
  Speaker 1: the moment. I take string theory to be such an area.
• 26:41
  Speaker 1: Some people believe in strings, some people don't, and it's
• 26:45
  Speaker 1: yet to be settled. Happy to take pure advice on this,
• 26:47
  Speaker 1: but that's my understanding of the current state of play there.
• 26:50
  Speaker 1: And why is it that people are not concerned about
• 26:54
  Speaker 1: other particles like electrons? Well, because electrons do a lot
• 26:59
  Speaker 1: of work in of theories and it's hard to imagine
• 27:02
  Speaker 1: any of our current physical theories functioning with our electrons
• 27:05
  Speaker 1: or something very much like them, whereas there are alternatives.
• 27:10
  Speaker 1: So some of the more speculative parts of physics now,
• 27:12
  Speaker 1: the speculty parts, very often get settled down the track somewhere,
• 27:17
  Speaker 1: but that's how we go about deciding whether things are real.
• 27:20
  Speaker 1: In physics, it seems right. Is it indispensable to sort
• 27:24
  Speaker 1: of greater physical theory? So people have turned to mathematics
• 27:27
  Speaker 1: with that same kind of view and thought, okay, for mathematics.
• 27:31
  Speaker 1: What would it take for mathematics to be real? Well,
• 27:33
  Speaker 1: perhaps if it's indispensable to our best science, that's a
• 27:38
  Speaker 1: clue that is real. It's not just some constructor of
• 27:41
  Speaker 1: the Human Mind, for instance. So this line of thought,
• 27:45
  Speaker 1: often called it indispensability argument. If mathematics is indispensable to science,
• 27:50
  Speaker 1: the bits of mathematics that are in fact indispensable should
• 27:54
  Speaker 1: have the same status as the science itself. It's certainly
• 27:59
  Speaker 1: a common feeling among physicists that mathematics is somehow the
• 28:04
  Speaker 1: language of the universe itself, because we can so effectively
• 28:08
  Speaker 1: describe these rules of physics in terms of this language.
• 28:12
  Speaker 1: You know, Stephen Weinberg said it's positively spooky how the
• 28:15
  Speaker 1: physicist finds a mathematicians has been there before him or her.
• 28:19
  Speaker 1: And you know there are often these cases, especially in
• 28:22
  Speaker 1: particle physics, where we struggle to understand something then we
• 28:25
  Speaker 1: discover this some bit of mathematics like group theory invented
• 28:28
  Speaker 1: just because of the curiosity of mathematicians playing games basically
• 28:32
  Speaker 1: in their minds, turns out to be applicable to the
• 28:34
  Speaker 1: world in a gorgeous way that clicks into place and
• 28:37
  Speaker 1: suddenly gives us insight. And those moments, I mean that
• 28:39
  Speaker 1: they're not religious moments or spiritual moments, but there are
• 28:43
  Speaker 1: moments where you feel like you've gained some deep insight
• 28:45
  Speaker 1: into the way the universe works, and it doesn't feel
• 28:48
  Speaker 1: like here's a useful description of the universe, it feels
• 28:51
  Speaker 1: like you're revealing the inner mechanisms of the universe itself.
• 28:55
  Speaker 1: But how can we know the difference right? How can
• 28:58
  Speaker 1: we tell whether these things are real, not just the
• 29:02
  Speaker 1: physical particles we're talking about, but the mathematics that describes them?
• 29:06
  Speaker 1: How do we distinguish between whether real or not before
• 29:09
  Speaker 1: we meet alien scientists? So I'm very interested in this
• 29:12
  Speaker 1: argument you suggest about the indispensability of mathematics, and I
• 29:17
  Speaker 1: read a book recently called Science Without Numbers, much, which
• 29:20
  Speaker 1: I'm sure you're familiar with, by hartree field, because he
• 29:22
  Speaker 1: suggests that math is very, very useful, you know, like
• 29:26
  Speaker 1: the way making it to do list is a good
• 29:28
  Speaker 1: way to organize your day, but you could probably get
• 29:30
  Speaker 1: through your day without it. But it's not actually necessary,
• 29:33
  Speaker 1: and he says in this book, and I'll quote him
• 29:35
  Speaker 1: because I find this outrageous, I am denying that numbers
• 29:38
  Speaker 1: or any similar entities exist. What a statement to make.
• 29:43
  Speaker 1: Can you help us wrap our minds around this opposite argument,
• 29:46
  Speaker 1: the one that suggests that we don't actually need math,
• 29:48
  Speaker 1: that it's useful but not indispensable. How do we make
• 29:51
  Speaker 1: sense of that? Yeah, I mean, let me say from
• 29:53
  Speaker 1: the get go I just think that's a fantastic book.
• 29:56
  Speaker 1: It's I disagree with heartree field on these issues, but
• 30:00
  Speaker 1: it is one of the absolute gems in philosophy of mathematics.
• 30:04
  Speaker 1: That book. It's, you know, outrageous, audacious, incredible project. I
• 30:09
  Speaker 1: mean so in response to this indispensability argument that mathematics,
• 30:14
  Speaker 1: you know, you can think of this argument of the
• 30:16
  Speaker 1: following form. We ought to believe in all of them,
• 30:19
  Speaker 1: only the entities that are indispensable to our best science.
• 30:22
  Speaker 1: So just should we believe in electrons? Well, just go
• 30:25
  Speaker 1: and see. Are they indispensable? Could you do science without electrons? No,
• 30:29
  Speaker 1: you can't. Could you do science without point masses? Well,
• 30:34
  Speaker 1: it would be difficult, but you could recast all talk
• 30:37
  Speaker 1: of point masses a little bit more carefully. Right. Could
• 30:40
  Speaker 1: you do science without coffee? No, for a coffee is three.
• 30:45
  Speaker 1: I think it was the old joke about the mathematician
• 30:48
  Speaker 1: being a machine that turns caffeine into theorems. You've solved
• 30:54
  Speaker 1: the question of the philosophy of math. What is mathematics exactly?
• 30:57
  Speaker 1: It's turning coffee into papers. We're telling us about the
• 31:00
  Speaker 1: argument about indispensability. Right, the indispensability argument says, you know
• 31:04
  Speaker 1: is something indispensable. That's really all you need to do.
• 31:07
  Speaker 1: So you look to your science. You don't need to
• 31:09
  Speaker 1: philosophize too much about this. In a way, just philosophizing
• 31:12
  Speaker 1: is done by recognizing that things that are indispensable to
• 31:16
  Speaker 1: your best scientific theories. That's what you ought to be
• 31:18
  Speaker 1: committed to. All right, but let's explore that a little
• 31:20
  Speaker 1: bit more deeply actually, before we get back to hartshred
• 31:23
  Speaker 1: field this indispensability argument, because there's some wrinkles there that
• 31:26
  Speaker 1: I don't really understand. I read like Putnam and quine
• 31:29
  Speaker 1: arguing that because our best theories are mathematical and those
• 31:34
  Speaker 1: theories are confirmed that's sort of like also confirms the
• 31:37
  Speaker 1: mathematics as you go along with it. Like if you
• 31:39
  Speaker 1: have electrons in your theory and your theory works, then
• 31:42
  Speaker 1: you believe in electrons. Well, if your theory also has
• 31:44
  Speaker 1: math as part of it, you're adding numbers, then you
• 31:47
  Speaker 1: that sort of like comes along with the proof. But
• 31:50
  Speaker 1: you know, that makes me wonder about things like infinity.
• 31:53
  Speaker 1: You know, we can do experiments in the universe to
• 31:55
  Speaker 1: explore particles. But as far as we know there's a
• 31:58
  Speaker 1: certain number of particles in the univer like tender to
• 32:01
  Speaker 1: eat or something, depending on how many you count. Does
• 32:03
  Speaker 1: that mean that only numbers up to tend to the
• 32:05
  Speaker 1: EAD are real and indispensable and numbers bigger than that,
• 32:09
  Speaker 1: like infinity, are not real or just parts of our mind?
• 32:12
  Speaker 1: You need to be careful there. I mean that's not
• 32:15
  Speaker 1: the only application of infinity. Right. So if you think
• 32:18
  Speaker 1: space time is continuous, as it is treated in general relativity,
• 32:22
  Speaker 1: at least quantum mechanics, it's argument that it's treated discreetly there,
• 32:27
  Speaker 1: but at least in general relativity spacetime is treated continuously.
• 32:32
  Speaker 1: So how many space time points are there right continuum? Many,
• 32:36
  Speaker 1: you know, not just the basic infinity there. That's, in
• 32:39
  Speaker 1: mathematical terms, to do the able of zero. That's the
• 32:42
  Speaker 1: infinity of the continuum. So it's not just a number
• 32:45
  Speaker 1: of particles. But are you going to need infinity in
• 32:48
  Speaker 1: other places? Try and do probability theory without continuous distributions, right.
• 32:55
  Speaker 1: So infinity crops up in all sorts of places in science,
• 32:59
  Speaker 1: not just think things. So that argument allows us to
• 33:02
  Speaker 1: believe in numbers and also believe in numbers like up
• 33:05
  Speaker 1: to infinity. They are real, but that's only if we
• 33:07
  Speaker 1: actually need them in our science. If we could do
• 33:10
  Speaker 1: the science without the numbers, then we wouldn't be able
• 33:12
  Speaker 1: to necessarily argue that the numbers are also real. So
• 33:15
  Speaker 1: help us understand Hartley field's argument that we don't need
• 33:18
  Speaker 1: numbers to do science right. So that's the starting point.
• 33:21
  Speaker 1: Is this this argument that you know science, you've committed
• 33:25
  Speaker 1: to everything that's indispensable to your best science. Taken for
• 33:29
  Speaker 1: granted that mathematics is indispensable for science. So the action
• 33:33
  Speaker 1: was really kind of on the first premise. Do you
• 33:35
  Speaker 1: really want to believe in everything in your best science?
• 33:37
  Speaker 1: We've got fictionalist planes. What about inertial restframes and so on?
• 33:41
  Speaker 1: And so forth. But heartrey field came along and said
• 33:44
  Speaker 1: maybe you could do science without numbers, maybe you could
• 33:47
  Speaker 1: just be realist about space time itself instead of having
• 33:51
  Speaker 1: the basic idea is, instead of treating space time in
• 33:55
  Speaker 1: this mathematical way, you just deal with space time itself.
• 34:00
  Speaker 1: So you're realist about the space time as an entity,
• 34:02
  Speaker 1: as it were, rather than treating space time as this
• 34:07
  Speaker 1: mathematical structure that has metrics and coordinate systems and so
• 34:11
  Speaker 1: on and so forth. Okay, so Hartley field hasn't smoked
• 34:13
  Speaker 1: so many bin in appeals that he doesn't believed in
• 34:15
  Speaker 1: the universe. He says space is real. Time is real,
• 34:19
  Speaker 1: but a mathematical description of that is not necessarily real.
• 34:22
  Speaker 1: Is that where we are? That's right, but in order
• 34:25
  Speaker 1: to say that you can't just I mean, you can
• 34:27
  Speaker 1: just say it, but for anyone to believe what you're
• 34:29
  Speaker 1: saying you've got to deliver the goods. You've got to
• 34:32
  Speaker 1: show how you can do something like newtonian mechanics. Is
• 34:36
  Speaker 1: that his case? Study show how you can do at
• 34:39
  Speaker 1: least the differential fragment of newtonian mechanics without talking about
• 34:45
  Speaker 1: anything mathematical. So just looking at relational properties of space
• 34:49
  Speaker 1: time points. And that's the basic trick and it's surprisingly
• 34:54
  Speaker 1: how far you can go with that. How is that
• 34:56
  Speaker 1: possible at all? I mean, if I think about Newtonian mechanics,
• 34:59
  Speaker 1: the first thing I comes to my mind is f
• 35:01
  Speaker 1: equals g MM over our squared. It's about relative distances,
• 35:05
  Speaker 1: it's about masses, it's about forces. If you throw that out,
• 35:08
  Speaker 1: what do you have left? Well, the thing is you
• 35:10
  Speaker 1: don't throw it out, you reconstruct it in a much
• 35:13
  Speaker 1: more direct or indirect, depending on how you're looking at it, way.
• 35:18
  Speaker 1: So instead of thinking about, for instance, a point in
• 35:22
  Speaker 1: space having a gravitational potential and the gravitational potential function,
• 35:28
  Speaker 1: then is this map from the space time to real numbers. Right,
• 35:32
  Speaker 1: that's the standard presentation. Newton would have talked in those terms,
• 35:35
  Speaker 1: of course, but now we think of it as a spacetime,
• 35:38
  Speaker 1: clearly in manifold, and you map from that to real
• 35:42
  Speaker 1: numbers and that's just your gravitational potential function. Archery field
• 35:47
  Speaker 1: shows that you can do that directly. Just think about
• 35:50
  Speaker 1: compare space time points with respect to their gravitational potential,
• 35:55
  Speaker 1: not having a gravitational potential function. That sits on top
• 35:58
  Speaker 1: of that and what shows is by doing it this
• 36:01
  Speaker 1: way you can recover the standard presentation. So you can
• 36:05
  Speaker 1: actually prove these results that show that you get everything
• 36:09
  Speaker 1: back that you would have had in the standard presentation.
• 36:12
  Speaker 1: So again, the take home message from field is not
• 36:16
  Speaker 1: you should be doing it this way rather than the
• 36:18
  Speaker 1: way everyone has done it. It's just big like the
• 36:21
  Speaker 1: story with sets. Right. The fact that you can do
• 36:24
  Speaker 1: it gives you evidence that the mathematics is not indispensable.
• 36:28
  Speaker 1: It's just a nice, quick and a much more elegant
• 36:33
  Speaker 1: way of doing it, but it's not indispensable and you
• 36:36
  Speaker 1: can recover everything. You can prove that you can recover
• 36:38
  Speaker 1: everything that you get in standard new Tony Mechanics this way.
• 36:42
  Speaker 1: I understand that it's a useful way to answer the
• 36:45
  Speaker 1: question do we need math? By proving that you could
• 36:48
  Speaker 1: do without it if you had to, doesn't mean that
• 36:50
  Speaker 1: you should do without it. Right, it's like asking a
• 36:52
  Speaker 1: question of could you live without jelly beans? You could
• 36:55
  Speaker 1: go without them for a year and you could prove
• 36:56
  Speaker 1: that you don't need to eat them. It doesn't mean
• 36:58
  Speaker 1: that nobody should eat jelly beans. But I'm still not convinced.
• 37:01
  Speaker 1: I mean your description here of his formulation of gravity
• 37:05
  Speaker 1: includes things like comparing potentials, and to me potentials are
• 37:09
  Speaker 1: numbers and comparing is a relationship. Are you seeing? Those
• 37:13
  Speaker 1: things are not mathematical or they're just not numbers. Yeah,
• 37:17
  Speaker 1: you've got to be realist about the points themselves, and
• 37:20
  Speaker 1: that's one of the criticisms of field is that you've
• 37:23
  Speaker 1: got to be realist about space time points and that
• 37:26
  Speaker 1: those things have properties. They have primitive properties like the
• 37:30
  Speaker 1: gravitational potential, electromagnetic potential and so on. So they have
• 37:33
  Speaker 1: those properties. So rather than those being a mathematical function
• 37:37
  Speaker 1: that lives on top of that, it's just these primitive
• 37:40
  Speaker 1: properties of the space time points. And so a lot
• 37:42
  Speaker 1: of people who are concerned about believing in mathematical objects,
• 37:46
  Speaker 1: because after all, that's kind of spooky, believing in the
• 37:49
  Speaker 1: spacetime points is also rather spooky. Right. It's not enough
• 37:54
  Speaker 1: for field to just believe in the manifold has actually
• 37:57
  Speaker 1: got to believe that individual points have these properties. But
• 38:01
  Speaker 1: whichever way you go on this, and as I said,
• 38:02
  Speaker 1: I disagree with him about the upshot of all this,
• 38:05
  Speaker 1: but the exercise itself is just incredible. You know, before
• 38:09
  Speaker 1: he did this, no one would have thought did you
• 38:11
  Speaker 1: even get started there. So you've got a lot of criticism.
• 38:14
  Speaker 1: People are saying, oh well, what about Hamiltonian formulations of
• 38:17
  Speaker 1: classical theories? What about chroantum mechanics where the underlying space
• 38:21
  Speaker 1: is infinite dimensional, Hilbert spaces, and so on and so forth,
• 38:24
  Speaker 1: and these are all fair and interesting criticisms. But before
• 38:28
  Speaker 1: he started no one thought you could do the differential
• 38:30
  Speaker 1: fragment of Newtonian mechanics either, you know. So for a
• 38:33
  Speaker 1: great deal of time, a great deal of the the
• 38:35
  Speaker 1: debate was about how far can you go with this
• 38:38
  Speaker 1: field style program because if you can go take it further,
• 38:42
  Speaker 1: then that's going to suggest that mathematics isn't indispensable after all.
• 38:47
  Speaker 1: So we have, on one hand, folks arguing that mathematics
• 38:51
  Speaker 1: is beautiful and elegant and unreasonably effective and, beyond that,
• 38:56
  Speaker 1: actually indispensable to understand in the universe and hurtlee field,
• 39:00
  Speaker 1: and some folks suggesting that maybe it's just useful but
• 39:03
  Speaker 1: not actually necessary. So where do you come down on that?
• 39:05
  Speaker 1: You're a philosopher of mathematics. You've thought deeply about these things.
• 39:09
  Speaker 1: Do you think that numbers are real? Are they just
• 39:12
  Speaker 1: part of our minds or are they something we found
• 39:14
  Speaker 1: in the universe? I'm a realist about mathematics, so I
• 39:17
  Speaker 1: come down on the former. So I think that mathematics
• 39:20
  Speaker 1: is in fact indispensable to our best scientific theories, and
• 39:23
  Speaker 1: that's why I take mathematical into at least some mathematics,
• 39:27
  Speaker 1: that I think they can still be speculative parts of
• 39:30
  Speaker 1: mathematics that we don't have reason to believe yet. And
• 39:34
  Speaker 1: so what convinces you? Just as there are, respectively, parts
• 39:36
  Speaker 1: of physics right that we don't believe yet. So you know,
• 39:40
  Speaker 1: you might think some of the higher reaches of set theory,
• 39:42
  Speaker 1: they're hard to go into the details here now, but
• 39:45
  Speaker 1: there are bizarre higher reaches of set theory that don't
• 39:49
  Speaker 1: look like they have any direct applications anywhere yet. You know,
• 39:52
  Speaker 1: maybe if they do, then then you'd be realist about those.
• 39:55
  Speaker 1: But it's not a blanket argument that, because mathematics is indispensable,
• 40:00
  Speaker 1: believe in all of it right. It's got to be
• 40:02
  Speaker 1: the bits that get applied, and that's one of the
• 40:05
  Speaker 1: criticisms of this line of thought. You don't get realism
• 40:09
  Speaker 1: about mathematics. You get realism about Calculus, you get realism
• 40:12
  Speaker 1: about Algebraic Topology, realism about differential geometry. You know you're
• 40:16
  Speaker 1: going to get realists about the bits that get used
• 40:19
  Speaker 1: and not all of it Um but that's for me.
• 40:22
  Speaker 1: That's as it should be. Just because physics is in
• 40:25
  Speaker 1: the business of describing the universe doesn't mean we should
• 40:28
  Speaker 1: believe all of physics. We should believe the bits of
• 40:31
  Speaker 1: physics that is really indispensable to our understanding of the universe.
• 40:34
  Speaker 1: And as you know, there are going to be speculative
• 40:36
  Speaker 1: parts of physics, not just things like string theory, but
• 40:39
  Speaker 1: nonphysical models. So, for instance, massless universes. People study things
• 40:44
  Speaker 1: like a universe with no mass. Can you have curvature
• 40:47
  Speaker 1: in the universe with no mass, for instance? Not that
• 40:49
  Speaker 1: we live in such a universe. We know we don't
• 40:51
  Speaker 1: mean one of those universes, but the question is that
• 40:54
  Speaker 1: will give us some understanding about our own universe if
• 40:56
  Speaker 1: we can study these non existent universes. So same with mathematics.
• 41:01
  Speaker 1: They're gonna be speculative parts of mathematics like that that
• 41:04
  Speaker 1: even someone like me is not going to be a
• 41:06
  Speaker 1: realist about. But I am not convinced at the end
• 41:09
  Speaker 1: of the day by heart tree fields project, despite the
• 41:12
  Speaker 1: fact that I think it's a fascinating and, you know,
• 41:15
  Speaker 1: beautiful technical exercise. And so I understand that being convinced
• 41:19
  Speaker 1: about the realism of sets doesn't mean that all of
• 41:21
  Speaker 1: mathematics is real. But what is it that convinces you
• 41:24
  Speaker 1: about the realism of sets? So what is the argument
• 41:26
  Speaker 1: that persuades you? Is it the usefulness of mathematics? Is that,
• 41:30
  Speaker 1: you know, seeing something beautiful in nature and seeing, you know,
• 41:34
  Speaker 1: mathematics in it, the FIBONACCI sequence or, you know, the
• 41:36
  Speaker 1: golden ratio? What is it that convinces you that mathematics
• 41:39
  Speaker 1: is real and not just a human construct? Well, it
• 41:42
  Speaker 1: comes back to this indispensability that we just I can't
• 41:45
  Speaker 1: see how we could do science without mathematics. And, moreover,
• 41:50
  Speaker 1: it's not just that it's this language of science, as
• 41:53
  Speaker 1: you often hear. I'm not quite sure what that really
• 41:56
  Speaker 1: even means. I mean it's not like, you know, once
• 41:59
  Speaker 1: upon a time all academic work had to be carried
• 42:01
  Speaker 1: out into Latin and Latin was the language of science.
• 42:04
  Speaker 1: It's not that that people are talking about it like
• 42:07
  Speaker 1: there's something much deeper about mathematics than Latin. No one
• 42:10
  Speaker 1: for a moment really thought that you couldn't do science
• 42:13
  Speaker 1: in English or whatever. It had to be in Latin.
• 42:16
  Speaker 1: That when people say mathematics is the language of science,
• 42:18
  Speaker 1: they mean something much deeper than that, I take it.
• 42:21
  Speaker 1: And one of the ways I think that mathematics is
• 42:24
  Speaker 1: indispensable is in offering up explanations. So this is very controversial.
• 42:29
  Speaker 1: Great deal of debate in philosophy mathematics about this at
• 42:32
  Speaker 1: the moment, about whether you can get explanations in physics.
• 42:38
  Speaker 1: Say That mathematically in character. So the mathematics is not
• 42:42
  Speaker 1: merely just his language, that is providing explanations for what's
• 42:47
  Speaker 1: going on in the physical world and, as I said,
• 42:51
  Speaker 1: very controversial. I do believe that. I do think that
• 42:54
  Speaker 1: mathematics is offering explanations and that's a really important way
• 42:58
  Speaker 1: in which you can be in to spend sable. If
• 43:01
  Speaker 1: something's it play an explanatory role in your best scientific theory,
• 43:05
  Speaker 1: then it really does look like you should be a
• 43:07
  Speaker 1: realist about it. If you'RE gonna be realist about anything,
• 43:10
  Speaker 1: and I mean there are anti realists about a great
• 43:12
  Speaker 1: deal of science as well, but I'm setting that aside.
• 43:15
  Speaker 1: If you're going to be realist about science, then it's
• 43:18
  Speaker 1: very odd to say that the reason for such and
• 43:21
  Speaker 1: such an event occurring was because of some entity, but
• 43:27
  Speaker 1: there's no such entity. You Haven't explained anything if you
• 43:30
  Speaker 1: say that. So if mathematics can played this kind of
• 43:32
  Speaker 1: explanatory role, then that looks like really good grounds for
• 43:37
  Speaker 1: thinking that it's indispensable to scientific explanation. It is really
• 43:41
  Speaker 1: compelling to me that mathematics can describe not just the
• 43:45
  Speaker 1: fundamental bits of the universe and the fundamental elements of mathematics,
• 43:50
  Speaker 1: but also that it seems like we can find these
• 43:53
  Speaker 1: fairly simple mathematical stories to describe emergent things. You can
• 43:57
  Speaker 1: imagine living in a universe where the basic thing are
• 44:00
  Speaker 1: strings and there's math about strings. And that doesn't mean that,
• 44:03
  Speaker 1: even if you understood how strings work, that you could
• 44:05
  Speaker 1: use them to predict the path of a hurricane. Right.
• 44:07
  Speaker 1: It's incredibly complicated to go from fundamental bits, even from
• 44:10
  Speaker 1: water drops, to a hurricane, not to mention from strings. Right.
• 44:14
  Speaker 1: So strings don't really like provide any explanation of the
• 44:17
  Speaker 1: path of a hurricane, even if you knew what all
• 44:19
  Speaker 1: the strings were doing. Right. But you can zoom out
• 44:21
  Speaker 1: and find some higher level laws. You know, you find
• 44:24
  Speaker 1: fluid mechanics and you find gravitational rotational theorems about how
• 44:29
  Speaker 1: galaxies move. We can seem to do physics at these
• 44:32
  Speaker 1: higher levels even without understanding the little bits underneath. And
• 44:35
  Speaker 1: the same way it seems like we can find mathematics
• 44:38
  Speaker 1: that describes the universe, even if it's not necessarily connected
• 44:42
  Speaker 1: to those fundamental little bits of the universe. Why do
• 44:44
  Speaker 1: you think that is? Why do you think it is
• 44:46
  Speaker 1: that mathematical descriptions of the universe emerge at all these
• 44:50
  Speaker 1: different levels, even when they're not necessarily connected to each
• 44:53
  Speaker 1: other or easily built from one to the other? Again,
• 44:56
  Speaker 1: one of the bright puzzles and philosophy of mathematics. It's
• 44:59
  Speaker 1: this often col the unreasonable effectiveness problem. How is it
• 45:02
  Speaker 1: mathematics just turns up in all of these places, not
• 45:05
  Speaker 1: just fundamental physics, but in Chemistry and biology and psychology. No,
• 45:10
  Speaker 1: at which level you know? If you think of science
• 45:12
  Speaker 1: as these levels, from the fundamental to the more complex,
• 45:16
  Speaker 1: at every level you've got relevant mathematics that appears there
• 45:20
  Speaker 1: and again. I wish I had a good answer to that.
• 45:23
  Speaker 1: One suggestion is that when you know the old, the
• 45:26
  Speaker 1: old sort of addage that if hammer is your only tool,
• 45:30
  Speaker 1: then the whole world looks like a nail. Right. So
• 45:33
  Speaker 1: we've got differential equations, damn it, we're going to use
• 45:35
  Speaker 1: them everywhere, right. But that just doesn't wash with me.
• 45:39
  Speaker 1: It's not just that we're forcing everything to be thought
• 45:43
  Speaker 1: of in this framework of a particular bit of mathematics
• 45:46
  Speaker 1: like differential equations. And to be fair, there were times,
• 45:49
  Speaker 1: I think, where physics was a bit like that. Everything
• 45:52
  Speaker 1: had to be well behaved. Differential Equations Linear First Order,
• 45:58
  Speaker 1: and you're try and do as much as you can
• 45:59
  Speaker 1: with is because they were well understood. But I just
• 46:02
  Speaker 1: don't think that's how we work now. I mean there's
• 46:04
  Speaker 1: so many different branches of mathematics that are turning up
• 46:07
  Speaker 1: in all different places. As you mentioned earlier, group theory.
• 46:10
  Speaker 1: A number of places where you need group theory grows
• 46:13
  Speaker 1: by the day. It doesn't seem to be simply we
• 46:16
  Speaker 1: have these tools and we're going to use it, damn it.
• 46:19
  Speaker 1: It's more like these are the very tools that we
• 46:22
  Speaker 1: would need to do any such science. And again, what
• 46:26
  Speaker 1: does that tell us about the mathematics? Well, like it's
• 46:29
  Speaker 1: intritically connected to the physical world in this kind of
• 46:32
  Speaker 1: way that you just can't understand the physical world without, yeah,
• 46:38
  Speaker 1: having the relevant mathematics under control. You know, as I said,
• 46:42
  Speaker 1: that's controversial. Is just try and flag the things that
• 46:45
  Speaker 1: are more controversial. But since we're talking philosophy here, we
• 46:49
  Speaker 1: can just have a general disclaimer. All of this is controversy. Well,
• 46:54
  Speaker 1: we like to get into the weeds on this show,
• 46:56
  Speaker 1: and so I have a lot more deep questions about
• 46:58
  Speaker 1: philosophy of math but when to take another quick break?
• 47:14
  Speaker 1: All right, so we're back and we're having a lot
• 47:16
  Speaker 1: of fun talking to Professor Mark Coldevan about whether mathematics
• 47:20
  Speaker 1: is inherent in the universe and is it real. And
• 47:23
  Speaker 1: I was wondering when you were talking earlier, since philosophy
• 47:26
  Speaker 1: of mathematics asks like what is mathematics and is it real,
• 47:30
  Speaker 1: is there a branch of philosophy called philosophy of philosophy
• 47:34
  Speaker 1: that asks like whether philosophy is real and what our
• 47:36
  Speaker 1: philosophers doing? Anyway, there is, there is. Very recently people
• 47:41
  Speaker 1: have been working on philosophy of philosophy. I think you know,
• 47:45
  Speaker 1: white philosophers have been doing this for a long time.
• 47:47
  Speaker 1: I just didn't come up with the phrase, but not
• 47:49
  Speaker 1: so much the idea of where the philosophy is real,
• 47:51
  Speaker 1: because you're not interested where the physics is real. You're
• 47:54
  Speaker 1: interested in whether the things that physics posits are real. Right,
• 47:58
  Speaker 1: and so, in such far as philosophy is positing entities,
• 48:03
  Speaker 1: are those things real? You could ask that sort of question.
• 48:06
  Speaker 1: But my understanding at least, the philosophy of philosophy is
• 48:08
  Speaker 1: more a kind of a systematic study of methodology, right,
• 48:12
  Speaker 1: which is kind of how I think of philosophy of
• 48:15
  Speaker 1: science is in many ways looking at science and trying
• 48:20
  Speaker 1: to discern useful things to say about methodology. And so
• 48:25
  Speaker 1: philosophy of philosophy is much more about methodological questions about
• 48:29
  Speaker 1: philosophy questions like does philosophy make progress? So philosophy cops
• 48:35
  Speaker 1: criticism because the questions were interested in the questions we're
• 48:39
  Speaker 1: talking about here now. Are Numbers Real? That goes back
• 48:42
  Speaker 1: to at least back to Plato, right. And have we
• 48:45
  Speaker 1: made much progress since then? Well, you know, I'd like
• 48:49
  Speaker 1: to think we've made some, but certainly if you look
• 48:51
  Speaker 1: at progress physics has made since such times to now,
• 48:55
  Speaker 1: physics has not much better firm a ground then. Right,
• 48:59
  Speaker 1: that's true. Maybe you guys just need more coffee, although
• 49:02
  Speaker 1: you could also say that physics is just an outgrowth
• 49:04
  Speaker 1: of philosophy. When a question becomes experimental, it decrimes its
• 49:07
  Speaker 1: own science and philosophy sort of loses control of it.
• 49:10
  Speaker 1: But speaking of concrete questions, I want to come back
• 49:13
  Speaker 1: to the framing we had earlier about aliens. Do you
• 49:16
  Speaker 1: think that, if aliens arrived, that we could use mathematics
• 49:21
  Speaker 1: as a sort of basis for building a mental connection
• 49:25
  Speaker 1: with them, of understanding whether or not we're thinking in
• 49:28
  Speaker 1: a similar way as them? Would you send of mathematicians
• 49:31
  Speaker 1: or philosophers of math out to meet the aliens? First thing,
• 49:34
  Speaker 1: I do think that there would be a good place
• 49:36
  Speaker 1: to start. Would be bits of mathematics do you think
• 49:41
  Speaker 1: likely to be universal? I must say I haven't given
• 49:43
  Speaker 1: a lot of thought to who I would send first
• 49:45
  Speaker 1: to meet the aliens. You don't realize that you're near
• 49:48
  Speaker 1: the top of the list. That should concern me for
• 49:51
  Speaker 1: all sorts of reasons. But but yeah, I do think
• 49:55
  Speaker 1: the look looking for bits of mathematics that you think
• 49:59
  Speaker 1: would be calm. Again, you wouldn't want decimal expansion of
• 50:02
  Speaker 1: pie based ten, but expansion of pie based two. That's
• 50:08
  Speaker 1: something that you might think would be recognizable fundamental theorems
• 50:12
  Speaker 1: suitably couched, because you know that the notation you use
• 50:16
  Speaker 1: is perhaps arbitrary in various ways. But the fundamental theorem,
• 50:22
  Speaker 1: fundamental theorem of Calculus, for instance. You think that any
• 50:26
  Speaker 1: reasonably advanced life forms who are capable of traveling to
• 50:31
  Speaker 1: Earth from great distance would have come across the fundamental
• 50:35
  Speaker 1: theorems of Calculus. So how do you express those in
• 50:38
  Speaker 1: a way that's not merely notational dependent too much on
• 50:42
  Speaker 1: the notation? You can't express it in English, obviously, but
• 50:46
  Speaker 1: the standard notation using integral signs and so forth. I
• 50:49
  Speaker 1: think that's kind of accidental that. How do you get
• 50:52
  Speaker 1: that idea across? That does seem like a good place
• 50:55
  Speaker 1: to start. You'd think that an intelligent, advanced race would
• 50:58
  Speaker 1: know the fundamental theorem of Calculus, but how would they
• 51:01
  Speaker 1: write it and how how should you convey it to them? Well,
• 51:04
  Speaker 1: we had Noam Chomsky on the podcast a few weeks
• 51:07
  Speaker 1: ago and we asked him this question and he said,
• 51:11
  Speaker 1: I'll quote, there's a good chance that arithmetic is universal.
• 51:14
  Speaker 1: It's a fair guess that at least the arithmetic would
• 51:16
  Speaker 1: be close enough to be absolute, so that anything we
• 51:19
  Speaker 1: might call intelligence that we would recognize this intelligence would
• 51:22
  Speaker 1: at least sit on that. And I suppose that he's
• 51:24
  Speaker 1: making the argument that you're making that mathematics is probably fundamental.
• 51:28
  Speaker 1: And in addition, he's drilling down and he's saying, let's
• 51:30
  Speaker 1: not start with something complicated, let's go down to the basics,
• 51:33
  Speaker 1: like you were saying earlier, set theory, let's find the
• 51:36
  Speaker 1: fundamental elements and see if we can begin from that.
• 51:39
  Speaker 1: Do you think that program is likely to be successful
• 51:41
  Speaker 1: if aliens arrived? Yeah, yeah, I think that's it's a
• 51:44
  Speaker 1: very good suggestion. You know, again back to the something
• 51:46
  Speaker 1: like pie not just any old numbers, because you might
• 51:50
  Speaker 1: think that nothing special about one to three, four in particular,
• 51:55
  Speaker 1: but it's really crucial to number theory. Be a concept
• 52:01
  Speaker 1: of prime number, for instance. You might think certain numbers
• 52:04
  Speaker 1: jump out at you like prime numbers, Pie, e. Some
• 52:09
  Speaker 1: of these numbers in particular, and so if you can
• 52:11
  Speaker 1: get a way of expressing those numbers, but fundamental parts
• 52:14
  Speaker 1: of arithmetic. But the notion of primness again, how do
• 52:18
  Speaker 1: you imagine an intelligent, advanced race not having that concept? Again,
• 52:24
  Speaker 1: just how do you convey it? But I do like
• 52:27
  Speaker 1: the suggestion. Yeah, well, I'd love to examine the sort
• 52:29
  Speaker 1: of counter idea. Like to think about what an intelligent
• 52:34
  Speaker 1: race might have to have in their minds in order
• 52:36
  Speaker 1: to not arrive at Arithmetic. You know, I can imagine
• 52:38
  Speaker 1: sitting across from their mathematicians and drawing a symbol for
• 52:41
  Speaker 1: one and pointing at one thing and an apple, and
• 52:44
  Speaker 1: bringing another apple and then writing the symbol for two
• 52:47
  Speaker 1: or something, and you know, then you have one plus
• 52:49
  Speaker 1: one equals to this kind of stuff. And you must
• 52:51
  Speaker 1: have thought about this more deeply than I have. What
• 52:53
  Speaker 1: assumptions are there inherent in that? You know, are we
• 52:56
  Speaker 1: assuming that the concept of abstraction to say, are these
• 53:00
  Speaker 1: two apples? They have the similar property. They're both apples.
• 53:03
  Speaker 1: Obviously they're not the same apple. They're different ones, darker,
• 53:05
  Speaker 1: one brighter or whatever. What if the aliens are like,
• 53:08
  Speaker 1: you know, that's one apple and that's one apple. We
• 53:10
  Speaker 1: don't know what you mean with this whole two business.
• 53:12
  Speaker 1: That's nonsense. Aren't there fundamental assumptions we're making there, even
• 53:16
  Speaker 1: with one plus one equals too? Yeah, I think so.
• 53:18
  Speaker 1: I think it's exactly right. It's got to be counting
• 53:21
  Speaker 1: the right sorts of things right. So one cloud and
• 53:24
  Speaker 1: another cloud is one big cloud right. So you don't
• 53:29
  Speaker 1: think one cloud plus one cloud equals one. That's a
• 53:33
  Speaker 1: falsification of basic arithmetic. You think, no, you're counting the
• 53:36
  Speaker 1: wrong kinds of things. They're one plus one is too
• 53:39
  Speaker 1: given that you're counting the right kinds of things, discreet
• 53:41
  Speaker 1: things that have a certain kind of property, you can
• 53:45
  Speaker 1: still count an apple and an orange and get to
• 53:48
  Speaker 1: but then you've got to have this overarching concept of
• 53:51
  Speaker 1: pieces of fruit or things before me or something. But
• 53:54
  Speaker 1: there's got to be some overarching concept there. So you're
• 53:57
  Speaker 1: absolutely right. There are some conditions for even understanding basic arithmetic.
• 54:03
  Speaker 1: I was reading about how Japanese people count and discovering
• 54:07
  Speaker 1: that Japanese counting words are actually quite different from English
• 54:10
  Speaker 1: counting words, that if you put a set of things
• 54:12
  Speaker 1: in front of them, they tend to group them by shape.
• 54:15
  Speaker 1: So those are not just two things, these are two
• 54:17
  Speaker 1: flat things and there's two tall things over there. It
• 54:19
  Speaker 1: strikes me that if, even across human cultures we count
• 54:23
  Speaker 1: and abstract things in different ways, it might be that aliens,
• 54:26
  Speaker 1: I have no idea what we're talking about when we're
• 54:28
  Speaker 1: demonstrating our basic arithmetic to them. There are cases in
• 54:31
  Speaker 1: English as well where you're not interested in the number
• 54:34
  Speaker 1: of things for various reasons. She's just interested in existence.
• 54:38
  Speaker 1: So we say it's raining, meaning that there are drops
• 54:41
  Speaker 1: of rain falling, and that's all we care about. Just
• 54:44
  Speaker 1: want to know whether I need my umbrellas today or not.
• 54:46
  Speaker 1: So you can imagine someone putting apples in front of them.
• 54:48
  Speaker 1: They're saying it's appling. Right, I have the concept of
• 54:54
  Speaker 1: multiple apples, but I don't need to three or four.
• 54:57
  Speaker 1: Who Cares how many apples? There's zero apples and there's
• 55:01
  Speaker 1: appling again, and we do that. There are lots of
• 55:04
  Speaker 1: instances of that, like raining is the obvious example, but
• 55:08
  Speaker 1: there are other cases as well where you're interested in
• 55:11
  Speaker 1: zero or existence. You know of something and you don't
• 55:15
  Speaker 1: particularly worry about counting, even though you could. There are
• 55:19
  Speaker 1: rain drops are discreet. You could worry about how many
• 55:23
  Speaker 1: rain drops there are, and it's not purely because it's
• 55:26
  Speaker 1: a difficult exercise to count them. It's more just no,
• 55:29
  Speaker 1: I'm not really interested in how many there are. I'm
• 55:31
  Speaker 1: just interested is it raining or not. So again, you know,
• 55:34
  Speaker 1: you could have this idea of zero things or many.
• 55:40
  Speaker 1: You know, zero, one and many. You can imagine an
• 55:43
  Speaker 1: intelligent civilization getting by with a very different kind of
• 55:46
  Speaker 1: sense of what many is as many? You know, more
• 55:49
  Speaker 1: than a thousand or you know, is it just a few?
• 55:51
  Speaker 1: I find that people have a different sense like effective infinity.
• 55:55
  Speaker 1: You know what is a lot and what needs to
• 55:57
  Speaker 1: be counted individually. Anyway, these are really fast and any
• 56:00
  Speaker 1: questions and I really thank you for answering them and
• 56:02
  Speaker 1: for exploring these questions with me. I think my last
• 56:05
  Speaker 1: question to you is do we expect any sort of
• 56:07
  Speaker 1: breakthroughs in philosophy of mathematics? I mean, you said we've
• 56:10
  Speaker 1: been struggling with a question of our numbers and circles
• 56:13
  Speaker 1: real since Plato. Do you think we're going to figure
• 56:15
  Speaker 1: that out, that we ever have a point where we're
• 56:17
  Speaker 1: like yeah, we proved that, now we can move on
• 56:19
  Speaker 1: to something else, or is philosophy of math basically going
• 56:21
  Speaker 1: to go forever until we meet the Aliens? I would
• 56:24
  Speaker 1: like to think that it would be solved, but I
• 56:27
  Speaker 1: don't think that's necessary for it to be worthwhile exercise. Um.
• 56:33
  Speaker 1: So why I think that is we learn a lot
• 56:36
  Speaker 1: along the way. Sometimes asking the right questions is more
• 56:40
  Speaker 1: interesting than finding answers to them, and I don't think
• 56:42
  Speaker 1: there's any special about philosophy. You know, I think you
• 56:44
  Speaker 1: would find that in physics as well. Not being able
• 56:47
  Speaker 1: to answer some questions, but those questions giving rise to
• 56:50
  Speaker 1: other questions and new areas is what motivates us and
• 56:53
  Speaker 1: what keeps the disciplines rolling. It sounds to an outside
• 56:57
  Speaker 1: of that might sound rather close shop. As you know,
• 57:01
  Speaker 1: you're only interested in these little questions. We're interested in
• 57:04
  Speaker 1: getting the exercise rolling. But I do think that we've
• 57:07
  Speaker 1: learned a lot about the relationship between mathematics and the
• 57:10
  Speaker 1: physical realm, a lot of understanding about the foundations of mathematics.
• 57:14
  Speaker 1: We don't have the answer to what the foundations of
• 57:17
  Speaker 1: mathematics are, but you have some interesting insights from set
• 57:22
  Speaker 1: theory and the like. So are we like to to
• 57:25
  Speaker 1: solve the problems in my lifetime? I don't think so.
• 57:28
  Speaker 1: I hope not. I'd be out of a job, you know,
• 57:30
  Speaker 1: but I don't think it's going to happen. But I
• 57:33
  Speaker 1: don't think that that means that it's all a waste
• 57:36
  Speaker 1: of time. I think we get in many interesting insights.
• 57:38
  Speaker 1: In particular, the insights that motivated me early on in
• 57:41
  Speaker 1: my career were this connection that why is that? The
• 57:44
  Speaker 1: mathematics is applicable. I read this paper, famous paper by
• 57:48
  Speaker 1: Eugene Vigna called the unreasonable effectiveness mathematics and natural sciences,
• 57:53
  Speaker 1: and I read that as an undergraduate and it just
• 57:57
  Speaker 1: captivated by that paper. And did he answer the questions? No.
• 58:01
  Speaker 1: Has Anyone answered those questions? No, but fascinating stuff to
• 58:04
  Speaker 1: think about and I think we have a much better
• 58:06
  Speaker 1: understanding of the relationship between Applied Mathematics and physics now
• 58:10
  Speaker 1: as a result of asking me sort of other questions
• 58:14
  Speaker 1: about his mathematics. Real absolutely no. I think it's very
• 58:17
  Speaker 1: useful and also, in a way, maybe you haven't even
• 58:19
  Speaker 1: anticipated that. You have spent your life preparing, I think,
• 58:22
  Speaker 1: to meet the aliens and when they ask me who
• 58:25
  Speaker 1: we should send, you know, in our first contingent to
• 58:28
  Speaker 1: chat with our alien technological friends, I'm going to nominate you. Good,
• 58:32
  Speaker 1: as long as they're friendly, you know. Well, we'll find
• 58:36
  Speaker 1: out right just will send the military. You know, make
• 58:40
  Speaker 1: sure they're friendly first. All right, sounds good. Well, thank
• 58:43
  Speaker 1: you very much for joining us and for talking to
• 58:45
  Speaker 1: us about these crazy ideas. I hope that one day
• 58:47
  Speaker 1: we do figure out our numbers real why math works.
• 58:51
  Speaker 1: And if, in fact, math is just a game we
• 58:53
  Speaker 1: invented in our minds or the fundamental code of the
• 58:56
  Speaker 1: universe itself. Thanks very much for joining us today. My pleasure,
• 59:00
  Speaker 1: thanks for having me, thanks for listening, and remember that,
• 59:10
  Speaker 1: Daniel and Jorge explained, the universe is a production of
• 59:12
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• 59:20
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