 Most people see skepticism of modern higher mathematics as kind of laughable. It's silly, couldn't possibly be any truth to it. And though people are very tolerant of entertaining skepticism and other disciplines, skepticism of math just seems too radical, too crazy, too illogical, and certainly I'm up to some funny business. I think this actually comes from a poor quality mathematics education, at least in the West. Poor quality because math students tend to believe the very first thing that they are taught without question and never think in any other way. They think the very first thing they were taught is certainly true and proven by some really smart people in the past. There's no alternative ways of thinking about it, and so they go on never realizing that in fact there's a very rich and controversial history in higher mathematics that leads to our present state. So if you've been listening to my work and maybe you like some of the political stuff, or maybe you like some of the stuff on the metaphysics of mind, but you really think I'm off base with the math, this video is made for you. What I'm going to do is present a history of mathematics that you simply don't get in high school, you don't get in college. I would venture to say the vast majority of PhD mathematicians don't actually understand what I'm going to tell you because they've never been taught it so they think the history simply doesn't exist. They're unaware of just how much room there is for skepticism, especially about the fundamentals of mathematics. So that might sound enticing to you, I hope it does, and what I'm going to do is break it into really two parts. There's no possible way I could do this subject justice. It's fascinating. I've been studying it for several years now. There's not one really good resource on this. This is a lot of knowledge from a lot of different areas of information. Eventually I of course will write significantly about this and synthesize it. But this I'm going to try to say is the context you need in order to have any sophisticated and reasonable opinion about the foundations of modern mathematics. So you can break the history of mathematics I think into two parts. You've got from let's say recorded history up until about let's say the 1870s, give or take. And then from the 1870s to like the 1850s you had a gigantic foundational crisis at the heart of modern mathematics, how that resolved brings us into the modern era. All of which is very, very interesting. So we'll talk about those different parts. Pre-1870s, 1880s, and then the crazy transition period and then our present state. So prior to modern history, mathematics was not built on top of you could call arithmetic or numbers. Numbers were not necessarily seen as fundamental. What was seen as fundamental was geometry, in particular Euclidean geometry. So the math that you think of when you're just thinking about like elementary math. Most mathematicians and most people would have been taught various mathematical principles that grew out of Euclidean geometry. There's a really, really good and interesting reason for that. The Greeks thought that geometry was more fundamental than numerical mathematics for a fascinating reason. So everybody's familiar with the Pythagorean theorem. A squared plus B squared equals C squared. The Greeks said, okay, well take a triangle that has the length of one unit, has the height of one unit. And then you have the hypotenuse. What is the length of the hypotenuse of that triangle? Well, if it's A squared plus B squared equals C squared, that's one squared, which is one, plus one squared, which is one. So that gives you two equals C squared. So that would mean the length of the hypotenuse is the square root of two. Well, what exactly is the square root of two? It's not something you can actually write out. You can't get to the square root of two by writing, you know, little squiggles. By writing numbers. It's something according to the story that has infinite decimal expansion. The Greeks didn't like this. I don't like it either. However, you could actually get the concept of the square root of two in Euclidean geometry. There's a really cool way with just a ruler and a compass that you can come up with the value, the length of the square root of two, but you can't put it into numbers. So the Greeks said, ah, this must mean that geometry is in a sense more precise. It's more fundamental. We can demonstrate with perfect precision. That is the square root of two, but we can't put it into numbers. Fascinating. Very smart. It should also be mentioned that Euclidean geometry was an axiomatic deductive system of mathematics, meaning it articulates a certain set of axioms which Euclid thought were self-evident, and many of them are self-evident or appear to be self-evident, and then it uses logical deduction to build more advanced structures of knowledge on top of those axioms. This is very important as we'll get to later. So for roughly 2,000 years, Euclidean geometry was the foundation of mathematics, essentially. Roughly speaking, it was the foundation of higher mathematics. It was seen as something that was perfect. In fact, mathematics education was synonymous with, for a long time, learning Euclidean geometry, at least getting your basics in Euclidean geometry. So that was considered the foundations of mathematics. Unshakable foundations of mathematics. And then what happened? Well, in the late 1800s, there was some mathematicians, some radicals, who actually doubted the universality of one of the axioms of Euclidean geometry. It's called the parallel postulate. The details are not relevant. But essentially they said, hmm, actually I think you could build an alternative coherent logical mathematical geometric structure and actually doubt this axiom. And if you do, you get all these other odd conclusions. And the mathematics profession at the time said, ah, utterly preposterous, impossible. There's no conceivable way that you can doubt Euclidean geometry. Any of these axioms, you guys are crazy. But they kept working on it. And they kept working on it. And sure enough, they were developing coherent alternative systems. You could call them non-Euclidean geometric systems. It's a bit of a misnomer because even the Greeks had things like spherical geometry, which is technically Euclidean. But it was a very big deal. So not only was it a big deal in geometry. Oh wow, we have Euclidean geometry and non-Euclidean geometry. No, this in fact cut to the very heart of the foundations of mathematics because if it's possible that we'd be assumed to be certainly true and certainly rigorous, could coherently and reasonably be doubted what actually is the foundation of mathematics. Why do we think anything is true? Does that mean 2 plus 2 might not actually equal 4? Could we go that far? The answer is no, you can't actually go that far. Anyway, so we had around this time the work of a German mathematician named Georg Cantor, who I'll talk quite a bit about. One of the most influential mathematicians ever, unfortunately. The man in my opinion was largely a fool. He had mental health issues. He died in an asylum. There's a phrase that if you want to make somebody crazy, have them think about infinity for too long. If you keep meditating on infinity, eventually you'll lose your mind. And that's probably started, I don't actually know the history of this, but it probably started around the time of Georg Cantor because he had quite a few crazy mathematicians who had these supposedly amazing discoveries with regards to infinity. But I'm getting ahead of myself. So, we now have let's say from the period of 1880 to about 1950 a foundational crisis in mathematics. What does it mean to say that some mathematical formula is true? Does mathematics tell you anything about the world? Does mathematics tell you the truth? Is mathematics just kind of us playing a game? There were different approaches to these questions, fundamentally different approaches. So as I said, around this time you have the German mathematician Georg Cantor working away and what is he working on? Well, among other things he's working on set theory. What is set theory? Well, it is as it sounds. It's about sets, sets of numbers. What was revolutionary and unique and dubious about Georg Cantor's work is that he came up with the theory of the transfinite set, the infinite set. So when we think of a set of objects, there's a set of all the grains of sand in the world. There's a very large set, all the little pebbles and grains of sand in the world, a very large set. Cantor was talking about infinite sets, infinite sets of numbers. So if you think about supposedly the natural numbers, one, two, three, four, five, six, seven, eight, nine, ten and so on. Supposedly these numbers go on forever. There's no largest number out there that you reach. And Cantor was talking about all of those numbers put together, taken as one. Can you do that? I and many others have said no, but Cantor said yes. And he said, in fact, not only is there an infinite set, there's an infinite number of sizes of infinite sets. You can have smaller infinities and bigger infinities. Wow, remarkable. So you can get something like the set of all the real numbers between zero and one is a larger infinity than the set of all the natural numbers, one, two, three, four, five and so on. Yes, there are more real numbers between zero and one than there are a regular old infinity. If that sounds remarkable, that's because it is. And in fact, at the time, it was mocked roundly, which I'll talk about in a bit. So why is this important? It's important because modern mathematics is built on top of set theory. The modern state of mathematics is not built on Cantorian set theory, which was shown to have some paradoxes in it. It is built on what's called Zermelo-Frankel set theory, which again I'll talk about later. So it would be impossible to overestimate the influence of Mr. Georg Cantor here and his sets. So during this time of mathematical upheaval that had never happened before, you had, roughly speaking, three schools of thought about the foundations of mathematics. Where is it that mathematical truth comes from? Is there such a thing as mathematical truth? What do we base our mathematics on? What is mathematics? What is it telling you about? When I say 2 plus 2 equals 4, what is it that I'm actually saying? Am I saying take the number 2, combine it with the number 2 when you get the number 4? Am I saying 2 of something and 2 of something together will give you 4 of something? What is the meaning in mathematics? So with three general schools you could talk about. One school was the logisist school. People like Bertrand Russell, Gottlieb Frege, and Alfred North Whitehead are the best known logisists. You had the school of constructivism, which you could also later call intuitionism. Constructivism, I'll talk about in more detail, but some of the prominent people were people like Chroniker, L.E.J. Brower, Henry Poincaré, these were mathematicians, took a radically different approach than the logisists. And then you had the formalists. The formalists headed by David Hilbert, one of the most influential modern mathematicians or maybe the most influential modern mathematician. And each of these three schools took radically different approaches and got radically different structures of knowledge based on their kind of philosophic understanding of the foundations of mathematics. And let me explain. So let's start. Well, I'll give you some of the pros and cons, because if you find this information out there, usually you won't find this information in mathematics textbooks. You have to look in either the history of mathematics or in the philosophy of mathematics. And in the history of mathematics, they don't give you opinion or analysis. They just give you historical facts. They don't say this was right and this was wrong. They don't say that in the philosophy of mathematics. But let me give you the overview of the positions and my analysis of them, which will demonstrate, in fact, there's room for new ways of thinking about the fundamentals of mathematics. So let's start with the logisist position, because it sounds like it's a good term. In fact, that's my favorite term, logicism. If you're familiar with my book, Square One, The Foundations of Knowledge, you know I'm rather partial to logic. The logisist said that mathematics is fundamentally an extension of logic. Why mathematics is true is because logic is true. You can build mathematical structures. Fundamentally, they reduce to the laws of logic. On this point, I agree. I think this is actually true. The trouble is, the way that the logisist went about doing this was building their theory, their logical theory, on set theory. Most famously, this happened with Gottlieb Frege, who was kind of the founder of logicism. He came up, he was in the process of publishing his big tome on logicism, grounding mathematics and logic. And he got a letter after it had been accepted and was going to be published. He got a letter from a young Bertrand Russell who pointed out that, in fact, there was a paradox in Frege's theory and it's now called Russell's paradox, which undermined the entire theory. And Frege acknowledged this and the story goes sank into a deep depression. He couldn't resolve Russell's paradox. He, I think, he wrote an addendum as the book was being released, saying, oh yeah, there's this paradox. I haven't resolved it. Well, that's the myth, at least. I don't know if that's actually, that's the story. That's literally what happened with his mental health. So Russell took up the project of logicism and he and a man named Alfred North Whitehead developed a consistent, coherent structure of mathematical knowledge. The tome that he came out with was called Prankipia Mathematica. And famously it takes like something like 100 pages to prove that two plus two equals four in their structure. So needless to say, that is not a very smooth mathematical structure. An interesting thing about logicism is that the logists were explicit mathematical Platonists, meaning they thought mathematics was not about the physical world. It wasn't about, it wasn't mental goings on. It was platonic. Numbers are non-physical things, they're out there in the platonic realm and that's absolutely essential to their thinking about the world. I think this is mistaken. And this is where the next school comes in and actually gets it right. The next school is the constructivists or the intuitionists. The constructivists said, no, no, no, mathematics is not external. It doesn't tell you about the world necessarily. Mathematics is all in the mind. Mathematics, when you have a mathematical truth it's just saying that this accords with the particular structure in my mind. It is constructed. Mathematics isn't out there, outside of our minds, it's within our minds. I agree with the metaphysics of this and mathematical objects are indeed within our minds. I'm not a Platonist and I think that's absolutely essential for developing a new foundation of mathematics. But as I said, people like Kroniker were big constructivists. Brower took it even farther into intuitionism which is kind of an extreme form of constructivism which says that not only is mathematics essentially about the mind but so are the laws of logic. The law of, let's say, the excluded middle which the details of which are irrelevant. A classical Aristotelian law of logic. The law of the excluded middle. No, Brower says. It's not something that's logically certain. It's something that's simply a human convention in the center of mind. And I believe he'd go so far as the other fundamental laws of logic like the law of identity and non-contradiction that this is a kind of convention. It doesn't tell you anything about the external world which is a shame because the constructivists get the metaphysics of mathematics right but they get logic completely wrong. Logic is indeed external even though the objects of mathematics are within our mind. Now one of the benefits of constructivism is that the proponents of it were explicitly dismissive. I shouldn't say dismissive. They rejected the concept of infinite sets of infinite totalities and there's an obvious reason for this. In the case that mathematics is within the mind we cannot conceive of an infinite totality within our mind. Therefore it doesn't exist. And so they rejected and had very damning things to say about cantorian set theory and this whole idea of the infinity of infinities. I'll give you some quotes later. So to give you the two I get to contrast constructivism with logicism. The logists got logic right they got infinity wrong and they got metaphysics wrong. They were Platonists and Cantorians. The constructivists got metaphysics right mathematical objects in our minds but they got logic wrong but they got the rejection of infinities right. And the very strong condemnation of Cantor and his nonsense. And people don't realize this is not an education that people get if they're a Ph.D. in mathematics. People like Henry Poincaré for example was a French mathematician very well respected so was Kroniker and so was Brower in fact. They were mathematicians and they didn't politely disagree with set theory, with cantorian set theory. They thought it was an abomination an intellectual abomination but thinkers like Ludwig Wittgenstein who is a super famous philosopher Wittgenstein considered his greatest contribution to be in the philosophy of mathematics and he mocked set theory. So it's actually not a radical position or wouldn't have been a radical position to have very strong feelings about cantorian set theory it's only over time now that we're so many years along that skepticism about set theory the cantorian set theory and this idea of the infinite totality is seen as totally crazy there's no way you can be rational and doubt that. Okay so let me give you the third and final school an approach to the foundations of mathematics the one that ultimately you could say ended up winning. The formalists most notable of which was David Hilbert the formalists say hey look don't worry about this truth thing don't talk about don't talk about it in terms of that mathematics is simply symbolic manipulation of symbols symbolic manipulation of symbols as opposed to what according to rules and axioms that we specify so remember in Euclidean geometry the axiomatic deductive method took axioms and used logical deduction to build structures of knowledge axioms back then in Euclidean geometry meant something like a self-evident truth something that was fundamental and true the formalists take the axiomatic approach but they don't worry about the truth of the axioms they just say you can explicitly state your axioms and your rules and rules from manipulation of symbols and then build your structures of knowledge and that's it you don't need any of the philosophizing on top of that now this believe it or not wound up winning and is massively influential the current as I said before the current foundation of mathematics probably speaking axiomatic foundation of mathematics is called Zermalo-Frankl set theory or sometimes ZFC set theory it's a set of explicit axioms that you are supposed to or that you can use to justify your mathematics the axioms were specifically designed to get around Russell's paradox so this is where we find ourselves but it just so happens that the axioms themselves because they make no attempt at being true can coherently and reasonably be doubted namely one of the explicit axioms in ZFC set theory is the so-called axiom of infinity which says at least one infinite set exists namely the set of the natural numbers the reason that that is taken as an axiom is it can't be proven you can't prove the existence of the infinite set so you have to assert it as a given it's not something you're seeing whether or not it's true or false you're just taking it as an axiom of your system this is not exaggeration folks if you've ever heard this before that might sound preposterous or at least open to doubt like maybe that's not the best way of building structures of knowledge especially when you can't prove something and so you take it as an axiom of the foundation of your theory so that gives you an overview and my commentary on formalism is that I don't see much merit in it I'm interested in truth I don't really care about empty formalisms ironically one of the spectacular benefits of formalism is that if you are a mathematician it enables you to do quite a lot of work you now have license to essentially create any old arbitrary structure you like regardless if it's true or not and make a little paper about it and publish it and look real smart you're not letting your symbols according to the rules that you specify so if you're an academic not interested in truth but interested in publishing words on paper formalism is just brilliant it's a license to write whatever the heck you want to write so one more thing on that is there's this idea of rigor in mathematics when you talk about mathematical rigor virtually everybody thinks oh well it's unquestionable unfortunately that's not the case especially not anymore mathematicians, bottom mathematicians are concerned with validity they're concerned with whether or not the conclusions that you state follow from the premises that you've stated they're not concerned with whether the premises that you've stated are true so when somebody says this is mathematically proven or this is mathematically rigorous that doesn't necessarily have any persuasive power because that might just mean according to a certain set of flawed premises okay so with the success of formalism, Bertrand Russell who kind of eventually gave up his project of logicism came up with a brilliant quip to describe the state of formalist mathematics he says quote mathematics may be defined as the subject in which we never know what we're talking about nor wither what we're saying is true it's brilliant and it's tongue in cheek and what he's saying is well we don't know what we're talking about meaning your mathematics isn't about truth it's not about the corresponding of your symbols to things in the world it's just symbols just symbols with no explicitly no meaning behind them they have explicitly taken meaning out of the equation because that suddenly brings you into the realm of philosophy so believe it or not there is in fact room for new foundations of mathematics here if you disagree with the logists the formalists the constructivists well there's a lot more to say about it but I'm still in the process of research to really get all of my ducks in a row to give you my educated thoughts on this topic so believe it or not constructivists well if you disagree for philosophic reasons you can coherently build different structures of mathematical knowledge something that most people consider impossible because they're ignorant on the subject I don't mean ignorant in a pejorative way I mean that in a literal sense people simply aren't taught this information and so they don't understand the context for repeating the beliefs that they think are certainly true what they think mathematics is but they don't have the knowledge to actually put that in context and know what they're talking about so I do, before I end this I'm going to give you some interesting quotes not from me about my crazy and heretical ideas in mathematics but from mathematicians especially around the time of this crisis in mathematics if it was perfectly reasonable then accomplished intellectuals to be skeptical of the claims of Kantorian set theory then it is perfectly reasonable for us to do so now especially when their criticisms were not satisfactorily responded to so this is what Leopold Kroniker said about Kantor's set theory quote, I don't know what predominates in Kantor's theory philosophy or theology but I am sure there was no mathematics there part of the reason for Kroniker saying this is because Kantor who I said before I think had mental issues was devoutly religious and he claimed in his proofs there is the first size of infinity another size of infinity an infinite number of sizes of infinity and then at the end you had the absolute infinity the infinity which no greater infinity could be conceived absolute infinity he called God he said that's what God is and he also claimed that God the absolute infinity spoke to him directly and told him the truths about his Kantorian set theory maybe it's true I'm not saying that can't happen but it should be a red flag this is a quote from Carl Gauss who was another massively influential mathematician he says infinity is nothing more than a figure of speech which helps us talk about limits the notion of a completed infinity does not belong in mathematics oh my gosh the heresy can you imagine Gauss said this David Hilbert by contrast the founder of formalism says that no one shall drive us from the paradise that Kantor has created from us to which the philosopher Wittgenstein responded well if one person can see it as a paradise of mathematicians why should another not see it as a joke Herman Weill wrote classical logic was abstracted from the mathematics of finite sets and their subsets forgetful of this limited origin one afterwards mistook that logic for something above and prior to all mathematics and finally applied it without justification to the mathematics of infinite sets this is the fall and original sin of Kantor's set theory so those are quotes from prestigious well-respected mathematicians who were deeply skeptical of Kantorian set theory I want to give a funny story as well on the history of mathematics that had to do with Isaac Newton Hume and the philosopher George Barclay and this actually took place a little bit before the fundamental crisis in mathematics but it was also about infinities in particular infinitesimals so Newton is known for being one of the co-discoverers of calculus but in his theory he had this concept of fluctuations and what was a fluctuation well you could think of it as an infinitesimal it was a dubious logical concept that he came up with to try to make the math work to give you practical consequences in in his theory of calculus the philosopher George Barclay had scathing and hysterical things to say and in fact wrote a book I think called The Analyst if I'm not mistaken where he essentially mocks mathematicians for their poor logical reasoning and he's a Bishop Barclay as a priest and he says well it takes far less faith to believe in God than it does the mathematicians Hume, David Hume also was very skeptical of the concept of infinite divisibility for good reason so I just want to give you a quote from an article from Dr. James Franklin who's a mathematician talking about Hume's conception of infinitesimals and a little bit of the history here which you might find interesting he says quote, Hume can claim something of the same success he had in the matter of infinitesimals Barclay said that the mathematicians use of infinitesimals was contradictory and that they regarded infinitesimals as non-zero when it suited them and zero when it suited them which is true the mathematicians ignored Barclay as one ignorant of the subtleties of their art until they solved the problem by eliminating infinitesimals in favor of multiple quantification at which point they said Barclay was right which demonstrates the excellence of our new answer Hume then repeats himself a number of times as usual compares mathematicians to papists and so on but adds no new argument of substance interestingly he does consider the strictly mathematical question of whether it is possible to develop the science of geometry without assuming infinite divisibility how about that David Hume's No Shlup I will say this issue of infinity really gets at the heart of mathematics it gets at the heart of what's called the continuum or what's considered the continuum which is if you think about a number line as we've been taught you've got zero here you've got one, two, three, four, five, six, seven in between zero and one you have point five in between zero and point five you have point two five unfortunately this is infinitely divisible so between any two numbers there is a third number this is you can think of the continuum it's where modern mathematicians think numbers come from, it comes from the continuum there's not a finite set of numbers out there to work with but unfortunately there's not been a lot of very rigorous ways of thinking about the continuum in a non-plotonic way you can have the generation of as many numbers as you need without appealing to actualized infinities or platonic mathematics this is what my work is doing it's what I'll be doing over the long run have a long ways to go but in fact there's a perfectly reasonable intuitive, logically rigorous method for preserving finitism and yet giving you as many numbers as you need it would also be hard to overestimate just how fundamental this concept of infinities within infinities and infinite cesses in modern mathematics in topology for example it's essential in real analysis it's essential or the story goes modern mathematics takes as biblical truth the idea that you can have an infinite totality nested within an infinite totality no problem so when people hear that there's reasonable skepticism about these ideas, their jaws drop their eyes wide and they think you're crazy because they don't know what the heck they're talking about and they don't understand the history of the subject matter geometry is another great one I had a little series on my youtube channel about investigating the fundamentals of geometry most people throughout history have taken for granted the supposed infinite divisibility of space, space is a continuum I reject that idea I think that space fundamentally is discreet you still can preserve a bunch of beautiful mathematics you don't need infinities you just need a different way of thinking about it so I hope that you guys have found that history lesson interesting the process of finding this information out has been a blast for me it has been like going on a murder mystery many years ago I found the story about infinity uncompelling and there's all kinds of trouble trying to resolve zeno's paradoxes with calculus which you can't do which I've explained before it comes up all the time an infinite causal regress is there a prime mover, is there not questions about infinity are absolutely fundamental and poorly, poorly resolved and so I've been on this journey for many years figuring out where did these terrible ideas come from where did these terrible ideas come from where did these terrible ideas come from and sure enough many of them if not most of them are rooted in mistakes in reasoning about the foundations of mathematics from about 1870 to about 1950 with gigantic implications and it's weird to come to that conclusion because prior to the investigation I thought just like you probably did that there's no room for skepticism in mathematics it's all logically wrigglers and certain really smart people who have figured this shit out but in fact they haven't they've done a poor job and there's lots of room for new foundations here somebody working on alternative structures of mathematics is Norman Weilberger who I've interviewed on Patterson to Pursuit I highly recommend his work even though I disagree with his metaphysics of mathematics he's still rejecting this nonsensical notion of the completed infinity and exploring the implications of that so I'd highly recommend it believe it or not there really truly is room for rational skepticism about modern mathematics