 What are the foundations of mathematics? Have those foundations been rigorously established? Is there any room for reasonable skepticism about the foundations of modern mathematics? How important is the concept of infinity to the foundations of mathematics? These are the questions I'm trying to answer on the 48th episode of Patterson in Pursuit. Hey everybody, got a very special episode for you today. As you guys know, my favorite topic in the world to talk and think about is logic, and very closely related to logic is the area of mathematics. Like most people, I assumed prior to investigation that the area of math was one of perfect logical certainty, but of any field of thought where skepticism would be kind of silly and unjustified, surely it's the area of mathematics. But what I found after investigation is actually there are some fundamental ideas in math that aren't necessarily that rigorously established, specifically this concept of infinity or the completed infinity. The completed infinity has become central to the way that orthodox modern mathematicians do their craft, and yet there is quite a lot of room for reasonable and rational skepticism. This is true not just now, but it also historically has been the case. Lots of other mathematicians and thinkers have thought that this idea of the actualized infinite is a bit dubious. But don't take my word for it. To help me answer these questions, I have traveled to Sydney, Australia to speak with Dr. Norman Wildberger, who's a professor teaching at the University of New South Wales and he also has a popular YouTube channel where he explains not only the orthodox way of thinking about mathematics, but his own new approach to conceiving of mathematics. He also has a suspicion that maybe the fundamentals and foundations need to be reworked. And unlike me, who's only written a couple of articles online about some foundational ideas in math, Dr. Wildberger has hundreds and hundreds of very detailed videos that he's produced for anybody that is interested in examining mathematics from a different perspective. I could not more highly recommend his YouTube channel if you're interested in these topics. I will have a link to it in the description of this podcast. And he also has a Patreon page, which if you are enthusiastic about his work, you can help contribute to his very laudable project financially. Now what's also special about this interview is not just the guest, but also that this was recorded with video. Dr. Wildberger on his YouTube channel is or is going to very shortly post the video of our conversation. So if you'd like to watch us talk instead of listen to it on this podcast, head over to YouTube and you can check out the video there. I don't think it's coincidental that a unorthodox thinker is choosing to produce a lot of really high quality work online and he's getting funding from a source like Patreon. That is also something that yours truly is in the process of doing. And there's a reason for it. That is because I think very strongly that the future of the world of ideas is going to belong to those individuals who use the internet. The market has already changed. This is the direction that everything is going and it is clear as day to those who are not already stuffed in the traditional establishment. And this shift isn't happening just in the world of ideas. The sponsor for the show is a company Praxis that is taking the next generation of young entrepreneurs out of the halls of academia and straight into the real world. The Praxis program is three months of a professional boot camp that is followed by six months of a paid apprenticeship which makes the net cost of the program to participants $0. The apprenticeship pays for the program. This is the kind of radical and dynamic shift that is taking place right now in the world. The world of commerce is changing. The world of the ideas is changing. And if you want to be part of it, head over to Steve-Patterson.com slash Praxis. That will take you to their website where you can get more information. You can sign up to get a module of their curriculum sent to your inbox and they were just featured on the Tucker Carlson show, a mainstream network because they are in the process of exploding in popularity while changing the paradigm of how we think about education. So I really think you're going to love this interview with Dr. Weilberger. I honestly could have sat down and spoke with him probably for four hours straight. I hope to eventually have a back on the show because the man is filled with an incomprehensible amount of knowledge both historical and theoretical on this topic. He's also written a book called Divine Proportions, Rational Trigonometry to Universal Geometry which if you're interested in his unique way of viewing mathematics, I recommend you pick up. Enjoy. So first of all I want to thank you very much for sitting down and speaking with me today. This is a great opportunity. I appreciate being able to use your studio here for some video content. Welcome to Australia and to UNSW, Steve. It's great to meet you and pleasure to have you here and I'm looking forward to an interesting chat. So I want to pick your brain about a couple of ideas. So from outside the world of mathematics, most people think that the conclusions in the world of math are all logically certain. From beginning to end A to Z, you start with some simple premises. You use airtight mathematical logical reasoning and then the conclusions necessarily follow. And so the idea of being skeptical of conclusions in mathematics, most people are like wait, you can't be skeptical of math, that doesn't make any sense. Do you think that this is a correct way to conceive of the area of mathematics, that it really is this area of completely airtight logical reasoning and there's no room for any kind of skepticism about some more philosophic questions? Well, I think it's pretty safe to say that it's not that safe, that historically there have always been contentious aspects of mathematics and the logical structure has often been challenged and alternative interpretations have been around. But I think it's important to make a little bit of a distinction between let's say foundational issues and more advanced mathematics. So typically what happens is that people put a lot of energy into the logical structure and rigor of more advanced arguments and they're looked at very carefully and generally thought about very deeply and mathematicians are rather careful thinkers generally. So sort of at the higher levels, there's often a lot of really solid and one can be pretty confident of what's going on there. But the difficulties are often at the more foundational issues at the bottom where it's actually sort of setting up the foundations of the subject. There things are not so clear cut, there is more interpretation and ambiguity possible. So it's really maybe even at the high school or even primary school level sometimes or maybe your first year or second year undergraduate level at those areas where we're introducing fundamental definitions and so on that problems can certainly arise and I think they certainly have. I think modern mathematics has severe foundational challenges and that probably we're seeing a time where there very well may be quite a lot of reversals in how we think about foundational things. So as somebody who's interested in philosophy, when I think of foundational issues I think of the really important stuff because if you get the foundations wrong then your conclusions are going to be dubious. The farther you go out on the theory you've built on your foundations, if that foundation is shaky you just may be almost wasting time or building up a theory that's catastrophically mistaken. So let's dive into some of those foundational claims. I know again from not being a professional mathematician, one of the areas that I find difficult to understand and yet foundational is this concept of infinity. It seems like infinity comes up all the time at least in modern mathematics and this notion of an infinite set or a completed infinity and it's hard to, let's put it this way, it's hard for me to wrap my mind around it. So if you can help me understand maybe where I'm misunderstanding this concept of an infinite set or maybe whether there's some meat to some skepticism of an infinite set, so help me work through this concept of infinity in modern math. Okay, so maybe I'd like to do that and I'm someone who shares your maybe reservations about infinite sets. I happen not to believe in infinite sets and not really an advocate of a lot of the structure built on that. But I think it's important to realize that although that's a very popular question especially amongst sort of amateurs and people outside the discipline, it's not actually as important inside mathematics as you might imagine. So the reality is that most working mathematicians don't have a lot of problem with the current framework of infinities because they don't really use them. They operate in an alternate system which is largely independent of the so-called axioms of modern set theory that were built up in the 20th century. So those axioms and that framework is really some kind of formal justification that people point to but it's not really something that's used on a daily basis. So most mathematicians like myself are not even really consciously aware of all of the axioms so we certainly don't use them in our daily work. So it's not the case that we use axiom five and then the next couple of lines we use axiom seven as you might imagine. So but in that direction I think it's really important to understand that the essential difficulties with mathematics although they're reflected these days in the discussion about infinities and so on are really about something else. The infinities come up because of our attempt to clarify the nature of the continuum. So it's really the continuum which is at the heart of the problems. And there we have really serious problems that go back historically a long ways which there really is contention. There really has been a lot of debate back and forth and that's really where the fundamental problems reside. So the continuum, what is that? Well that's a very difficult question. And if you go back to Euclid, if you go back to Euler, if you go back to Newton they would have had a very physical approach to that question. So their understanding of the continuum would have been based on a physical intuition about the space in which we actually live. So Euclid when he was developing his geometry and we have to remember that Euclid was really the Bible for mathematics for several thousand years when people studied mathematics they were really studying Euclid. This geometry is firmly based on an understanding of the geometry of physical space, in particular the geometry of the plane. So he's concerned about drawing lines on a piece of paper or maybe in the sand drawing circles, what you get when you make various constructions and make various arguments about physical space. Now what's happened is that in the 17th century the European mathematicians discovered an alternate way of thinking about geometry and that was really the transformative point and it was also a place where some of the difficulties of Euclid now started becoming clearer. So Euclid does not adopt coordinates but in the 17th century Fermat and Descartes and Newton and people like this started thinking about geometry in a more cartesian, in a more coordinate kind of framework. And this allowed them to really get a completely different view of what the actual fundamental basis of space is. So we move to a now a position where space itself is encoded by essentially n-tuples or let's say pairs in the plane situation, pairs of numbers. So our conception of space becomes subordinate to a conception of arithmetic to the number system that's available to us. And so the problems of the foundations of geometry which Euclid had largely been able to avoid because he was able to point to points and point to lines and everybody kind of knew what he was talking about. So those ambiguities were then displaced by a more accurate understanding of what a point is in terms of pairs of points, a pairs of numbers, what a line is in terms of an equation and so on. So we were able to make this shift where the foundational problems started to now be associated with arithmetic. So the question was what is our number system and how does that work? And the problem is that when people started investigating that deeper and deeper and as the century sort of passed, they had to start probing more and more carefully into what exactly the number system really was. What do we really mean when we talk about a number? So it turns out to be a very subtle and difficult point and it really came to a head in the 19th century when people really tried to get a grip on what we now call real numbers and it's that attempt to try to establish a theory of real numbers which is really at the core of almost most of the foundational problems that we currently have. Okay, so let's go a little bit deeper into that. When you say there are these foundational, potentially foundational problems with understanding real numbers, first of all what do you mean by a real number and does that have a precise definition in terms of what the standard mathematical orthodox answer would be? Okay, that's a good question, like what exactly is a real number? It's not an easy question to answer because there are actually several sort of competing different theories but it's probably a good to orient your viewers with the initial position that a naive kind of number system, one that everyone is familiar with and is very solid is that of the rational numbers, otherwise fractions, so we all learn fractions in primary school and we're comfortable with the arithmetic and then in the modern world we have this alternate system where we have floating point numbers, that's what our computers largely use, where we have decimal numbers maybe to eight digits or these days to sixteen or something bigger and our computers work in this framework of these floating point numbers with an arithmetic which parallels the arithmetic of integers but has to worry about the truncations and round off that are necessarily involved when you talk about floating point but that's sort of the arithmetic, the arithmetic system of our computers. But in a lot of situations we want to actually go further and we want to create or think about numbers which are not given by finite decimals but essentially by infinite decimals and this is where the problem comes about, what do we mean by an infinite decimal, how do we do arithmetic with infinite decimals, how do we verify the laws of arithmetic with infinite decimals, so this turns out all these questions are highly problematic and challenging in fact to set up. Okay so would you accept this kind of reframing of what's going on here, correct me if I'm wrong, that this issue of the continuum is central to really how we conceive of the foundations of mathematics. Another essential concept in how we conceive of the continuum is essentially what are numbers, what are, we use integers one, two, three, four, five but when we start talking about the floating point numbers, these decimal expansions then maybe right there we start running into the first issue because we're dealing, that's where infinity comes in, potentially infinity comes in. So why are, is there a potential issue with infinity in trying to express real numbers? Well maybe it's clear if you think about just basic arithmetical properties, basically arithmetical operations, so if you have a finite decimals, let's say you have two finite decimals and you want to add them, school kids know that what you have to do is you have to start on the right hand side and you sort of have to add from the, starting from the right and then the carries go to the left and then you proceed from right to left. So you can kind of see what the difficulty would be if you're thinking about the idea of having two infinite decimals and you're asking how are we going to define what addition is here. Right, where do you start? So where do you start? And well the answer is you can't really start at the end so you have to start at some perhaps arbitrary point and then you do that calculation and then you have to go further and do a more refined calculation and you have to keep doing this and it's clear that really an infinite amount of work is in general what you can expect to have to do in order to add these two, these two infinite decimals. So that's certainly a key problem is that the arithmetic is something that it's even hard to define actually how you're actually going to do that when you're working at the level of infinite decimals and people I guess in the 19th century were aware of that and so they were loath to frame a theory of real numbers in terms of infinite decimals because it was kind of obvious that there were these difficulties. And then someone called Richard Dedekin came along and discovered an alternate possible way of proceeding to set up real numbers which didn't involve these infinite decimals and for which the actual arithmetic was much easier to set up but the price that he had to pay was that the objects themselves were much more complicated than the infinite decimals that we are at least roughly familiar with in terms of intuition. But what he did is he said well let's think about something like a square root of two. Let's suppose for the moment that there is such an object a square root of two. Now I happen to believe that there is no such thing but most people think there is. So a square root of two is some point on the number line around 1.414. So just imagine for a second that there is such a thing there. Then that divides the rational numbers, it's like a wedge that divides the rational numbers to the right of root of two and to the left of root of two into two sets. We can imagine painting them red on the left and say green on the right. And then if you take that irrational number away and just look at the rational numbers that are left over, now you have this subdivision of the rational into these two groups. All the red ones over here and all the green ones over there. And Dedekin said that now we can actually think that what the real number root of two is, the actual definition of it is it's this subdivision of the rational numbers into this red and this green set. So that's actually the definition of the square root of two. Okay. That's called a Dedekin cut. Okay. Now that's a little bit maybe hard to get your head around if you think about it for the first time but it turns out to have a very big advantage is that when you want to define operations like addition and multiplication and so on, it's actually relatively easy to do in that context. The problem is that even though in the root two case, it's relatively easy to specify those sets because the condition is really just whether a rational number is square is bigger than two or less than two or negative largely. But if you have a general real number then getting hold of that Dedekin cut itself is a process that requires an infinite amount of work. In other words, if you wanted to, if Dedekin wanted to create his coloring for say pi, he would have to first get a sequence of nested intervals that surround pi and that sort of converge towards it and use that sequence to successively paint more and more infinite quantities of rational numbers. So the Dedekin cut approach appears to put real numbers on a kind of a set theoretical foundation because now real numbers are sets or pairs of sets of rational numbers but ultimately effectively it doesn't actually escape the problem of having to involve or invoke an infinite process in order to actually create it. So in some sense it's a bit of a linguistic sleight of hand that allows you to make the theory of real numbers seemingly dependent on a prior theory of infinite sets. And that's actually the reason, that's really the core reason why the infinite set story is so important in Mathematics to justify Dedekin's creation of real number is using this particular system of ideas that he had. Okay, so let's get into that story of the infinite sets. I think that's an excellent preface to see just how fundamental really that this concept is. There are some, let's say counterintuitive conclusions that emerge from thinking of infinite sets, you get, let's put counterintuitive in quotes. For anybody that's not familiar, one of the claims, and I can correct them if I'm wrong, is that we could conceptually have a, let's say, the set of all positive even integers, 2, 4, 6, 8, 10, and so on. And we can say all of them, so we take every single one and we put them into a set, we somehow put a boundary out and say we're talking about all of them. So if you do that, if you play by this line of reasoning, say, okay, how many elements are in that set while there's an infinite amount of elements? But then you say, okay, now imagine a set of all the positive even and odd integers, 1, 2, 3, 4, 5, and so on, which seems like there would be more elements in that set. Intuitively you think, of course, there are new numbers in this set that there weren't in that set. And yet the conclusion is they actually contain the same amount of elements. Is that correct? Well, I think a mathematician wouldn't say it that way because we don't want to talk about the number of elements when we're talking about so-called infinite sets. You have to understand I don't really believe in infinite sets, so I have to take on the position of the majority here in defending it. So they would say, okay, we're not allowed to talk about the number of elements in an infinite set, but what we can do is we can talk about a corresponding, making a correspondence between the number of elements in this set and the number of elements in this set. So if you can kind of match them up, you know, one boy for every girl, you know, then that's some kind of reasonable replacement of the idea that they have the same number of elements. We may not be able to count them, but we can see that we can match them up. Okay, so that's the way a modern mathematician would want to rephrase that argument. Okay. Now is that kind of a euphemistic way of rephrasing it, or is that an incorrect way of thinking of what an infinite set would imply? Is that, for example, the concept of cardinality would, is really talking about the number of elements, the distinct elements in the set. So is that, so is it a mistake to think of cardinality in the same way that we think of it applied to finite things when you apply them to infinite things? I wouldn't say it's a mistake, but I think you have to be very careful, and it's, you know, it's easy to distort things by using the language of finite things and the intuition that one has for finite things and trying to apply them to the so-called infinite situation. I think it was Hermann Baal, or maybe it was Poincare, but maybe it was Hermann Baal who said something about, you know, in the lines of that, that this is one of the weaknesses of the modern approach with set theory is that we try to invoke the same arguments for the infinite case as we do for the finite case, but the laws of logic are really different. But look, my point of view is actually a much more simple minded in the sense that, you know, I believe in writing things down. I have a very computational point of view, and not a very philosophical point of view, in fact. So maybe, you know, I'm a little bit naive from a philosophical point of view, but so for me, a mathematical object is specified when you write it down. And that's, you know, that's really necessary. So one can have fancy thoughts and so on, that's great. But ultimately, if I want to convey an object or the idea of an object to you, I'm obligated to write it down. And that's really the problem with thinking about infinite sets. That, you know, we can say things like the set one, two, three, four and so on, but you can't actually express that set. In fact, that problem emerges long before we get to infinite sets. It starts appearing when we get to very large sets. Maybe this isn't appreciated as much as it could be, but that the difficulties with sets theory are not just at the infinite level, they're also at the very big level. Once we get to the idea of numbers that are so big that we need some kind of towers of exponential kind of notation to get at them, then we're already in a realm where a lot of our arguments break down. In other words, there are limits to computation. There are limits to what computers can do. And not because our technology is necessarily limited or because we're not smart enough yet or we haven't built enough being a machine. It's just because the universe is organized in such a way that there seems to be some smallest limit to how much you can compress things, to how much, how you can subdivide space. There's a plank scale sort of limit to how small you can progress. And there seems to be some large scale limit to how big the universe is. So if you put those two together, it means that there's really only a finite number of possibilities that are inherent in the world in which we live. And you can quantify that, at least by, if you make some estimates that the physicists can give you. And you can quantify and see that once you reach numbers that are of the order of 10 to the 10 to the 10 to the 10 to the 10, something like that, then you're talking about numbers which now no longer can represent real collections in our world, no matter how small you created them. Now you say that's not a philosophic position, but that is very much a philosophic position. You're saying that numbers don't exist really outside of our conception of them or outside of us writing them down, is that fair to say? Let's say that that's my position. So my position is I'm kind of down to earth person I want to see things written down. To me that's the litmus test of whether something is real or not. You may have all kinds of ideas, but until you can express them in some kind of way that we can all look at them and compare notes and then I can really study them, then to me it's at some kind of abstract level. Would you say this, that's also my position as well, would you say that if it's true that numbers don't exist separate of their being conceived and neither do mathematical formulas, then that necessarily entails a kind of logical finitism that you can't get to infinity if all the objects you're talking about must be objects of your conception because you can't conceive of infinity. I guess I would rather not even discuss existence. To me that's a philosophical question, the question of whether ghosts exist or angels exist and so on. I don't know, it's a philosophical question. I'm a mathematician and I'm very concrete and down to earth, so the question for me is what can we write down? What can we compute? What formulas can we come up with that we can take the left hand side and we can take the right hand side and we can verify that the left hand side is on the right hand side. So you're agnostic for the other stuff? Yeah, I'm agnostic. I'm uncertain as to what the word exist really means. To me, the relevant thing is what can one write down? Right. Okay, so for anybody that's not familiar with some of the claims about infinite sets and infinity, I think it's helpful. I remember watching a video of yours where you gave an example to the audience with some intuitive conclusions about whether or not you can actually complete an infinite process or I believe the example you gave was something like, you know, there are an infinite number of balls over here and you take a ball and you put it here if you know what I'm talking about. So if there are some, what are some examples of where you get profoundly counter-intuitive conclusions when you try to actually complete an infinity? There are quite a lot of them in modern mathematics. There's quite a lot of unintuitive results that people believe are true and they're really dependent on a prior belief system with respect to real numbers and infinite sets. One could mention the Banach-Tarski paradox which you're probably familiar with. It's a well-known chestnut that you can take a ball of this size and rearrange it into a finite number of pieces, maybe five or six, and then rearrange those same pieces, and then go around and rearrange them to get a solid ball exactly twice the size, let's say. There's things like non-measurable sets, non-measurable functions which in modern mathematics are somewhat unintuitive, counter-intuitive. There are things like space-filling curves, curves that ostensibly go through every point in a square in the plane. I think those are all in some sense sort of peripheral and ultimately really rather unimportant because they're results with no consequences. They're sort of G-Wiz, isn't that interesting things, but they're completely devoid of consequences because the mathematics behind them invokes these things that you actually cannot do computationally. That's sort of hidden behind the scenes. I think a much better and more interesting example is a belief that most mathematicians have that I think most lay people would find rather astounding. This is actually a belief that actually has a lot of core implications. So if we look at the interval from zero to one, and we think about the numbers in there as infinite decimals, so we may have to convert from a Dedekind Kot or a Cauchy sequence point of view to infinite decimals, but they're largely equivalent. So if we imagine infinite decimals between zero and one for a second, then there's this really fundamental dichotomy that was pointed out strongly by Wittgenstein, who you will know is sort of one of the leading philosophers of the 20th century, not the leading philosopher most influential. And he said, look, there's really a fundamental distinction between two kinds of infinite decimals. At the one kind, you can think about an algorithmically defined infinite decimal, one that's generated by a computer program or maybe a function or some kind of algorithm. And on the other hand, there are, at least one is potentially able to think about, sequences which are not generated by a finite process, for which you might imagine some kind of oracle, maybe on another universe that has an infinite brain and an infinite time. And this oracle just spews out digits for this infinite decimal, but without any prior finite algorithm or pattern. So on the one hand, we have these sort of algorithmic decimals, on the other hand, we have these sort of infinite choice decimals, you might say. Now, if you actually look in the real world, when you make an integral or when you compute an arc length of a curve or you have some infinite sum or you do one of the many things that mathematicians actually do involving some infinite process, if you do believe and you allow them to get some result, you will always or almost always get one of these first kind of infinite decimals. They're necessarily algorithmic decimals. But the belief system that modern mathematicians almost universally share is that most of the numbers in interval 01 are actually not of the algorithmic kind. They are of this infinite choice kind. Which means that in fact the numbers that we use actually in practice are a miniscule fraction. In fact, measure 0 is a terminology that's used. So a measure 0 subset of all of the real numbers. So this is some kind of fantastical belief system really that most of the numbers that are actually there are ones that are completely inaccessible to us and our machines. Now can you give the devil's advocate from your perspective? Because from outside the profession that seems whimsical. Can you even reference these numbers? Are they literally like behind a veil that we can never even access that are out there? What is the justification for that kind of position? Well so the terminology that we use are computable numbers and uncomputable. So the algorithmic ones are the computable numbers and the other ones are uncomputable. And the claim is that most numbers are uncomputable. Most in a very strong sense in the sense that the measure of the computable numbers is 0, the measure of the uncomputable numbers is 1 in the interval from 0 to 1. So that's the claim and yes it's quite astounding but I mean most computer scientists would be very skeptical about that because of course they want to deal with algorithmic numbers and they have no appreciation for the existence of so-called uncomputable numbers which in fact don't figure in practice at all either. And many people have said, well why don't we just stick with computable numbers? Wouldn't that make life a lot easier? But it turns out that there's this thing called the axiom of choice which is one of these set theoretical things that people like to believe in and if you want to buy the axiom of choice you basically are forced to buy all these uncomputable real numbers and sort of they come as a package. So it's an unpleasant situation and people kind of shrug their shoulders and say, well let's just carry on. Now from my perspective that position strikes me as imminently philosophic in the sense that what it seems like the modern mathematical orthodoxy is claiming is that mathematical Platonism is true, that there exists these numbers that are out there that are inaccessible and computable and I guess we can engage with them in some kind of limited fashion but does that not require a very strict metaphysical claim that there exists these numbers that are out there separate of them being conceived? I think you're absolutely right, I think that's correct. I think it is highly idealistic, I think it is highly philosophical and in fact in my view 20th century mathematics really does have a strong philosophical foundation. The foundations of set theory and logic that were introduced in the early part of the 20th century really are coming from classical logic and the philosophical side of mathematics and that's really dominated the thinking at least in a sort of formal sense. It's like going to church once a year. Mathematicians will say they believe in this theory but they say it just to keep people happy and they don't actually really effectively use it but it's a kind of a justification, it's a framework that will allow them to continue this belief in these real numbers, in these basically being able to complete infinite processes. Okay, so let me ask you a couple of short questions on this. When talking with people who have some kind of training in mathematics or people who don't have training in mathematics I actually was having a conversation with some engineers about this and they said Steve, how can you claim that there's no such thing as these completed infinities because in calculus for example we use completed infinities all the time, central to calculus is this idea of the completed infinity and they're conceiving of how limits work. So how do you respond to something like that that if you don't have the completed infinity will you just undercut a huge amount of mathematical thinking? Well yes, you do undercut a lot of current thinking but this thinking has not always been this way. So in the course of the last three or four centuries since the 17th century revolution there has been quite a wide range of opinions and thoughts in this direction. So the current belief system is not really representative of the historical record, one has to appreciate that. The way we think now is as a 20th century construction that would have been very alien to 19th, 18th and 17th century mathematicians even though it may be almost universal these days. But in earlier times people did think about calculus problems and how to resolve them and they realized there were issues and there was a lot of discussion and I wouldn't say they were resolved entirely but just prior to the 19th century I think Euler and Lagrange were on a very positive direction. They had mapped out a direction that was leading to a more algebraic approach to things and I think this is really where the potential for the future lies is to go back to the Euler and Lagrange who probably understood calculus better than anybody has before or since and really build on their thinking. And it's possible to do that and it's possible I think to go quite a long ways but we need kind of a more diverse pool of people thinking about other ways of thinking, other alternatives to the limit-based orientation that we have now which is basically a 19th century invention. So do you think it would be a mistake to think that calculus can't work or that there's no way of conceiving of an effective way of doing calculus without including the actual infinities? Oh absolutely, it's a very big mistake. In fact I hope in my YouTube channel in the next year or two to create a course on algebraic calculus and to demonstrate that in fact one can do calculus in a completely down-to-earth, cut-and-dried, right-downable fashion. So I hope to demonstrate this sort of directly that these things are possible. Okay, now what about something like geometry? Because at least from the outside it seems like infinity also comes up in geometry. Is that something where you think we don't really need, you really don't need any kind of infinities in geometry either? Yeah, you don't need infinities in geometry. You can get around that. I mean it's not that you don't want to talk about things that one might say are infinite or infinitely far or something like that. It's just that you have to find a way of doing it in a finite, right-downable way. And people have wrestled with that in projective geometry. So in projective geometry one introduces points which are infinity. And formally that was considered rather mysterious and what does it mean to have points that are infinitely far away and so on. But the 19th century people, people like Mobius and Pluker, understood really how to do this in a very concrete, beautiful way without any resorting to infinite processes. So one can do things, there are alternate ways of doing things that allow us to get at infinities but in a finite, concrete, computable way. And let me add that ultimately our computers, what our computers do is sort of almost by definition finite mathematics. So if you can get a computer to do something, it's almost necessarily based on ultimately finite thinking because that's what a computer is. And so computers are known to be very good at calculating lots of interesting things and I think they're going to become increasingly important in the decades to come even in terms of doing pure mathematics. And we're going to see that their limited, seemingly limited non-infinity point of view will ultimately swamp our philosophical current approach. So are there areas of mathematics which can't be rescued? If it's true that this idea of the completed infinity is a bit wonky, are there areas which that really does fundamentally deflate? Look, I think a lot of them deflate to a certain extent, but I wouldn't go so far to say they can't be rescued. I think every subject can be rescued and can be rebuilt and can be restructured and rethought. And I think there's a huge challenge that awaits 21st century mathematicians is to really get down, roll up the sleeves and really get down to the foundations and figure out how to do these things. So even modern set theory, I believe that what they're doing is not just out in the left field, that there really is content there but that they have to find some way of saying what that content is in a way that's concrete and finite and computable. So from your perspective, this finitist perspective is very much a minority, especially in the modern world. In my own discussions, in my own elementary understanding of mathematics, I get a lot of flack from people. I haven't produced nearly as much content as you have just a fraction of a fraction, but I get a lot of flack from people who say you can't get rid of infinities in modern math. You're just a crank. This has been settled 100 years ago. How do you respond to that? I mean, obviously you're in the process of creating a ton of finitary mathematics that is very powerful and can be understood without resorting to these grandiose stories and abstractions about infinity. Do you think that there is a, in the mathematics profession, do you think that there is a kind of, maybe dogmatism is too strong a word? Do you think that there is an overconfidence, let's put it that way, in not establishing these foundational issues before going and doing their craft? So one more concrete question that you can answer. Somebody who goes through the mathematics education and they get their PhD, they go through the system, are they going to be grounded in sound foundational reasoning about their subject matter, or is that kind of overlooked to do the more interesting stuff? We just assume this is all correct, the foundational stuff, and then we do the interesting craft. Sure, that's certainly true. Yeah, absolutely. I mean, the foundational problems are deep problems. There's a whole range of concepts which are not really spelled out clearly. Most of the elementary words in mathematics from function to sequence, to number, to set, these are just used loosely and kind of intuitively, and everyone kind of agrees that we kind of know what a function is and so we don't really have to spend a lot of time and effort saying exactly what we're talking about. And if we do say exactly what we're talking about, we tend to use the same or synonyms. We play kind of a circular game. A function is a rule and a rule is a process and the process is a function. But look, I think that it's true the modern PhD has been, we are elements of our culture. I have been too. I think it's important, and I think it's really great, and maybe your channel and your podcast are doing this kind of thing, enlarging people's views, showing them possibilities that, yes, there's an orthodoxy, but maybe it's not the whole story. In fact, maybe it's not even the best part of the story. And certainly in my investigations in the last dozen years ago, so I developed this thing called rational trigonometry, some 10, 12 years ago, and I've used that to reconfigure hyperbolic geometry. So I've discovered that this new point of view, I wouldn't call it finitis so much as concrete. I don't really mind infinity if you can write it down in some concrete way. But this concrete approach really opens up so many beautiful new avenues for investigation. The subject is so much richer to me now. Having looked at these things with these concrete point of view. You have to appreciate that the foundational difficulties that are encoded in the imprecision of the way we use words means that a lot of mathematicians can't really get up close to the things that they're ostensibly studying. Like you can't really get a real number, for example. You can't actually possess, an infinite decimal. So you end up saying things like, let alpha be a real number. That's different from let alpha be a decimal with eight digits. That I can exhibit. I can get it right up close and we can all have a look at it. But if you say let alpha be a real number, then there's a distance. And there's so many things in modern mathematics more complicated and more sophisticated things that are of that kind where there's a separation between the practitioners and the actual objects. There's like a curtain or a veil. And getting rid of that, I think really will bring so many things more into focus. We can actually see what we're actually studying and things would be more concrete, more precise. It'll be a completely different and I think more vivid mathematical world. So you think that something like a real number. This concept absolutely elementary when conceiving about modern mathematics. You would make the claim that there is no actual clear, full conception of such a thing. Absolutely. That's absolutely my position. Well, I happen to agree with you, but you are doing some fantastic work explaining why that is the case. I really appreciate this conversation. This has been great. I hope other people discover that maybe there's a bit more room for skepticism with these issues than what immediately meets the eye. Thanks, Steve. It's been a pleasure talking with you. Thanks. Alright, that was my conversation with Dr. Norman Wildberger. I hope you guys enjoyed it. There is plenty of room for new ways of thinking about mathematics. Not just the foundations of mathematics, but also other areas like geometry, trigonometry, calculus, and any other area, just because there's one standard orthodox approach or way of thinking doesn't mean that is the only approach. And thinking otherwise leads to dogmatism. In all the areas of thought that I've studied, I must say I find an extreme amount of dogmatism in mathematics because people chafe the idea that there are unorthodox or new approaches to these old and important subjects. There is a huge amount more to say on this topic, and I hope you'll join me for those interviews and conversations. Thanks, everybody. Have a great week.