 Resting at the heart of modern mathematics is the concept of infinite sets. The idea that you can put all of an infinity into a set and treat it as a completed totality. That's a difficult concept to wrap your mind around. And from the outside, it seems like it might be a little bit contradictory. But is it? Is there a way to make sense of an infinite set? Surely, with such a foundational concept, these ideas have been firmly established for the last 150 years, right? This is the question I'm trying to answer on the 45th episode of Patterson in Pursuit. Hello, my friends and enemies. Welcome to Patterson in Pursuit. I'm your host, Steve Patterson, and I know I say it every time. For real, this is an awesome episode, and it's got a really cool story behind it, too. Occasionally, I get emails from listeners and supporters who say, hey, I said you're going to be in this particular location. You want to get some lunch or something like that, and it's always a blast. A couple of weeks ago, I got an email from a guy who said, hey, I listened to some of your stuff on mathematics. I read some of your work on mathematics, and I'd like to talk to you about this. I'm an ex-mathematician, got my PhD from Oxford in mathematics. I specialize in set theory, which is the thing that I've been speaking about, and I thought, wow, this is a great opportunity. The guy's in Auckland. Let's do it. And I said, hey, is it cool if I bring my microphone along, because this conversation might be awesome, but it might be so awesome and so relevant that my listeners might want to listen to it. He said, yeah, that's cool. So what you're about to listen to is a conversation with a guy that I've never met. I knew nothing about, but he emailed me. He's got a background in math. He's just super cool. We hit it off right away. He's very knowledgeable, and the conversation was about as relevant to my work as possible. Depending on how closely you follow my written work, steve-patterson.com, if you're interested. I get a lot of heat from people who say, oh, Steve, you just don't understand the basics of set theory or whatever the topic I'm talking about is. And in all circumstances, that's not true, because I don't write about subjects that I don't know anything about. But I'm always open and eager to learn what is the error that I'm making in my understanding of set theory in particular, which I'm so critical of modern set theory. And so whenever I get an opportunity to have a conversation with somebody that actually knows what they're talking about, as you'll discover, maybe, just maybe, my criticism and conceptual confusion is not so unjustified after all. What makes the conversation especially fantastic is that my sticking point has always been infinite sets. I like the idea of sets, but when you throw infinity into it, my brain kind of explodes. I think there's some funny business going on. Well, the man I'm talking with specifically has got this background in mathematics, specifically in set theory, and he supports and believes in the coherence of the idea of infinite sets. So just an epic recipe for a conversation. Now, one of the drawbacks is because he's not in academia, he's outside of academia, though he's got his PhD. We didn't have an office or anything, so this was recorded outside of a coffee shop in downtown Auckland, New Zealand, so you're going to hear a lot of background noise, and it'll just be part of the ambience. Near the end, you'll hear some background music turns on, and then we have to stop the interview because of that. But other than that, I'm sure if you guys are remotely interested in this topic, you are going to love this conversation, and if you're not interested in this topic, listen to it anyway, because you're going to discover something that I'm sure you'll find fascinating. But before we start, I want to give a shout out to another supporter and a company that hooked me and my wife up for a little bit while we were here in Auckland. We're leaving now, but one of the listeners of the show reached out and said, hey, Steve, you're in Auckland. I'm at this co-working space. If you want to come down, you know, I'll hook you up in a few days at this co-working space. If you're tired of, you know, working in your apartment. And I was like, yep, let's do that. So a special thank you to Max for helping my wife and I out. The co-working space, if you're in Auckland, go check it out. It's called GridAKL, and it's a really sweet co-working space. My wife said it's the best co-working space that she's ever been in, and it's a really cool atmosphere, and I got to give a little talk to a group of people down there about podcasting, and they have a stable internet connection, which down here is fantastic. So a special thanks to you guys. And also, before we start the show, let me tell you about the sponsor of the show, Praxis, because yet again, it is immediately relevant. There's a growing movement of professionals and even intellectuals now who are explicitly working outside of academia, because the cost is astronomical, take it to your degree, and the benefit is marginal and sometimes negative. I'm somebody who's interested in the world of ideas. I'm working outside of academia. My guest this week, Gareth, has a PhD in mathematics from Oxford, and he's chosen to work outside of academia. He thinks he can make a bigger, more positive impact in the world outside the academy. And the sponsor of the show, Praxis, is a company that specializes in taking young people and jump-starting their career without the college degree. It's a nine-month program which starts with three months of a professional boot camp, where you learn actually relevant real-world information that will help you on your job, and then it's followed by six months of a paid apprenticeship. After you complete the program, you can actually guarantee you a job offer, and the net cost of the entire program is zero dollars. Yes, that's right. That is the kind of real education and training that you can get outside the academy in the modern world. So if that sounds like something you're interested in, go to steve-patterson.com. That'll take you to their website where you can put in your email address, get a free Praxis module, you can schedule a call with them, and it is the wave of the future, and I highly recommend it. So without further ado, we venture into the world of mathematics, dealing with questions in the foundations of mathematical reasoning, with my new friend, Gareth. I'd like to pick your brain on a few issues. Maybe we can lay out some basics, some of the foundational issues and topics in mathematics, and maybe some of your own experiences in academia, if you're interested in math. So there's a very popular idea for people who don't have any kind of formal education in mathematics, that mathematics is kind of the perfect intellectual discipline. Everything is crystal clear, logically explained from beginning to end. All of the theories are rigorously proved. There's no room for doubt or discussion, especially when we're talking about foundational issues. And then the more you learn about mathematics, the more you realize, oh, that's not the case. In fact, that's never really been the case. There have always been discussions about foundational issues in mathematics. So one of the issues, which is my sticking point, that I know modern math is supposed to have resolved 150 years ago, is about infinities, which might seem abstract to people, but really the issue of infinities is kind of absolutely central in mathematics and how you generate the continuum of the number line. But also historically, it's kind of a central issue because modern set theory, which is a huge part of modern mathematics, is built on the idea of what we call the completed infinity, or the infinite set, if that's a thing. So help me work through the logic of an infinite set. Because when I think of the concept of infinity, I think never ending, endless. And then I think of set and I think of complete. So it seems straight off the bat, like that's kind of like a contradictory thing to say an infinite. It's like a square circle. So help me work through that. So I guess my first thought on the idea that the infinite sets that mathematicians are studying and playing with this actualized infinity, I don't see that the same kind of immediate contradiction drops out when considering that concept because if it were, then you'd be able to take, say, the axioms of set theory and constructs a quite short, neat, simple proof that it's broken. Russell's paradox, but for the existing rules of set. And let me interject for people who are aware of that. Russell's paradox is the set of all sets which do not contain themselves. This is supposed to be a logical contradiction that shows that naive set theory, as it's been called, is internally inconsistent and yields a contradiction. Okay. So I feel, my first thought is feeling that that kind of argument, what the mathematician who's the set theorist, let's say to be more precise, means when they talk about an infinite set, what we might be thinking of when you say something which is infinite and which is a set contained. One or both of those are not quite talking about the same thing. I can't quite, I certainly can't imagine that all these people that could easily be building on some sort of wonky foundation, that it would be that wonky that actually has just this obvious three or four line proof that breaks all of set theory as it stands. It's not the first thing I'm thinking about it. So let me respond to that just as we wanted. So I've spoken with some mathematicians who say that. Well, they say, well, if there's a logical contradiction, then show the inconsistency. Show the proof where you yield a contradiction. My claim is kind of one step prior to that. It's not even a mathematical proof that yields a contradiction. It's that the foundational concept itself is contradictory. So it's like, is an infinite set of contradiction, is not something that requires mathematical proof, just conceptual analysis, something that is never-ending, never-complete, never-founded with something that is... Okay, but then would it be your contention then that this is some sort of fundamental logical contradiction which cannot be reduced to something like p and not p? Something that a mathematician would respond to and say, yes, that can't be. I mean, we're not physicists. That's very... Okay, so put it this way. Insofar as you could have a contradiction that would look like a square circle. It's not necessarily p and not p, but packed into the concept of square and packed into the concept of circle, there is mutual exclusivity. If it can't be elicited necessarily in that kind of form... So then it feels as though packed into the definition of set, you have this... you could add an axiom, let's say, and this axiom is say that any proper subset of it must be not equinumerable there would be no bijection or something to say that it must be finite. But if you build that axiom in it will be consistent with everything else but it will be inconsistent with the infinite set axiom. Okay, so I guess maybe it comes down to fundamental concepts of what we mean by set. Yeah, okay. So what do we mean by that? What do you mean by that? What do I mean by that? I guess... Well, okay, so let's just go through the story again. I grow up, I learn English. I know what a set is but actually if you look up in the Oxford English dictionary there are many different meanings. People mean different things by it. It's usually many different contexts. It has some sort of, you know, flavor to it. That's fine. When I come up to mathematicians that are talking about a set, a mathematical set, I say, well, what precisely do you mean by that? There's some sort of definitional process going on there. You know, they say it's a thing which satisfies these postulates. Okay, so then I start from there and I say, okay, well, let's take that as a definition for now and work from there and see if I can understand more about this thing in the set, okay? Now, as I try to make sense of those definitions I'm piecing in my mind some sort of conception. My own idea, I fill in the blanks. Later I can find out, oh, actually that was the wrong picture or oh, that was a good picture and it also helps me to predict future results that I might be able to prove with the axiom system. But yeah, I'm basically treating the axioms that they set as a definition of set. And so if you take one of the axioms away you're getting a different kind of set. Now, that strikes me like a bit of a dangerous methodology. Okay, why is that? Because imagine I would have tried to say, try to find what a circle is or a square with some particular formal criteria. And embedded in them I say, satisfies the criteria, let's say I have a square, it has four equivalent sides. Yeah. And also nestled somewhere in there as a, it has no sides. Yeah. Circle. Now, I guess that might look like a formal contradiction, but that seems like a kind of backwards way of going about defining something. So if I say what is a set? Well, yes. Because here's what happens. Yeah. In common language, we talk about sets pretty straightforward. A set can be understood as a collection of discrete elements. Yeah. And if that's all that it is, I think that's what most people mean by it. Yeah. And even mathematicians, they use that kind of language because they talk about the cardinality of a set. The cardinality of a set is how many objects are... Certainly use other terms like cardinality to fit in with this intuitive notion of set. Right, exactly. So is the claim then that the intuitive notion of set is something that we should get away from and we should just try to formally have the definition that is just formally defined within the mathematics itself? Well, I don't... Firstly, I wouldn't suggest that the axioms as written would fully define this thing set anyway. Okay, so you've got this in completeness anyway. But I guess when I wanted to jump in with the square and say yes and no, I feel that when we're talking about... I guess it's almost like a difference between applied and pure in a sense, that firstly there are the rules of the game. And as a pure mathematician, I like to just take the rules to the game and play with them. That's the part of maths that's very pure and I'm not making any errors. If errors have been made, they've been made by maybe people that have set these axioms along. So long as they're not logically contradictory. And then there's the other approach of the other important components of that, which is to look at the real world and then come up with some rules or some idea about what maths we might see a physical thing that looks a lot like a square and then have this idea about equal angles and formulate that and say okay, so if that's a square and that's a circle, what more can you tell me about these objects and then I can do some pure mathematics and then they can see, does that actually correspond to these objects in the real world? The thing is in the real world, I don't see that there are any things like sets. Though I don't see that there is a logical reason to say that it's impossible for them conceptually to exist. I don't believe they do exist. What about something like this though? A set exists as a mental construction, as abstract boundaries around discrete things. So I could say we've got a chair here, we've got a cup of coffee here, the set of the chair and the cup of coffee. So it's got two elements in it. In a sense the set exists in my mind. It's a conceptual tool. But that's no problem. I could make the set as large as I want. I could say the set includes all of the individual atoms which compose those two things. Okay, well that's still a discrete thing. But in so far as that's the concept, that must be mutually exclusive with the idea of a set that has a never-ending amount of atoms. I can't possibly conceive of all of that. I think I'll go with you on that. In so far as we're talking about this kind of set, we're talking about a set of things in the universe, in space of some kind. For our universe, I don't believe in the existence of infinitely many things if that makes sense or that things exist without end. Although I'm sure there are plenty of people who disagree with me. It's either space is infinite or time is infinite or something. I'll go so far as to say that metaphysically, I don't have a logical problem with it, like a universe existing in that way. I just don't happen to believe that. But yeah, so if we take that concept, I don't think that what the mathematicians are studying, when they're studying sets in set theory, is the same thing as what you and I talked about when we say the set containing this and that. We're talking about finite sets. As a mathematician set theory, we'll call that finite sets. And there's an entire theory of finite sets and it stacks up quite well against the truth. So do you think, I think this gets to one of these fundamental metaphysical differences in mathematics. If it's true that sets exist, let's say, in our mind and they don't exist separate of our mind, then how could you have any set, even if it's not in this world, let's say it's in the platonic mathematical world, in the mathematical universe, could it be that you were talking about a set that is a construction and yet contains an infinitely large amount of elements? Or does that necessarily presuppose it's separate of our mind? It's not a construction anymore? Well, I guess I would be, it depends precisely what you mean by construction, but my intuition would say at the moment that, yes, there's no particular problem with constructing, completing an actualized kind of infinite thing. Really? But yeah, I just don't, I mean, it depends on, it's like if we try to construct this thing, it's saying, okay, we'll take an empty bag, and we use a new word now just to, we've got set so many times in collections, and I'll put a ball in the bag, and I'll put another ball in the bag, and I'll put another ball in the bag, and I'll, right? There's no way this is ever actualized. This is only a potentially real infinity. But if I could somehow conceive of the concept of a bag containing infinitely many, if you like, or more than any finite number of balls in the system. What does it mean to say infinitely many, as if it's an actualized, doesn't that imply that it's an actualized amount? Well, I'm saying, I'm actualizing it in a kind of a constructing in a different way. It's not construction in the way of constructiveness mathematics, but I'm just saying it depends what you mean by constructed. Could you have a full and complete conception of the bag containing an infinitely many number of elements? A full and complete conception. Yeah. Yeah, I think so. If it's a bag, it has these balls in them. You can see that it has balls in them. You can take a ball out. But you know also that no matter how many times you take a ball out, there are still balls remaining in the bag. Are there the same amount of balls in the bag after you take one out? Each time. Yeah. Now does that make sense to you to say that we have an amount? Well, it depends what you mean by amounts, because if we're talking about amount, meaning kind of finite amount, then no, it wouldn't make sense. Well, then what amounts don't do that? So what is the infinite amount? So that's a concept I don't understand. Infinite amount, because you can take one ball out, but the same amount remains. So how does that work? Yeah, I guess amounts is probably a very poor choice of word, let's say. It's got the same cardinality as all I can really jump back to. It's now maybe becoming circular. But that's the thing, right? So I agree that that mathematician has the same cardinality. What's cardinality? I understand cardinality. Yeah, I know you get the bijections. Like in the intuitive sense, cardinality totally crystal clear, but then when it goes into the infinite world, that concept explodes in my mind. Because then I think amount, if you remove a bowling ball from the bag, there is exactly one less amount of things in your bag. But that somehow doesn't work when we're talking about cardinality. Yeah, I don't know. I'll have to admit at this point, there is an element to which I was in Oxford for nine years studying. So as you know, you learn the environment that you're in and whatnot. And cardinality is very much one of those things that I can quite happily take my one away from Aleph Zero and be left with Aleph Zero. So it's difficult for me to fight past that. But at the same time, I think it doesn't fit amount very well. I think amount is the fourth choice of word for what it is. I want to clarify for the listeners. Sure. When you said Aleph Zero, there's this idea and set theory that you have different sizes of infinity. And the smallest size of infinity is Aleph Zero, which is supposed to be the set of all... Is it all natural numbers? And it's got a bunch of weird quirks, which is you can subtract individual elements from Aleph Zero, and you're still left with Aleph Zero. You can even add Aleph Zero to itself, and it's still Aleph Zero. You have all these, what appear to me, logically contradictory things. And when I've investigated this, and I research it, a lot of people say something like, well, it doesn't fully make sense, but you get used to it as a mathematician. Everybody kind of agrees that this is how things are done. Well, I wouldn't go so far as to say it doesn't really make sense. One thing for me, it doesn't. Okay, fine. I will admit that going through, I think there have been other mathematicians around me that have made precisely this kind of comment before, and on many other areas as well. But it's not something you ever really understand. It's just something that you get used to. That's the phrase. And I was kind of quite shocked and appalled almost at the first time someone said it to me, because I very much felt like I was understanding. Okay, I don't feel as though it's contradicting itself, but I think that amount doesn't stack up quite well. I quite like those early thought experiments, trying to bring myself back to sort of secondary school of, you know, like, I forget all of the classic examples. I guess one of the old ones is, you know, just considering the natural numbers, the counting numbers, one, two, three, four, and then writing just below them, or you're hitting it in all of them, of course. You wrote the first eight or so. And below them, you write their squares. This is something that Calalay, I think, was considering at some point. And just considering that, actually, there was a correspondence, a one-to-one correspondence between each number and its square, and therefore it seems that for some sense of amount, if it were to apply to infinity, we'd have to have that the square numbers were the same as the natural numbers. But doesn't that imply... Despite the fact that there are clearly fewer. But doesn't that imply that those numbers exist separate of their being conceived? Because any one of those... any one of those lists that you make, it terminates wherever it terminates. Yeah, it terminates down this to eight, for the purposes of this. But in that circumstance, so like with Cantor's proof, for example, where supposedly you have the list going infinitely this direction, infinitely that direction, if somebody were to say, well, numbers don't exist out there, numbers are constructions on our minds, how do you get to that, that leap to say, oh, well, it's just like the correspondence between numbers and their squares, but that correspondence exists when it's being constructed. All right, well, I guess, I guess there's a few different ways to look at it. I guess when I'm pushed to it, because I look at these conceptualizations of, say, Cantor's diagonal argument and whatnot, and often I find it's quite... it's not presented in a way that's particularly, you know, that would be a place where you can see all kinds of... dubious things being done with infinity. The first time you come across that, as I was going on here, it doesn't quite add up. I guess the way that I argue something like that actually doesn't really have to leave, doesn't have to actually do actualized infinities at all. You can use the whole thing with potential. Really? Yeah, I can say it. I feel so. I mean, okay, so we can have a... I mean, maybe we can't actualize the infinity of all the natural numbers, but we can have a procedure which enumerates, and we can say one and then add one, and add one could be our procedural operation, and we can learn to do this repeatedly, and do as many as we like. So there's a procedure for generating the sequence, one, two, three, four, and so on, I quote, I quote's there. And equally, we can speak of procedures as finite objects. In some sense, we can talk of a procedure as being a rule written down in a finite amount of space, so we can communicate. And we can come up with other procedures for generating numbers, for example, the even numbers, or the square numbers, or the prime numbers. There's lots of different procedures out there for an increasing sequence of numbers. So then I think, well, can we enumerate procedures? And we can indeed, we can, you know, do the, for example, the natural numbers, and then the even numbers, and then the multiples of three, and then the multiples of four. We can do a more advanced kind of enumeration of procedures. But even when you say that, you say the even numbers. Yeah, I say that. I'm airquesting some of these. Yeah. But even the way, and I see this throughout mathematical thinking and literature, is they... It's like assuming that they exist already. Assuming that they exist rather than... So, when we write seven down, I don't think it's the case that we're referencing some entity out there, just like if I were to say, I don't know, the table. No, I don't think that, really. But if you don't, I still don't... How do you get to the infinity? Everything is just constructed, and all that mathematics is, is a description of finite rules and what the finite rules generate. How do you ever get to an actualized infinity? I mean, you can't conceive of all of an infinity at once. That would be... No, you don't conceive of the individual... No, that's not what I'm seeing at all. But rather an object, which is like a... a limit of some kind, if you will, or off the procedure. I mean, something to fill a gap. But if you don't conceive of all the entities, if you never conceive of all the entities, wouldn't we say, therefore, there's no such thing as an actualized infinity? Well, yeah, I think that if you're pushing me to the point of saying that these things are you actually conceptualizing all of the points, if not, then what you've got there is just a finite object with a particular kind of flavor or relationship to other finite objects. Yes, let me give you an analogy. Then that's fine. I mean, I'm quite happy with considering Omega, which is the set of all natural numbers and set theories, they say, to be a finite object in that, in the sense that when you look back and you just look at it from a kind of formless perspective of symbols and manipulation, I think it's a finite length proof, everything's a finite term. But it wouldn't have an actually infinite amount of elements to that. No, but what would happen is that there would be a concept called infinity within the finite language and structure. But it's not a concept of the actualized infinity, it's a concept of being able to produce more finite elements, right? It would probably look more like that, yes, than off. Well, again, it's like, it's not actualized, and if I, again, think of the logical symbols, it's not as if I can say 1 and 2 and 3, sorry, p of 1 and p of 2, where p is a statement of plane, and p of 3 and p of 4 and p of, you know, an infinite length, whatever that might mean. I'm kind of with you on that, it's got to be a finite length statement for it to even, yeah. You can't reference it. Right, exactly, exactly. No, I get that, but I feel like sitting here and talking with our finite language and finite concepts, we can talk about this concept of infinity and have a sensible discussion about it without breaking into contradictions somehow. I feel like there's a sense of infinity embedded in that finite constructual framework that is useful, and it might be much closer to a kind of potential infinity than a realized one. So, let me give you an analogy that I like to use and see if you think this breaks down when I import it into the math. Okay, I like to use the analogy of language, of natural language, that I could use the concept of, I could say something like, there is no inherent limit to the size of sentence that I can create. For any given sentence, I could add more words. In a sense, I could say, oh, there's an infinite number of sentences out there. In one sense, I could say that, but that doesn't mean that there exists some sentence of an actually infinite sign. That certainly doesn't matter. So I would import this into the mathematics conversation, I would say there is no number, which is so big that you cannot conceive of a greater number, but that doesn't mean there is no actualized infinite number or set or anything, really, that's out there. I see where the problem that you're coming in with is, and I do have the same kind of a version to that. Okay. And it's kind of a conceptual problem I found particularly because, of course, when you're doing PhD, you're teaching students as well. And sometimes they come along with this notion and they end up doing something which looks like they're trying to do infinitely many steps to make their argument work. I want to say, no, your proofs have got to be finite. So please phrase this differently. Okay. So one more question on this, and then I want to talk to you about the academia. Let's return to this notion. You said you don't like the idea of a mount because a mount seems to imply, embedded in the concept of a mount, I think is finite. I don't think you can rescue the concept. So what is the higher mathematical alternative to a mount that will include infinity? Well, I mean, I guess we keep touching on that word cartonality, but really I kind of want to stick with that and suggest that, okay, there is this logical framework with rules that is ZFC effectively in it. And in that we have a whole bunch of objects and we can do arbitrarily many, consider arbitrarily many of these objects at any one particular time. And how do I phrase this elegantly? But cartonality in particular. So say, okay, we're replacing a mount with cartonality. Okay, unpack the concept of cartonality. I want to suggest that cartonality, within the system, I mean, within the system we have objects which we call finite and other objects which we call infinite. And cartonality, which is this term that we've defined applies equally to both. Okay. And if we apply the concept of cartonality to these objects which are finite, okay, then it very much looks like what we're talking about when we talk about numbers and amounts. Okay. Which is another abstraction that people talk about. Now, but when you say you're applying the concept of cartonality. Yeah. It sounds like you're... I have conceptualized cartonality in some sense, yes. Right, so what is the concept that you're applying? I can understand and say there is this kind of mathematical rule that we symbolize as cartonality, but that's not the concept. No, no, no. I'm talking about the fact that when I do mathematics, and I'm coming back to this point again, yes, there are the formal symbols, but I don't just think in terms of the symbols that I'm applying. I actually try to visualize, if you like, certain objects in this space, in my head, in certain ways, and try to build an intuition slowly with months or years about what they might or might not do, how they interact with each other, how you can take two infinite sets and intersect them, what does that mean? In terms of this intuition that I have, and I can see my fellow mathematician across the way, and he also has spent a long time building up this intuition, and he can play with these things in his head, and he'll come up with some idea. I think that this is a true statement, and I can think in my own head about my own particular, and I can say, no, I think it's a false statement. And we can actually debate this point, and we can actually go back to the axioms and try to work out who's correct, but actually the statement is neither provable nor it's negation. It's provable in the axioms. Actually, you both have just different models of these particular objects. So I don't want to go out there and suggest that there is sets as like an objective thing, but rather you build a model yourself of something that begins to satisfy these properties. So I can't really describe sets much more than they have these properties, and maybe I can draw some pictures of the sorts of things that I visualize, and you'll see the same kinds of pictures in most mathematical texts, with magic ellipses and things going on. But with the cardinality in particular, how is it that you conceived that? Because it's crystal clear to me. When I read about cardinality, when I read about cardinality as applied to anything finite, it is a crystal clear concept. But like I said, it gets exploded when I'm talking about infinity. So something in the concept of cardinality changes when we're moving from finite to infinite. How is it that that concept gets... We can use it, everybody, even if they don't have a background in mathematics, you have this person on the street can understand cardinality in about 15 seconds. Certainly for finite things, yeah. So then how is it that you... How is it that that concept gets imported coherently, and how can somebody who's struggling with it understand if it even makes sense when applied to infinite things? Okay. How could I help somebody who is struggling to work out what cardinality could even mean? I felt that it was totally nonsensical when applied to infinite stuff. How could I explain to them what I actually meant by that? I feel almost as though if they're having a conceptual issue with cardinality in that they don't feel like it could possibly logically make sense, then I'd encourage them to try to not pre-load cardinality with the intuitive sense of amount too much. But rather, let's just leave that to one side and come up with a new property of cardinality or something. That's just a property that we apply to each set. It's just a label. It has no particular relationships. And then from that, I would start to build up and let's have a look at these interesting relationships we can build from them. And hey, look at that. When I apply it to this subclass which we'll call finite sets, it looks just like cardinality. So I would always encourage them to come up with this new concept, which really isn't amount. It's almost like the word cardinality is misleading. But I don't feel like it's contradicting itself. I feel like it's... Well, it's not... The cardinality isn't contradicting itself unless you apply it, mind from my perspective, to this notion of infinity. But what you said is interesting. You're saying there is a higher-level concept of cardinality and you can... It's got two parts. It's got the finite part and it's got the infinite part. When you import that concept into the infinite part, it makes sense everybody on the street would get it. When you import it into the infinite discussion, that's where things start to get much less intuitive. But the trouble that I have with that is it seems circular to me. And I've spoken with many mathematicians about this and they say, essentially, it almost comes down to, well, look, it works. It works in the calculation. You can do things with it. It's so massively practical. And if you don't grasp it, well, that's just your problem. Well, I wouldn't go that far. There's an awful lot of different ways you can go about establishing most of what is useful mathematics. You can do it constructivist style. You can do it formalist style. You don't need to conceive of infinite sets. I mean, heck, as you say, they were introduced 150 years ago. It's not as if we had no mathematics. We had plenty of mathematics. And a lot of the stuff that's been done, it's almost as though a lot of the very high-level abstracts, we're talking about much larger cardinalities, compactness, interactive middle-earthness, and doing all weird things, much of that, you can wind it back and look at it from a, well, here's a finite sentence. We're doing these particular manipulations and kind of sidestep the whole issue. Sometimes you do go deeply enough into Ramsey theory. You actually find that you're really using the assumption of their existence in infinite sets to prove something that you were otherwise would not be able to prove about finite things, which is weird. Now, can you explain that? Because I don't know very much about that. So what are the things in mathematics that you're proving about finite things that presuppose the axiom of infinity? Okay. Now, unfortunately, these things get a bit complicated, so I'd love to give you a very simple example, but I'll try the simplest one I can, which I think would be something like the, I'll call it the strong finite Ramsey theorem, which I guess you can call the strong Ramsey theorem. Okay. But unfortunately, it's one of those things where the statement is, for all x there exists a y, for all z there exists a w such that something. So it's of that form, okay, but all of the x, y, z, and w are finite things, and something is a finite thing. Okay. And it's a statement which is, has been shown to be not provable in Iano arithmetic, in the sense that that statement in its own right would be able to prove the consistency of pre-Iano arithmetic and thereby through that. So this is kind of like a girdle. Like a girdle kind of thing, yeah. So you can prove that this thing cannot be proved from just the axioms of Iano arithmetic. However, it's claimed, you can find people say, but it is true. I want to say it's true. It means they derived it from ZFC. So that's how they say something's true, just because it's derived from ZFC, which is obviously true. So the point is that if you add in that axiom of infinity, and actually I followed the proof, it looks like you also want to throw choices to make this proof work, right? But infinity plus another axiom to do with infinity, then you can show this thing follows. But it doesn't follow just from Iano arithmetic, which suggests to me that there is another model of Iano arithmetic in which the statement is false. So whether or not you can say it's true or false, I guess most mathematicians would say it is true because in my conception of what the natural numbers mean, and I believe in these actualized entities and various things, that this statement is true. You know, I can see that it's true. I can reason that it's true. But I guess it's quite possible that a person that has enough of this finite and formalist mind view would actually analyze his statement and say, you know what, this is a false statement. And the reason your proof doesn't work is because you've assumed that there exists an actualized infinitive. Okay. Right? But because it's like for all there exists, for all there exists kind of thing, it's not like a simple goal-back conjecture thing where you can just say, well, a hit-case is 9 and 10, and there your counter-example or something. Okay, so with that concept of the actualized infinity, this is something I'm still trying, in my own research, I'm trying to wrap my head around. Yeah. It seems like, as you said, you can build a huge amount of mathematics without infinitives. Yeah, yeah. However, it also seems like, at least modern mathematics, there's also a huge amount of mathematics that presupposes the actualized infinity. I know you said you have a background on topology. Yeah, very much so. It seems to be an area where you have the... And the actualized infinity. You've got to have, well, for those who are unfamiliar, topology, how would you describe it? It's about the study of spaces. Yeah. It's like geometry, but without the angles and distances. It's just how things are kind of connected to each other or not connected. So, how much, if it's the case, that this idea of the infinite set maybe is a little wonky. I think you can rescue calculus. Some people think you need the actualized infinity to get it to make calculus work. I don't think that's the case. No, I think you can do an awful lot without it. I think you can. But how much higher mathematics do you think would have a foundational error if it is true that my suspicions about an actualized infinity or an infinite set are correct? For analytic topology, if we reject the absolute infinity and just try to rescue as much as we can, it's almost as though either you go with A, you can rescue almost none of it, or B, you invent a new game and pull it formalism and make it all finite and say, well, here's the rules, and then we just import it into that rule. And it's like, okay, here's a game that doesn't actually mean much. I mean, of course, if you can prove a contradiction from the axioms, then, yeah, the whole thing falls apart and there's totally unrescual. But in terms of rescuing it, in terms of an intuition, then, yeah, I guess it's very little. I mean, I'll take an example. We're talking about points and sets of points. And indeed, sets of points contain infinitely many points. There's this particular question that's asked quite a long time ago, about 100 years ago, I guess. Which is whether or not you can have a two-point set. Now, a two-point set, not just a set with two points, but it's a subset of the plane. There's a subset of the reels across the reels. We have these actualized infinities such that it meets every straight line in exactly two points. And you might start to play around with the title and think of maybe I can do a parabola with some extra bit on the side or a circle or a line. You can think in geometric terms, is there a solution to this that any straight line meets it in two points? Can you prove it constructively? There is no such constructive proof. But using these axioms in infinity and particularly the axiom of choice, you can do a kind of transfinite induction recursive thing that says all of the lines of which there are continuum many and induct all the way through this, like, infinite upon infinite sequence. On every line, we put two points if there aren't points already there and you get this complete mess of points. And then you say, okay, so we've got these things. What can we learn about them? Can you show that the complement is path connected? Can you show that this thing is necessarily zero-dimensional? There's no such zero-dimensional. So there's an awful lot of people oh, this is exciting, interesting object. We study all these things. You know, obviously, if there's no concept of an actualized infinity, most of this stuff in an intuition sense is in the frying pan. It's gone. You can only rescue it in a sense that actually all our proofs of finite will make a new game of finite-left proofs and play with that. That's all you can really do with it. A huge part of my own work is investigating foundations of fields, philosophy, economics, religion, mathematics, geometry, all of these things. That's what I'm interested in. And it seems like, maybe this is my own ignorance on the topic, that there's a great number, a huge amount of work about what follows from square two onward. Oh, yeah. Square two, three, four, five, six. But not as much about square one. This strikes on the earliest observation you made about mathematicians and people have that mathematicians have got this kind of perfect art form of immaculate, clear logic. It's not clear if you start worrying about step one, because if you have step one and somebody's made a mistake at step one, that's their problem. Now, you just get to do ands and aurs and nots and aurs. You even put, say, the law of excluded middle in those base assumptions and you can do your non-constructive proofs. It's all the game then, and you can see when somebody else has gone wrong. I mean, that's the pure part of mathematics. But yeah, if you start worrying about step one, then it all goes to pot. So people, the mathematicians don't like to, they don't like to. Now, do you think that's a reasonable position to take and say, hey, how about we start investigating? What do you mean by reasonable? I mean, I play go for two years. I mean, it's not reasonable in the sense it is doing something productive in the real world. But it's a fun thing to do. If it turns out that the rules of go are somehow self-configured, I don't think that they are. Or if the people give up, I mean, the thing is, I mean, studying go, you could be the only person in the university who could study go. It's just as pointless as most other academic, you know, you want to feed yourself and close yourself to things. But go is somehow rewarding to study because lots of other people out there also play by the same rules. And then you can interact with them and have fun with them and converse. And for me, set theory is very, very similar. I mean, I studied set theory for many years. I went and studied go. I came back to set three. I was studying go while I was doing set theory. It's very similar. The motivation for learning and studying this field is because of the people who can talk to them about it. And I feel there's an awful lot of that going on in mathematics. A lot of people just don't care. They just take in their check and they play their game. That is also my suspicion, but I can't say that because I don't have a PhD in Mathematics in Oxford. So people say, Steve, you're a crank if you make these kind of criticisms. You're not allowed to say that until you actually go through the system. And I have tried to demonstrate and many areas will know the one that I think resonates with people is in theology. There's a massive amount of work that theologians have reduced based on the assumption that a God exists. And I actually have the belief that there's something like a God which is actually what that means. But if that assumption is incorrect... That's a lot of wasted work. A lot of wasted work. And it's not... I don't have to have a degree in theology to be able to see that that is an obvious fact, especially when something like the existence of God or something like the existence of an infinite set is really hard to wrap your head around. It seems like it doesn't seem like what a proper mathematical education would consist of is working from the foundations. Yeah. Right? Maybe working, you know... Well, I can handle that because you're just, you know, down to the idea of working from the foundations. You've got to study and consider the foundations of mathematics and philosophy. Well, yeah. I guess, again, this is, again, is my bias. I think in order for somebody to have a genuine understanding of mathematics or whatever field is, they have to start with some philosophy. Because I can see people dedicate their entire lives to some theological idea that's foundation equality. They have it all the time. And my suspicion, much to my surprise before examination, is that mathematicians are doing the same thing. Okay. Because there's fundamental assumptions that don't seem to be challenged. I still feel like a lot of mathematicians, it's not really a case of, again, it comes more down to, at that level, it definitely seems to come more down to being consistency. In fact, I recall one particular lecture I was attending where something about foundational stuff was at one point in there. One of her, because I had two supervisors, technically. I mean, one that was my actual supervisor, and one that was more of an austere person that was associated with the university, eight-year-old, really nice guy. But yeah, he was quite clear on his, what we care about is consistency, whether or not these things particularly reflect any other thing, I mean, modeling it and applying something else. So I feel like a lot of mathematicians would still say, you can say that in pure mathematics, I think it is still kind of understood and believed in mathematical mathematics. So a lot of it, you can say it's just not useful, you know, because is there anything that you can, which applies to all of those axioms? Well, if there isn't, then the results are not so useful. If there is, then there are limited use. So this is a great segue to your own experience. It's not so much as saying that, you know, what we find out, we discover that the sets are wrong, but rather just, oh, okay, so what we mean is the thing that is defined by these axioms. And I say the thing, as I say, many people would have different conceptions of to what a set is. So this is a great segue to your own experience in academia. So you have a PhD from one of the most prestigious universities in the world. One mathematics, why are you not in academia? Why am I not in academia? What was your experience that would lend you to not be there? Well, as I say, I mean, I was attracted into it, mainly because I like mathematics. I went through school, I enjoyed maths, I enjoyed the fact that I could solve these problems, you know, I could... There was something very attractive and intuitive about these problems, and I have to say, even including things like the diagonalization arguments and the scrolls of tuition rational and Pythagoras theorem. All these kinds of things were intuitive and interesting to take the assumptions and work on with them. I went through the academic system. When you're coming out the other end, when you're getting out to the PhD and trying to justify grounds and whatnot, you know, all of a sudden you find that the puzzles that are being set are no more just, you know, testing that you can think and reason and improve your ability to think and reason, you know, that kind of exercise and fun. But rather, you're setting the problems yourself or rather other mathematicians are setting the problems and you're answering them and to what end you very much get this kind of feeling of futility coming around when, you know, you're much like if you took a computer game which had 30 preset problems and then a level editor, you play the preset problems and then you'd lose interest quite quickly unless you had a large group of people doing the same thing. For me, it just seemed very in the end it just seemed very incestuous. The whole motivation, the whole idea of you're writing a paper because some person asked a question and you dedicate, you know, 50 hours of your life trying to solve that and you write something up and then you pose some other idle thoughts and somebody else starts to answer them and actually, if you were to really try and unpack it and unpin it you could find there are probably 8 or 9 different people that are actually answering the same fundamental question just with different terms because they haven't really communicated properly with each other and again, the motivation is not to learn more about mathematics it's to convince funding bodies that you are doing something worthwhile and that's why it's nice to maintain this image of mathematics is so austere and august and above, you know, just so that I'm just giving money and they'll do clever things, mathematicians, you know I suspect the same kind of thing is happening in theoretical physics for a large degree and in philosophy and so I think it's shot throughout actually it's not something that I ever really conceived of when I went in to do the degree and even with the PhD I was still, you know, I was very much enjoying the degree, but from the PhD I didn't, I just, you know at the end of the day you can only spend so many years studying a game or a game like Go or I don't pay so much anymore or studying ZFC until you realize you want to make some money and live your life I just wanted to do it productively I wanted to do something that helps other people and for me that's not mathematics not mathematics In your actual education when you were getting your PhD how much was how much of it was the philosophy of mathematics how much of it was dealing with foundational questions like how do we generate when you're actually in school like a question like how do we get the continuum does the continuum make sense, how do we work through the logic Well obviously when you do something like real analysis what they'll do is they'll start from the first lecture and they'll give you the axioms and say these are the axioms of real analysis and then the very first thing you're doing is trying to prove things like 1, 9, 10, just using the axioms not using the intuition you might have had from before in school about what numbers are axioms but not studying why Is there any explanation of the axiom it seems like we should try to explore why we would assert different axioms and why they are axioms From my memory from the first, second and third year there was effectively no exploration of the issues of why or how these particular axioms are motivated In the fourth year, four-set theory people were motivating things like the reason we added the axiom of foundation or rather Von Neumann added the axiom of foundation was because we had problems like the Russell experiment so we talked on it a little bit but honestly any of the foundational stuff that I learned about mathematics was more when I went to philosophy lectures just in my own time not part of the official course I had a friend that was doing math and philosophy who was particularly interested in the foundations of mathematics but also of language but yeah he would often come to me and say look at these cool ideas and puzzles which are interesting to study a bit more so I went to a few lectures but as I said I didn't study it formally it's not encouraged This reminds me of in physics, you mentioned physics earlier, I was having a conversation with a guy from Oxford about this and specifically in quantum physics there are as I'm sure you're aware lots of remarkable claims that have been made about the nature of quantum mechanics but we shut up and calculate when we try to ask the question what are we actually claiming about the world things start getting a little fuzzy and so the response is well just do the calculations this is the game that we're playing this is the game that we're studying we've got all these problems that we've set up I would suspect that I have a close friend who did physics he did the degree, PhD he did a post-doc in America that there's very much this impression particularly from first year onwards that you kind of have to do away with all of these issues that are for loftier more philosophical people and just spend each week learning and studying how to do the calculations so that you can get to that front of the field and be productive if you spent the first two years doing the philosophical well trodden ground that all the other people have already done you don't accept these things that face value then you'll just end up being two years behind everyone else it's true because the philosophers have discovered it so it's outsourced so it's more about the career of being a physicist and it is about exploring the ideas well on that note I want to thank you for this conversation this music has started on so thanks, this has been great thank you very much alright that was my interview with my new friend Gareth, I hope you guys enjoyed it I hope it piqued your curiosity and I hope it made very clear to you as it will be in the coming years that it is okay to be skeptical about some foundational claims in mathematics, there is plenty of room for doubt here the orthodoxy should be questioned maybe they're right maybe they're wrong if it's the case that you only think it's clear cut obvious that infinite sets are logically airtight and there's no problems with them that is a reflection of your own ignorance on the topic my friend if you want to truly understand the subject matter you can't simply repeat what you've been told in class or what you see in your textbooks I'm afraid that amount of information is way way too limited and it won't expose you to the real conversations that have been going on historically and should be going on at the present but in my opinion it is a scandal that these ideas are not discussed within the halls of academia and you've got a bunch of PhDs all around the world who aren't as intellectually curious as Gareth who go around thinking that they're masters in some particular area of mathematics that include presupposing ideas and set theory all the while having absolutely no idea about the foundational challenges of some base level concepts in their own field naturally I've got a million more things to say on this topic I really do hope that you start this journey with me in examining the foundations of mathematics if you've valued this conversation and I hope you did check out patreon.com slash Steve Patterson and you can become a patron of the show just chipping a buck or two whenever I release new content and it helps keep the show and the journey continuing alright that's all for me today I hope you guys enjoy the rest of your week