 Hello, everybody, and welcome to the 22nd episode of Patterson in Pursuit. I am your host, Steve Patterson, and today we are going to infinity and beyond. Well, that's not actually quite true. My brain can't even go to infinity, so we're trying to just get to infinity much less beyond it. This week's interview was conducted in Oxford, England, where I spoke with a philosopher of mathematics about infinity. If you're familiar with my position on this topic or any of my writing you know, I have a tough time with infinity. To help me sort out my confusion, I had a conversation with Dr. Daniel Isakson, who teaches the philosophy of mathematics at Oxford. This is his area of expertise. Interesting note about Oxford, of all the places that I've gone so far, of all the universities I've been to, the individuals I've spoken with, I was by far most impressed with Oxford. I have a very negative opinion of academia and academicians in general, but my skepticism was pleasantly overcome just through the conversations that I had with people at Oxford. That includes my previous interview with Dr. Timothy Williamson, we talked about logic, and Dr. Simon Saunders, where we talked about quantum physics. Unfortunately, I do think that Oxford might be one of the few exceptions to the rule, especially given that I only spoke with a few academicians there. But if you are like me and didn't get to go to Oxford and you're currently enrolled in a college that you're just not impressed with, maybe your fellow students are disappointing you and the professors are disappointing you, I strongly urge you to check out a program called Praxis. They're the sponsor for the show and they specialize in taking young and independent minds and putting them straight into the real world, skipping college altogether. Praxis lands you a paid apprenticeship, they teach you real world job skills and after you complete their program, they guarantee you a $40,000 a year job offer. And the net cost to you of all of this is $0. So if that sounds like an alternative that you want, then head over to discoverpraxis.com and click on their schedule a call button. You can talk to them, see if it's a fit. And I know the founders of this company, I give them the highest marks, I know several people who have gone through the program and everybody is raving about it. So give them a call and see what you think. One of the more general criticisms that I have of academia is kind of a methodological one. It's not unique to any particular branch of thought, but you see it in mathematics, you see it in physics, you see it in economics, you see it in political theory and it has to do with what might be an overlooking of the foundations of a particular discipline. I think if you get the foundations inaccurate then the rest of your entire theory built on the inaccurate foundations is also going to be inaccurate. What's interesting is that I suspect in mathematics there is more work that needs to be done in regards to the foundations. There was a utter existential crisis around the turn of the 20th century in mathematics. There were a lot of different competing schools of thought offering different mutually exclusive foundations for mathematics and I think the group that won made a couple of important errors. One of them has to do with infinity. The other has to do with the metaphysics of mathematics. What are numbers? Mathematicians talk about numbers all the time. What are those types of things? So I love this interview that you're about to hear with Dr. Isaac Siniff Oxford because at the end after we have a good back and forth for about a half an hour or so. He actually said that if my conception of mathematics is correct then pretty much all professional mathematics for the last century has been foundationally flawed. And most people because they're reasonable would say, oh well therefore that probably means my own idea is wrong. It couldn't be the case that an entire profession has been making errors for a century. But because I'm completely unreasonable and bullheaded my suspicion is I think actually it might be the case that professional mathematicians have been wrong for a century. But you'll have to listen to it and decide for yourself. There's a great deal more to say on this particular topic but I really hope you enjoy my interview with Dr. Isaac Sin who teaches the philosophy of mathematics at Oxford University. In the world of mathematics a central concept is that of a infinite set or a completed infinity. What I struggle with is this idea just purely conceptually when you think of a completed infinity it smacks me of an impossibility that you could never have a completed infinity. So what I'd like to do is lay out just my own understanding of what we mean by the term infinite. And if that's wrong then please correct me and then I'll ask you a question. So when I use the term infinite or infinity what I mean is never ending or never completed or without boundaries something like that. Is that a fair? Which is rather the Greeks had that notion of infinity. Do you think that that is a good way, a healthy way, a proper way of thinking about it? Unbounded, yeah. Yes. So how could it be then that you have... It's a conception of infinity, it's not the only one. Right. Okay, maybe we can talk about that too. So with that conception of infinity it seems like simply by analyzing our concepts you couldn't have a infinite set because a set implies it's something that's boundary that it is something that's definite. So can you help me, how could it be that you have an infinite set in the first place? Right. Well I mean that's why we need different notions of infinity. On that conception of infinity, yes, there are no completed infinites. I mean this was one of Aristotle's very great insights. I mean Aristotle says that all infinity is potential, not actual. And that's a very important idea which is persisted to the present day and there are people who still take that view. For example I mentioned intuitionism, the approach to mathematics developed by in particular the Dutch mathematician L.E.J. Brouwer. But that's not the only way to understand infinity. Another way to understand infinity or rather one should say to understand infinite structures or infinite sets is in terms of the structure that is infinite. What does that mean? So if you take for example the natural numbers, if you think of the natural numbers as generated by producing the next one and the next one and the next one, that will never get you to a completed infinite. But it will get you to infinity. I mean a child who is learning arithmetic has this great moment when they realize that it goes on and on and that there is no limit. There are infinitely many numbers. That's a great moment of discovery. So what if I challenge that and I say there aren't infinity. So what if I say this? What I would actually say, not just playing devil's advocate or what I would actually say is, I think that is a confusion about the nature of numbers. It's not that there are an infinite amount of numbers out there. Numbers are something that have a conceptual existence. They exist when we conceive of them. So to say there are infinite amount of numbers implies that they're out there. No it doesn't. Oh it doesn't. I mean it's just what you said. I mean you said infinity was when you can go on and on and that's the nature of numbers. Okay. So if we say you can go on and on, that's what I meant. But that's a very, yeah that is a very, very different conception. I'm saying that's not the only conception but that's your conception of infinity and that's a perfectly okay one. So what's the alternative though? The alternative or an alternative is to say I'm thinking about the natural numbers and they are not some bunch of things. I'm not, in fact I think it's totally, totally misguided to think of natural numbers as a bunch of things. That would be, I mean that's indefensible ultimately but that one is thinking about a certain structure and what is the nature of that structure? It's a structure in which there is a two-place relation of next and that two-place relation is one to one so that no two things have the same next thing. Yes, but doesn't that imply then that these numbers are out there? No. No. No. We're talking about conceptual properties. Okay, so let me try to rephrase that and if this is an incorrect way of what you're saying then please correct me. What you're saying is that another way of thinking about infinity is that it is almost a statement about the generation, let's say, of sets. So for example we can come up with what an infinite set means has to do with how you generate the set and how there's no- No, that's not what I'm saying. That's not what you're saying. That's what you're saying. I'm not saying that. Okay. I'm not saying that a different way to think about the natural numbers is to think about the structure that they, if you like, constitute. That's the part that I'm not, I guess I don't understand. What is it? I'm trying to tell you. Okay. So there are certain fundamental properties that characterize that structure. First of all we need a language in which to express these properties and then with that language we can express the conditions that characterize it. But when you use the term structure I don't even have my head wrapped around what you mean by structure. No, that is a difficult notion, I agree, but just bear with me. So suppose we have a language, well not suppose we just give ourselves a language in which we have a symbol for, which we'll call zero, which is just is the, let's it be, the usual symbol of zero, and let S be a function symbol, that is it applies to something that gives us something. And the intended meaning is that S of X is the next natural number. But we don't talk about natural numbers, we just have zero and successor. And then we write down as an axiom that for every X and Y, if the successor of X equals the successor of Y, then X equals Y, which is to say that S is one to one mapping, right? Okay. And then we also say that zero is not the successor of anything. Okay. You can just write that in the language for all S of X not equal zero. And then we add another axiom which says that, and this is a second order axiom because we're quantifying over the elements of the domain, the collections of elements of the domain. For any collection of elements in the domain, if it contains zero and is closed under the S operation, then it contains everything. What do you mean by everything though? It's just I use a quantifier. I say for every X, well, for every C, let's say C is a collection of things, if zero is in C and for every Y, if Y is in C, then S of Y is in C, I don't think it's closed under successor, then for every X, X is in C. Yes, but I don't understand the connection between that and infinity. All right, I'll explain it. I'm jumping ahead here. So those three axioms characterize the structure of the natural numbers. Okay, understood as products of our conception. You can think of that as a product of our conception. Yes, I'm not saying there are things. I'm just saying this is what it is to be that structure. It starts with zero and it goes on and on and on and it has nothing else in it. That's what we're saying. So the first, if you have the successor operation, it goes from one thing to another, right? The first axiom that zero is, so if you have a successor operation, you start with zero. You have the successor of zero, the successor of the successor of zero and so on, right? Those things might not be the structure of the natural numbers in two ways. Well, in three ways they might not be and that's what exactly these three axioms guarantee doesn't happen and then you get exactly that in structure of the natural numbers. Something that starts with zero and goes on and on and on step by step. And the first condition is that, well, how could it end up not being the intended infinite structure? Well, it might circle back on itself to zero or it might circle back on itself to some non-zero element. The axiom that says zero is not the successor of anything guarantees that it doesn't circle back to zero and the one to one condition for all x, x, y, if s of x equals s of y, then x equals y, tells you that you don't get two things that both get their successors taken to the same thing, i.e. that sequence doesn't circle back to some other point, some non-zero point. Okay. And then the last condition is that there shouldn't be anything extraneous, right? Those two axioms tell you that you have zero and you have the successor and the successor and they're all different. And so that's, anything that satisfies those conditions contains the structure that we think of as the natural numbers. But then we need something that tells us that there's nothing else in that structure and that's what the last axiom does. That's the principle of induction. And with those properties, we then know what it is to be the structure of natural numbers and we can take the natural numbers to be what is characterized by those axioms. So, so let me- And there's nothing potential about it. It's those three axioms at once. Tell us what we are talking about. Okay, so for me in listening to that, it sounds like what you've done is accurately described or characterized the nature of a certain group of numbers. Well, not a group of numbers. I mean, a certain structure. Or you define the natural. That structure of the natural numbers. Yes, but so to me that sounds like a very precise definition. Like that's what we're talking about when we're talking about the natural numbers. But can you make the connection? So if I agree with that, I say, okay, let's say that conceptually speaking, when we talk about the natural numbers, this is what it is. Can you help me see the connection between that and an infinite set? When you're talking about an infinite set, are you simply referencing like this type of thing, which is the natural numbers? So it's not that it's actually some set that has a non-finite amount of elements in it, it's just you're talking about one group of things, which are the natural numbers. Anything that satisfies these axioms is an infinite collection. An infinite in what sense? So. Well, all right, in what sense? Well, in the sense that, well, of course, we have to say what we mean by infinite. I mean, so one definition of infinite and a slightly it's not exactly plain sailing to prove that this satisfies it. But that can be done would be to say that a set is infinite if there's one to one correspondence between it and a proper subset of itself. Okay, but all of the proofs that I've seen, the diagonal correspondence proofs, and they have to do with this one to one correspondence. But in all of them, I think they incorporate my definition of infinity, which is, and this continues on at infinite. But what I'm saying is that if we acknowledge that you can't actually have an infinite set with that conception of infinity, what I'm saying is what is this alternative conception of infinity? So it seems like what we established at the beginning is if it's the case that what an infinite means is actually never ending, then you can't have an infinite set. But so you said there's an alternative. You can't have an actually infinite set. Of course, you have infinite sets. They're just potentially infinite. Okay, what I would say is I don't think- Or you're a strict finetist. Do you think that there's- Well, it's not that I'm a strict finetist. I mean, I mean- They're only finite in many things. Well, what I would say is this. When we're talking about a set, we are talking about something that is definite, that is defined. It's not talking about a set. It's not something and then more than what's in it. It is something that is concrete. And it seems like- Concrete. Well, maybe concrete isn't a precise word. But it is boundaryed. There's boundaries around it when we're talking about a set. Okay. So and if that's the case, if that is built into the concept of set, then it seems like you definitely cannot have an infinity that is incorporated into that set. So when we're talking about one-to-one correspondence, that seems to incorporate this actually never ending idea into it. And so what I'm trying to do is identify what is this alternative definition of infinity. Because when you're, is there a way that you can explain not my definition of infinity, but your definition of infinity, without including like one-to-one correspondence? No. Okay. Well, that seems, so that this is maybe my sticking point is that I, when looking at one-to-one correspondence, it seems to incorporate my definition of infinity. I know what you mean by your definition of infinity. I mean, you have a notion of infinite as potentially infinite. Nothing you've said says there's no other notion of infinity. I don't, and I don't reject your notion of infinity. Right, no, what I'm, I'm not saying- But I don't know why you're rejecting anything else. It's not that I'm rejecting it. It's that I'm trying to fully wrap my head around it. So I still, I still for whatever reason can't do that. So it seems like, so, so let's, let's, let's walk through just piece by piece again exactly just for my own sake to try to grasp this. We've established that when we're talking, when you use the term infinite in the sense that actually never ending, you don't have any infinite sets. I would say it doesn't even- Any actually infinite sets. Maybe I could put it this way. The idea of a potentially infinite set doesn't really make that much conceptual sense because it's not actually, it's not really potentially infinite because at no point would you ever say, ah, you have an infinite set. So it's not, there's not even the potentiality for it. Right? Well, I mean Aristotle makes the distinction that the potential infinity of the numbers is not like the potential existence of the statue and the block of marble. So the block of marble is there and the potential statue inside it, as it were, exists as concretely as the block of marble, you might think. Okay. It's not like that. Right? It's a, the potentiality is something that is in progress. It is in progress, but not really because in progress implies that at some point you've generated an infinite set, which is not even something that can be done. No, why should it possibly, it doesn't mean that at all. I mean, it just means you can go on and on and on. That's the nature of infinity. Yes, I agree. I agree you can go on and on and on, but that doesn't mean that you're in progress of generating an infinite set. Maybe we're across purposes here actually from your questions. I mean, maybe you're a strict scientist. Maybe you think that everything is finite and there's a limit to how big anything can be. Now, just a minute. Sure, sure. There are strict finetists. I think they have a pretty miserable time of it, but you know, you can with heroic measures try to be a strict finetist. There's a Russian mathematician, Yesenin Volpin, who advocated this, and indeed a colleague of mine, Robin Gandy, explored it. I don't think he was quite an advocate of it, but you can take the idea that we, you could, for example, it's a fact that there is, that if you think of numbers as given by numerals, strings of symbols, right? A numeral is a concatenation of a certain number of digits. There is a biggest numeral that's ever, in terms of, well, even in terms of length, there's a biggest numeral that's ever been written down and probably, I mean, indeed, I mean, as far as that goes, I mean, we know the universe, or anyway, we know the earth is going to be destroyed at some point, probably we'll be destroyed long before then, but anyway, so there's going to be a biggest one, right? At any point in time, yes. No, but even ultimately, at all times, if we're in terms of human beings anyway. Well, sure, I think that's the same thing, to say, at any point in time, at all times, yeah. We'll come to an end. So at the same time, I don't know, maybe you accept that, but you don't accept that for every natural number, there's a larger natural number. But here, it's the way, it's the phrasing of statements like that, which I think I'm very, I can't get my head around, or I think there's an error there. So to say, for any natural number, there is a larger natural number. Right. That implies, at least in the way that I hear those words, that it's out there, prior to our conception of it, it's out there, versus, here's how I would like to rephrase it, for any number you conceive of, you can conceive of a larger number. All right, do it like that. But if that's the case, then that means there is no such thing as like an infinite cardinality. Like that concept wouldn't even make sense. Look, I mean, you accept that principle as you just formulated it? That. For every number that you can conceive of, you can conceive of a larger one. And I would make an addition to say, and that number that you conceive of does not exist prior to its conception. That's what I would say. So all sets are necessarily fine. You use the very dangerous word there, can, right? That's a modal word. Okay. You didn't say, does exist, you said can exist. Because what I'm saying is sets do not exist separate of our conception, therefore, when you conceive of a set, it is fine. But this notion can conceive, what do you mean by that? What I mean is that, just like any other concept, I'll take this as a totally non-mathematical analogy. Analogies are dangerous to a prior. So when J.K. Rowling was writing her book about Harry Potter, she had the ability to conceive of alternative characters that were in the book. She had the ability to say Harry Potter had a different eye color than he did. There was the potential to have that kind of conception. Prior to that conception, and if she never had that conception, those characters wouldn't have any type of existence. That's what I mean. So importing that back into mathematics, that you can conceive of a set of any given size, but you cannot conceive of a set of an infinite size, that the idea of an infinite size itself. I'm happy with that. That's perfectly okay. That's the notion of potential infinity. I don't know what we're arguing about here. So what we would be arguing about is the idea that you could have a completed infinity. You keep jumping to, I'm- Yeah, do you see what I'm- No, I don't. I mean, I'm giving an account of a completed infinite, that is the structure of the natural numbers, that as a structure, in a way that does not in any way turn on changing your conception. Your conception is fine. It's a certain conception infinity, which you can make use of as much as you like. But I don't see anything in your advocacy of that conception that prevents someone else from having another conception infinity. So let's go on to this question, because maybe it will elicit where I'm getting hung up on. There is an idea that people talk about different sizes of infinite sets. Can we talk about that with your conception? Not with my conception, because my conception, I guess they obviously this is an impossibility to even say. Well, actually that's not even true. That's not even true. With the notion of potential infinity, but not strict finetism. That, I think, is, I hope, off the table. Not necessarily. OK, well, then I may have to go back to that. But if you treat infinity as potential infinity, then that is the position, for example, that was advocated by Brower in developing intuitionism. And he, I mean, initially he thought that there was only countable infinity, potential countable infinity. And so no different, no two sizes of infinity. But he was also, in fact, a great mathematician. And he realized however much he was willing to change mathematics to give up the calculus, to give up the continuum, to give up the properties of mathematical analysis, was just too much. And he- I'm sorry if I may interrupt you. Just so our listeners understand, well, it seems to be that you have to have some particular conception of infinity when you're talking about calculus, when you're talking about curves, because you kind of almost incorporate infinities into curves, right, in order to get to generate a proper, a perfect curve, if you will. So I just want to say that, so continue. So you can think of a curve in a two dimensional curve, let's say, as a function that takes a point to another point. If you have an x, y axis, let's just think of it like that. And let's think of it as something like a parabola, x equals y squared, or y equals x squared, I guess is the more usual way we use the variables. So y equals x squared. So for every x, you square it, and you get another point. And that gives you a parabola in the plane. Now, what that is is a mapping from something that is, well, we have to ask ourselves, what is a point? What is a point? What is a point? Well, so one way to think of it, what is classically considered to be, is a point. A real number, which is a point given by a real number, is an infinite decimal expansion, or infinite binary expansion, or whatever. That's one way of thinking of it. Now, that would be anathema to Brouwer. He couldn't take it, because that makes a real number an infinite object. Right, which doesn't seem to make sense to my point. No, not he rejected that idea. So what he realized was that he could develop the theory of functions on the real line, functions from the real numbers into the real numbers, by taking real numbers, which we think of as infinite objects, as potentially infinite. So that what it was to be a real number was to be an infinitely proceeding sequence of digits, or an infinitely proceeding series of rational numbers that approximate it. OK, those are two very different things. No, they're not. So for example, if we're talking about curves, it's one thing to say that, like we were talking about calculus. This is actually my position, which is I'm not aware of anybody that agrees with me, but do you need to incorporate infinities into calculus in order to be able to use calculus? And what I would say is no, because all of the calculations ultimately are finite and terminate. So you can have a curve as smooth as you like. That's very different from saying you have a perfect curve, if you will. So when we're talking about approximation, my position is in reality, everything terminates. So there is in reality, it's not the case that space is infinitely divisible or anything like that. So all of the calculations ultimately do terminate, though they terminate at whatever point you like, at whatever degree of specificity that you like. All right. What do you think about that? By the way, we're talking about terminating the degree of specificity, where do you terminate it? Wherever you like. That's what we were saying in approximation. I'm saying you can be as precise. This is my conception of if we want to rescue the concept of infinity, it simply means that you can go as large as you like or as strong as you like. That's the right word. But anyway, if you want to hold to yours. Yes. So what did Brower do? I mean, so Brower had this idea of a real number, which we would think of as an infinite thing. By itself, it's an infinite object. A natural number, accounting number, is a finite thing. But a real number is an infinite thing. On the classical conception. Now, Brower's idea was you can make the real number manageable on his, well, as one would say, constructivist conception, by making it something that you are in the process of generating at all times. It's never finished. It's never a completed infinite. Now, I mean, that's a perfectly manageable notion. Now, what does that do to our development of the calculus? Well, it does something very specific. It means that all the functions that we can ever develop in our theory will be continuous. Now, classically, I mean, by that, I mean the mathematics that everybody did before Brower and since, there are discontinuous functions. Of course, like step functions. I don't know if you studied the calculus. If you did the Riemann integral, then you use step functions to approximate the integral. You divide it up into sections. And then you take the step that gives you the value within that range. They take the value at the leftmost end of the interval and you get a step. And then you take finer and finer gradations and you generate the integral for a given function under the area under the curve. Now, Brower and step functions are discontinuous. So that looks like a problem. But Brower's, well, I mean, there's how you develop the calculus without that. But at any rate, what you get out of Brower's approach is that every function is continuous. From every function from the real numbers into the real numbers is continuous. And why is that? Because in order to generate a value of the function from the input to the function, you need to be able to start approximating the output while you're approximating the input. If you have to wait till the whole input has been generated, then you'll never start generating the output. And you won't have a function on that conception. And that condition of generating the output to any degree of accuracy by generating the input to whatever it takes to generate that degree of accuracy, that is the epsilon delta definition of continuity. So Brower's development of the calculus works with the conception that all infinity is potentially not actual. And it gives you a very different mathematics because you only have continuous functions. You have no discontinuous functions. You don't get step functions. And you certainly don't get functions that people in the 19th century worried about, which are, say, zero on the rational numbers and one on the irrational numbers as completely non-continuous as you could possibly have. However, on that conception, the continuum is not equinomorous with the natural numbers. It's an uncountable infinity. There is no one-to-one correspondence between the real numbers and the natural numbers. Now, on Brower's conception, there is no other infinity beyond that. But there are two sizes of infinity. So let's dive into this notion of the different sizes of infinity. For me, because it's very hard for me to think about infinity in any other way than the way that I've been thinking about, in terms of never-ending. So the idea of different sizes. Well, that's Brower's conception. So what are the other conceptions then? So what I want to explore is Cantor's conception or the Cantor's. But I would like to get back to this point about how, on that conception, there are different sizes of infinity. There can be no one-to-one correspondence between natural numbers and the infinitely preceding sequences that generate the real numbers. The thing is the infinitely preceding sequence, that's the thing that I would have issue with. So maybe this is where the strict finalizing. Are they perfectly good potential infinity? Well, I'm not sure, because it depends on what we mean by potential infinity. Oh, this is the thing where we were talking about earlier, where it seems like in all of these circumstances when we're talking about one conception of infinity and another conception of infinity, and maybe I'm missing something here, but it seems like the conception of infinity, which implies never-endingness, is seemed to always be smuggled in. So when we talk about, regardless, and maybe this will be elicited as we keep talking, but it seems like I have not grasped a full and complete conception of infinity that does not include or does not presuppose this concept of never-endingness, actual, like, never-endingness. But I thought you were, that was your notion of infinity, never-ending. What I'm saying is that is my conception, and what I don't understand is any other conception that doesn't include. Okay, well, never mind that. But I'm just saying, pointing out that if you accept that notion of infinity, you can't bulk at the idea that there are different sizes of infinity. I can't. No, because Brower has exactly that. He has developed the theory of the continuum on the basis of infinity as potential and never-actual, and he has two different sizes of infinity, the natural numbers and the real numbers are not equineumers with each other. But I don't, what I would say is, I don't even think conceptually, it makes sense to talk about sizes of infinity, right? Well, let's not talk about sizes of infinity. There is no one-to-one correspondence between the natural numbers and the real numbers. And see that, again, this is my own, maybe this is radical, strict, finiteism. I don't think numbers work that way. I don't think when you're saying, well, when you're saying one-to-one correspondence, yes, I think that it is the case that all numbers are conceptually generated. And so, in other words, it's not, like when we're talking about numbers, it's not that they're out there, right? They're not out there separate of our conception of them. Yeah. Right, so when you're saying- But every single number has to be thought about? By somebody? By whom? There is no existence of a number outside of that number's conception. By whom? Whoever's doing the conceiving. Right, so- Well, who does the conceiving? Whoever's thinking about the topic. So if there were no- Well, how many numbers have you conceived of? I don't know. We're getting into the metaphysics of mathematics, which is really important here. So I would say the same analogy to the Harry Potter and J.K. Rowling that applies. Well, outside of anybody's conception, her conception, our conception of her characters, they have no existence whatsoever. And so that's, it doesn't sound like that would be- Fictionalism. Is that fictionalism? I think that's what it's labeled as a philosophy of mathematics, isn't it? Oh, oh, is that the case? That this is part of the plot? Because fictionalism has a very specific meaning in philosophy, and I didn't think it was that, but I didn't realize that that term applied to mathematics as well. Yeah, there are people who say that sort of thing. So is it the case then that Brower, when we're talking, did not have this conception of numbers not existing outside of when somebody is thinking about them? Well, he has a theory of what he called the creating subject. But it's idealized. You don't think, he didn't think it was him, and he didn't think it was you, of course he didn't know about you, but it wasn't some particular person who was doing the imagining. Okay, well, so does it sound, this is, maybe this is the underlying issue here. Wouldn't this be interesting? Perhaps this comes down to the philosophy of mathematics in terms of metaphysics, is your conception of mathematics, and perhaps Brower's, and the majority of mathematicians' conception, some type of Platonism, that there is some kind of, No, I'm opposed to Platonism. But don't you believe that there's numbers have some kind of existence separate of our conception of, No. Okay, well, it sounded like that's what you were just saying. No, I'm not. Okay, so what was your, you're not objecting to what I was saying before about the idea of numbers having no kind of existence outside of when we think of them? Well, the question is, what are the ways of thinking about them? And I've given you a way of thinking about, for example, the natural numbers, which you have balked at, but I'm telling you how to think about the natural numbers. A way to think about the natural numbers in such a way that you can see what you can call the infinity of the natural numbers. Yes, but you're not thinking of all of the infinities, right? You're thinking about what is the structure of this type of thing. And that does it. That does it. That's the way mathematics is. As Hilbert said, mathematics is a symphony of the infinite. How can we have this symphony? Well, by understanding these properties, we're not limited to a step-by-step generating this and this and this. We wouldn't get anywhere. We would never have had mathematics. If the conception you're talking about is good for six-year-olds when they're learning to count, but it's not good for mathematics. We would never have had professional mathematics on that conception. Well, I don't disagree with that, but that doesn't necessarily mean... So you think all professional mathematics is misguided? If that is the conclusion, I mean, that's what the position I would be forced into, I suppose. Doesn't that worry you? Well, I don't think it's the case that just because there's a large discipline of people engaging in a certain practice that, therefore, their method of approaching their area of study is correct. I think you can have fundamental misconceptions in different areas of thought. Okay. But is that the stakes that we're talking about here? If what I'm saying is true about the kind of the metaphysics of mathematics, does it imply, then, that professional mathematicians have been... Sure. Absolutely, then, twice. Okay. Well, so when you said, well, let's explore that just for a little bit more. When you said you wouldn't have professional mathematicians, are you saying that... Is this a modern development? So is the way that I'm conceiving about infinity, is this something that would have been entertained prior, let's say, to the 20th century or maybe prior to the 19th century, or are we talking for the last 2,000 years we wouldn't have development? No, I mean, your conception is fine with Aristotle. That's what he was saying. And arguably, mathematics until the first part of the 19th century was fine with it, because that's how people thought. So... But mathematicians got beyond that. So, what I need to do is just go back to the 19th century, essentially. Well, I mean, if you study what they did. Well, it was. I don't know. I think a very important aspect of the philosophy of mathematics is to be sensitive to what mathematicians do. That's the guiding... And do you think it's presumptuous to say, well, given this error in conceptions of infinity that I perceive or conceive of, the way that mathematicians have been doing their practice for the last 100, 150 years is fundamentally wrong? Do you think that is just completely presumptuous? Do you know of anybody, for my own reference here, do you know of anybody who would say that in terms of modern mathematics? No, I mean, the thing is that the great example of doing that is Brower. I don't know how much you know. Very little. Just actually from the book I was talking about, Morris Klein's book, he talks about Brower's book. But Brower was an intuitionist, was he not? Yes, yes, yes. I mean, that's what he called his position. So you might find this interesting or maybe ridiculous, and maybe this will be kind of been wrapping it up because I don't want to keep you too long. What I find very persuasive about intuitionism is the rejection of infinite sets. Sure, sure, sure. What I think they get wrong is they conceive of logic as being something that is perhaps you could say non-universal or in the sense that it's not... My own position, if I could try to historically put it somewhere, is somewhere between logisticism and intuitionism. Because I think that everything can ultimately be reduced to the laws of logic. I do think that is true. But I don't think intuitionists wouldn't say that. No, I mean Brower says that... Explicitly. Yes, he says logic is not a means by which to establish a truth which is not establishable in some other way. Right, and so my position is the merging of the two. It's the logicism with the metaphysics of intuitionism. All right. And as far as I know, there's a group of one of us. Okay, good luck. But on that note, I really want to thank you for sitting down and speaking with me. This has been great. Very interesting. Okay, so that was my interview with Dr. Isaacson of Oxford University. I hope you enjoyed that. And at the very least, maybe you think I'm a complete crank. And if that's the case, you're not alone. I think there's room for discussion in talking about the foundations of mathematics. I don't think it's completely unreasonable to say that maybe what numbers are is just ideas. They're just concepts in our head. Mathematical relationships are logical relationships. They're conceptual relationships. That's it. They don't have any kind of existence outside of our mind. That doesn't mean mathematics doesn't apply to the real world. In fact, quite the contrary. It just means that the actual mathematical units as they exist are fundamentally conceptual in nature. And if that's so, it restricts what we can and cannot say about them. At the very least, I think that's a reasonable and entertainable position, but it stands radically opposed to professional mathematicians of the last century. Make of that as you will. And as always, eventually I will be doing an interview breakdown of this episode because I think there's just so much meat here to talk about. So that's it for me. Hope you guys enjoyed it and have a fantastic week.