 Hi everybody, welcome to episode number 8 of Patterson in Pursuit, and I'll be honest, I don't think this interview could have gone better. I didn't know what to expect at first, but at the end of it, I was just flabbergasted that it went so well. So as many of you know, I'm kind of a contrarian when it comes to the philosophy of mathematics. I seem to be rejecting the mainstream conclusions in mathematics for the last century, and I'm honestly trying to get to the bottom of why I just can't wrap my head around some of the claims of modern mathematics. So I thought, well, I'm going to be in Dublin, Ireland, and I need to talk to a mathematician about this. So my intention going into this in all sincerity is the expectation that no mathematician is going to agree with me because I disagree with the foundations of mathematics for the last century. So I sent out an email to Professor Gary McGuire, who is a professor at the University College of Dublin. He is the head of the mathematics and statistics department saying, hey, look, I'm struggling with this concept. Will you be able to help me out? I said, yeah, that'd be great. So I had absolutely no idea that in the very short period of time in our interview, turns out he agrees with me on almost all points, which was absolutely flabbergasting. In the world of philosophy of mathematics, there is a very, very, very small school that's called the Finitist School of Mathematics, which essentially rejects the idea of any completed infinities. Now, if I had to put myself in a school, that's the school I'd be a part of. And it sounds like that's also the school that Gary would be a part of. This was entirely coincidental. I'm expecting in the future that the vast majority of mathematicians I'm going to speak speaking with won't agree with the ideas that you're about to hear, but it's really awesome that off the get go, I get to talk to somebody that seems to understand some of the hesitation that I have when it comes to infinity. The stuff that we're talking about in this interview you can find at the show notes page this week is steve-patterson.com slash eight. As I said earlier, Professor Gary McGuire is the head of the math and statistics department at the University College Dublin. And like everybody else that lives on this side of the Atlantic Ocean, he's got a very charming accent. I just love the interview I had with Dr. Westcott. I love the English accent. This interview, I love the Irish accent. It just makes these conversations that much more enjoyable when you're talking about heavy material. I really hope you guys enjoy this interview. First of all, Professor Gary McGuire, thank you very much for speaking with me today. Thank you, you're welcome. So I have a lot of questions for you because I have been struggling with one particular concept in mathematics for quite a while. And that's the conception of infinity. And I know that infinities are accepted into modern mathematics. And to a large degree, the fiery debates happened a century ago, but I still fully can't wrap my head around some things considering infinity and mathematics. So I was thinking to start off with it would be helpful to lay out exactly what we mean by the term infinite or infinity and kind of just get the basic conceptual definition out there. And I'll give you my understanding of what I think the term means. And if I'm wrong, please correct me. So I always understood infinite or infinity to mean something like never ending or without boundaries or limitless or never completed something like that. Is that fair? I think, yes, I think that's a very good description of infinity. I think that increasing without bound is probably the way to think about it rather than the fact that infinity is some kind of a number or some kind of an object, a number which is bigger than any other number or something like that. Because that implies that infinity is a particular thing. It's an object, I feel like, at the top of a ladder or something. But we should think of it as a never ending ladder. So you can always go one row and higher on the ladder. I know that that conception is very foundationally linked with modern mathematics in set theory in particular, the idea of infinite sets at the very core of the foundations of mathematics. So here's what I'm struggling with in particular. The idea that there can be such a thing as a completed infinity or an actual infinity, an infinity that is realized at some point. That, just for me, being outside of mathematics strikes me as contradictory to think that there could be an infinite set, for example, seems like something that couldn't even be just by laying out the basic concept. So can you help me explain what I'm missing there? Not really, but in some ways I think of it as it's an abstract idea. It's a mathematical concept. So does infinity exist in the real world is an interesting question. And I think probably not. Now, you might find people who disagree with me, but I think it probably doesn't. However, mathematicians have gotten used to thinking about infinity as the famous mathematician John von Neumann said, when you're understanding mathematics, you don't understand mathematics, you just get used to it. So in some ways the concept of infinity is like that. To me, we kind of get used to it and we think of infinity as an object, as we get used to it as a mathematical object. And so we get used to thinking about it and even different kinds of infinity. But do they really exist? Personally, I think not. Okay, so one question is kind of the relationship between mathematics and the real world to say. I think I definitely agree with you that there can't be in the real world some kind of infinite set. But my issue is kind of one step beneath that is that even the concept itself of an infinite set, whether or not it exists in the real world, strikes me as something that's contradictory. Because what we were saying earlier, it never ending is the definition or never completed is the definition of infinity or infinite. And yet, when we say there is an infinite set, doesn't that mean that it's like a strict contradiction in terms? So a set being understood as a collection of elements in a set, a definite collection, but how could you have a definite collection that is actually never ending? It seems like you would never actually get any kind of infinite set. Yes, yeah. I mean, yes, you're right. And that's why, as you mentioned earlier, it's one of the axioms of mathematics that such an object exists. So the classic example of an infinite set is the positive whole numbers, the positive integers 1, 2, 3, 4, and so on and so on. And we think of that set just goes on forever. And that's if you like, everybody I think is somehow used to thinking of that set. However, you can never write them all down. That's it's never ending. So does the set of natural numbers, we call them, does the set of natural numbers exist? Well, that's the mathematical axiom. That's an assumption we make. We have to make that assumption in order to go on. So for somebody that is not a mathematician and would look at it, like I said, well, that seems like a very dicey way of reasoning to think that let's just take this as an axiom and go from there when this doesn't seem like it's a clearly analyzed possibility. So if we were to go outside of mathematics, if we were to say something like I believe in x and somebody says, oh, well, why do you believe in x? Well, I take it as assumption so that I can work from there. I mean, we wouldn't accept that as being a satisfactory reason. So surely there's more justification than just taking it as an assumption. Do you think that's fair? Well, to do completely rigorous mathematics and set theory, I think it has been shown, as you said, as an axiom of infinity. So has been shown to be necessary. But so that's not a good answer as far as you're concerned. Well, as somebody outside of mathematics, my interest is in discovering truth as a philosopher. That's really what I'm interested in. And I'm very interested in methods of reasoning for why people end up at certain conclusions they do. And it seems like for something so foundational, a lot of conclusions are derived from set theory and from incorporating infinities into modern mathematics. It seems like every axiom would need to be scrutinized to make sure that it's a true axiom and not just a convenient one so we can do calculations from it. Every axiom is, in mathematics, is carefully analyzed, very carefully analyzed and scrutinized to see if it's necessary. So we would certainly not have any axiom that wasn't absolutely necessary for the development of the entire theory of mathematics. So in that sense, I mean, if any axiom can be derived, another famous example would be in geometry. You have Euclid's, the fifth postulate. And for a long time it was assumed and people tried to derive the fifth postulate from the other four. And people thought that it wasn't necessary that it could be derived from the other four but it turned out it was necessary because these non-Euclidean geometries were discovered, which satisfied the first four but not the fifth. So that fifth one was absolutely necessary in the Euclidean geometry. It could not be derived from the other four. The same thing happens in set theory. Set theory has been reduced, if you like, to these axioms and each one is absolutely necessary. That's what mathematicians, the foundational mathematicians have done. Okay, that's something I would love to, maybe we can get into Euclid a little bit later because I think that's actually a really important thing that you bring up. So can we revisit this idea of the, let's say the set of the positive integers? I think is what you brought up this. We assume that this is an infinite set. There's different ways that you can conceptualize numbers. So for me, for example, this is how I think this is the most accurate way of conceiving of what numbers are and then thinking of numbers this way completely resolves any questions of infinity. So I would say something like numbers aren't abstract objects that are separate of our conception of them. Numbers are concepts. They're created by human minds and their existence is entirely dependent on our conception of them. And if that's true, then it would mean that there is no such thing as any infinite sets because you can't conceptualize all of an infinity. And the question of what are all of the positive integers is kind of a mistaken question. It's not what they are. It's whatever ones you've specified. So it's one, two, three, four, five, six, seven, eight. As far as you go, that's as big as the set is going to be, but it's not gonna be beyond what you haven't conceptualized. Mm-hmm. Yeah, I think you're right. I think that we can conceptualize up to a million or up to a billion or we can conceptualize any particular number. Something I like to do with my students is I go to the board and I write down some random string of numbers, one, three, five, zero, eight, seven, so on, so on, all the way across the board. And I end up with 50 digits on the board and I say, here's a number. Before today, none of you have ever thought about this number, but it's a natural number and this is, we can go this far, we can conceptualize this number, but there's always gonna be a bigger number. And no matter what number we think of, there's always a bigger one. That's the concept of infinity. That's an excellent example. And what I would say is even the way we phrase it though, I think is important. So it's one thing to say, there is always a bigger number versus there's always a number we can think of that's bigger than this. So it's one is like kind of the platonic way of thinking of mathematics where out there in the world, there's this infinite amount of numbers that we're drawing from that we're referencing and there's always bigger ones out there versus you know, you can always, your mind can always add a zero to whatever number you conceive of and that strikes me as a clearer, more accurate way of thinking of infinity but the trouble is in, and I'm very persuaded, I'm very persuaded by my own idea as well. In thinking of things this way, it rejects the axiom of infinity. It says, oh well if that's the case then there are no infinite sets and therefore everything that follows from that axiom is, there's a flaw in the whole thing which strikes me as kind of a radical position to take. Yeah, I would, I don't know, we're kind of thinking, we're talking about in a way mental constructions here. Aren't we? We're not talking about things that exist in the real life. These are all things that exist in our mind. So when you take, if you think of this set of all positive integers, one, two, three, four and so on forever, you, that's a concept. It's concept in our mind. We think of, you can think of it as being each number and then throw all the numbers together. But I don't think you can't have all the numbers if it's a non-finite amount. So let me give you an example that I think illustrates what I'm getting at that it's not even that it's a mental construction, it's that the concept itself is something that is a logically contradictory and so it can't even be a mental construction. So do you think this is a fair analogy? And I've seen this lots of places in mathematics where they talk about a circle whose radius is infinite. Is that something that is a fair analogy to think that, oh, we can conceptualize a circle whose radius is infinite? No, I would say you need to go back a step. If you're gonna conceptualize that, you have to be able to conceptualize infinity in the first place. Yeah, and what I'm saying is I think it can be demonstrated that the idea of a circle with an infinite radius is something that we can't conceptualize. So for example, I was just talking to somebody about this and they were saying, yeah, you can think of a circle with an infinitely sized radius. And the question would be this, what would the curvature be of a circle with a radius of an infinite size? So if it's the case that there is actually a curvature, some curvature, then it necessarily means the circle is finite, because you just follow the curvature around and you have a finite circle, whatever it is. But if there's no curvature, which is the only other option, then it's a straight line. And straight lines certainly aren't circles and so I would say, well, therefore, it can't be that you could even have, you can't even conceptualize an infinitely sized circle. Yeah, I see that there's a sort of common in mathematics, yes, to think of a straight line as an infinite circle. If you, it's a way some people conceptualize the real line, the number line with infinity is you imagine taking it at either end and bringing the two ends or wrapping them around and joining them up together and you get a circle. And one point on the circle is infinity. Personally, I don't think I agree with you. I think I can't conceptualize that because I can't conceptualize infinity. So it's the same thing as thinking about infinity is thinking about this circle. Doesn't really help me. But to go back to your earlier point of the logical contradiction, I don't see a logical contradiction here between, could you elaborate on the logical contradiction? I don't see a logical contradiction in the shorty. Yeah, so that was, I was trying to give an example of the logical contradiction that I think we might actually be on the same page with the infinite circle thing. I gave that example to say it is certainly not the case that circles are straight lines or that parts of circles are straight lines. Those concepts are mutually exclusive and therefore it can't be the case. I would say the concept of an infinite circle is a logical contradiction because what we mean by circle and what we mean by straight line are mutually exclusive. And I would say that the reason that's a logical contradiction is the same reason that an infinite set is a contradiction where we're saying it is a well-defined set or a well-defined thing but it actually isn't a well-defined thing because the boundaries are never ending. So you never actually get the boundaries because they can't ever be anywhere. Yeah, yes, I think we are on the same page. I think I agree with you. Yeah, but I don't think, I still don't think there's a logical contradiction here. I think we, as we were saying earlier, you postulate the existence of the infinite set. We assume the existence of the positive integers. We just kind of assume they exist and we assume that we can put them into a set that's mathematics does. But that doesn't imply that that's not logically contradictory, right? So we could say the same thing about square circles. We say, well, we just assume the existence of square circles. And what I would say is, well, if you unpack the concept of a square circle, it's a logical contradiction. And wouldn't you say that's the same thing with the infinite circle and a straight line that when you unpack the concepts, it is something that's impossible and therefore it shouldn't be an axiom. And wouldn't that also then apply to infinite sets that it's like saying we have a square circle at the bottom of our, we just take it as an axiom? Well, not sure, I'd have to go in to ask you what you mean by square circle, but you can, yes, you can take anything you want as an axiom, yes. But it can't contradict the previous axioms or each axiom should be, if you like, independent of all the previous axioms. So you can't, they should be necessary. This is what I was saying earlier when mathematicians study the axioms a lot and they make sure that each axiom is really necessary. It's independent of the others. And yet it's necessary for us to develop a theory. If we have too few axioms, we can't develop a mathematical theory. So we need somewhere to start and the axioms are where we start. And we can't, if we throw in stupid axioms, we're gonna get contradictions. So we're also not gonna get a good theory out of it. So let's explore if we can a little bit more about this idea of the logical contradiction because it sounds like we're in agreements a lot of the way, but not quite completely. And so maybe this is the piece of the puzzle that I'm missing here. You would, we agree that in actuality there's no infinite set, is that right? Yes, I think so. Then if that's the case, is, so that's, what is the reason that you would say there is no infinite set? What I would say is it's because the concept is logically contradictory. The concept of a set and the concept of infinity are mutually exclusive. And I would say that's a logical thing. What would you, why do you think there's no infinite set? Well, I can't because I think, I mean, in the world we live in, is that what you mean? Well, it depends. That's what I mean if that's what you mean. So what is your, what is your objection? In the world I live in, of the things that I can really conceive of, I think everything I can conceive of is finite. Is my personal belief. But I'm happy to accept for the purposes of developing a mathematical theory, I can accept the existence of an infinite set. Because I can think of, you know, the numbers one, two, three, four, five, and so on forever. And I can think of putting all of them into a bag. Some kind of a bag, which I call a set. But not all of them, right? Not if there's an infinite amount, you can't put all of them in. Well, exactly, I can't, in the real world I can't. But in my head I can, I have to accept that as an axiom if I'm gonna develop mathematics. Couldn't there be an alternative? I mean, wouldn't it be so, so if I were to try to construct a theory based on axioms, I would say one of the, it should be a principle or maybe an axiom that the concepts that we're using shouldn't be so impossible in reality. I mean, it seems like by saying there's an infinite set, we're saying we are condemning this to never be applicable to the real world. Or even, not even just to be applicable to the real world, but never actually be conceptualized. That there's an axiom here, which when we really break it down, doesn't even work in the mental world because we can't fully conceive of the infinite set and putting all the numbers in a bag because by definition, if it's infinite, you can't fit them all in. Yes, I agree. I mean, there are actually, I don't wanna talk too much maybe about foundational mathematics, but there is a school of thought in mathematics where everything is finite. So you don't use any infinite sets and you try and develop mathematics simply using only finite sets and finite objects. And that's quite interesting to do and it's perfectly valid. And it just, I don't know, it just turns out that especially for applications of mathematics in the real world, it's more useful to us to allow infinite sets. We're coming to the sort of applicability of mathematics in the real world, which is somewhat surprising, but it turns out that mathematics is useful. We can go to the moon and so on using mathematics, but that mathematics uses these infinite sets. That is very counter-intuitive for sure. Can we, I do wanna talk about that a little bit because it's something I don't know that much about. Where, so can you give some examples of where the use of infinite sets is something that's necessary to be applicable to the real world? Because in my mind, being ignorant on this topic, I would say, well, if there's two competing schools of mathematics, you have the finiteist school and you have the standard school that accepts infinities. It seems like whatever you can do in the standard world accepting infinities, you can probably do without infinities. So where's the benefit that you get when you accept the infinities? Well, actually, that's a really good point. You're absolutely right in that. Everything that you can do, everything that we've done could be done with, let's call it the finite mathematics because when you actually go to implement things, you write computer programs and you build machines and everything is actually finite. Everything has to be approximated. But the theories that we use in order to design those machines are somehow easier if we allow the infinite mathematics. We use the infinite mathematics, we come up with these theories and then we build things. Yes, those things could have been built only using finite mathematics. It just, I don't know why we don't do it that way but we don't, it's the way the world has evolved, we don't. Wow, that's fascinating. So are you saying that conceptually speaking, there's nothing that infinity adds to the equation that you just can't get from finite mathematics but let's say the calculations and the formulations are a lot easier when you allow infinities? That's exactly right, yeah. In my opinion, that's a perfect way of putting it. Well, that is, this is fantastic. If I do want to talk just a little bit, you mentioned Euclid, I was just reading a book. Actually, this is now one of my favorite books is called The End of Certainty by Morris Klein and it talks specifically about the history of mathematics moving from supposedly certain foundations to less certain foundations and now modern mathematics, the certainty is lost and it comes back a lot to the conceptualizations of infinity and Euclid. So we don't have to go in any great detail but can you give a little bit more explanation? Maybe, I don't know if you know this off the top of your head so if not, that's totally fine. For the, you said there was five postulates in Euclid when we're talking about geometry and I don't know them off the top of my head and the fifth postulate is one that everybody's been taking for granted apparently for a few thousand years and in the, I believe it was the late 18th century and in the 19th century, people started challenging that this fifth postulate was something that was, you could derive from the other four and they took kind of as an axiom that let's say this isn't true and let's see what we get from there and there's the development of non-Euclidean geometry so can maybe we talk a little bit about that like what was the denial of the fifth postulate and then what was the result with the non-Euclidean geometries? Well, sure, I don't know if I can recall all the details off the top of my head but the fifth postulate, roughly speaking it's if you have two straight lines we're talking about in the plane now, in the plane so if you've got two straight lines drawn on a piece of paper and they're not parallel then they will meet at some point. I think, roughly speaking, I've got it right but it's something like that, that's the fifth postulate which it sounds, when I say it like that it sounds completely, I think you'd accept it you'd probably accept that it's true so it makes a lot of sense and everybody thought, yeah, the controversy was that it maybe was this postulate necessary or could it be derived? There were four preceding postulates or axioms and people were wondering could it be derived from the other four and I think over centuries many people tried to prove that it could be derived but the advent of non-Euclidean geometries meant that this was a geometry which satisfies the first four but not the fifth so that means that implies it cannot be derived so you can have things which satisfy the first four and not the fifth but if it could be derived then anything which satisfied the first four would also satisfy the fifth so is the claim that it is at least logically possible that you could have non-Euclidean geometry be existent so there's nothing internally inconsistent about non-Euclidean geometry even if that's not the world we live in right now? Yes, it's entirely possible that there is a world or a universe that could have non-Euclidean geometries in fact people have tried to measure our universe to see if our universe could be non-Euclidean in some very tiny way all the indications are that our universe is Euclidean but when I say an example of a non-Euclidean might be the idea would be that you live on some kind of a curved surface so if I think of like for example living on a sphere if you think of two lines on a sphere they could be, if they're not parallel they would intersect in two points so there might be one point on one side of the sphere and another point on the other side of the sphere where they would both intersect so in the plane that doesn't happen if they're not parallel they're only gonna meet in one point so that kind of thing that's the kind of thing that could happen in a non-Euclidean universe but I think all the indications are that our at least spatially I'm not going into space time or relativity but spatially our universe is Euclidean So for an example like that it seems like intuitively the idea of a sphere like we're saying well imagine you're living on a sphere that to me presupposes Euclidean geometry so the only, our conception of what a sphere is is only in the conception of Euclidean space so it doesn't even seem like you could when we say something is curved that implies that it's not straight and so it seems like it's kind of like the axiom infinity in the sense that it was like well let's just assume, let's just take that as an axiom and see what follows versus whether or not that concept is even possible I would so tentatively say something like the Euclidean space is something which is like conceptually inescapable that even when we talk about any other try to talk about non-Euclidean spaces it stills within the framework of like a meta Euclidean space Yeah I see, I think I know what you're getting at and I remember struggling with this for a long time as a student but the point to think about it as if you lived on a sphere but you didn't know that you lived on a sphere and you couldn't leave the sphere how would you know that you live on a sphere? It's in a way it's what happened many centuries ago and people used to think the earth was flat because if you look around us the earth seems to be flat and if you can't leave the earth and how do you know that our earth is not flat? Yes even in that circumstance though we're saying in reality the thing that you live on is curved but even saying that would mean it's still in the context what we mean by curved is still in the context of Euclidean geometry so the only, in fact I vaguely remember this but I'm not confident enough to say this conclusively I vaguely remember one of the postulates of non-Euclidean geometry is something along the lines of like ultimately lines would have to be understood as circles, as small sections of really really big circles in order to make it work and that seems like that seems unsatisfactory it seems like a logical contradiction let's assume that this is the case let's assume that lines are just small sections of really big circles meaning that ultimately the space we're dealing with is curved and so therefore lines are circles but it would seem like well that's then it's not a line if it's curved it's certainly not a line even if it's very close to it it's not actually that Yeah, yes, yeah I understand what you mean still, I mean two points I could make one of the classic theorems of Euclidean geometry is that if you add up the three, take a triangle and you add up the three angles in a triangle the sum of the angles is 180 degrees and that can be derived from the axioms and in a non-Euclidean geometry that's not true but if you think of a I think this is the same point you're making think of a triangle drawn on a sphere and the lines will be curved and the angles will therefore be different and the three angles won't add up to 180 degrees so that's a way of in principle you could tell if the earth that the earth was curved and not flat if you drew a big triangle and measured the three angles and added them up so there are ways of this is my second point is I think there are intrinsic ways of telling whether the surface you live on is curved or not curved and I know what you mean by you know curved is a Euclidean concept curved or not curved this is a Euclidean concept but it actually isn't there actually are ways of doing it intrinsically Oh, see that's interesting that you say that because I would intuitively think the opposite that when we say oh we're drawing a triangle on a curved surface what I would say is well if it turns out that the lines are actually curved then it's not a triangle and you're not dealing with a triangle you're dealing with something you know something else Yeah, no, I see what you mean I see what you mean and maybe you're right but I still I stick to my point that there are ways like that of telling intrinsically determining whether you live on a Euclidean or any Euclidean world or in a non-Euclidean world Okay, and so the last concept that I want to talk to you about is very closely related to these two and it's about the application of infinities in calculus it's specifically the concept of convergence which I think is immensely practical and I think we can accept convergence in like a finiteist mathematics at least how I'm conceiving of it but I have a coming from my interest in philosophy there's these very famous paradoxes I'm sure you're aware of Zeno's paradoxes paradoxes of motion where in order to get from point A to point B you have to go through the middle point and then you get to that middle point you have to go through another middle point another middle point at infinitum and his argument was well therefore it's impossible because you can't move over an infinite amount of points and the standard resolution to Zeno's paradoxes oh well calculus solves that and that's the argument that if we give the calculus solves that and I have a resolution for how calculus does solve that but it's not including infinities but most people say oh well there's this concept of convergence so if we can let's just take maybe not the standard Zeno example I just wanna take another example and then you can help me maybe solve the problem here let's say that we're trying to construct a pi and the way we do that is by taking halves of pi in succession so we start with half a pi and then we add a quarter of a pi and then we add an eighth of a pi and then a sixteenth the standard idea would be at some point you actually complete the pi by adding successive halves together eventually yes these things converge and you get a whole pi is that a fair would you say that that is the standard resolution to something like that that eventually it converges into a whole pi yes I think that's a very good way of thinking about it I think that but when you say eventually you mean in some sense this is again infinity so when you say eventually you mean after an infinite amount of time has elapsed you will have a complete pi but at any particular finite time you won't have a complete pi you might be very, very, very close as close as you like to a complete pi but after any finite time you won't have a complete pi but couldn't we say then if it's the case that after so maybe we can rephrase it this way at no point in time would you ever have a complete pi is that true? Yes, that's true so but isn't that mutually exclusive with the idea that at some point you do get a complete pi if it's at no point you ever get a complete pi? The depend what do you mean by at some point you get a complete pi or eventually you get a complete pi eventually means are at some point that means in the limit there's this mathematical idea of a limit so in the limit as time approaches infinity or as the number of slices you add approaches infinity that's when you get a complete pi but it's not something you can approach I mean it seems like by definition we're saying we're saying that the pi at no point in time ever would you ever get a completed pi because that would mean that it would be a finite that the calculations of divisions would stop and if that means at no point you would ever get a complete pi doesn't that mean that you never get a completed pi? Yes, I agree yeah, you'll never get a complete pi it's an infinite process I mean it wouldn't be an actually infinite process because what it seems like and this I totally agree with what it seems like we're saying is if what you're doing, if you have a process of actually putting together a pi this way then you're never actually going to get a whole pi you're always gonna have a little bit left over Yes, exactly Now if that's true then does that not mean that Zeno's paradoxes are not solved by calculus because the claim is not that the runner will get ever so close to the final point but that the runner will actually complete the race that ultimately, or the pi will ultimately be completed doesn't that mean that Zeno had a was making a good point there? Yes, yeah, yeah, no a really good point I mean, I'm agreeing with you not disagreeing with you, so So what do you think of this potential resolution that the reason calculus does work in the real world is because reality is finite it's not infinitely divisible and therefore at some point the calculations terminate and then you can complete the whole pi and you would complete the race? Well in the real world as we were saying earlier we don't get into the infinite so we have to approximate everything by a finite number and so in the real world we would get, we wouldn't be able to I don't know if we were adding smaller and smaller and smaller pieces of pi we'd eventually have to stop somewhere we can't get ever smaller and smaller and smaller pieces we just can't do that so we have to stop at some smallest possible piece and then we add that in and we finish the pi And what about with something like distance so could we say the same thing that ultimately that this is what I think the resolution is to Zeno's paradoxes is that there is like a base distance unit in the universe that you can't actually divide in half because otherwise it seems like motion would be impossible but if there's like a base unit of distance then everything seems to resolve itself just like a base unit of pi I kind of agree with you I think in the real world in practice there is a base unit of distance and what we get into you almost get into theoretical physics I mean you get down to nanometers and you get down to pines constant and quantum mechanics and all that sort of stuff if you want to start the uncertainty principle you'll get down to if you want to get ever smaller and smaller and smaller you're going to get down to distances where the laws of physics are different but you know we don't do that in everyday life we certainly don't do that right well on that note I won't take any more of your time but I really really appreciate this conversation I think I really think this has been great I know I've learned a lot well thank you very much I really enjoyed it too so that was my interview with Professor Gary McGuire I don't know what else to say I mean really after that interview I was just flabbergasted that that the one in a million happened where I found somebody that seems to agree with my own personal perspective on this I'd recommend if this sounds interesting to you check out the show notes page steve-patterson.com slash eight and I'll have a link there where you can get to the book that I was talking about The End of Certainty by Morris Klein absolutely essential book to read if you're interested in these ideas make sure to tune in next week as well because just like I was assuming to disagree with Professor Gary McGuire and we turned out agreeing I had an interview at Columbia University when I was in New York City with a professor that I thought I would agree with and I ended up disagreeing with in the most radical way possible my raison d'etre is logic that's what my upcoming book square one is about is about logic and the necessity of logic and my interview at Columbia was with a gentleman who is a logical pluralist which means that he believes there's competing logics out there that logic there is ultimately no bedrock for epistemology which I went into the interview thinking the opposite was gonna be the case and it just turned into a absolutely magnificent interview so make sure to tune in next week to hear my interview with somebody that I disagree with 110% on but we remain cordial throughout and at the end I feel like I made a friend so that's the show thanks everybody for listening and I hope you have a great day