Jason Bardi - The Great Math War: How Three Brilliant Minds Fought for the Foundations of Mathematics

33:05and because there were German laws basically forbidding, you know, forbidding women from holding those positions. There was a pernicious belief that women had a corrupting influence and that young men would be led astray. And, you know, so it's because of her and it's because of Sofia Kowalewski. I mean, I really wanted to tell the stories of some of the women in this, who were involved in this. Because I think that we still have a ways to go today. I mean, people still sometimes hold on to this belief that girls can't do math, that men are better at math. So I was going to say, we go to Cantor next, because he's obviously a big character in this story.

33:53Why does he draw a distinction between potential infinity and actual infinity? Yeah, so Cantor was another interesting figure. He was a German mathematician who was kind of in his prime in the 1880s to early 1900s. He invented set theory, which is considered one of the great breakthroughs. It's a cornerstone of modern mathematics. And he kind of came up with these ideas out of thin air. Nobody had ever done anything. Nobody had conceived of mathematics in the way that he did. And the big thing he took on was infinity. And so I talk a lot about infinity in this book. And, you know, I could talk about infinity forever.

34:45It's an interesting subject. and one of the most fascinating things about infinity is the idea that there are multiple types of infinity. And so this idea had been around for a long time. So the ancient Greeks actually struggled with how to understand infinity and what to do about it. And Aristotle, the philosopher, came up with this idea originally of there being a potential infinity and an actual infinity where he talks about, you know, the idea of infinity and then, you know, an actual infinity in the world. So he considered the universe to be infinite. People did. We didn't realize it was finite back then.

35:34And so that was an actual infinity. A potential infinity was, you know, the sort of infinity that you can contemplate, it's the idea of infinity as opposed to infinity itself. The whole notion that we can conceive of infinity suggests that, you know, this sort of thing is real. So essentially he punted. What Aristotle did was he sort of avoided the problems with infinity by explaining it away in this philosophical way. And it kind of stayed that way for, you know, for the better part of 2 ,000 years. But Cantor came along and he realized, you know, we can actually talk about actual infinities and not just talk about them, but we can actually deal with them mathematically.

36:26And so his work came up with a lot of, I mean, there were all kinds of crazy things that came out of his work defining infinity. But the most profound was this concept of two types of infinities not being actual and potential. But he defined a countable infinity and an uncountable infinity. And those are represented by the whole numbers and the real numbers. And so the idea of a countable infinity is one that we're very familiar with. I mean, you know, one, two, three, four, five, six, go on and on and on. I have infinity pieces of gold, that sort of thing. This idea that you could count forever and you'll eventually – you have an infinite set in that counting.

37:14Well, there's another type of infinity which is uncountable. And so this is where, for instance, if you take the number line, you look at the endpoints, 0 and 1. So all the numbers between 0 and 1, how many are there? Well, there's an infinite number. But the point is it's not countable because you can take one and divide it in half. You've got 0.5 divided in half again, 0.25, 0.01, 0.01, 0.01. You can have an infinity of steps in the first tiny little fraction of a step in that number line. And so that's where the idea of uncountable infinity was. And, you know, it's interesting to note, if any of your listeners are fascinated by this kind of thing, Cantor really thought that there were religious implications to this idea.

38:08In fact, he was convinced that he was sort of a Moses figure among mathematicians, that God was revealing to him some of the truths of the universe, most notably this idea of the two types of infinity, the absolute infinity and the countable infinity. And so he actually, for many years, he spent time trying to foster relationships with cardinals and bishops and high officials in the Catholic Church to sell them on this idea of the two types of infinity, one infinity being the domain of man, one infinity being the domain of God. I was going to say, you mentioned the cardinal responded to him in one case, or is it the cardinal or bishop, and he says, yeah, I find that interesting, but never send me any mail ever again.

39:01Thanks, but no thanks. Yeah. I love reading stuff like that because you see how people were polite as they were brushing somebody off then as we are now. You can imagine somebody writing a letter like that today. Thank you very much for your inquiry. We will keep it on file. We hope you're enjoying the podcast. You know, we work hard putting together this show, but we work even harder for our investors at Smead Capital Management. At Smead, we believe in disciplined investing, which is why the Smead funds have a proven track record of long-term outperformance. If you're an investor who plays the long game and want to invest in wonderful companies to build wealth, we invite you to visit smeedcap.com.

39:47Past performance is not indicative of future results. Investing involves risks, including loss of principle. Please refer to the prospectus for important information about the investment company, including objectives, risks, charges, and expenses. Read and consider it carefully before investing. Smeed funds distributed by Smeed Funds Distributors, LLC, not affiliated. So what was Kroniker's view in comparison? So Kroniker was another mathematician in Germany who was a contemporary of George Cantor. And he was sort of an older, almost father figure to George Cantor in a way. They had a falling out over their work.

40:28Kroniker didn't buy Cantor's ideas at all. In fact, he was vehemently opposed to them. and they kind of fought a nasty dispute and there was a lot of gossip. And Cantor fared poorly in this. I mean, it was the stress of this and the stress of the work and he felt like somewhat of a failure in his career because he was stuck at a small university. He couldn't get a job at the big university. He was convinced that Kroniker was keeping him back, damaging his reputation, holding back his career. He also had apparently some psychological problems. It's hard to diagnose and assess exactly what was going on de post facto, as it were.

41:24But there are articles out there by psychologists who look at him and who talk about his mental health woes. He was in and out of clinics, out of nerve clinics for most of his adult life after the 1880s. He had several mental breakdowns. And partly this was due to the stress of work, thinking about infinity. You really lose your mind a little bit when you start really contemplating infinity. And then it was also partly because of personal tragedies. He lost his brother. His son died. So Cantor suffered these things and Kroniker was no big help in his mental woes. But Kroniker had a different view of the world.

42:16His view was that nothing mathematically was real unless it could be constructed. So the idea is if you can take one and you take two and you add those together, you take three. What is that? That's real. To Kroniker, that was what mathematics was. That encapsulated it. And so in order to do anything in mathematics, you needed to have this way of sort of constructing using actual numbers, using real reason, your argument. And that was very much flying in the face of the convention at the time in mathematics, which was people were looking more and more at doing these very unconventional proofs where you would do things like you would say this is true because if it weren't true, everything would be impossible.

43:17So therefore it's true. Sure. You know, these kind of indirect proofs and things like that. And Kroniker was not a fan. How did the Piano postulates come about? The Piano postulates. Yeah, so those were basically – so Pino was an Italian mathematician and he was working in logic. and he came up with some ideas for how to represent statements logically using mathematical symbols and came up with postulates. And he was sort of a visionary too. I mean he bought a printing press and he created his own journal. He became the media for himself and decided to promote his ideas. Because, you know, if Pino were alive today, there would be a Pino podcast, I have no doubt.

44:11But he – but he – you know, his ideas influenced others, you know, primarily Bertrand Russell. So when Bertrand Russell discovered Pino's work, you know, incidentally at the same conference in 1900 in Paris that Hilbert was at, he was blown away. And he went back and he started working on a really big problem, which was can we establish the fundamental solid foundations of mathematics? And that's really what my book is about. That's where it starts. And it starts with Hilbert and Bertrand Russell. You bring this up and there's a current event that I think in light of this. So you talk about the fallacy of pure process data.

45:04You use the Boer War as an example of this fallacy. In other words, the data was all correct, and yet the outcomes are pretty perverse. Right. Or the desired effects aren't received. Right. I mean, we just watched Meta go through a court trial where they considered to have an addictive product and it affected children. Some would argue those kind of platforms have incredible vast troves of data and the ability to manipulate that data, et cetera. Is it somewhat similar in that the outcomes from some of that stuff might not be what we want despite the data being so pure? Yeah, yeah. That's a great question.

45:48I had never really thought about it in terms of meta. I was thinking the contemporary example I have thought about related to this are the ice raids and homeland security. So let's go – let's step back to 1900. So Britain is fighting its biggest, most expensive war in a century. Not since the times of Napoleon has Britain put together this many soldiers on the field. And they're fighting a war in South Africa. So at that time, almost every country in Africa is ruled by a European colonial power. And Britain has several countries including the Cape Colony, which is where Cape Town is today. But to the north of the British territories are two independent nations ruled by the Afrikaner, the Boer.

46:49These are the Orange Free State and the Natal. So these are countries that are populated and ruled by basically white European descendant settlers but who had been there for many years, decades, centuries. They first came when the Dutch were exploring around the tip of Africa in the 1600s and had been colonizing and living there ever since. So these two nations were sitting on a mountain of wealth. The largest gold strike ever discovered was in the Natal. And, you know, within a few years of its discovery, that country went from being a landlocked, backwater, dirt farming nation to one of the richest countries in the world.

47:51And it was on the basis of this incredible wealth. Now, historians argue, you know, was the war about gold? Was it not about gold? Britain had fought a very expensive, you know, war in Asia just a few years before. There was economic malaise all around London. And so, you know, there was a need for wealth. And so, you know, to me, it seems like that was part of the issue, even though a lot of historians will tell you it was a much more complex thing. Okay, that being aside, what happened in this war was the Boer Army numbered around, I don't know, 50 or 60 ,000 soldiers. Britain was able to send in an initial expeditionary force of around 40 ,000 soldiers.

48:50And they were sending in more and more soldiers every single month on troop transports. Eventually, Great Britain had something like 400 ,000 boots on the ground in South Africa fighting this war, which took place – it started in late 1899. It ended around 1902. So in 1900, they had essentially subdued the Boer army. They captured its major general, its field marshal. They captured a third of all the soldiers. They were in military prisons. And what was left, however, was the war changed into a guerrilla war. And so Britain was forced to fight against these kind of, you know, hit and run guerrilla tactics.

49:45And so they changed their strategy and their strategy was, and this was all, let me take a step back. This was all, this was all based on numbers. It was a very expensive war and the British commanders had a mandate. You've got to end this thing soon. We have to end it. This is so expensive. You've got to end this war. And so, but they knew they couldn't do it as long as these commandos were out there. So they started, you know, exploring other tactics and they started looking at, you know, what can we do? Well, let's start burning farms. You know, scorched earth was an approach to warfare that was, you know, common in the 19th century.

50:25The Germans had done it. The French had done it. And so the British did it there in South Africa. Any time there was a raid by commandos, they would locate that raid and they would go to all the farms in the area and burn them down and kill the livestock and displace the farmers, largely women and children. And the reason that they were doing this was to deprive the commando soldiers from having a safe house, having a place to go and crash. And so they started doing this and they were – this was a numbers game. They were looking at the number of farms we burn every day, the number of people we displace.

51:11But very quickly it became apparent that there's another problem. You're displacing all these people. Where do they go? They're suddenly homeless. You're suddenly creating a refugee crisis. And so what the British did was they set up these so-called camps of refuge where people could go and they would put people in these camps. We know them now as concentration camps. They were the first concentration camps in the 20th century. No, they were not anything like the Nazi concentration camps in terms of the sheer horror. But they were nevertheless, you know, brutal and dehumanizing places. And in my book, I, you know, I talk a lot about it.

51:52And there's references at the end. You can go and read about this history. It's fascinating. But here's the issue. So there's something I call the fallacy of pure process data. And what that says is that when you – it's very similar to an idea that I love in economics called Goodhart's principle, which says that when a measure becomes the goal, it ceases to be a good measure. And so, for instance, if you say how popular is this, let's look at web hits, you know, to take a contemporary example. How many clicks do we get on this post or this story? And then what can we do to increase that? And if you start focusing only on increasing the hits, it ceases to be a useful metric for your outcome.

52:52And that's the sort of thing that is similar to the fallacy of pure process data. But the fallacy of pure process data is much more fundamentally psychological. So it's not just saying that an outcome is worthless or useful. What it's saying is we get fooled into thinking that the process outcomes are the things that we want to measure. And so, you know, in this case, in the Boer War, what they were measuring were the number of farms burned, the number of people displaced, the number of soldiers captured. These are process measures. Ending the war is the outcome. But there's no connection. There's no mechanism that says you're necessarily going to get there from here by pursuing these process measures.

53:44Nor is there even a sense that you're going to get there successfully by doing those things. And so – Well, we run that all the time in our industry. I mean, for example, like to your point, if someone says, you know, what are sell-side analysts on Wall Street really good at? They're really good at predicting quarterly earnings. To your point, like that is their metric, if you will. And some of these pod shops that are out there for these big multi-strat hedge funds, they love this idea of like I predicted the number. And they're good at it, by the way. Like they're the kid when, you know, they're sitting there as a child and they drop like, you know, gummy bears in the ground.

54:24like a whole box. And the kid like Rain Man style says like, 54. And the parent picks them up all up. They're like, Oh, my gosh, that was 54. You should be a sell side analyst, right? That's the kind of kid, right? It's kind of funny. But I say that because to your point, that they might be right in their goal. But the outcome that's received from that goal can be vastly different than what was hoped for. In other words, they succeeded in that. And yet the outcome for say, the investor might have nothing to do with that. Right. Exactly. Exactly. And so in the case of the Boer War, what happened was, you know, they won the war, but they lost the battle in some ways for public opinion because they created this human horror, a refugee crisis that resulted in concentration camps and unbelievable mortality.

55:20I read a statistic that infant mortality in one of the camps was 92%. I mean, that's unheard of in any record that I've ever heard of. And it And it really turned a lot of people in the British public, Bertrand Russell especially, against the war effort. Sure. Let me pivot a little bit on that because I want to get to this idea that you had next in your book because this is like – it's philosophical but it's mathematical too. If I am a Cretan and I say all Cretans are liars, how can I say that while not being a liar? Right. Yeah, that's the famous liar's paradox. And there was a philosopher named Epimenides, who I think is the most famous liar in history.

56:17And he's actually described in the Bible in the New Testament, you know, they describe him in this quote, basically saying all Cretans are liars. And it's a famous paradox because, as you point out, if he's telling the truth, then he's lying. And if he's lying, then he's telling the truth, right? But how do you work that out? And so Bertrand Russell was fascinated with this. And he came up with something very similar called Russell's paradox, which was really at the heart of mathematics. And it was a flaw fundamental flaw that showed that mathematics is not on solid ground because it has a paradox at its very foundation.

57:07Now, a paradox is not something where it's an ambiguity. I don't know, it could be this, it could be that. A paradox is not an uncertainty. A paradox is a certainty. A paradox says that two things that shouldn't both exist both do. And that's, you know, a lot of a lot of problems arise out of this mathematically if you start, you know, thinking about it. So, you know, to take an easy example, 1 plus 1 equals 2. 1 plus 1 does not equal 0. But what if I told you 1 plus 1 equals 2 and 0, right? That would be a paradox. Those two things can't possibly coexist. And what Russell discovered was that there was a paradox at the heart of mathematics, you know, which he explains in various ways.

58:01But it sort of butts up on this idea of the liar's paradox. And what he did to solve it was he came up with a new way of sort of thinking about what Epimenides said all those thousands of years ago based on, you know, classes. So he said, OK, it's not that Epimenides is talking about himself. He's talking about other Cretans. So when he says, you know, all Cretans are liars, what he's really saying is all other Cretans are liars. And therefore, it's not a contradiction. It's not a paradox. And that's how he worked out the system. And this is, I mean, I'm simplifying this. This was really years of Russell's hard work along with, he worked with Alfred North Whitehead, who was another famous mathematician.

58:51Correct. They did Principia together. Yeah, yeah, exactly. So Russell and Whitehead spent years working on a series of three books that they published that are considered the most groundbreaking work in logic to that date. and that was really at the heart of what allowed him to move forward. What they were trying to do in these books was establish finally, solidly, the fundamental foundations of mathematics. They were going to base mathematics on logic. They were going to derive all of mathematics on purely logical means. And the books, surprisingly, were a success. They sold well. They continue to sell well.

59:42I bought my copy. They're still on sale today. The books succeeded, but Russell failed. He was not able to solve the foundations of mathematics based on logic. He couldn't get there. Hey, I want to give a big shout out to everyone who's been working so hard on this show. You know, we recently hit the top 10 in investing podcasts on Apple Podcasts and even number one in the business category in several countries. As you may know, this show is brought to you by Smead Capital Management. Smead Capital Management understands how frustrating and illogical the stock market can be. If you're searching for funds with a proven track record, give the Smead funds a look or better yet, reach out at SmeadCap.com.

1:00:24And don't forget to mention you're a fan of the podcast. Past performance is not indicative of future results. Investing involves risks, including loss of principle. Please refer to the prospectus for important information about the investment company, including objectives, risks, charges, and expenses. Read and consider it carefully before investing. Smead funds distributed by Smead Funds Distributors, LLC, not affiliated. And then the next character, obviously, that presents itself with that point right there is Brewer. Brewer reminds me, when you're talking about where this guy's coming from and the Bericom and where he's coming from, this swampy place like Amsterdam, he's like the grisly Adams of Amsterdam, right?

1:01:09Just kind of coming out of nowhere. Not necessarily the person you'd expect to be this earth-shattering mathematics person, but yet at the same time, he was terribly drawn to this. He said that mathematics is independent of logic, right? Back to your point of like, you tried to solve mathematics with logic, and he said, no, it's independent of logic. And his early success didn't even come from mathematics. It was like topology, if I remember correctly. That's right. Yeah, yeah. He was famous. He did some groundbreaking work in topology, which is, you know, the field that looks at shapes and morphing shapes into other shapes and that sort of thing and surfaces.

1:01:55He published the, excuse me, the very first article ever published with color pictures was one of his articles in topology. He also developed some of the most groundbreaking cornerstones of that field, and he became famous because of it. He was something of a genius, and I read that he graduated high school at the age of 12, I believe. He was in the Guinness, and he may still be in the Guinness Book of World Records for the youngest person in the Netherlands ever to graduate high school. And he was an interesting person, too. He had a lot of different quirks. He had strange dietary practices. He had strange exercise practices.

1:02:49He liked sunbathing in the nude, for instance. He was a nudist, which, you know, in those days seems pretty uncommon, although, So, you know, in the area in Holland where he lived, there were a lot of, you know, sort of hippie colonies, if you want to call them that, at the time. And so he fit right in. Um, but, uh, but mathematically he had this idea of the foundations of math and an approach to math that was based on, um, sort of, uh, building things up in a countable way, kind of like what we were discussing with Kroniker earlier. Um, this, you know, fundamentally different approach to math that he took and that ultimately put him at odds with David Hilbert.

1:03:38So after World War I, Brewer and Hilbert come into conflict with each other because they have sort of diametrically opposed ideas for how to solve the foundational crisis. So what happened right before World War I, Bertrand Russell finished his books realizing he hadn't solved this problem. But it was more than that. He was falling in love. His marriage was falling apart. He started having an affair. He fell in love with a woman. He's like a Jerry Springer episode. I mean, the guy is chaotic. From 1912 to 1914, he's kind of a hot mess. And he's also sort of figuring things out. He's at the peak of his field.

1:04:33He's like he's considered a hero in philosophical circles. Harvard in 1914, early in 1914, they invite him to come give a series of lectures and they're wooing him. They're trying to recruit him. They want to hire him. He has all kinds of possibilities, potentials for jobs. He's really at the height of his career and he gives it all up essentially because when World War I breaks out, he becomes a peace activist and stakes out a very unpopular position. by doing so and eventually loses his job at Cambridge University and he loses some of his friends. And, you know, it's a really tumultuous time for him.

1:05:17He's eventually put in jail, in fact, for his anti-war activities. So he's kind of out of the game. When World War I ends, Bertrand Russell is still sort of, he still has a toe in the mathematical world, but not really. He's moving on to other things. But Brewer and Hilbert are, you know, very much ready to pick up this issue. Hilbert during World War I discovers Bertrand Russell's book and his work and he starts thinking about it. He starts working on it. And he quickly convinces himself that Russell is on the wrong track, that this isn't the way to solve the foundations of math at all, that we don't need to think about logic.

1:06:04What we need to think about is our rules and games. And so Hilbert's idea is to treat mathematics like one big game. And it's just a matter of figuring out what are the rules, define those rules, and then work out the proofs that you need to according to the rules. So again, this is somebody who thinks that we'll be able to solve anything. Any problem in the world can be solved. And so he starts working on that. He develops an idea called metamathematics, which is sort of treating mathematics as a game, an abstract game with just rules. He starts approaching mathematics through what we call now today formalism, which is defining these rules and finding ways to algorithmically establish the foundations of math.

1:07:03And he couldn't be happier. He's nearing retirement. He feels that he's getting close. He thinks that he'll be able to do it. there's only one problem, and that problem is Brewer. His former friend, who's younger, Brewer's younger than him, but Brewer looks up to him. Brewer idolizes Hilbert. He thinks Hilbert is a genius. They work together, by the way. They're both editors on the same mathematics journal. It's the most prestigious journal in mathematics in the whole world. Hilbert is the chief editor. Brewer is the section editor for topology. And in some ways, you know, they're colleagues, they're friends.

1:07:50When there's an opening at Hilbert's University for a new math professor, he turns first to Brewer. He tries to recruit Brewer into that position. Brewer says no. And then a couple years later, they fall into this nasty dispute over the right approach to mathematics because Brewer has a completely different idea of how to approach it. He develops a concept called intuitionism, which I won't go into now. It's a little bit complicated. I describe it in the book, but the basic idea is to develop mathematics kind of like what I was saying with Kroniker, where you're thinking about one plus one equals two, two plus two equals four.

1:08:33Find a way for any mathematical question to develop an actual series of choice sequences that are going to define that. And that's his approach. He actually, you know, he doesn't think that, he doesn't think much of Bertrand Russell's work either. He says, you know, mathematics is not based on logic. Logic is based on mathematics. So in the 1920s, those two, you know, fight a nasty battle between each other. And that I describe in the later part of the book. Well, you also, because this is really the war, if you will. And I, you know, it seems like a game of intellectual minds among friends that are battling.

1:09:25I thought of it as kind of like a term sometimes used is frenemies, right? They are friends. They are peers. But they have great disagreements over very unique things. I mean, these are not what you're typically fighting over your buddy, like what kind of beer are we going to drink? This is like you don't understand how the world works, my friend. And here's how my world works. And so I say that because one of my favorite parts of the book is you're talking about Brewer had rejected some advice, so he wrote to Blumenthal several times that someone had complained about Lebesgue, who's another person on the other side of this.

1:10:02And you have a poem in here. So here's my interpretation of Brewer's perspective in a spike of a few lines. Lebesgue, oh, Lebesgue. And you put in parentheses, rhymes with tsk, tsk. While he's in motion, his brain's at rest. You can barely even call this stuff proof more like a dumb goof, okay? Which I love that because you know that there's times that they're interacting with each other. They're smiling. They are trying to push back while trying to be respectful and walking away thinking, that guy just – he's never going to get it. And your poem really touches this idea of what's going on in the mind, maybe not what's going on in the relationship.

1:10:46Yeah. And I think that, you know, it's interesting. I'd love, I wish we had a psychologist here who could talk about human emotions and interactions because I think sometimes, you know, we hear what we want to hear. Other times we hear something we think we hear, but it's often never exactly what the other person is saying. And so somebody who is trying to hold out an olive branch, trying to meet you halfway, it may seem to you like they're doubling down. They're twice as obstinate as they were before the apology. And I think that this is – that's a case here where Brewer is somebody who's – he's given to kind of lavish, hand-wavy arguments.

1:11:38People love to go and listen to him talk because he's somewhat of a revolutionary in terms of mathematics. And so I think that sometimes he says some things that get him in trouble that are perceived as much, much harsher than they should be. Well, the other thing with Brewer, too, is I think you talk about his writing. His writing is just not fun to read. Like, it's tough. And so also like you could have a great idea if you can't communicate that well. It's just not going to be accepted by some because of the style of communication. That's right. People either won't read it or they won't want to read it.

1:12:19And that was the case with Brewer. I mean people describe his papers as being inconsistent at the very least, awful at the worst. You know, he would – some years he would publish several papers. Some years none at all. he would he was kind of all over the map he he would dive into really arcane obscure subjects and and wouldn't really spend the time to kind of explain it to a way in a way that generalizes it for a lot of people that makes the ideas accessible that was part of the problem another part of the problem was that his approach to mathematics was just fundamentally complicated There was something about what Hilbert was doing that is basically what the modern mathematician does, developing proofs to demonstrate ideas.

1:13:13And it was – you know, Hilbert was all about brevity. He was about beauty, in fact. Well, so you called it simple elegance. Yeah, yeah, yeah. I mean the idea that math should be elegant is one that he really embraced. whereas I think Brewer, he didn't really care about the elegance. He wanted, you know, sort of ideological purity in our approach. This is how we need to approach it and this is how we should do it. And so Brewer didn't care if that meant we throw away some of the greatest mathematical results of the last 30 or 40 years. He was fine with that. He said, if you can't prove it using my methods, then it's not proven.

1:14:00He rejected it. Sure. Obviously, Hilbert – you pointed this out a second ago, but Hilbert ends up – on that journal was also – Einstein was an editor to that journal. There's this whole – and not for our discussion, but they effectively take Brewer off the journal, causing this big blow up. Lawyers get involved. Can we fire them? Well, you've been paying him, but he hasn't had a contract. I mean, it was like labor law being like thrown out in Germany in the early 20th century, which I thought was terribly interesting. And then you make the case that really Hilbert kind of wins, at least in the setting of the moment.

1:14:37And then Kurt Goodell shows up and just kind of rolls that. Right. Right. That's right. Yeah. So Hilbert wins. But right on the heels of him, you know, essentially ousting Brewer and Brewer is suffering in silence and he sort of gives up mathematics for a number of years. And Hilbert is entering his retirement. He feels like he's won. And right when that happens, he publishes an article just a year or two before he retires that's basically laying out some of the things that need to be done still in mathematics, this idea of here are some problems that need to be solved. And he calls for we need proof of, you know, some of the fundamental ideas and logic.

1:15:29And then a graduate student actually does exactly what he wants. Within a year, this young graduate student in Austria, Kurt Gödel, comes out with the proof of, you know, some of the basic logical ideas. and it seems like a step in the right direction. It seems like we're getting ever closer to establishing the foundations of math and being able to essentially solve any problem that can be solved. And then right on the heels of that, the same young graduate student does something that is completely mind-blowing. So Kurt Gödel comes up with this idea of incompleteness And he actually proves that there are things which we cannot prove, questions we cannot answer, and problems we cannot solve.

1:16:25He proves that. And it basically shuts the book on Hilbert's dream of having a fundamental foundation of mathematics that's going to allow us to solve any problems. Gödel shows that there are actually indeed problems that we cannot solve. Sure. Sure. I was going through my notes here. There's so much that we didn't bring up. I mean, Klimt was in your book. I was just at the Museum of Modern Art and got to see Klimt's work there. So I was just like, gosh, I feel like this is just such a fun book to read right on the back side of that. We didn't really talk about Wittgenstein. He was interesting in this.

1:17:08Again, came from a very successful family. I was going through other parts of this. We didn't really talk about Hilbert's role in relativity and that he was right there with Einstein, but we associate the theory of relativity with Einstein, not Hilbert at all, even though they were effectively, Hilbert was in between the days, if you will, according to your work. So I would throw that out to our, let's see, other paradoxes. We did not talk about the Burale 40 paradox. We did not talk about Zermelo's paradox. I mean, there's a bunch of stuff in this book that I think is totally great and also just really good breadcrumbs that I want to leave out there.

1:17:43Where can people follow you going forward, Jason? You can find me on LinkedIn. I have a Twitter account. What's your handle? At Jason Barty. Okay, at Jason Barty. And I'm actually starting to toss around ideas for an X-Book, and so I'm starting to work on that as well. Okay. Well, this has been a total treat. Jason, your book, The Great Math War, proves that life isn't necessarily logical or purposeful, as we've learned. Some of these things these characters did in their life was incredibly logical, and some of the things they did were just whatever. Humans are drawn to interests and attributes that they intensely love but may not be a blessing to you in relationship or them or even their family.

1:18:36Madness and pride haunts the edges of us all. If you enjoy this podcast, go to Apple, Spotify, YouTube, or wherever you listen to A Book With Legs. Give us a review. Tell others about the books and great authors like Jason Barty that we have the opportunity to understand and study the world with and through. For our tribe, if you have a great book that you'd like to recommend, email podcast at smeedcap.com. That's podcast at smeedcap.com. You can also send your suggestions to us on X. Our handle is at smeedcap. Thank you for joining us for A Book With Legs podcast. We look forward to the next episode.

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1:19:38Thank you.