1 00:00:08,680 --> 00:00:11,840 Speaker 1: We know a lot of things about the universe. We 2 00:00:11,920 --> 00:00:15,400 Speaker 1: know that everything around us is made of tiny little 3 00:00:15,480 --> 00:00:18,919 Speaker 1: particles that obey strange quantum rules. We know that our 4 00:00:18,920 --> 00:00:22,439 Speaker 1: planet moves through space curred by the mass of the Sun. 5 00:00:22,800 --> 00:00:24,560 Speaker 1: We know that the earth is four and a half 6 00:00:24,640 --> 00:00:27,520 Speaker 1: billion years old and that the universe is almost fourteen 7 00:00:27,600 --> 00:00:31,120 Speaker 1: billion years old, and all of that knowledge has something 8 00:00:31,240 --> 00:00:35,800 Speaker 1: in common. It's all expressed in terms of mathematics. Our 9 00:00:35,880 --> 00:00:40,120 Speaker 1: quantum theories, our ideas about gravity, our understanding of the 10 00:00:40,159 --> 00:00:43,559 Speaker 1: age of the earth and the universe all depend deeply 11 00:00:43,760 --> 00:00:46,280 Speaker 1: on math. And if we're going to dig deep into 12 00:00:46,320 --> 00:00:50,160 Speaker 1: the foundations of reality and see if we understand what's there, 13 00:00:50,440 --> 00:00:53,360 Speaker 1: shouldn't we do the same thing and ask ourselves some 14 00:00:53,479 --> 00:00:58,040 Speaker 1: hard questions about the mathematics, asking what it is, why 15 00:00:58,080 --> 00:01:02,200 Speaker 1: it works and whether it's even necessary? If math is 16 00:01:02,200 --> 00:01:05,080 Speaker 1: a language of physics, then how certain are we that 17 00:01:05,160 --> 00:01:09,200 Speaker 1: it reflects something true about the universe rather than something 18 00:01:09,240 --> 00:01:28,200 Speaker 1: about our minds? Hi, I'm Daniel. I'm a particle physicist 19 00:01:28,200 --> 00:01:31,360 Speaker 1: and a professor at UC Irvine, and I love speaking 20 00:01:31,480 --> 00:01:35,000 Speaker 1: the language of math. Sometimes, when I'm confused about how 21 00:01:35,120 --> 00:01:38,520 Speaker 1: something works, it's the math that leads me through and 22 00:01:38,640 --> 00:01:43,520 Speaker 1: shows me the answer. There's something wonderfully crisp about mathematics. 23 00:01:43,640 --> 00:01:46,960 Speaker 1: I love how the Patterns Click together with an exactness 24 00:01:46,959 --> 00:01:50,480 Speaker 1: and reliability. I love that it doesn't zag or break 25 00:01:50,600 --> 00:01:53,680 Speaker 1: or wilt, that six times nine is the same today 26 00:01:53,720 --> 00:01:57,440 Speaker 1: as it is tomorrow and will be forever and welcome 27 00:01:57,480 --> 00:02:00,800 Speaker 1: to the podcast Daniel and Jorge Explode in the universe, 28 00:02:00,840 --> 00:02:03,760 Speaker 1: where we dig deep into the universe around us and 29 00:02:03,840 --> 00:02:08,160 Speaker 1: try to find some answers. We have an unquenchable thirst 30 00:02:08,200 --> 00:02:12,600 Speaker 1: to understand and an insatiable appetite for asking questions. My 31 00:02:12,720 --> 00:02:15,080 Speaker 1: friend and Co host, Jorge is on vacation and so 32 00:02:15,160 --> 00:02:18,080 Speaker 1: today we are going to ask some of the deepest 33 00:02:18,320 --> 00:02:22,080 Speaker 1: of questions. Regular listeners know that we mostly talk about 34 00:02:22,120 --> 00:02:25,360 Speaker 1: the physics of the universe, but that sometimes we dig 35 00:02:25,400 --> 00:02:28,280 Speaker 1: a bit deeper and ask about the philosophy of it. 36 00:02:28,600 --> 00:02:32,079 Speaker 1: We don't just want to know what the fundamental particles are, 37 00:02:32,240 --> 00:02:35,160 Speaker 1: but we want to know why those particles, what does 38 00:02:35,200 --> 00:02:38,440 Speaker 1: it mean that it's these particles, and also what is 39 00:02:38,480 --> 00:02:41,360 Speaker 1: a particle and why is the universe made out of 40 00:02:41,400 --> 00:02:47,000 Speaker 1: them instead of something else? That's the philosophical side of physics. 41 00:02:47,320 --> 00:02:51,320 Speaker 1: Today we're gonna follow our noses all the way down 42 00:02:51,400 --> 00:02:55,320 Speaker 1: the philosophical rabbit hole and ask questions about what lies 43 00:02:55,440 --> 00:02:59,360 Speaker 1: underneath all of that. If you dig far enough into physics, 44 00:02:59,440 --> 00:03:02,880 Speaker 1: you always end up face to face with math. Our 45 00:03:02,919 --> 00:03:07,400 Speaker 1: equations are written in Math, our predictions and calculations are mathematical. 46 00:03:07,639 --> 00:03:12,160 Speaker 1: So math provides the bricks for building our castle of physics. 47 00:03:12,160 --> 00:03:16,120 Speaker 1: But that should intrigue us, that should inspire our curiosity. 48 00:03:16,400 --> 00:03:19,800 Speaker 1: What are these bricks, the numbers and shapes and functions 49 00:03:19,800 --> 00:03:22,320 Speaker 1: and sets that we use to build up our physics? 50 00:03:22,400 --> 00:03:26,280 Speaker 1: Where do these mathematical bricks come from? Where do they live? 51 00:03:26,560 --> 00:03:28,960 Speaker 1: Can we smash them together to learn about them? How 52 00:03:28,960 --> 00:03:32,320 Speaker 1: do we know what rules they follow? Is this the 53 00:03:32,400 --> 00:03:35,760 Speaker 1: only way to build physics, or could we have done 54 00:03:35,760 --> 00:03:40,560 Speaker 1: it without math? Would aliens use math in their theories? 55 00:03:40,800 --> 00:03:43,400 Speaker 1: And if you're a regular listener, you'll hear me saying 56 00:03:43,440 --> 00:03:47,040 Speaker 1: this all the time. It's amazing that math does describe 57 00:03:47,080 --> 00:03:50,080 Speaker 1: the universe, that it works so well, that we can 58 00:03:50,120 --> 00:03:54,600 Speaker 1: devise these beautiful and simple mathematical stories about the universe 59 00:03:54,640 --> 00:03:58,840 Speaker 1: and all different scenarios. Tiny particles seem to follow group theory, 60 00:03:59,200 --> 00:04:03,000 Speaker 1: rushing rivers obeyed differential equations and massive galaxies are bound 61 00:04:03,000 --> 00:04:05,840 Speaker 1: together by geometry. What does it mean that it works 62 00:04:05,880 --> 00:04:09,040 Speaker 1: so well? Is it something about how our mind works, 63 00:04:09,520 --> 00:04:12,880 Speaker 1: or is it something deep and true about the universe itself? 64 00:04:13,400 --> 00:04:16,400 Speaker 1: So today on the podcast will be answering the question. 65 00:04:21,560 --> 00:04:25,560 Speaker 1: Is Math the language of the universe? And to help 66 00:04:25,600 --> 00:04:28,159 Speaker 1: me soart through some of the slippery issues at the 67 00:04:28,200 --> 00:04:32,200 Speaker 1: heart of this deep question, is our guest, professor, Mark Coldy. 68 00:04:32,279 --> 00:04:35,400 Speaker 1: Van Mark is a professor of philosophy at the University 69 00:04:35,440 --> 00:04:38,960 Speaker 1: of Sydney in Beautiful Australia, where he thinks deeply about 70 00:04:39,000 --> 00:04:42,280 Speaker 1: these questions all day long. He's also an accomplished writer, 71 00:04:42,440 --> 00:04:46,720 Speaker 1: publishing the indispensability of mathematics and an introduction to the 72 00:04:46,720 --> 00:04:50,080 Speaker 1: philosophy of math, which I read recently cover to cover 73 00:04:50,320 --> 00:04:54,400 Speaker 1: and found to be very compelling and accessible. The title 74 00:04:54,440 --> 00:04:56,760 Speaker 1: makes it sound a little bit like a textbook from 75 00:04:56,760 --> 00:05:00,560 Speaker 1: an introductory philosophy course, but it's very conversation national and 76 00:05:00,800 --> 00:05:03,279 Speaker 1: very easy to read. I learned a lot and it 77 00:05:03,360 --> 00:05:05,520 Speaker 1: inspired me to invite mark to join us on the 78 00:05:05,560 --> 00:05:08,039 Speaker 1: podcast to chat about some of the questions at the 79 00:05:08,080 --> 00:05:14,080 Speaker 1: heart of mathematics. So it's my pleasure, then, to welcome 80 00:05:14,080 --> 00:05:17,240 Speaker 1: Professor Mark Colevin to the podcast. Mark, thanks very much 81 00:05:17,279 --> 00:05:19,880 Speaker 1: for joining us, thanks for having me, and I understand 82 00:05:19,920 --> 00:05:22,039 Speaker 1: while I have never been to your part of the world, 83 00:05:22,080 --> 00:05:24,640 Speaker 1: you have actually spent some time here in Irvine. Is 84 00:05:24,680 --> 00:05:27,479 Speaker 1: that right? That's right. Yeah, I had a visiting fellowship 85 00:05:27,640 --> 00:05:31,400 Speaker 1: in Irvine back in two thousand and one. So you 86 00:05:31,440 --> 00:05:33,880 Speaker 1: can compare for us then, the glorious weather of Orange 87 00:05:33,880 --> 00:05:37,440 Speaker 1: County with the weather of your local Sydney. Nothing compares 88 00:05:37,480 --> 00:05:41,280 Speaker 1: with the weather of Orange County. Is it's faculous every day? 89 00:05:41,440 --> 00:05:44,039 Speaker 1: Correct answer. Correct answer. All right, now that we have 90 00:05:44,120 --> 00:05:46,800 Speaker 1: your qualifications sorted out. Hell me, you're a philosopher of 91 00:05:46,880 --> 00:05:49,640 Speaker 1: math and I've never spoken to a philosopher of math before. 92 00:05:49,720 --> 00:05:52,479 Speaker 1: So tell me. What does a philosopher of math do 93 00:05:52,680 --> 00:05:55,039 Speaker 1: all day? I mean, is it reading and writing and 94 00:05:55,080 --> 00:05:58,760 Speaker 1: coffee and emails? What got you excited about philosophy of math? Well, 95 00:05:58,760 --> 00:06:01,680 Speaker 1: I started out in mathematics. The usual story for a 96 00:06:01,800 --> 00:06:05,279 Speaker 1: philosopher of mathematics. You start out in mathematics, you start 97 00:06:05,279 --> 00:06:11,760 Speaker 1: getting interested in certain questions in mathematics. That leads you 98 00:06:11,920 --> 00:06:15,880 Speaker 1: to more philosophical pondering and at some stage then you 99 00:06:16,080 --> 00:06:18,840 Speaker 1: defect to the dark side and become a philosopher, which 100 00:06:18,880 --> 00:06:23,200 Speaker 1: is what happened with me mathematics, that undergraduate an honors 101 00:06:23,279 --> 00:06:28,479 Speaker 1: level and then that PhD, switched to philosophy, mainly because 102 00:06:28,480 --> 00:06:32,000 Speaker 1: I was interesting questions about what counts as the right 103 00:06:32,120 --> 00:06:35,680 Speaker 1: logic for mathematics and what is a proof in mathematics, 104 00:06:35,800 --> 00:06:40,120 Speaker 1: and these are questions that mathematicians have a good handle on, 105 00:06:40,800 --> 00:06:44,080 Speaker 1: but mainly by doing them, I mean you're trained in 106 00:06:44,120 --> 00:06:48,160 Speaker 1: mathematics to do proofs, by just doing proofs, and the 107 00:06:48,279 --> 00:06:50,520 Speaker 1: question of what, why is this a proof and that 108 00:06:50,640 --> 00:06:53,400 Speaker 1: not a proof, is mostly given to you by way 109 00:06:53,400 --> 00:06:56,360 Speaker 1: of example. Right, there's a flaw in this proof, there's 110 00:06:56,400 --> 00:06:58,479 Speaker 1: a gap in this proof, or this one is a 111 00:06:58,520 --> 00:07:00,600 Speaker 1: good proof, and so on and so for but as 112 00:07:00,600 --> 00:07:02,720 Speaker 1: a philosopher of mathematics you're much more interested in a 113 00:07:02,800 --> 00:07:06,240 Speaker 1: systematic answer to such such questions. What is the correct logic? 114 00:07:06,320 --> 00:07:09,800 Speaker 1: Is that classical logic? Is it something other alternative logic 115 00:07:09,960 --> 00:07:12,800 Speaker 1: mathematicians are using, and so on and so forth. So 116 00:07:13,160 --> 00:07:15,480 Speaker 1: these are sorts of questions that I was interested in, 117 00:07:15,680 --> 00:07:18,840 Speaker 1: or became interested in by studying mathematics and found that 118 00:07:18,920 --> 00:07:22,800 Speaker 1: the answers really weren't in the mathematics department so I 119 00:07:23,400 --> 00:07:26,520 Speaker 1: straight over to the philosophy department occasionally and they didn't 120 00:07:26,520 --> 00:07:29,560 Speaker 1: have the answers either, but at least they recognized that 121 00:07:29,880 --> 00:07:33,000 Speaker 1: these were interesting questions. So that was my particular path 122 00:07:33,040 --> 00:07:35,360 Speaker 1: into the philosophy of mathematics at least. And so why 123 00:07:35,400 --> 00:07:39,000 Speaker 1: do you think it is that mathematicians aren't that interested 124 00:07:39,080 --> 00:07:43,720 Speaker 1: in like why proofs work or whether proofs should work? 125 00:07:44,240 --> 00:07:46,640 Speaker 1: You know. Well, why is it that it's the philosophy 126 00:07:46,720 --> 00:07:49,480 Speaker 1: department to ask those kinds of questions? I mean, are 127 00:07:49,480 --> 00:07:51,440 Speaker 1: there folks in the mathematics side of it that do 128 00:07:51,480 --> 00:07:54,640 Speaker 1: that and just don't call it philosophy? Yeah, I think so. 129 00:07:54,760 --> 00:07:57,600 Speaker 1: I wouldn't say that mathematicians not interested in this. I 130 00:07:57,640 --> 00:07:59,920 Speaker 1: mean one of the things that I think is interesting 131 00:08:00,040 --> 00:08:04,520 Speaker 1: about philosophy of x, whatever X is. For me it's 132 00:08:04,560 --> 00:08:07,880 Speaker 1: philosophy of mathematics primarily. But if you're doing philosophy of 133 00:08:08,000 --> 00:08:11,960 Speaker 1: something rather, then you need to engage with there's something 134 00:08:12,040 --> 00:08:14,440 Speaker 1: or other. So if you're doing philosophy of quantum mechanics, 135 00:08:14,520 --> 00:08:17,320 Speaker 1: you've need to talk to folks doing quantum mechanics. If 136 00:08:17,320 --> 00:08:20,200 Speaker 1: you're doing philosophy of biology, you need to speak to 137 00:08:20,240 --> 00:08:23,200 Speaker 1: folks doing biology and you learn a great deal about 138 00:08:23,240 --> 00:08:26,560 Speaker 1: the other discipline as well. So for me philosophy of 139 00:08:26,560 --> 00:08:29,040 Speaker 1: mathematics was an excuse to kind of do a bit 140 00:08:29,080 --> 00:08:33,480 Speaker 1: more mathematics, talk to mathematicians. Mathematicians, some are interested in 141 00:08:33,520 --> 00:08:36,040 Speaker 1: such questions, some are not. That's as you would expect. 142 00:08:36,120 --> 00:08:39,360 Speaker 1: Some are interested in Topology, some are not. So it's 143 00:08:39,480 --> 00:08:42,800 Speaker 1: just a particular bunch of questions that some mathematicians are 144 00:08:42,800 --> 00:08:46,480 Speaker 1: interested in and as a philosophy of mathematics it's good 145 00:08:46,480 --> 00:08:48,400 Speaker 1: to talk to them about these things. You know, I'm 146 00:08:48,440 --> 00:08:52,360 Speaker 1: interested in mathematical intuitions about such things, not just sitting 147 00:08:52,400 --> 00:08:55,840 Speaker 1: back in the philosophical armchair, as it were, and coming 148 00:08:55,880 --> 00:08:58,080 Speaker 1: up with my own theories of these things. Right. Do 149 00:08:58,120 --> 00:09:00,440 Speaker 1: you feel like there's something of an asymmetry there, that 150 00:09:00,520 --> 00:09:04,640 Speaker 1: maybe philosophers of mathematics are more interested in what mathematicians 151 00:09:04,679 --> 00:09:08,199 Speaker 1: are thinking about than mathematicians are interested in what philosophers 152 00:09:08,240 --> 00:09:10,240 Speaker 1: are thinking about? In the case of the physics department, 153 00:09:10,280 --> 00:09:12,000 Speaker 1: for example, we have a lot of people over here 154 00:09:12,040 --> 00:09:14,040 Speaker 1: who are doing physics and a few of us are 155 00:09:14,080 --> 00:09:17,160 Speaker 1: interested in what philosophers of physics are saying about what 156 00:09:17,200 --> 00:09:19,400 Speaker 1: we're doing, but a lot of people seem to subscribe 157 00:09:19,400 --> 00:09:22,160 Speaker 1: to you know, fine, men's approach. Philosophy of science is 158 00:09:22,200 --> 00:09:25,600 Speaker 1: about as useful to scientists as ornithology as the birds right. 159 00:09:25,720 --> 00:09:28,520 Speaker 1: Do you have that same reaction from mathematicians? They're like look, 160 00:09:28,559 --> 00:09:31,160 Speaker 1: proofs work. Why do we care why they work? It 161 00:09:31,240 --> 00:09:35,160 Speaker 1: depends again on the mathematicians, as you rightly point out. 162 00:09:35,160 --> 00:09:38,880 Speaker 1: Amongst physicists you've got things like trying to sort out 163 00:09:39,080 --> 00:09:44,080 Speaker 1: the interpretation of quantum mechanics. That's a deeply philosophical question 164 00:09:44,640 --> 00:09:47,160 Speaker 1: that a lot of physicists are engaged with. You know' 165 00:09:47,160 --> 00:09:49,840 Speaker 1: you've got to just dismissed that. As you know, that's philosophy, 166 00:09:49,880 --> 00:09:52,520 Speaker 1: that's not physics. It's crucial to quantum mechanics to have 167 00:09:52,679 --> 00:09:56,240 Speaker 1: an appropriate interpretation of what's going on there. So there's 168 00:09:56,240 --> 00:09:59,680 Speaker 1: a place where some physicists, not all physicists, are interested 169 00:09:59,679 --> 00:10:02,080 Speaker 1: in the reputation of quantum mechanics, but those who are 170 00:10:02,800 --> 00:10:07,240 Speaker 1: recognized that as a philosophical problem and interested in well 171 00:10:07,280 --> 00:10:12,400 Speaker 1: informed opinions from suitable philosophers. Not every philosopher has an 172 00:10:12,440 --> 00:10:15,920 Speaker 1: opinion on that either, and so in mathematics I would 173 00:10:15,960 --> 00:10:19,440 Speaker 1: say most mathematicians are not particularly interested in the philosophy 174 00:10:19,440 --> 00:10:22,360 Speaker 1: of mathematics. But there are some. Well, in physics it 175 00:10:22,400 --> 00:10:25,120 Speaker 1: seems sort of natural to ask these questions. We discover 176 00:10:25,280 --> 00:10:27,600 Speaker 1: the universe is this way and then we can ask 177 00:10:27,640 --> 00:10:29,680 Speaker 1: like what does that mean, or why is it this way, 178 00:10:29,679 --> 00:10:32,439 Speaker 1: in that some other way? In the case of mathematics, 179 00:10:32,880 --> 00:10:35,560 Speaker 1: what are the sort of foundational questions here? What are 180 00:10:35,600 --> 00:10:39,280 Speaker 1: the questions? The philosophy of math is answering basic questions 181 00:10:39,320 --> 00:10:41,600 Speaker 1: about what the subject matter is. I think that one 182 00:10:41,640 --> 00:10:44,320 Speaker 1: of the interesting things about philosophy of mathematics is the 183 00:10:44,320 --> 00:10:47,319 Speaker 1: problems start really early on. So if you say you 184 00:10:47,360 --> 00:10:50,760 Speaker 1: know someone who gives you a scientific discipline, biology, what 185 00:10:50,920 --> 00:10:53,800 Speaker 1: is biology? It's not always easy to answer such questions. 186 00:10:53,800 --> 00:10:57,720 Speaker 1: But you can say something helpful like it's the study 187 00:10:57,720 --> 00:11:02,120 Speaker 1: of living organisms or it's the study of evolution, and 188 00:11:02,200 --> 00:11:06,000 Speaker 1: say what that is. What's physics? Well, it's the study 189 00:11:06,040 --> 00:11:10,719 Speaker 1: of the fundamental particles and large scale structures, theories of 190 00:11:10,800 --> 00:11:15,080 Speaker 1: spacetime and so forth. Mathematics is the study of dot 191 00:11:15,080 --> 00:11:18,120 Speaker 1: dot dot, filling the dots right. It's not easy to 192 00:11:18,160 --> 00:11:21,560 Speaker 1: answer that question, attempting to say that it's about the 193 00:11:21,600 --> 00:11:25,559 Speaker 1: study of numbers, functions, sets and the like, but that 194 00:11:25,640 --> 00:11:29,080 Speaker 1: immediately raises a question of what are they, then these 195 00:11:29,080 --> 00:11:31,920 Speaker 1: are not the sorts of things that one can gain 196 00:11:32,000 --> 00:11:36,160 Speaker 1: access to. It's not like fundamental particles. You can build 197 00:11:36,280 --> 00:11:39,400 Speaker 1: accelerators and you can smash things into one another and 198 00:11:39,480 --> 00:11:44,600 Speaker 1: you can find traces of fundamental particles, but numbers are 199 00:11:44,640 --> 00:11:46,719 Speaker 1: not the sorts of things that even leave traces. So, 200 00:11:47,440 --> 00:11:50,480 Speaker 1: you know, if the mathematics is the study of numbers, 201 00:11:50,800 --> 00:11:55,080 Speaker 1: then how is it that mathematicians gain access to this 202 00:11:55,280 --> 00:11:57,880 Speaker 1: mathematical realm? And there's some of the you know, the 203 00:11:58,080 --> 00:12:00,920 Speaker 1: questions you get right from the get I think this 204 00:12:01,000 --> 00:12:03,480 Speaker 1: is one of the most fascinating questions. Is this question 205 00:12:03,480 --> 00:12:06,760 Speaker 1: of like, what our numbers? Are they things in our 206 00:12:06,800 --> 00:12:10,000 Speaker 1: minds or are they things in the universe? You know, 207 00:12:10,040 --> 00:12:12,400 Speaker 1: what are the rules by which they operate? Are they 208 00:12:12,480 --> 00:12:14,679 Speaker 1: rules that we invented the way we like invented the 209 00:12:14,760 --> 00:12:18,000 Speaker 1: rules of checkers, or are they rules that we've discovered 210 00:12:18,080 --> 00:12:20,640 Speaker 1: that are true and deep in the universe? But it's 211 00:12:20,679 --> 00:12:24,240 Speaker 1: amazing to me that we can do math without knowing 212 00:12:24,280 --> 00:12:27,000 Speaker 1: these things right, that we can calculate one plus one 213 00:12:27,040 --> 00:12:29,040 Speaker 1: equals two and one plus two equals three and all 214 00:12:29,040 --> 00:12:32,040 Speaker 1: sorts of complicated into rules in multiple dimensions without knowing 215 00:12:32,040 --> 00:12:34,959 Speaker 1: the answers about what it is we are doing. How 216 00:12:35,000 --> 00:12:37,679 Speaker 1: do we reconcile that? How do we understand why it's 217 00:12:37,679 --> 00:12:39,760 Speaker 1: possible to do it without knowing what it is we 218 00:12:39,840 --> 00:12:43,000 Speaker 1: are doing? Right? That's the really, I think the heart 219 00:12:43,040 --> 00:12:48,040 Speaker 1: of philosophy of mathematics is to understand mathematical progress in 220 00:12:48,520 --> 00:12:51,840 Speaker 1: light of these difficult questions. So it's not like you 221 00:12:51,840 --> 00:12:53,920 Speaker 1: know as a philosopher mathematics. So I'm going to the 222 00:12:53,920 --> 00:12:56,480 Speaker 1: mathematics department and telling them to hang on, stop, stop, 223 00:12:56,559 --> 00:12:59,480 Speaker 1: until we sort these things out. Right, mathematics is business 224 00:12:59,520 --> 00:13:01,640 Speaker 1: as usually. With a few exceptions, there have been some 225 00:13:01,720 --> 00:13:04,600 Speaker 1: interesting cases in the history of mathematics. So early in 226 00:13:04,600 --> 00:13:10,240 Speaker 1: the twentieth century there was a movement called intuitionism or 227 00:13:10,280 --> 00:13:13,839 Speaker 1: constructive ism, and that came from within mathematics. So one 228 00:13:13,880 --> 00:13:17,640 Speaker 1: of the Great Mathematicians of the Twentieth Century, Alie J Brauer. 229 00:13:18,760 --> 00:13:23,360 Speaker 1: Many theorems named after Brauer, became concerned that if mathematics 230 00:13:23,440 --> 00:13:26,840 Speaker 1: is a kind of construct, a mental construct, then you 231 00:13:26,960 --> 00:13:30,840 Speaker 1: can't just assume that every mathematical proposition is either true 232 00:13:30,920 --> 00:13:34,000 Speaker 1: or false until it's constructed. So if you think about 233 00:13:34,000 --> 00:13:36,520 Speaker 1: the fiction, for instance, in this way, what's true in 234 00:13:36,559 --> 00:13:40,320 Speaker 1: a fiction is all that's said to be true in 235 00:13:40,360 --> 00:13:43,640 Speaker 1: the fiction, plus a bunch of natural implications. Right. So, 236 00:13:43,679 --> 00:13:46,079 Speaker 1: for example, she collins lives that twenty two and a 237 00:13:46,120 --> 00:13:48,680 Speaker 1: half Baker Street or whatever. That's true in that story. 238 00:13:48,840 --> 00:13:53,760 Speaker 1: That's right. And implied by that that he lived near 239 00:13:54,679 --> 00:13:57,560 Speaker 1: other streets that are nearby London. The geography of London 240 00:13:57,679 --> 00:14:00,400 Speaker 1: is supposed to be held fixed. So without even saying that, 241 00:14:00,400 --> 00:14:03,320 Speaker 1: that's a natural implication of that. So all of that 242 00:14:03,360 --> 00:14:06,959 Speaker 1: can be taken be true. But Sherlock Holmes walked down 243 00:14:07,040 --> 00:14:11,080 Speaker 1: Good Street exactly fourteen times in his life. Neither true 244 00:14:11,080 --> 00:14:14,119 Speaker 1: nor false. Surely right? There's not stated in any of 245 00:14:13,720 --> 00:14:17,080 Speaker 1: the books. It's not stated that he didn't. So it 246 00:14:17,120 --> 00:14:20,400 Speaker 1: seems like that's neither true nor false. That's very different 247 00:14:20,440 --> 00:14:23,080 Speaker 1: to the actual world, the actual real world. Even if 248 00:14:23,120 --> 00:14:26,120 Speaker 1: you don't know whether something is true or false, it's 249 00:14:26,120 --> 00:14:29,920 Speaker 1: true or false nonetheless, or at least that's a natural position. 250 00:14:30,080 --> 00:14:34,280 Speaker 1: I don't know how many pairs of gray socks Napoleon owned, 251 00:14:34,720 --> 00:14:36,640 Speaker 1: but there is a fact of the matter about that 252 00:14:36,720 --> 00:14:40,320 Speaker 1: and we'll probably never know that. Right. So brow became 253 00:14:40,400 --> 00:14:43,280 Speaker 1: concerned that if mathematics was a kind of construct like this, 254 00:14:44,040 --> 00:14:48,920 Speaker 1: then you can't use certain proof methods, proof methods that 255 00:14:49,000 --> 00:14:52,480 Speaker 1: require that the proposition in question is either true or false, 256 00:14:52,800 --> 00:14:57,760 Speaker 1: in particular reductive of proof methods which proceed by assuming 257 00:14:57,760 --> 00:14:59,600 Speaker 1: the negation of the thing that you want to prove 258 00:15:00,640 --> 00:15:04,120 Speaker 1: and then derive a contradiction from that and therefore include 259 00:15:04,280 --> 00:15:09,040 Speaker 1: that the proposition unnegated, is true. But if it's neither 260 00:15:09,040 --> 00:15:12,280 Speaker 1: true nor false to start with, that neither negation nor 261 00:15:12,360 --> 00:15:16,440 Speaker 1: the proposition itself are true or false. So brow became 262 00:15:16,520 --> 00:15:20,360 Speaker 1: concerned about particular proof methods and wanted to restrict mathematics 263 00:15:20,360 --> 00:15:24,360 Speaker 1: too purely constructive methods, and this movement continues. There are 264 00:15:24,400 --> 00:15:29,480 Speaker 1: still mathematicians who stand by this very, very strongly and 265 00:15:29,520 --> 00:15:33,320 Speaker 1: think that classical mathematics, which uses these kinds of non 266 00:15:33,360 --> 00:15:38,960 Speaker 1: constructive methods, problematic. That's saying that you want to reconstruct 267 00:15:39,000 --> 00:15:43,680 Speaker 1: some of the important theorems of mathematics and provide alternative 268 00:15:43,760 --> 00:15:46,040 Speaker 1: proofs that are constructive. There may be that all of 269 00:15:46,080 --> 00:15:48,680 Speaker 1: mathematics is just a big castle built on sand and 270 00:15:48,720 --> 00:15:50,880 Speaker 1: then the end none of it is real. Is that 271 00:15:50,960 --> 00:15:52,960 Speaker 1: the idea? Yeah, so the idea is if it is 272 00:15:53,080 --> 00:15:56,360 Speaker 1: some sort of construction, mental construction. That doesn't mean that 273 00:15:56,400 --> 00:15:59,600 Speaker 1: it's nothing, it's still kind of you know, Shakespeare is 274 00:15:59,640 --> 00:16:02,560 Speaker 1: a meant. You know, shakes, works of Shakespeare Mental Constructions. 275 00:16:02,720 --> 00:16:06,240 Speaker 1: That great, great things, but you just need to be 276 00:16:06,240 --> 00:16:09,040 Speaker 1: careful about the logic that you're using. So, according to 277 00:16:09,080 --> 00:16:13,200 Speaker 1: this line of thought, then some proofs are in fact 278 00:16:13,240 --> 00:16:15,280 Speaker 1: not proofs at all. I think this touches on a 279 00:16:15,320 --> 00:16:18,680 Speaker 1: really interesting question here, which is like, why does it 280 00:16:18,800 --> 00:16:22,040 Speaker 1: matter whether or not these things are real or whether 281 00:16:22,200 --> 00:16:24,480 Speaker 1: not these things are just constructed? And I think it 282 00:16:24,520 --> 00:16:26,440 Speaker 1: goes to the heart of sort of what we're trying 283 00:16:26,440 --> 00:16:28,920 Speaker 1: to do, at least in physics, which is learned about 284 00:16:28,960 --> 00:16:31,840 Speaker 1: the universe. Right, I am interested in physics because I 285 00:16:31,840 --> 00:16:33,440 Speaker 1: want to know what's out there and what's real, not 286 00:16:33,520 --> 00:16:37,280 Speaker 1: because I want to build a complicated mathematical construct that 287 00:16:37,320 --> 00:16:39,400 Speaker 1: I can use to play around with in my head 288 00:16:39,440 --> 00:16:41,200 Speaker 1: and with my friends. Right, I want to know what's 289 00:16:41,200 --> 00:16:44,560 Speaker 1: actually out there. So, in the question of mathematics, that 290 00:16:44,640 --> 00:16:47,320 Speaker 1: brings up this issue like our numbers real, are these 291 00:16:47,360 --> 00:16:49,960 Speaker 1: proofs real, or are they just a game that we 292 00:16:50,040 --> 00:16:52,680 Speaker 1: have invented? And to make it more concrete, I like 293 00:16:52,760 --> 00:16:55,560 Speaker 1: to think about it in terms of like aliens. You know, 294 00:16:55,720 --> 00:16:59,280 Speaker 1: if alien scientists showed up here. Could we talk to 295 00:16:59,320 --> 00:17:02,160 Speaker 1: them about out the things in physics that we've discovered? 296 00:17:02,240 --> 00:17:04,240 Speaker 1: If the things we've discovered are real and part of 297 00:17:04,240 --> 00:17:07,399 Speaker 1: the universe, then yes, if they're just like in our minds, 298 00:17:07,520 --> 00:17:09,560 Speaker 1: then no, they would have different ways of thinking about it. 299 00:17:09,680 --> 00:17:12,040 Speaker 1: So can we take that same sort of question and 300 00:17:12,080 --> 00:17:15,480 Speaker 1: ask it about mathematics and ask like what would aliens 301 00:17:15,560 --> 00:17:19,320 Speaker 1: have developed mathematics if it's part of the universe, or 302 00:17:19,320 --> 00:17:21,399 Speaker 1: where they have some other way of like putting together 303 00:17:21,520 --> 00:17:24,400 Speaker 1: structured thought to figure out the universe? If, in fact, 304 00:17:24,440 --> 00:17:26,600 Speaker 1: mathematics is just part of our minds and the way 305 00:17:26,640 --> 00:17:28,280 Speaker 1: we think, is that a reasonable way to think about 306 00:17:28,320 --> 00:17:30,560 Speaker 1: the questions of the philosophy of math? Yeah, I think 307 00:17:30,600 --> 00:17:32,720 Speaker 1: that's a very good way of putting it. You might 308 00:17:32,760 --> 00:17:36,239 Speaker 1: think so. For instance, some things in mathematics are just 309 00:17:36,640 --> 00:17:38,960 Speaker 1: artifacts of the way we are. So you might think 310 00:17:39,480 --> 00:17:42,560 Speaker 1: using based ten, for instance, that's you know, to do 311 00:17:42,640 --> 00:17:45,480 Speaker 1: with numbers of fingers and so on and so forth. 312 00:17:45,480 --> 00:17:48,000 Speaker 1: But something very natural about base two. And I don't 313 00:17:48,160 --> 00:17:50,080 Speaker 1: believe you would know more about this than me, but 314 00:17:50,080 --> 00:17:52,320 Speaker 1: I believe that it's thought that if you were going 315 00:17:52,359 --> 00:17:57,639 Speaker 1: to contact extraterrestrials, then the initial segments of the international 316 00:17:57,680 --> 00:18:02,080 Speaker 1: expansion of Pie in base two would be something that 317 00:18:02,160 --> 00:18:05,560 Speaker 1: an intelligent life form would recognize. That's assuming that sometimes 318 00:18:05,560 --> 00:18:09,200 Speaker 1: there's something really objective about Pie. It's hard to imagine 319 00:18:09,240 --> 00:18:11,960 Speaker 1: that that's just the kind of construct of ours. You know, 320 00:18:12,040 --> 00:18:15,480 Speaker 1: you think that surely pie turns up in the most 321 00:18:15,560 --> 00:18:19,040 Speaker 1: unexpected places. It's not just about the ratio of the 322 00:18:19,119 --> 00:18:24,080 Speaker 1: circumference to the diameter of a circle. That's the initial definition, 323 00:18:24,160 --> 00:18:26,119 Speaker 1: but you know, as you know, it turns up in 324 00:18:26,680 --> 00:18:31,199 Speaker 1: it's just about everything right everywhere, in complex analysis, in geometry. 325 00:18:31,600 --> 00:18:33,920 Speaker 1: So yeah, so that the thought that that's something that 326 00:18:33,960 --> 00:18:38,560 Speaker 1: would be recognized by another intelligent life form seems reasonable. 327 00:18:38,640 --> 00:18:41,639 Speaker 1: But that pushes you to this sort of objective point 328 00:18:41,720 --> 00:18:45,400 Speaker 1: about mathematics, that it seems to be something that's objectively true, 329 00:18:45,480 --> 00:18:50,080 Speaker 1: not just mental construction. You wouldn't expect another an alien 330 00:18:50,119 --> 00:18:55,320 Speaker 1: life form too recognize facts about Sherlock Holmes, for instance. 331 00:18:55,359 --> 00:18:59,919 Speaker 1: You know, the detective novel could be a universal kind 332 00:19:00,000 --> 00:19:03,320 Speaker 1: throughout that could be exists and all intelligent beings everywhere. 333 00:19:03,480 --> 00:19:06,159 Speaker 1: You know, I imagine sitting across the table from alien 334 00:19:06,200 --> 00:19:09,920 Speaker 1: mathematicians and introducing them to ours, and I can imagine 335 00:19:09,920 --> 00:19:12,600 Speaker 1: that it might be that the kind of elaborate constructs 336 00:19:12,600 --> 00:19:15,840 Speaker 1: that we've built calculus and geometry, they might have very 337 00:19:15,840 --> 00:19:18,040 Speaker 1: different ways of doing these kind of things. I mean, 338 00:19:18,240 --> 00:19:20,600 Speaker 1: even in the history of our mathematics, our path to 339 00:19:20,680 --> 00:19:23,080 Speaker 1: these sorts of things have been varied and could have 340 00:19:23,119 --> 00:19:25,560 Speaker 1: gone differently. But it feels like maybe at the heart 341 00:19:25,560 --> 00:19:27,479 Speaker 1: of it there could be something in common that if 342 00:19:27,480 --> 00:19:30,560 Speaker 1: you drill down to the core of mathematics, of fundamental 343 00:19:30,680 --> 00:19:33,240 Speaker 1: ideas on which everything else is built, maybe we could 344 00:19:33,240 --> 00:19:36,360 Speaker 1: compare those with alien mathematicians. How well have we done 345 00:19:36,359 --> 00:19:40,119 Speaker 1: in terms of like examining the foundations of our own mathematics, 346 00:19:40,119 --> 00:19:43,760 Speaker 1: of understanding what our castle is built on? You know, 347 00:19:43,800 --> 00:19:48,160 Speaker 1: what are the basic rules of mathematics? Great Question. I mean, 348 00:19:48,280 --> 00:19:51,639 Speaker 1: when we think about the foundations of mathematics, a lot 349 00:19:51,680 --> 00:19:54,920 Speaker 1: of it is this program of trying to construct other 350 00:19:54,960 --> 00:19:59,199 Speaker 1: bits of mathematics from some other mathematics. So set theory 351 00:19:59,640 --> 00:20:03,879 Speaker 1: or category, if the theory for prefer, but set theoryt'st 352 00:20:03,880 --> 00:20:07,720 Speaker 1: stick for that for the moment. There's beautiful constructions in 353 00:20:07,800 --> 00:20:10,400 Speaker 1: set theory where you can construct the natural numbers out 354 00:20:10,440 --> 00:20:13,359 Speaker 1: of sets and then you can construct ordered pairs of 355 00:20:13,440 --> 00:20:16,480 Speaker 1: natural numbers out of sets and then you can get 356 00:20:16,760 --> 00:20:19,000 Speaker 1: functions and so on and so forth. So you can 357 00:20:19,040 --> 00:20:22,399 Speaker 1: get a great deal of mathematics built just out of 358 00:20:22,400 --> 00:20:24,280 Speaker 1: set theory. Can you explain that to me? Like, how 359 00:20:24,280 --> 00:20:26,440 Speaker 1: do you get natural numbers out of sets? What does 360 00:20:26,440 --> 00:20:29,199 Speaker 1: that mean? So you just have a series of sets. 361 00:20:29,200 --> 00:20:31,359 Speaker 1: So you start with the empty set, right. So the 362 00:20:31,400 --> 00:20:33,760 Speaker 1: empty set is the set that has nothing in it. 363 00:20:33,960 --> 00:20:38,120 Speaker 1: You identify that with zero. Just call that zero. It's 364 00:20:38,119 --> 00:20:40,240 Speaker 1: not zero. That's an empty set, but let's let's just 365 00:20:40,920 --> 00:20:44,280 Speaker 1: humor me call that zero. Then you have the set 366 00:20:44,320 --> 00:20:47,679 Speaker 1: that contains the empty set that has one member, you know. 367 00:20:47,880 --> 00:20:50,320 Speaker 1: So it's the set that has the empty set inside it. 368 00:20:50,400 --> 00:20:53,480 Speaker 1: So it has one member. Let's call that one. This 369 00:20:53,600 --> 00:20:55,560 Speaker 1: is just arbitrary. We're just making this up as we go. 370 00:20:56,119 --> 00:20:58,399 Speaker 1: And then you collect all of the sets from the 371 00:20:58,440 --> 00:21:02,800 Speaker 1: previous stages and collect them together. So the next stage 372 00:21:02,800 --> 00:21:06,040 Speaker 1: you take the empty set plus the set that contains 373 00:21:06,080 --> 00:21:10,240 Speaker 1: the empty set that has two members in it. Suggestive 374 00:21:10,320 --> 00:21:13,880 Speaker 1: name for that one too. This is a construction due 375 00:21:13,920 --> 00:21:17,119 Speaker 1: to the mathematicians John Von Neuman, and they're called the 376 00:21:17,200 --> 00:21:20,560 Speaker 1: von Neuemen ordinals. So you can construct the natural numbers. 377 00:21:20,600 --> 00:21:23,639 Speaker 1: In this way, you can then define, you know, I 378 00:21:23,680 --> 00:21:25,879 Speaker 1: won't go into details, but you can define edition and 379 00:21:25,880 --> 00:21:28,840 Speaker 1: so forth in this set theoretic way. And what does 380 00:21:28,840 --> 00:21:31,600 Speaker 1: that accomplished for you? Now, instead of having zero, one 381 00:21:31,640 --> 00:21:34,360 Speaker 1: and two, you have these weird sets. Why is that 382 00:21:34,440 --> 00:21:37,040 Speaker 1: better or more foundational, or what have you learned from 383 00:21:37,080 --> 00:21:39,880 Speaker 1: doing that? That's the kind of really interesting question here. 384 00:21:40,000 --> 00:21:43,240 Speaker 1: In one sense, it's now no longer sort of transparent, right, 385 00:21:43,359 --> 00:21:46,199 Speaker 1: we're familiar with the natural numbers and sort of building 386 00:21:46,200 --> 00:21:49,399 Speaker 1: it out of these these sets. They get really is 387 00:21:49,400 --> 00:21:52,199 Speaker 1: you can imagine. They get really ugly really quickly. So 388 00:21:52,280 --> 00:21:55,640 Speaker 1: once you start talking about numbers like seventeen, it's hard 389 00:21:55,680 --> 00:21:57,920 Speaker 1: to write it down on the one page what that is. 390 00:21:58,680 --> 00:22:01,840 Speaker 1: But no one suggests thing that you need to use 391 00:22:01,960 --> 00:22:05,520 Speaker 1: these things instead of natural numbers. But it's an interesting 392 00:22:06,200 --> 00:22:10,000 Speaker 1: exercise that you can construct the natural numbers out of sets. 393 00:22:10,040 --> 00:22:13,560 Speaker 1: So you might think in a way sets are all 394 00:22:13,600 --> 00:22:17,400 Speaker 1: you need. Sets really like the fundamental particles of physics. Right, 395 00:22:17,600 --> 00:22:20,960 Speaker 1: no one says there are no tables and chairs and 396 00:22:21,000 --> 00:22:23,560 Speaker 1: there are no people. Well, some people say such things, 397 00:22:23,560 --> 00:22:26,960 Speaker 1: but just because we can show that people are made 398 00:22:26,960 --> 00:22:31,000 Speaker 1: out of fundamental particles no one says stop doing biology 399 00:22:31,080 --> 00:22:35,359 Speaker 1: or sociology or psychology handed all over particle physicists, because 400 00:22:35,359 --> 00:22:38,880 Speaker 1: it's all particles. It's an interesting discovery that we're all 401 00:22:38,920 --> 00:22:41,960 Speaker 1: made up out of these fundamental particles. So in that 402 00:22:42,080 --> 00:22:44,600 Speaker 1: sort of vein you might think it's interesting that you 403 00:22:44,640 --> 00:22:49,080 Speaker 1: can reconstruct almost all of mathematics out of set theory 404 00:22:49,160 --> 00:22:51,800 Speaker 1: like this. Not suggesting you do it that way, but 405 00:22:52,040 --> 00:22:54,840 Speaker 1: it's an interesting construction. And that may be that the 406 00:22:54,880 --> 00:22:59,120 Speaker 1: fundamental mathematical particles, as it were, our sets, I see. 407 00:22:59,160 --> 00:23:02,320 Speaker 1: So it's like reduct Shinism to say what bits are 408 00:23:02,320 --> 00:23:06,040 Speaker 1: fundamental and what bits emerge. If we can figure out 409 00:23:06,080 --> 00:23:08,000 Speaker 1: which bits are fundamental and then we can ask questions 410 00:23:08,040 --> 00:23:10,720 Speaker 1: just about those and try to get some insight into, 411 00:23:10,760 --> 00:23:15,800 Speaker 1: like the actual nature of mathematics. So does that work? 412 00:23:15,840 --> 00:23:17,879 Speaker 1: I mean, can you say I'm going to start with 413 00:23:17,920 --> 00:23:21,760 Speaker 1: sets and from that build everything geometry and into roles 414 00:23:21,920 --> 00:23:25,359 Speaker 1: and differential equations? Can you base all of mathematics on 415 00:23:25,640 --> 00:23:29,879 Speaker 1: these weird sets? Yes, I mean with certain caveats. Are are 416 00:23:29,800 --> 00:23:32,400 Speaker 1: are a couple of little areas of mathematics that don't 417 00:23:32,400 --> 00:23:36,040 Speaker 1: succumb to this primarily category theory. But but set that 418 00:23:36,119 --> 00:23:39,280 Speaker 1: aside all of the mathematics that most of us know 419 00:23:39,359 --> 00:23:42,560 Speaker 1: and love. You can build out of sets in this 420 00:23:42,640 --> 00:23:45,399 Speaker 1: kind of way, and that's, you know, again, that's just 421 00:23:45,440 --> 00:23:50,760 Speaker 1: an interesting fact about mathematics. It demonstrates, firstly, the power 422 00:23:50,840 --> 00:23:55,359 Speaker 1: of set theory, but really it's such a versatile tool 423 00:23:55,480 --> 00:23:58,720 Speaker 1: set theory. Secondly, you know, it does lend support to 424 00:23:58,720 --> 00:24:01,480 Speaker 1: this idea that sets are the equivalent of the fundamental 425 00:24:01,520 --> 00:24:05,399 Speaker 1: particles and mathematics. And again, business is usual for topology 426 00:24:05,520 --> 00:24:08,239 Speaker 1: and all the other areas of mathematics. Not suggesting they 427 00:24:08,320 --> 00:24:11,160 Speaker 1: quit and go and do set theory instead, but rather 428 00:24:11,400 --> 00:24:14,320 Speaker 1: it's an interesting fact that their area can be reduced 429 00:24:14,720 --> 00:24:19,639 Speaker 1: in this, you know, admittedly cumbersome way, just as reducing 430 00:24:19,960 --> 00:24:23,120 Speaker 1: a table or a chair to fundamental particles. Try and 431 00:24:23,160 --> 00:24:26,040 Speaker 1: do that in particle physics, give the full description of 432 00:24:26,119 --> 00:24:29,720 Speaker 1: what a table is in particles. Right, if it's possible 433 00:24:29,760 --> 00:24:32,439 Speaker 1: at all, it's going to be incredibly cumbersome and not 434 00:24:32,600 --> 00:24:36,600 Speaker 1: terribly useful to you furniture removalists and other people working 435 00:24:36,600 --> 00:24:38,600 Speaker 1: with furniture. All right, well, I have a lot more 436 00:24:38,680 --> 00:24:41,960 Speaker 1: questions about the foundations of mathematics, but first let's take 437 00:24:42,040 --> 00:24:57,200 Speaker 1: a quick break. Okay, we are back and we're talking 438 00:24:57,200 --> 00:25:00,320 Speaker 1: to professor Mark Collivan about the fundamental part of goals 439 00:25:00,400 --> 00:25:04,840 Speaker 1: of mathematics and he is suggesting that if aliens arrive 440 00:25:05,040 --> 00:25:07,600 Speaker 1: and we are sitting across the table from their mathematicians, 441 00:25:07,720 --> 00:25:09,760 Speaker 1: that we might be able to talk to them about 442 00:25:09,880 --> 00:25:14,120 Speaker 1: the foundations of mathematics, which may be built out of sets. 443 00:25:14,320 --> 00:25:16,679 Speaker 1: We understand now that these sets follow some of the 444 00:25:16,760 --> 00:25:19,840 Speaker 1: rules that we identify with, for example basic arithmetic, and 445 00:25:19,840 --> 00:25:22,800 Speaker 1: that from that you can build everything else. So then, 446 00:25:23,000 --> 00:25:26,200 Speaker 1: what does it mean that the fundamental units of math 447 00:25:26,280 --> 00:25:28,880 Speaker 1: are sets? Does that mean that sets are real in 448 00:25:28,920 --> 00:25:31,399 Speaker 1: some way, or does it just mean that if sets 449 00:25:31,400 --> 00:25:34,359 Speaker 1: are real, then everything else is real? Or if those 450 00:25:34,440 --> 00:25:38,600 Speaker 1: rules about sets are real, they're from the universe, then 451 00:25:38,680 --> 00:25:41,320 Speaker 1: we can rely on everything else being true? Is that 452 00:25:41,400 --> 00:25:44,439 Speaker 1: the situation? Yes, so I think there are realists in 453 00:25:44,480 --> 00:25:49,000 Speaker 1: mathematics and those people will say maybe there's just sets, 454 00:25:49,040 --> 00:25:51,320 Speaker 1: but the sets are at least real. You've got to 455 00:25:51,359 --> 00:25:54,520 Speaker 1: sort of think that the fundamental furniture of the universe includes, 456 00:25:55,119 --> 00:25:57,199 Speaker 1: you know, all of the things that your particle for 457 00:25:57,359 --> 00:26:02,280 Speaker 1: physicists tell us about, plus sets. And the anti realists 458 00:26:02,280 --> 00:26:05,480 Speaker 1: about mathematics say no, the sets are some sort of 459 00:26:05,640 --> 00:26:11,879 Speaker 1: construction and their their role is to build this edifice 460 00:26:11,920 --> 00:26:15,359 Speaker 1: of mathematics upon sets. But that doesn't tell us anything 461 00:26:15,400 --> 00:26:18,920 Speaker 1: about the nature of sets, whether they're real or not. 462 00:26:19,119 --> 00:26:22,840 Speaker 1: The action, I think, in modern debates in philosophy mathematics 463 00:26:22,840 --> 00:26:26,920 Speaker 1: turns to applications of mathematics pretty quickly. Then you can 464 00:26:26,960 --> 00:26:29,840 Speaker 1: think about it in parallel with physics. Why is it 465 00:26:29,880 --> 00:26:33,560 Speaker 1: that we believe in certain bits of physics and not others? 466 00:26:33,600 --> 00:26:37,440 Speaker 1: So take bits of physics that are more speculative at 467 00:26:37,440 --> 00:26:40,960 Speaker 1: the moment. I take string theory to be such an area. 468 00:26:41,200 --> 00:26:44,960 Speaker 1: Some people believe in strings, some people don't, and it's 469 00:26:45,040 --> 00:26:47,439 Speaker 1: yet to be settled. Happy to take pure advice on this, 470 00:26:47,560 --> 00:26:50,560 Speaker 1: but that's my understanding of the current state of play there. 471 00:26:50,800 --> 00:26:53,840 Speaker 1: And why is it that people are not concerned about 472 00:26:54,200 --> 00:26:59,439 Speaker 1: other particles like electrons? Well, because electrons do a lot 473 00:26:59,480 --> 00:27:02,120 Speaker 1: of work in of theories and it's hard to imagine 474 00:27:02,280 --> 00:27:05,719 Speaker 1: any of our current physical theories functioning with our electrons 475 00:27:05,960 --> 00:27:09,480 Speaker 1: or something very much like them, whereas there are alternatives. 476 00:27:10,040 --> 00:27:12,840 Speaker 1: So some of the more speculative parts of physics now, 477 00:27:12,880 --> 00:27:16,720 Speaker 1: the speculty parts, very often get settled down the track somewhere, 478 00:27:17,240 --> 00:27:20,000 Speaker 1: but that's how we go about deciding whether things are real. 479 00:27:20,359 --> 00:27:24,720 Speaker 1: In physics, it seems right. Is it indispensable to sort 480 00:27:24,760 --> 00:27:27,800 Speaker 1: of greater physical theory? So people have turned to mathematics 481 00:27:27,800 --> 00:27:31,320 Speaker 1: with that same kind of view and thought, okay, for mathematics. 482 00:27:31,320 --> 00:27:33,480 Speaker 1: What would it take for mathematics to be real? Well, 483 00:27:33,840 --> 00:27:38,040 Speaker 1: perhaps if it's indispensable to our best science, that's a 484 00:27:38,119 --> 00:27:41,119 Speaker 1: clue that is real. It's not just some constructor of 485 00:27:41,119 --> 00:27:45,200 Speaker 1: the Human Mind, for instance. So this line of thought, 486 00:27:45,400 --> 00:27:50,560 Speaker 1: often called it indispensability argument. If mathematics is indispensable to science, 487 00:27:50,800 --> 00:27:54,679 Speaker 1: the bits of mathematics that are in fact indispensable should 488 00:27:54,680 --> 00:27:59,119 Speaker 1: have the same status as the science itself. It's certainly 489 00:27:59,160 --> 00:28:04,040 Speaker 1: a common feeling among physicists that mathematics is somehow the 490 00:28:04,160 --> 00:28:07,920 Speaker 1: language of the universe itself, because we can so effectively 491 00:28:08,000 --> 00:28:12,000 Speaker 1: describe these rules of physics in terms of this language. 492 00:28:12,040 --> 00:28:15,480 Speaker 1: You know, Stephen Weinberg said it's positively spooky how the 493 00:28:15,520 --> 00:28:19,320 Speaker 1: physicist finds a mathematicians has been there before him or her. 494 00:28:19,960 --> 00:28:22,000 Speaker 1: And you know there are often these cases, especially in 495 00:28:22,000 --> 00:28:25,800 Speaker 1: particle physics, where we struggle to understand something then we 496 00:28:25,880 --> 00:28:28,880 Speaker 1: discover this some bit of mathematics like group theory invented 497 00:28:28,960 --> 00:28:32,320 Speaker 1: just because of the curiosity of mathematicians playing games basically 498 00:28:32,359 --> 00:28:34,720 Speaker 1: in their minds, turns out to be applicable to the 499 00:28:34,720 --> 00:28:37,159 Speaker 1: world in a gorgeous way that clicks into place and 500 00:28:37,240 --> 00:28:39,800 Speaker 1: suddenly gives us insight. And those moments, I mean that 501 00:28:39,920 --> 00:28:43,040 Speaker 1: they're not religious moments or spiritual moments, but there are 502 00:28:43,120 --> 00:28:45,560 Speaker 1: moments where you feel like you've gained some deep insight 503 00:28:45,640 --> 00:28:48,680 Speaker 1: into the way the universe works, and it doesn't feel 504 00:28:48,680 --> 00:28:51,680 Speaker 1: like here's a useful description of the universe, it feels 505 00:28:51,720 --> 00:28:55,640 Speaker 1: like you're revealing the inner mechanisms of the universe itself. 506 00:28:55,840 --> 00:28:58,720 Speaker 1: But how can we know the difference right? How can 507 00:28:58,800 --> 00:29:02,280 Speaker 1: we tell whether these things are real, not just the 508 00:29:02,320 --> 00:29:06,160 Speaker 1: physical particles we're talking about, but the mathematics that describes them? 509 00:29:06,240 --> 00:29:09,560 Speaker 1: How do we distinguish between whether real or not before 510 00:29:09,560 --> 00:29:12,680 Speaker 1: we meet alien scientists? So I'm very interested in this 511 00:29:12,800 --> 00:29:17,040 Speaker 1: argument you suggest about the indispensability of mathematics, and I 512 00:29:17,080 --> 00:29:20,280 Speaker 1: read a book recently called Science Without Numbers, much, which 513 00:29:20,320 --> 00:29:22,600 Speaker 1: I'm sure you're familiar with, by hartree field, because he 514 00:29:22,680 --> 00:29:25,960 Speaker 1: suggests that math is very, very useful, you know, like 515 00:29:26,320 --> 00:29:27,960 Speaker 1: the way making it to do list is a good 516 00:29:28,000 --> 00:29:30,480 Speaker 1: way to organize your day, but you could probably get 517 00:29:30,480 --> 00:29:33,320 Speaker 1: through your day without it. But it's not actually necessary, 518 00:29:33,520 --> 00:29:35,120 Speaker 1: and he says in this book, and I'll quote him 519 00:29:35,120 --> 00:29:38,440 Speaker 1: because I find this outrageous, I am denying that numbers 520 00:29:38,560 --> 00:29:43,200 Speaker 1: or any similar entities exist. What a statement to make. 521 00:29:43,280 --> 00:29:46,240 Speaker 1: Can you help us wrap our minds around this opposite argument, 522 00:29:46,280 --> 00:29:48,560 Speaker 1: the one that suggests that we don't actually need math, 523 00:29:48,640 --> 00:29:51,440 Speaker 1: that it's useful but not indispensable. How do we make 524 00:29:51,480 --> 00:29:53,520 Speaker 1: sense of that? Yeah, I mean, let me say from 525 00:29:53,560 --> 00:29:56,640 Speaker 1: the get go I just think that's a fantastic book. 526 00:29:56,760 --> 00:30:00,640 Speaker 1: It's I disagree with heartree field on these issues, but 527 00:30:00,960 --> 00:30:04,440 Speaker 1: it is one of the absolute gems in philosophy of mathematics. 528 00:30:04,440 --> 00:30:09,680 Speaker 1: That book. It's, you know, outrageous, audacious, incredible project. I 529 00:30:09,680 --> 00:30:14,440 Speaker 1: mean so in response to this indispensability argument that mathematics, 530 00:30:14,720 --> 00:30:16,240 Speaker 1: you know, you can think of this argument of the 531 00:30:16,240 --> 00:30:18,880 Speaker 1: following form. We ought to believe in all of them, 532 00:30:19,000 --> 00:30:22,040 Speaker 1: only the entities that are indispensable to our best science. 533 00:30:22,440 --> 00:30:25,800 Speaker 1: So just should we believe in electrons? Well, just go 534 00:30:25,880 --> 00:30:29,480 Speaker 1: and see. Are they indispensable? Could you do science without electrons? No, 535 00:30:29,840 --> 00:30:34,040 Speaker 1: you can't. Could you do science without point masses? Well, 536 00:30:34,080 --> 00:30:37,320 Speaker 1: it would be difficult, but you could recast all talk 537 00:30:37,400 --> 00:30:40,720 Speaker 1: of point masses a little bit more carefully. Right. Could 538 00:30:40,760 --> 00:30:43,840 Speaker 1: you do science without coffee? No, for a coffee is three. 539 00:30:45,680 --> 00:30:48,240 Speaker 1: I think it was the old joke about the mathematician 540 00:30:48,320 --> 00:30:54,760 Speaker 1: being a machine that turns caffeine into theorems. You've solved 541 00:30:54,760 --> 00:30:57,640 Speaker 1: the question of the philosophy of math. What is mathematics exactly? 542 00:30:57,640 --> 00:31:00,760 Speaker 1: It's turning coffee into papers. We're telling us about the 543 00:31:00,840 --> 00:31:04,360 Speaker 1: argument about indispensability. Right, the indispensability argument says, you know 544 00:31:04,560 --> 00:31:07,040 Speaker 1: is something indispensable. That's really all you need to do. 545 00:31:07,160 --> 00:31:09,080 Speaker 1: So you look to your science. You don't need to 546 00:31:09,080 --> 00:31:12,520 Speaker 1: philosophize too much about this. In a way, just philosophizing 547 00:31:12,600 --> 00:31:16,320 Speaker 1: is done by recognizing that things that are indispensable to 548 00:31:16,320 --> 00:31:18,520 Speaker 1: your best scientific theories. That's what you ought to be 549 00:31:18,600 --> 00:31:20,760 Speaker 1: committed to. All right, but let's explore that a little 550 00:31:20,760 --> 00:31:23,080 Speaker 1: bit more deeply actually, before we get back to hartshred 551 00:31:23,120 --> 00:31:26,200 Speaker 1: field this indispensability argument, because there's some wrinkles there that 552 00:31:26,240 --> 00:31:29,520 Speaker 1: I don't really understand. I read like Putnam and quine 553 00:31:29,680 --> 00:31:34,160 Speaker 1: arguing that because our best theories are mathematical and those 554 00:31:34,200 --> 00:31:37,120 Speaker 1: theories are confirmed that's sort of like also confirms the 555 00:31:37,160 --> 00:31:39,160 Speaker 1: mathematics as you go along with it. Like if you 556 00:31:39,160 --> 00:31:42,600 Speaker 1: have electrons in your theory and your theory works, then 557 00:31:42,640 --> 00:31:44,800 Speaker 1: you believe in electrons. Well, if your theory also has 558 00:31:44,960 --> 00:31:47,200 Speaker 1: math as part of it, you're adding numbers, then you 559 00:31:47,520 --> 00:31:50,360 Speaker 1: that sort of like comes along with the proof. But 560 00:31:50,400 --> 00:31:53,000 Speaker 1: you know, that makes me wonder about things like infinity. 561 00:31:53,240 --> 00:31:55,720 Speaker 1: You know, we can do experiments in the universe to 562 00:31:55,800 --> 00:31:58,400 Speaker 1: explore particles. But as far as we know there's a 563 00:31:58,400 --> 00:32:01,000 Speaker 1: certain number of particles in the univer like tender to 564 00:32:01,120 --> 00:32:03,400 Speaker 1: eat or something, depending on how many you count. Does 565 00:32:03,440 --> 00:32:05,680 Speaker 1: that mean that only numbers up to tend to the 566 00:32:05,720 --> 00:32:09,680 Speaker 1: EAD are real and indispensable and numbers bigger than that, 567 00:32:09,760 --> 00:32:12,400 Speaker 1: like infinity, are not real or just parts of our mind? 568 00:32:12,720 --> 00:32:15,000 Speaker 1: You need to be careful there. I mean that's not 569 00:32:15,040 --> 00:32:18,320 Speaker 1: the only application of infinity. Right. So if you think 570 00:32:18,440 --> 00:32:22,600 Speaker 1: space time is continuous, as it is treated in general relativity, 571 00:32:22,640 --> 00:32:27,000 Speaker 1: at least quantum mechanics, it's argument that it's treated discreetly there, 572 00:32:27,080 --> 00:32:31,240 Speaker 1: but at least in general relativity spacetime is treated continuously. 573 00:32:32,000 --> 00:32:36,640 Speaker 1: So how many space time points are there right continuum? Many, 574 00:32:36,720 --> 00:32:39,000 Speaker 1: you know, not just the basic infinity there. That's, in 575 00:32:39,080 --> 00:32:41,960 Speaker 1: mathematical terms, to do the able of zero. That's the 576 00:32:42,040 --> 00:32:45,560 Speaker 1: infinity of the continuum. So it's not just a number 577 00:32:45,560 --> 00:32:48,480 Speaker 1: of particles. But are you going to need infinity in 578 00:32:48,560 --> 00:32:55,040 Speaker 1: other places? Try and do probability theory without continuous distributions, right. 579 00:32:55,160 --> 00:32:59,200 Speaker 1: So infinity crops up in all sorts of places in science, 580 00:32:59,240 --> 00:33:02,320 Speaker 1: not just think things. So that argument allows us to 581 00:33:02,360 --> 00:33:05,320 Speaker 1: believe in numbers and also believe in numbers like up 582 00:33:05,360 --> 00:33:07,920 Speaker 1: to infinity. They are real, but that's only if we 583 00:33:07,960 --> 00:33:10,040 Speaker 1: actually need them in our science. If we could do 584 00:33:10,080 --> 00:33:12,680 Speaker 1: the science without the numbers, then we wouldn't be able 585 00:33:12,720 --> 00:33:15,680 Speaker 1: to necessarily argue that the numbers are also real. So 586 00:33:15,840 --> 00:33:18,480 Speaker 1: help us understand Hartley field's argument that we don't need 587 00:33:18,560 --> 00:33:21,760 Speaker 1: numbers to do science right. So that's the starting point. 588 00:33:21,840 --> 00:33:25,640 Speaker 1: Is this this argument that you know science, you've committed 589 00:33:25,680 --> 00:33:29,640 Speaker 1: to everything that's indispensable to your best science. Taken for 590 00:33:29,680 --> 00:33:33,200 Speaker 1: granted that mathematics is indispensable for science. So the action 591 00:33:33,280 --> 00:33:35,240 Speaker 1: was really kind of on the first premise. Do you 592 00:33:35,280 --> 00:33:37,600 Speaker 1: really want to believe in everything in your best science? 593 00:33:37,640 --> 00:33:41,440 Speaker 1: We've got fictionalist planes. What about inertial restframes and so on? 594 00:33:41,480 --> 00:33:44,080 Speaker 1: And so forth. But heartrey field came along and said 595 00:33:44,200 --> 00:33:47,360 Speaker 1: maybe you could do science without numbers, maybe you could 596 00:33:47,400 --> 00:33:51,000 Speaker 1: just be realist about space time itself instead of having 597 00:33:51,720 --> 00:33:55,440 Speaker 1: the basic idea is, instead of treating space time in 598 00:33:55,520 --> 00:33:59,920 Speaker 1: this mathematical way, you just deal with space time itself. 599 00:34:00,000 --> 00:34:02,840 Speaker 1: So you're realist about the space time as an entity, 600 00:34:02,960 --> 00:34:07,000 Speaker 1: as it were, rather than treating space time as this 601 00:34:07,200 --> 00:34:11,160 Speaker 1: mathematical structure that has metrics and coordinate systems and so 602 00:34:11,239 --> 00:34:13,680 Speaker 1: on and so forth. Okay, so Hartley field hasn't smoked 603 00:34:13,719 --> 00:34:15,680 Speaker 1: so many bin in appeals that he doesn't believed in 604 00:34:15,719 --> 00:34:18,839 Speaker 1: the universe. He says space is real. Time is real, 605 00:34:19,400 --> 00:34:22,719 Speaker 1: but a mathematical description of that is not necessarily real. 606 00:34:22,960 --> 00:34:25,120 Speaker 1: Is that where we are? That's right, but in order 607 00:34:25,160 --> 00:34:27,319 Speaker 1: to say that you can't just I mean, you can 608 00:34:27,360 --> 00:34:29,719 Speaker 1: just say it, but for anyone to believe what you're 609 00:34:29,719 --> 00:34:32,239 Speaker 1: saying you've got to deliver the goods. You've got to 610 00:34:32,239 --> 00:34:36,560 Speaker 1: show how you can do something like newtonian mechanics. Is 611 00:34:36,600 --> 00:34:39,680 Speaker 1: that his case? Study show how you can do at 612 00:34:39,760 --> 00:34:45,520 Speaker 1: least the differential fragment of newtonian mechanics without talking about 613 00:34:45,520 --> 00:34:49,920 Speaker 1: anything mathematical. So just looking at relational properties of space 614 00:34:49,960 --> 00:34:54,319 Speaker 1: time points. And that's the basic trick and it's surprisingly 615 00:34:54,320 --> 00:34:56,520 Speaker 1: how far you can go with that. How is that 616 00:34:56,640 --> 00:34:59,440 Speaker 1: possible at all? I mean, if I think about Newtonian mechanics, 617 00:34:59,440 --> 00:35:01,640 Speaker 1: the first thing I comes to my mind is f 618 00:35:01,800 --> 00:35:05,320 Speaker 1: equals g MM over our squared. It's about relative distances, 619 00:35:05,400 --> 00:35:08,719 Speaker 1: it's about masses, it's about forces. If you throw that out, 620 00:35:08,760 --> 00:35:10,880 Speaker 1: what do you have left? Well, the thing is you 621 00:35:10,920 --> 00:35:13,879 Speaker 1: don't throw it out, you reconstruct it in a much 622 00:35:13,880 --> 00:35:18,160 Speaker 1: more direct or indirect, depending on how you're looking at it, way. 623 00:35:18,280 --> 00:35:22,200 Speaker 1: So instead of thinking about, for instance, a point in 624 00:35:22,320 --> 00:35:28,640 Speaker 1: space having a gravitational potential and the gravitational potential function, 625 00:35:28,719 --> 00:35:32,680 Speaker 1: then is this map from the space time to real numbers. Right, 626 00:35:32,760 --> 00:35:35,839 Speaker 1: that's the standard presentation. Newton would have talked in those terms, 627 00:35:35,880 --> 00:35:38,320 Speaker 1: of course, but now we think of it as a spacetime, 628 00:35:38,719 --> 00:35:42,680 Speaker 1: clearly in manifold, and you map from that to real 629 00:35:42,800 --> 00:35:46,680 Speaker 1: numbers and that's just your gravitational potential function. Archery field 630 00:35:47,080 --> 00:35:49,800 Speaker 1: shows that you can do that directly. Just think about 631 00:35:50,520 --> 00:35:55,080 Speaker 1: compare space time points with respect to their gravitational potential, 632 00:35:55,640 --> 00:35:58,759 Speaker 1: not having a gravitational potential function. That sits on top 633 00:35:58,800 --> 00:36:01,839 Speaker 1: of that and what shows is by doing it this 634 00:36:01,880 --> 00:36:05,880 Speaker 1: way you can recover the standard presentation. So you can 635 00:36:05,880 --> 00:36:09,480 Speaker 1: actually prove these results that show that you get everything 636 00:36:09,520 --> 00:36:12,040 Speaker 1: back that you would have had in the standard presentation. 637 00:36:12,600 --> 00:36:16,000 Speaker 1: So again, the take home message from field is not 638 00:36:16,719 --> 00:36:18,880 Speaker 1: you should be doing it this way rather than the 639 00:36:18,920 --> 00:36:21,520 Speaker 1: way everyone has done it. It's just big like the 640 00:36:21,560 --> 00:36:24,120 Speaker 1: story with sets. Right. The fact that you can do 641 00:36:24,200 --> 00:36:28,320 Speaker 1: it gives you evidence that the mathematics is not indispensable. 642 00:36:28,800 --> 00:36:33,000 Speaker 1: It's just a nice, quick and a much more elegant 643 00:36:33,000 --> 00:36:36,400 Speaker 1: way of doing it, but it's not indispensable and you 644 00:36:36,440 --> 00:36:38,760 Speaker 1: can recover everything. You can prove that you can recover 645 00:36:38,840 --> 00:36:42,439 Speaker 1: everything that you get in standard new Tony Mechanics this way. 646 00:36:42,760 --> 00:36:45,520 Speaker 1: I understand that it's a useful way to answer the 647 00:36:45,600 --> 00:36:48,279 Speaker 1: question do we need math? By proving that you could 648 00:36:48,280 --> 00:36:50,480 Speaker 1: do without it if you had to, doesn't mean that 649 00:36:50,520 --> 00:36:52,520 Speaker 1: you should do without it. Right, it's like asking a 650 00:36:52,600 --> 00:36:55,160 Speaker 1: question of could you live without jelly beans? You could 651 00:36:55,160 --> 00:36:56,759 Speaker 1: go without them for a year and you could prove 652 00:36:56,800 --> 00:36:58,279 Speaker 1: that you don't need to eat them. It doesn't mean 653 00:36:58,280 --> 00:37:01,760 Speaker 1: that nobody should eat jelly beans. But I'm still not convinced. 654 00:37:01,760 --> 00:37:05,440 Speaker 1: I mean your description here of his formulation of gravity 655 00:37:05,760 --> 00:37:09,839 Speaker 1: includes things like comparing potentials, and to me potentials are 656 00:37:09,960 --> 00:37:13,160 Speaker 1: numbers and comparing is a relationship. Are you seeing? Those 657 00:37:13,200 --> 00:37:17,719 Speaker 1: things are not mathematical or they're just not numbers. Yeah, 658 00:37:17,719 --> 00:37:20,239 Speaker 1: you've got to be realist about the points themselves, and 659 00:37:20,280 --> 00:37:23,000 Speaker 1: that's one of the criticisms of field is that you've 660 00:37:23,080 --> 00:37:25,800 Speaker 1: got to be realist about space time points and that 661 00:37:26,000 --> 00:37:29,799 Speaker 1: those things have properties. They have primitive properties like the 662 00:37:30,000 --> 00:37:33,799 Speaker 1: gravitational potential, electromagnetic potential and so on. So they have 663 00:37:33,920 --> 00:37:37,800 Speaker 1: those properties. So rather than those being a mathematical function 664 00:37:37,840 --> 00:37:40,200 Speaker 1: that lives on top of that, it's just these primitive 665 00:37:40,239 --> 00:37:42,920 Speaker 1: properties of the space time points. And so a lot 666 00:37:42,960 --> 00:37:46,360 Speaker 1: of people who are concerned about believing in mathematical objects, 667 00:37:46,400 --> 00:37:49,640 Speaker 1: because after all, that's kind of spooky, believing in the 668 00:37:49,719 --> 00:37:54,640 Speaker 1: spacetime points is also rather spooky. Right. It's not enough 669 00:37:54,680 --> 00:37:56,959 Speaker 1: for field to just believe in the manifold has actually 670 00:37:57,000 --> 00:38:00,880 Speaker 1: got to believe that individual points have these properties. But 671 00:38:01,040 --> 00:38:02,879 Speaker 1: whichever way you go on this, and as I said, 672 00:38:02,920 --> 00:38:05,759 Speaker 1: I disagree with him about the upshot of all this, 673 00:38:05,920 --> 00:38:09,520 Speaker 1: but the exercise itself is just incredible. You know, before 674 00:38:09,600 --> 00:38:11,360 Speaker 1: he did this, no one would have thought did you 675 00:38:11,719 --> 00:38:14,640 Speaker 1: even get started there. So you've got a lot of criticism. 676 00:38:14,680 --> 00:38:17,480 Speaker 1: People are saying, oh well, what about Hamiltonian formulations of 677 00:38:17,480 --> 00:38:21,080 Speaker 1: classical theories? What about chroantum mechanics where the underlying space 678 00:38:21,200 --> 00:38:24,080 Speaker 1: is infinite dimensional, Hilbert spaces, and so on and so forth, 679 00:38:24,239 --> 00:38:28,320 Speaker 1: and these are all fair and interesting criticisms. But before 680 00:38:28,360 --> 00:38:30,600 Speaker 1: he started no one thought you could do the differential 681 00:38:30,600 --> 00:38:33,440 Speaker 1: fragment of Newtonian mechanics either, you know. So for a 682 00:38:33,480 --> 00:38:35,560 Speaker 1: great deal of time, a great deal of the the 683 00:38:35,719 --> 00:38:38,239 Speaker 1: debate was about how far can you go with this 684 00:38:38,400 --> 00:38:41,520 Speaker 1: field style program because if you can go take it further, 685 00:38:42,200 --> 00:38:47,440 Speaker 1: then that's going to suggest that mathematics isn't indispensable after all. 686 00:38:47,840 --> 00:38:50,879 Speaker 1: So we have, on one hand, folks arguing that mathematics 687 00:38:51,000 --> 00:38:55,880 Speaker 1: is beautiful and elegant and unreasonably effective and, beyond that, 688 00:38:56,000 --> 00:38:59,960 Speaker 1: actually indispensable to understand in the universe and hurtlee field, 689 00:39:00,000 --> 00:39:03,000 Speaker 1: and some folks suggesting that maybe it's just useful but 690 00:39:03,120 --> 00:39:05,719 Speaker 1: not actually necessary. So where do you come down on that? 691 00:39:05,800 --> 00:39:09,560 Speaker 1: You're a philosopher of mathematics. You've thought deeply about these things. 692 00:39:09,560 --> 00:39:12,080 Speaker 1: Do you think that numbers are real? Are they just 693 00:39:12,200 --> 00:39:14,399 Speaker 1: part of our minds or are they something we found 694 00:39:14,440 --> 00:39:17,239 Speaker 1: in the universe? I'm a realist about mathematics, so I 695 00:39:17,360 --> 00:39:20,480 Speaker 1: come down on the former. So I think that mathematics 696 00:39:20,520 --> 00:39:23,440 Speaker 1: is in fact indispensable to our best scientific theories, and 697 00:39:23,520 --> 00:39:27,600 Speaker 1: that's why I take mathematical into at least some mathematics, 698 00:39:27,680 --> 00:39:30,359 Speaker 1: that I think they can still be speculative parts of 699 00:39:30,400 --> 00:39:34,080 Speaker 1: mathematics that we don't have reason to believe yet. And 700 00:39:34,120 --> 00:39:36,759 Speaker 1: so what convinces you? Just as there are, respectively, parts 701 00:39:36,760 --> 00:39:39,920 Speaker 1: of physics right that we don't believe yet. So you know, 702 00:39:40,000 --> 00:39:42,320 Speaker 1: you might think some of the higher reaches of set theory, 703 00:39:42,600 --> 00:39:45,440 Speaker 1: they're hard to go into the details here now, but 704 00:39:45,520 --> 00:39:49,040 Speaker 1: there are bizarre higher reaches of set theory that don't 705 00:39:49,080 --> 00:39:52,440 Speaker 1: look like they have any direct applications anywhere yet. You know, 706 00:39:52,760 --> 00:39:55,400 Speaker 1: maybe if they do, then then you'd be realist about those. 707 00:39:55,480 --> 00:39:59,680 Speaker 1: But it's not a blanket argument that, because mathematics is indispensable, 708 00:40:00,040 --> 00:40:02,360 Speaker 1: believe in all of it right. It's got to be 709 00:40:02,400 --> 00:40:05,960 Speaker 1: the bits that get applied, and that's one of the 710 00:40:05,960 --> 00:40:09,040 Speaker 1: criticisms of this line of thought. You don't get realism 711 00:40:09,080 --> 00:40:12,440 Speaker 1: about mathematics. You get realism about Calculus, you get realism 712 00:40:12,480 --> 00:40:16,920 Speaker 1: about Algebraic Topology, realism about differential geometry. You know you're 713 00:40:16,920 --> 00:40:19,040 Speaker 1: going to get realists about the bits that get used 714 00:40:19,080 --> 00:40:22,560 Speaker 1: and not all of it Um but that's for me. 715 00:40:22,640 --> 00:40:25,840 Speaker 1: That's as it should be. Just because physics is in 716 00:40:25,840 --> 00:40:28,360 Speaker 1: the business of describing the universe doesn't mean we should 717 00:40:28,400 --> 00:40:31,000 Speaker 1: believe all of physics. We should believe the bits of 718 00:40:31,040 --> 00:40:34,240 Speaker 1: physics that is really indispensable to our understanding of the universe. 719 00:40:34,400 --> 00:40:36,160 Speaker 1: And as you know, there are going to be speculative 720 00:40:36,160 --> 00:40:39,440 Speaker 1: parts of physics, not just things like string theory, but 721 00:40:39,640 --> 00:40:44,439 Speaker 1: nonphysical models. So, for instance, massless universes. People study things 722 00:40:44,560 --> 00:40:47,080 Speaker 1: like a universe with no mass. Can you have curvature 723 00:40:47,160 --> 00:40:49,719 Speaker 1: in the universe with no mass, for instance? Not that 724 00:40:49,760 --> 00:40:51,439 Speaker 1: we live in such a universe. We know we don't 725 00:40:51,440 --> 00:40:53,960 Speaker 1: mean one of those universes, but the question is that 726 00:40:54,000 --> 00:40:56,600 Speaker 1: will give us some understanding about our own universe if 727 00:40:56,600 --> 00:41:01,240 Speaker 1: we can study these non existent universes. So same with mathematics. 728 00:41:01,239 --> 00:41:04,239 Speaker 1: They're gonna be speculative parts of mathematics like that that 729 00:41:04,480 --> 00:41:05,920 Speaker 1: even someone like me is not going to be a 730 00:41:06,000 --> 00:41:09,400 Speaker 1: realist about. But I am not convinced at the end 731 00:41:09,400 --> 00:41:12,040 Speaker 1: of the day by heart tree fields project, despite the 732 00:41:12,080 --> 00:41:14,640 Speaker 1: fact that I think it's a fascinating and, you know, 733 00:41:15,239 --> 00:41:19,080 Speaker 1: beautiful technical exercise. And so I understand that being convinced 734 00:41:19,080 --> 00:41:21,320 Speaker 1: about the realism of sets doesn't mean that all of 735 00:41:21,400 --> 00:41:24,440 Speaker 1: mathematics is real. But what is it that convinces you 736 00:41:24,600 --> 00:41:26,840 Speaker 1: about the realism of sets? So what is the argument 737 00:41:26,880 --> 00:41:30,719 Speaker 1: that persuades you? Is it the usefulness of mathematics? Is that, 738 00:41:30,920 --> 00:41:33,759 Speaker 1: you know, seeing something beautiful in nature and seeing, you know, 739 00:41:34,200 --> 00:41:36,799 Speaker 1: mathematics in it, the FIBONACCI sequence or, you know, the 740 00:41:36,800 --> 00:41:39,480 Speaker 1: golden ratio? What is it that convinces you that mathematics 741 00:41:39,600 --> 00:41:41,960 Speaker 1: is real and not just a human construct? Well, it 742 00:41:42,000 --> 00:41:45,960 Speaker 1: comes back to this indispensability that we just I can't 743 00:41:45,960 --> 00:41:50,200 Speaker 1: see how we could do science without mathematics. And, moreover, 744 00:41:50,320 --> 00:41:53,000 Speaker 1: it's not just that it's this language of science, as 745 00:41:53,000 --> 00:41:56,319 Speaker 1: you often hear. I'm not quite sure what that really 746 00:41:56,480 --> 00:41:59,040 Speaker 1: even means. I mean it's not like, you know, once 747 00:41:59,120 --> 00:42:01,440 Speaker 1: upon a time all academic work had to be carried 748 00:42:01,440 --> 00:42:04,560 Speaker 1: out into Latin and Latin was the language of science. 749 00:42:04,760 --> 00:42:07,000 Speaker 1: It's not that that people are talking about it like 750 00:42:07,040 --> 00:42:10,640 Speaker 1: there's something much deeper about mathematics than Latin. No one 751 00:42:10,800 --> 00:42:13,440 Speaker 1: for a moment really thought that you couldn't do science 752 00:42:13,480 --> 00:42:15,880 Speaker 1: in English or whatever. It had to be in Latin. 753 00:42:16,080 --> 00:42:18,480 Speaker 1: That when people say mathematics is the language of science, 754 00:42:18,480 --> 00:42:20,920 Speaker 1: they mean something much deeper than that, I take it. 755 00:42:21,160 --> 00:42:24,480 Speaker 1: And one of the ways I think that mathematics is 756 00:42:24,520 --> 00:42:29,680 Speaker 1: indispensable is in offering up explanations. So this is very controversial. 757 00:42:29,880 --> 00:42:32,480 Speaker 1: Great deal of debate in philosophy mathematics about this at 758 00:42:32,480 --> 00:42:37,840 Speaker 1: the moment, about whether you can get explanations in physics. 759 00:42:38,000 --> 00:42:42,600 Speaker 1: Say That mathematically in character. So the mathematics is not 760 00:42:42,760 --> 00:42:47,799 Speaker 1: merely just his language, that is providing explanations for what's 761 00:42:47,840 --> 00:42:50,759 Speaker 1: going on in the physical world and, as I said, 762 00:42:51,920 --> 00:42:54,680 Speaker 1: very controversial. I do believe that. I do think that 763 00:42:54,719 --> 00:42:58,919 Speaker 1: mathematics is offering explanations and that's a really important way 764 00:42:58,920 --> 00:43:00,960 Speaker 1: in which you can be in to spend sable. If 765 00:43:01,000 --> 00:43:05,640 Speaker 1: something's it play an explanatory role in your best scientific theory, 766 00:43:05,760 --> 00:43:07,719 Speaker 1: then it really does look like you should be a 767 00:43:07,719 --> 00:43:10,040 Speaker 1: realist about it. If you'RE gonna be realist about anything, 768 00:43:10,040 --> 00:43:12,520 Speaker 1: and I mean there are anti realists about a great 769 00:43:12,520 --> 00:43:14,800 Speaker 1: deal of science as well, but I'm setting that aside. 770 00:43:15,200 --> 00:43:18,759 Speaker 1: If you're going to be realist about science, then it's 771 00:43:18,840 --> 00:43:21,880 Speaker 1: very odd to say that the reason for such and 772 00:43:21,920 --> 00:43:27,520 Speaker 1: such an event occurring was because of some entity, but 773 00:43:27,560 --> 00:43:30,120 Speaker 1: there's no such entity. You Haven't explained anything if you 774 00:43:30,160 --> 00:43:32,960 Speaker 1: say that. So if mathematics can played this kind of 775 00:43:32,960 --> 00:43:37,080 Speaker 1: explanatory role, then that looks like really good grounds for 776 00:43:37,320 --> 00:43:41,879 Speaker 1: thinking that it's indispensable to scientific explanation. It is really 777 00:43:41,880 --> 00:43:45,400 Speaker 1: compelling to me that mathematics can describe not just the 778 00:43:45,440 --> 00:43:50,560 Speaker 1: fundamental bits of the universe and the fundamental elements of mathematics, 779 00:43:50,600 --> 00:43:53,000 Speaker 1: but also that it seems like we can find these 780 00:43:53,040 --> 00:43:57,640 Speaker 1: fairly simple mathematical stories to describe emergent things. You can 781 00:43:57,680 --> 00:43:59,840 Speaker 1: imagine living in a universe where the basic thing are 782 00:44:00,000 --> 00:44:03,040 Speaker 1: strings and there's math about strings. And that doesn't mean that, 783 00:44:03,080 --> 00:44:05,399 Speaker 1: even if you understood how strings work, that you could 784 00:44:05,440 --> 00:44:07,520 Speaker 1: use them to predict the path of a hurricane. Right. 785 00:44:07,520 --> 00:44:10,919 Speaker 1: It's incredibly complicated to go from fundamental bits, even from 786 00:44:10,920 --> 00:44:14,000 Speaker 1: water drops, to a hurricane, not to mention from strings. Right. 787 00:44:14,040 --> 00:44:17,080 Speaker 1: So strings don't really like provide any explanation of the 788 00:44:17,120 --> 00:44:19,239 Speaker 1: path of a hurricane, even if you knew what all 789 00:44:19,280 --> 00:44:21,600 Speaker 1: the strings were doing. Right. But you can zoom out 790 00:44:21,680 --> 00:44:24,680 Speaker 1: and find some higher level laws. You know, you find 791 00:44:24,840 --> 00:44:29,120 Speaker 1: fluid mechanics and you find gravitational rotational theorems about how 792 00:44:29,160 --> 00:44:32,359 Speaker 1: galaxies move. We can seem to do physics at these 793 00:44:32,440 --> 00:44:35,600 Speaker 1: higher levels even without understanding the little bits underneath. And 794 00:44:35,640 --> 00:44:38,760 Speaker 1: the same way it seems like we can find mathematics 795 00:44:38,800 --> 00:44:42,080 Speaker 1: that describes the universe, even if it's not necessarily connected 796 00:44:42,120 --> 00:44:44,600 Speaker 1: to those fundamental little bits of the universe. Why do 797 00:44:44,640 --> 00:44:46,120 Speaker 1: you think that is? Why do you think it is 798 00:44:46,600 --> 00:44:50,120 Speaker 1: that mathematical descriptions of the universe emerge at all these 799 00:44:50,160 --> 00:44:53,120 Speaker 1: different levels, even when they're not necessarily connected to each 800 00:44:53,120 --> 00:44:56,080 Speaker 1: other or easily built from one to the other? Again, 801 00:44:56,160 --> 00:44:59,120 Speaker 1: one of the bright puzzles and philosophy of mathematics. It's 802 00:44:59,200 --> 00:45:02,200 Speaker 1: this often col the unreasonable effectiveness problem. How is it 803 00:45:02,400 --> 00:45:05,279 Speaker 1: mathematics just turns up in all of these places, not 804 00:45:05,360 --> 00:45:09,960 Speaker 1: just fundamental physics, but in Chemistry and biology and psychology. No, 805 00:45:10,080 --> 00:45:12,440 Speaker 1: at which level you know? If you think of science 806 00:45:12,480 --> 00:45:15,839 Speaker 1: as these levels, from the fundamental to the more complex, 807 00:45:16,239 --> 00:45:19,840 Speaker 1: at every level you've got relevant mathematics that appears there 808 00:45:20,600 --> 00:45:23,080 Speaker 1: and again. I wish I had a good answer to that. 809 00:45:23,840 --> 00:45:26,560 Speaker 1: One suggestion is that when you know the old, the 810 00:45:26,640 --> 00:45:30,640 Speaker 1: old sort of addage that if hammer is your only tool, 811 00:45:30,719 --> 00:45:32,759 Speaker 1: then the whole world looks like a nail. Right. So 812 00:45:33,040 --> 00:45:35,879 Speaker 1: we've got differential equations, damn it, we're going to use 813 00:45:35,920 --> 00:45:39,080 Speaker 1: them everywhere, right. But that just doesn't wash with me. 814 00:45:39,280 --> 00:45:43,040 Speaker 1: It's not just that we're forcing everything to be thought 815 00:45:43,080 --> 00:45:46,520 Speaker 1: of in this framework of a particular bit of mathematics 816 00:45:46,520 --> 00:45:49,160 Speaker 1: like differential equations. And to be fair, there were times, 817 00:45:49,200 --> 00:45:52,000 Speaker 1: I think, where physics was a bit like that. Everything 818 00:45:52,040 --> 00:45:57,080 Speaker 1: had to be well behaved. Differential Equations Linear First Order, 819 00:45:58,160 --> 00:45:59,680 Speaker 1: and you're try and do as much as you can 820 00:45:59,719 --> 00:46:02,040 Speaker 1: with is because they were well understood. But I just 821 00:46:02,080 --> 00:46:04,160 Speaker 1: don't think that's how we work now. I mean there's 822 00:46:04,239 --> 00:46:07,080 Speaker 1: so many different branches of mathematics that are turning up 823 00:46:07,080 --> 00:46:10,280 Speaker 1: in all different places. As you mentioned earlier, group theory. 824 00:46:10,840 --> 00:46:13,480 Speaker 1: A number of places where you need group theory grows 825 00:46:13,520 --> 00:46:16,600 Speaker 1: by the day. It doesn't seem to be simply we 826 00:46:16,680 --> 00:46:19,040 Speaker 1: have these tools and we're going to use it, damn it. 827 00:46:19,040 --> 00:46:22,239 Speaker 1: It's more like these are the very tools that we 828 00:46:22,280 --> 00:46:26,200 Speaker 1: would need to do any such science. And again, what 829 00:46:26,280 --> 00:46:29,319 Speaker 1: does that tell us about the mathematics? Well, like it's 830 00:46:29,360 --> 00:46:32,880 Speaker 1: intritically connected to the physical world in this kind of 831 00:46:32,920 --> 00:46:38,080 Speaker 1: way that you just can't understand the physical world without, yeah, 832 00:46:38,600 --> 00:46:42,520 Speaker 1: having the relevant mathematics under control. You know, as I said, 833 00:46:42,560 --> 00:46:45,440 Speaker 1: that's controversial. Is just try and flag the things that 834 00:46:45,480 --> 00:46:49,360 Speaker 1: are more controversial. But since we're talking philosophy here, we 835 00:46:49,360 --> 00:46:54,600 Speaker 1: can just have a general disclaimer. All of this is controversy. Well, 836 00:46:54,680 --> 00:46:56,720 Speaker 1: we like to get into the weeds on this show, 837 00:46:56,840 --> 00:46:58,759 Speaker 1: and so I have a lot more deep questions about 838 00:46:58,800 --> 00:47:01,600 Speaker 1: philosophy of math but when to take another quick break? 839 00:47:14,200 --> 00:47:16,080 Speaker 1: All right, so we're back and we're having a lot 840 00:47:16,120 --> 00:47:20,080 Speaker 1: of fun talking to Professor Mark Coldevan about whether mathematics 841 00:47:20,160 --> 00:47:23,160 Speaker 1: is inherent in the universe and is it real. And 842 00:47:23,360 --> 00:47:26,359 Speaker 1: I was wondering when you were talking earlier, since philosophy 843 00:47:26,360 --> 00:47:29,960 Speaker 1: of mathematics asks like what is mathematics and is it real, 844 00:47:30,440 --> 00:47:33,960 Speaker 1: is there a branch of philosophy called philosophy of philosophy 845 00:47:34,000 --> 00:47:36,719 Speaker 1: that asks like whether philosophy is real and what our 846 00:47:36,719 --> 00:47:41,040 Speaker 1: philosophers doing? Anyway, there is, there is. Very recently people 847 00:47:41,080 --> 00:47:45,040 Speaker 1: have been working on philosophy of philosophy. I think you know, 848 00:47:45,040 --> 00:47:47,000 Speaker 1: white philosophers have been doing this for a long time. 849 00:47:47,080 --> 00:47:49,640 Speaker 1: I just didn't come up with the phrase, but not 850 00:47:49,719 --> 00:47:51,879 Speaker 1: so much the idea of where the philosophy is real, 851 00:47:51,960 --> 00:47:54,080 Speaker 1: because you're not interested where the physics is real. You're 852 00:47:54,120 --> 00:47:57,680 Speaker 1: interested in whether the things that physics posits are real. Right, 853 00:47:58,360 --> 00:48:03,160 Speaker 1: and so, in such far as philosophy is positing entities, 854 00:48:03,440 --> 00:48:06,040 Speaker 1: are those things real? You could ask that sort of question. 855 00:48:06,080 --> 00:48:08,840 Speaker 1: But my understanding at least, the philosophy of philosophy is 856 00:48:08,880 --> 00:48:12,720 Speaker 1: more a kind of a systematic study of methodology, right, 857 00:48:12,760 --> 00:48:15,000 Speaker 1: which is kind of how I think of philosophy of 858 00:48:15,080 --> 00:48:19,960 Speaker 1: science is in many ways looking at science and trying 859 00:48:20,000 --> 00:48:24,960 Speaker 1: to discern useful things to say about methodology. And so 860 00:48:25,040 --> 00:48:29,600 Speaker 1: philosophy of philosophy is much more about methodological questions about 861 00:48:29,800 --> 00:48:35,240 Speaker 1: philosophy questions like does philosophy make progress? So philosophy cops 862 00:48:35,239 --> 00:48:39,160 Speaker 1: criticism because the questions were interested in the questions we're 863 00:48:39,160 --> 00:48:42,680 Speaker 1: talking about here now. Are Numbers Real? That goes back 864 00:48:42,719 --> 00:48:45,520 Speaker 1: to at least back to Plato, right. And have we 865 00:48:45,600 --> 00:48:48,920 Speaker 1: made much progress since then? Well, you know, I'd like 866 00:48:49,000 --> 00:48:51,400 Speaker 1: to think we've made some, but certainly if you look 867 00:48:51,480 --> 00:48:54,879 Speaker 1: at progress physics has made since such times to now, 868 00:48:55,040 --> 00:48:59,279 Speaker 1: physics has not much better firm a ground then. Right, 869 00:48:59,360 --> 00:49:02,240 Speaker 1: that's true. Maybe you guys just need more coffee, although 870 00:49:02,360 --> 00:49:04,480 Speaker 1: you could also say that physics is just an outgrowth 871 00:49:04,520 --> 00:49:07,920 Speaker 1: of philosophy. When a question becomes experimental, it decrimes its 872 00:49:07,920 --> 00:49:10,600 Speaker 1: own science and philosophy sort of loses control of it. 873 00:49:10,760 --> 00:49:13,319 Speaker 1: But speaking of concrete questions, I want to come back 874 00:49:13,360 --> 00:49:16,400 Speaker 1: to the framing we had earlier about aliens. Do you 875 00:49:16,440 --> 00:49:21,480 Speaker 1: think that, if aliens arrived, that we could use mathematics 876 00:49:21,520 --> 00:49:24,960 Speaker 1: as a sort of basis for building a mental connection 877 00:49:25,000 --> 00:49:28,200 Speaker 1: with them, of understanding whether or not we're thinking in 878 00:49:28,200 --> 00:49:30,960 Speaker 1: a similar way as them? Would you send of mathematicians 879 00:49:31,080 --> 00:49:34,440 Speaker 1: or philosophers of math out to meet the aliens? First thing, 880 00:49:34,680 --> 00:49:36,880 Speaker 1: I do think that there would be a good place 881 00:49:36,920 --> 00:49:40,200 Speaker 1: to start. Would be bits of mathematics do you think 882 00:49:41,080 --> 00:49:43,520 Speaker 1: likely to be universal? I must say I haven't given 883 00:49:43,560 --> 00:49:45,480 Speaker 1: a lot of thought to who I would send first 884 00:49:45,480 --> 00:49:48,480 Speaker 1: to meet the aliens. You don't realize that you're near 885 00:49:48,520 --> 00:49:51,840 Speaker 1: the top of the list. That should concern me for 886 00:49:51,880 --> 00:49:55,200 Speaker 1: all sorts of reasons. But but yeah, I do think 887 00:49:55,200 --> 00:49:59,319 Speaker 1: the look looking for bits of mathematics that you think 888 00:49:59,360 --> 00:50:02,480 Speaker 1: would be calm. Again, you wouldn't want decimal expansion of 889 00:50:02,600 --> 00:50:08,600 Speaker 1: pie based ten, but expansion of pie based two. That's 890 00:50:08,800 --> 00:50:12,920 Speaker 1: something that you might think would be recognizable fundamental theorems 891 00:50:12,960 --> 00:50:16,160 Speaker 1: suitably couched, because you know that the notation you use 892 00:50:16,520 --> 00:50:22,160 Speaker 1: is perhaps arbitrary in various ways. But the fundamental theorem, 893 00:50:22,600 --> 00:50:26,000 Speaker 1: fundamental theorem of Calculus, for instance. You think that any 894 00:50:26,200 --> 00:50:31,000 Speaker 1: reasonably advanced life forms who are capable of traveling to 895 00:50:31,120 --> 00:50:35,400 Speaker 1: Earth from great distance would have come across the fundamental 896 00:50:35,440 --> 00:50:38,520 Speaker 1: theorems of Calculus. So how do you express those in 897 00:50:38,520 --> 00:50:42,880 Speaker 1: a way that's not merely notational dependent too much on 898 00:50:42,920 --> 00:50:45,960 Speaker 1: the notation? You can't express it in English, obviously, but 899 00:50:46,080 --> 00:50:49,640 Speaker 1: the standard notation using integral signs and so forth. I 900 00:50:49,640 --> 00:50:52,399 Speaker 1: think that's kind of accidental that. How do you get 901 00:50:52,400 --> 00:50:55,440 Speaker 1: that idea across? That does seem like a good place 902 00:50:55,480 --> 00:50:58,479 Speaker 1: to start. You'd think that an intelligent, advanced race would 903 00:50:58,680 --> 00:51:01,960 Speaker 1: know the fundamental theorem of Calculus, but how would they 904 00:51:01,960 --> 00:51:04,839 Speaker 1: write it and how how should you convey it to them? Well, 905 00:51:04,880 --> 00:51:07,520 Speaker 1: we had Noam Chomsky on the podcast a few weeks 906 00:51:07,520 --> 00:51:10,600 Speaker 1: ago and we asked him this question and he said, 907 00:51:11,360 --> 00:51:14,719 Speaker 1: I'll quote, there's a good chance that arithmetic is universal. 908 00:51:14,760 --> 00:51:16,759 Speaker 1: It's a fair guess that at least the arithmetic would 909 00:51:16,760 --> 00:51:19,000 Speaker 1: be close enough to be absolute, so that anything we 910 00:51:19,080 --> 00:51:22,239 Speaker 1: might call intelligence that we would recognize this intelligence would 911 00:51:22,239 --> 00:51:24,919 Speaker 1: at least sit on that. And I suppose that he's 912 00:51:24,960 --> 00:51:28,160 Speaker 1: making the argument that you're making that mathematics is probably fundamental. 913 00:51:28,360 --> 00:51:30,759 Speaker 1: And in addition, he's drilling down and he's saying, let's 914 00:51:30,760 --> 00:51:33,400 Speaker 1: not start with something complicated, let's go down to the basics, 915 00:51:33,760 --> 00:51:36,200 Speaker 1: like you were saying earlier, set theory, let's find the 916 00:51:36,239 --> 00:51:39,040 Speaker 1: fundamental elements and see if we can begin from that. 917 00:51:39,400 --> 00:51:41,560 Speaker 1: Do you think that program is likely to be successful 918 00:51:41,600 --> 00:51:43,960 Speaker 1: if aliens arrived? Yeah, yeah, I think that's it's a 919 00:51:44,080 --> 00:51:46,880 Speaker 1: very good suggestion. You know, again back to the something 920 00:51:46,920 --> 00:51:50,239 Speaker 1: like pie not just any old numbers, because you might 921 00:51:50,320 --> 00:51:55,200 Speaker 1: think that nothing special about one to three, four in particular, 922 00:51:55,560 --> 00:52:01,080 Speaker 1: but it's really crucial to number theory. Be a concept 923 00:52:01,080 --> 00:52:04,560 Speaker 1: of prime number, for instance. You might think certain numbers 924 00:52:04,640 --> 00:52:08,920 Speaker 1: jump out at you like prime numbers, Pie, e. Some 925 00:52:09,080 --> 00:52:11,120 Speaker 1: of these numbers in particular, and so if you can 926 00:52:11,160 --> 00:52:14,640 Speaker 1: get a way of expressing those numbers, but fundamental parts 927 00:52:14,680 --> 00:52:18,880 Speaker 1: of arithmetic. But the notion of primness again, how do 928 00:52:18,920 --> 00:52:24,520 Speaker 1: you imagine an intelligent, advanced race not having that concept? Again, 929 00:52:24,680 --> 00:52:27,239 Speaker 1: just how do you convey it? But I do like 930 00:52:27,320 --> 00:52:29,759 Speaker 1: the suggestion. Yeah, well, I'd love to examine the sort 931 00:52:29,760 --> 00:52:33,960 Speaker 1: of counter idea. Like to think about what an intelligent 932 00:52:34,080 --> 00:52:36,480 Speaker 1: race might have to have in their minds in order 933 00:52:36,520 --> 00:52:38,920 Speaker 1: to not arrive at Arithmetic. You know, I can imagine 934 00:52:38,960 --> 00:52:41,840 Speaker 1: sitting across from their mathematicians and drawing a symbol for 935 00:52:41,960 --> 00:52:44,400 Speaker 1: one and pointing at one thing and an apple, and 936 00:52:44,480 --> 00:52:46,920 Speaker 1: bringing another apple and then writing the symbol for two 937 00:52:47,320 --> 00:52:49,600 Speaker 1: or something, and you know, then you have one plus 938 00:52:49,680 --> 00:52:51,759 Speaker 1: one equals to this kind of stuff. And you must 939 00:52:51,760 --> 00:52:53,680 Speaker 1: have thought about this more deeply than I have. What 940 00:52:53,800 --> 00:52:56,400 Speaker 1: assumptions are there inherent in that? You know, are we 941 00:52:56,480 --> 00:53:00,560 Speaker 1: assuming that the concept of abstraction to say, are these 942 00:53:00,560 --> 00:53:03,240 Speaker 1: two apples? They have the similar property. They're both apples. 943 00:53:03,440 --> 00:53:05,840 Speaker 1: Obviously they're not the same apple. They're different ones, darker, 944 00:53:05,880 --> 00:53:08,400 Speaker 1: one brighter or whatever. What if the aliens are like, 945 00:53:08,640 --> 00:53:10,520 Speaker 1: you know, that's one apple and that's one apple. We 946 00:53:10,600 --> 00:53:12,800 Speaker 1: don't know what you mean with this whole two business. 947 00:53:12,840 --> 00:53:16,280 Speaker 1: That's nonsense. Aren't there fundamental assumptions we're making there, even 948 00:53:16,320 --> 00:53:18,600 Speaker 1: with one plus one equals too? Yeah, I think so. 949 00:53:18,840 --> 00:53:21,160 Speaker 1: I think it's exactly right. It's got to be counting 950 00:53:21,160 --> 00:53:24,080 Speaker 1: the right sorts of things right. So one cloud and 951 00:53:24,120 --> 00:53:29,520 Speaker 1: another cloud is one big cloud right. So you don't 952 00:53:29,560 --> 00:53:32,640 Speaker 1: think one cloud plus one cloud equals one. That's a 953 00:53:33,160 --> 00:53:36,600 Speaker 1: falsification of basic arithmetic. You think, no, you're counting the 954 00:53:36,600 --> 00:53:39,000 Speaker 1: wrong kinds of things. They're one plus one is too 955 00:53:39,360 --> 00:53:41,840 Speaker 1: given that you're counting the right kinds of things, discreet 956 00:53:41,960 --> 00:53:45,120 Speaker 1: things that have a certain kind of property, you can 957 00:53:45,160 --> 00:53:47,840 Speaker 1: still count an apple and an orange and get to 958 00:53:48,480 --> 00:53:50,680 Speaker 1: but then you've got to have this overarching concept of 959 00:53:51,000 --> 00:53:54,560 Speaker 1: pieces of fruit or things before me or something. But 960 00:53:54,600 --> 00:53:57,799 Speaker 1: there's got to be some overarching concept there. So you're 961 00:53:57,800 --> 00:54:02,880 Speaker 1: absolutely right. There are some conditions for even understanding basic arithmetic. 962 00:54:03,080 --> 00:54:07,120 Speaker 1: I was reading about how Japanese people count and discovering 963 00:54:07,160 --> 00:54:10,560 Speaker 1: that Japanese counting words are actually quite different from English 964 00:54:10,560 --> 00:54:12,640 Speaker 1: counting words, that if you put a set of things 965 00:54:12,680 --> 00:54:15,200 Speaker 1: in front of them, they tend to group them by shape. 966 00:54:15,320 --> 00:54:17,080 Speaker 1: So those are not just two things, these are two 967 00:54:17,120 --> 00:54:19,279 Speaker 1: flat things and there's two tall things over there. It 968 00:54:19,360 --> 00:54:23,000 Speaker 1: strikes me that if, even across human cultures we count 969 00:54:23,040 --> 00:54:26,360 Speaker 1: and abstract things in different ways, it might be that aliens, 970 00:54:26,440 --> 00:54:28,600 Speaker 1: I have no idea what we're talking about when we're 971 00:54:28,600 --> 00:54:31,560 Speaker 1: demonstrating our basic arithmetic to them. There are cases in 972 00:54:31,640 --> 00:54:34,480 Speaker 1: English as well where you're not interested in the number 973 00:54:34,480 --> 00:54:37,520 Speaker 1: of things for various reasons. She's just interested in existence. 974 00:54:38,360 --> 00:54:41,800 Speaker 1: So we say it's raining, meaning that there are drops 975 00:54:41,840 --> 00:54:43,960 Speaker 1: of rain falling, and that's all we care about. Just 976 00:54:44,040 --> 00:54:46,160 Speaker 1: want to know whether I need my umbrellas today or not. 977 00:54:46,239 --> 00:54:48,839 Speaker 1: So you can imagine someone putting apples in front of them. 978 00:54:48,840 --> 00:54:54,600 Speaker 1: They're saying it's appling. Right, I have the concept of 979 00:54:54,680 --> 00:54:57,759 Speaker 1: multiple apples, but I don't need to three or four. 980 00:54:57,800 --> 00:55:01,640 Speaker 1: Who Cares how many apples? There's zero apples and there's 981 00:55:01,640 --> 00:55:04,200 Speaker 1: appling again, and we do that. There are lots of 982 00:55:04,560 --> 00:55:08,000 Speaker 1: instances of that, like raining is the obvious example, but 983 00:55:08,040 --> 00:55:10,640 Speaker 1: there are other cases as well where you're interested in 984 00:55:11,680 --> 00:55:15,560 Speaker 1: zero or existence. You know of something and you don't 985 00:55:15,560 --> 00:55:18,719 Speaker 1: particularly worry about counting, even though you could. There are 986 00:55:19,719 --> 00:55:23,279 Speaker 1: rain drops are discreet. You could worry about how many 987 00:55:23,400 --> 00:55:26,120 Speaker 1: rain drops there are, and it's not purely because it's 988 00:55:26,120 --> 00:55:29,000 Speaker 1: a difficult exercise to count them. It's more just no, 989 00:55:29,120 --> 00:55:31,080 Speaker 1: I'm not really interested in how many there are. I'm 990 00:55:31,160 --> 00:55:34,719 Speaker 1: just interested is it raining or not. So again, you know, 991 00:55:34,760 --> 00:55:40,040 Speaker 1: you could have this idea of zero things or many. 992 00:55:40,440 --> 00:55:43,080 Speaker 1: You know, zero, one and many. You can imagine an 993 00:55:43,080 --> 00:55:46,400 Speaker 1: intelligent civilization getting by with a very different kind of 994 00:55:46,440 --> 00:55:49,040 Speaker 1: sense of what many is as many? You know, more 995 00:55:49,040 --> 00:55:51,120 Speaker 1: than a thousand or you know, is it just a few? 996 00:55:51,480 --> 00:55:55,560 Speaker 1: I find that people have a different sense like effective infinity. 997 00:55:55,600 --> 00:55:57,799 Speaker 1: You know what is a lot and what needs to 998 00:55:57,840 --> 00:56:00,239 Speaker 1: be counted individually. Anyway, these are really fast and any 999 00:56:00,360 --> 00:56:02,640 Speaker 1: questions and I really thank you for answering them and 1000 00:56:02,680 --> 00:56:04,920 Speaker 1: for exploring these questions with me. I think my last 1001 00:56:05,000 --> 00:56:07,160 Speaker 1: question to you is do we expect any sort of 1002 00:56:07,239 --> 00:56:10,200 Speaker 1: breakthroughs in philosophy of mathematics? I mean, you said we've 1003 00:56:10,200 --> 00:56:13,120 Speaker 1: been struggling with a question of our numbers and circles 1004 00:56:13,200 --> 00:56:15,720 Speaker 1: real since Plato. Do you think we're going to figure 1005 00:56:15,760 --> 00:56:17,200 Speaker 1: that out, that we ever have a point where we're 1006 00:56:17,239 --> 00:56:19,080 Speaker 1: like yeah, we proved that, now we can move on 1007 00:56:19,120 --> 00:56:21,880 Speaker 1: to something else, or is philosophy of math basically going 1008 00:56:21,920 --> 00:56:24,560 Speaker 1: to go forever until we meet the Aliens? I would 1009 00:56:24,600 --> 00:56:26,520 Speaker 1: like to think that it would be solved, but I 1010 00:56:27,600 --> 00:56:33,000 Speaker 1: don't think that's necessary for it to be worthwhile exercise. Um. 1011 00:56:33,040 --> 00:56:36,040 Speaker 1: So why I think that is we learn a lot 1012 00:56:36,120 --> 00:56:40,279 Speaker 1: along the way. Sometimes asking the right questions is more 1013 00:56:40,320 --> 00:56:42,640 Speaker 1: interesting than finding answers to them, and I don't think 1014 00:56:42,680 --> 00:56:44,840 Speaker 1: there's any special about philosophy. You know, I think you 1015 00:56:44,840 --> 00:56:47,399 Speaker 1: would find that in physics as well. Not being able 1016 00:56:47,440 --> 00:56:49,960 Speaker 1: to answer some questions, but those questions giving rise to 1017 00:56:50,040 --> 00:56:53,759 Speaker 1: other questions and new areas is what motivates us and 1018 00:56:53,880 --> 00:56:57,280 Speaker 1: what keeps the disciplines rolling. It sounds to an outside 1019 00:56:57,280 --> 00:57:01,600 Speaker 1: of that might sound rather close shop. As you know, 1020 00:57:01,920 --> 00:57:04,440 Speaker 1: you're only interested in these little questions. We're interested in 1021 00:57:04,440 --> 00:57:07,399 Speaker 1: getting the exercise rolling. But I do think that we've 1022 00:57:07,480 --> 00:57:10,320 Speaker 1: learned a lot about the relationship between mathematics and the 1023 00:57:10,360 --> 00:57:14,319 Speaker 1: physical realm, a lot of understanding about the foundations of mathematics. 1024 00:57:14,320 --> 00:57:17,000 Speaker 1: We don't have the answer to what the foundations of 1025 00:57:17,080 --> 00:57:22,880 Speaker 1: mathematics are, but you have some interesting insights from set 1026 00:57:22,920 --> 00:57:25,160 Speaker 1: theory and the like. So are we like to to 1027 00:57:25,200 --> 00:57:28,600 Speaker 1: solve the problems in my lifetime? I don't think so. 1028 00:57:28,880 --> 00:57:30,520 Speaker 1: I hope not. I'd be out of a job, you know, 1029 00:57:30,640 --> 00:57:33,480 Speaker 1: but I don't think it's going to happen. But I 1030 00:57:33,520 --> 00:57:35,960 Speaker 1: don't think that that means that it's all a waste 1031 00:57:36,000 --> 00:57:38,560 Speaker 1: of time. I think we get in many interesting insights. 1032 00:57:38,800 --> 00:57:41,680 Speaker 1: In particular, the insights that motivated me early on in 1033 00:57:41,680 --> 00:57:44,840 Speaker 1: my career were this connection that why is that? The 1034 00:57:44,880 --> 00:57:48,200 Speaker 1: mathematics is applicable. I read this paper, famous paper by 1035 00:57:48,240 --> 00:57:53,120 Speaker 1: Eugene Vigna called the unreasonable effectiveness mathematics and natural sciences, 1036 00:57:53,480 --> 00:57:56,440 Speaker 1: and I read that as an undergraduate and it just 1037 00:57:57,320 --> 00:58:01,360 Speaker 1: captivated by that paper. And did he answer the questions? No. 1038 00:58:01,600 --> 00:58:04,440 Speaker 1: Has Anyone answered those questions? No, but fascinating stuff to 1039 00:58:04,440 --> 00:58:06,560 Speaker 1: think about and I think we have a much better 1040 00:58:06,640 --> 00:58:10,480 Speaker 1: understanding of the relationship between Applied Mathematics and physics now 1041 00:58:10,880 --> 00:58:14,080 Speaker 1: as a result of asking me sort of other questions 1042 00:58:14,120 --> 00:58:17,000 Speaker 1: about his mathematics. Real absolutely no. I think it's very 1043 00:58:17,080 --> 00:58:19,040 Speaker 1: useful and also, in a way, maybe you haven't even 1044 00:58:19,080 --> 00:58:22,720 Speaker 1: anticipated that. You have spent your life preparing, I think, 1045 00:58:22,840 --> 00:58:25,520 Speaker 1: to meet the aliens and when they ask me who 1046 00:58:25,560 --> 00:58:28,400 Speaker 1: we should send, you know, in our first contingent to 1047 00:58:28,480 --> 00:58:32,120 Speaker 1: chat with our alien technological friends, I'm going to nominate you. Good, 1048 00:58:32,160 --> 00:58:36,880 Speaker 1: as long as they're friendly, you know. Well, we'll find 1049 00:58:36,920 --> 00:58:40,720 Speaker 1: out right just will send the military. You know, make 1050 00:58:40,760 --> 00:58:43,960 Speaker 1: sure they're friendly first. All right, sounds good. Well, thank 1051 00:58:43,960 --> 00:58:45,640 Speaker 1: you very much for joining us and for talking to 1052 00:58:45,760 --> 00:58:47,920 Speaker 1: us about these crazy ideas. I hope that one day 1053 00:58:47,960 --> 00:58:50,920 Speaker 1: we do figure out our numbers real why math works. 1054 00:58:51,120 --> 00:58:53,680 Speaker 1: And if, in fact, math is just a game we 1055 00:58:53,760 --> 00:58:56,640 Speaker 1: invented in our minds or the fundamental code of the 1056 00:58:56,720 --> 00:58:59,840 Speaker 1: universe itself. Thanks very much for joining us today. My pleasure, 1057 00:59:00,000 --> 00:59:09,960 Speaker 1: thanks for having me, thanks for listening, and remember that, 1058 00:59:10,080 --> 00:59:12,840 Speaker 1: Daniel and Jorge explained, the universe is a production of 1059 00:59:12,920 --> 00:59:16,280 Speaker 1: I heart radio. For more podcast from my heart radio, 1060 00:59:16,440 --> 00:59:20,000 Speaker 1: visit the I heart radio APP, apple podcasts or wherever 1061 00:59:20,120 --> 00:59:21,800 Speaker 1: you listen to your favorite shows.