 Hello my friends and welcome to the 84th episode of Patterson in Pursuit. This is part two of my discussion with Dr. Graham Priest. Last week we talked about logic and contradiction, paradoxes and metaphysics. This week we're talking about the history of logic and its relationship with the history of mathematics, which probably sounds like it's the most boring thing that two people could talk about. But take my word for it, it's actually really interesting and really important. Most people, prior to investigation, think that mathematics is something like a logically certain discipline that has made linear progress ever since the Greeks. The Greeks discovered some foundational truths and we've just been building on the same kind of mathematical structure since then. It's actually not the case. Mathematics has undergone radical revision, especially around the turn of the 20th century. So this is the first time I've really stepped into the waters of talking about the history of mathematics and logic on this podcast. If you're interested in the truth, even if you don't see it yet, take my word for it. These ideas are very big and there's a lot of room for new thinking here. So I hope you enjoy my conversation, part two with Dr. Graham Priest. So I was listening to an interview before where you said you kind of started in mathematics and mathematical logic. You mentioned in the earlier part of this conversation that there was kind of a revolution around the turn of the 20th century in mathematical logic. I was hoping that we could put a little more meat on the bones for what exactly went on and then what exactly the contemporary orthodox resolutions are to the problems that arose in the beginning of the 20th century at the end of the 1800s. So let me give you a famous example that I'm guessing you think is incorrect and then maybe you can correct me and it'll bring us into the history of kind of logic. I believe it was Immanuel Kant who said that logic has essentially made no progress since Aristotle. Kant was, I think, in the 1700s and Aristotle was 400 BCE or something like that. So what do you think about that? Do you think that that's an accurate thing? That's just false. Just false. Kant is ignorant of medieval logic. Medieval logic. So one of the great periods of logic in the West and it's normally sophisticated. As sophisticated as anything you get to the 20th late 19th century in logic, Kant didn't know this stuff because it had been forgotten. I mean, medieval logic died with the death of scholasticism. So when most people think of logic, I think maybe they have an intuitive idea of something like Aristotle's laws of logic, you know, identity, non-contradiction, a lot of the excluded middle. Okay. Identity is not a logical law for Aristotle. It doesn't come into logic until lie minutes. Excluding middle and non-contradiction are metaphysical principles for him. They're not logical principles. He deals with those in the metaphysics. He deals with logic in the analytics, the prior and posterior analytics, which is syllogistic. Exactly. So you have syllogistic logic, which was developed with the Greeks. From my understanding, medieval logic was essentially like advanced syllogistic logic. Is that not the case? Did they go away from that? It's much more complicated than that. They take over syllogistic. They take over the other stream of ancient logic in the West, which is stoic logic. Stoic logic is kind of propositional logic, and what the medievalists do is try to meld these two things, but then they add very important theories on top of these things, such as they have a theory of meaning called supposition. In modern times it's sort of truth conditions. So they have the theories of truth conditions. They have theories of logical consequence. They have theories of the rules of debate. These are enormously sophisticated theories, and they just die with the death of scholasticism in the West. In the 20th century, we've actually started to find out how good these guys were, people like Ruben, Ockham, Paul of Venice, Bradford, and Scotus. So what happened? So we have Aristotelian logic, we have the development with the scholastics. That fell off, and then what was the cause of the big changes that happened around the turn of the 20th century? Okay. So there are two things. The first is this. Sometimes the 19th century in mathematics is thought of as the age of rigor, because people had all these kinds of different numbers, and they didn't really know how they worked properly, infinitesimal, complex numbers, and so on. And they wanted to understand how these things work, and so what you get is a number of the great mathematicians, like Dedekin, Nikantov, and Weishtaz, giving the first modern definitions of different kinds of numbers. Okay, but to establish what the properties they wanted, they needed a canon of logic, and syllogistic was not up to the job. So they needed a more modern canon of logic, and that's what Frege developed in his Becquishrith. So that's where the impetus came from. Now these guys were mathematicians, Frege was a mathematician, people like Bull and Schroeder were mathematicians. So when they were looking for this new canon of logic, it was natural to them to deploy the mathematical techniques they knew about. For example, axiomatization, which they knew about from the nonstander geometry, algebraicization, because algebra is taking off big time in the 19th century, late 18th century. So they applied the techniques for axiomatization and algebraicization to the canon of logic they're developing. And then 30 years later along comes someone like Tarski and says, well now let's apply mathematical techniques to something like the theory of meaning, semantics. So you get all these bits of new logic being developed by the application of mathematical techniques. So all these guys only thought that these techniques could be applied in one way, and they applied them to give us classical logic, which has nothing to do with classical civilizations such as Greece or India or Egypt. Classical logic was invented at the end of the 19th century, and these guys got it by applying the mathematical techniques they knew to logical problems, and they came up with a system called classical logic. Now what we now know is that these mathematical techniques have an enormously wide range of application, and you can apply them, you can apply the techniques of algebraicization, semantics, axiomatics to get many, many different kinds of logic. These are now called non-classical logics. And so we now know that the mathematical techniques are great, this is what modern logic thrives on, but they don't determine a unique solution to what the correct logic is, assuming that notion to make sense. So the development, the radical shift in logic or progress that was made in logic from an institutional standpoint was really driven by mathematics. You said that the logic that they were using wasn't up to the job, it wasn't strong enough. From a mathematical perspective, what does that mean? What were you unable to do with the old logic that this newer logic helped you do? So the obvious example is Frager's progress shift. So by the time that Frager is working, people like Weierstrass have given a plausible account of irrational numbers, they've shown how to get rid of infinitesimals, they've axiomatized the natural numbers, not 1, 2, 3, 4, 5, but what they don't have is an account of what the natural numbers are, not 1, 2, 3, 4, 5, and this was Frager's big project. He wanted to be able to define the natural numbers in terms of sets, elections. So given certain assumptions about the nature of set, you can give definitions, but then you need to show that from these definitions follow the axioms that were known to hold for natural numbers, produced by a guy called Dedekind. And there's no way using syllogists that you can do this. The principles of inference are just not powerful enough. That's why Frager developed the growth shift, the concept script. He developed it as a tool to allow him to deduce the fact about numbers from the definitions of numbers in terms of sets that he thought were correct. So correct me if my history is wrong here. That project, that Freguian project, was also undertaken by Bertrand Russell in the general school of thought called the logists. Right. Okay. So what happened there? So again, from my understanding, Frager developed this logic and had already taken his book to print and then Bertrand Russell shot him an email, essentially, said a letter, said, hey, I found there's a little paradox here. What about the set of all sets that don't contain themselves? Which is a contradiction in the theory. Again from my understanding, they still published the Bergers-Schrift or however you say it. But Frager had to write an addendum. And also from my understanding, Frager kind of gave up the project. His whole project that Russell tried to undertuck Russell and this other guy, Whitehead, tried to kind of preserve the structure that he was going with. But that eventually petered out as well, like the project of logicism kind of died out. Is that generally correct? That's essentially right. So the Bergers-Schrift is quite early. It's 1870s somewhere. So this is just developing the technical machinery of the logic. The application to mathematics comes into some later books. Okay. And the one where the technical project is carried out in detail is the Gungesetze. And I think it was just as the second volume of Gungesetze was going to press. And Frager received this letter from Russell, where Russell said, hey, essentially, the assumptions you're making about sets give you this paradox. Frager was devastated. He admits this and the letter is replied to Russell. And the Gungesetze was published with an appendix where Frager jisted it away. He thought he might be able to solve the problem. And we now know that that doesn't work. But so that's 1890s, I think. Okay. But Russell and Whitehead are working on this thing, well, 1900, 1910. They believe in Frager's logicism. But they also know they need to solve the problem of what's called Russell's paradox. And then Russell discovers that Russell's paradox is just one of these large family of paradoxes, self-referenced by the liar and zillions of others. Okay. So he has to come up with a solution to the paradox itself, reference, and he does this by a strategy of kind of constructing a hierarchy. And for various reasons, it doesn't really work. So logicism kind of, people are attracted to it until the 1930s, but in the end, another approach to number theory takes over. Now what was that approach? So I'm going to give you my rudimentary understanding, and then you fill the details here because this is really interesting stuff that I wish people realized how much of a radical reformulation of the fundamentals of mathematics and logic took place around this time period from like 1888 and 1945 or something like that. You just have a total upturning, from my understanding, of the fundamentals in this area. So you have logicism, which is one proposed school of thought. But you also have the intuitionists, right? You have the intuitionists have a totally different kind of metaphysical perspective on what numbers are. You have people like Brower, I think, was the guy who came up with his theory. He said, well, you see, the laws of identity and non-contradiction may be good, but this law of the excluded middle, that's nonsense. We're going to exclude that from our logic. And so he kind of built a counter structure or tried to build a counter structure of the foundations of mathematics and system of mathematics and logic from an entirely different set of principles. Is that right so far? Correct. Okay. And then you have another school, which is the formalists, where you have somebody named David Hilbert. This is where I'm not, my history is not quite as good. So where does Hilbert and formalism play into this? And then how does it spill over to where we are kind of in the modern world today? Okay. So, let's start with Brower. So he has a different metaphysics of mathematical objects. He doesn't think that, I mean, buildings and people and countries live out there in objective reality. There's a certain kind of view about mathematical objects, which says that they do too. This is something called plagiarism. And Brower thought this was crazy. Mathematical objects are simply mental constructions. To exist in mathematics is simply to be constructable. And if you subscribe to that view, then the law of it's good and middle just falls aside because there are some things that you can't construct and you can't do that, you can't construct them. So, you know, A.R. Nautilus disappears. And he attempted a reconstruction of mathematics or at least part of it because you can't get all of it. This never appeals to many people. It's never really taken up outside of the Netherlands. Intuitionism has made a comeback in the 1960s and 70s just because it's been shown to be closely connected with a lot in computation theory, topology, category theory, but that's much later part of the story. Hilbert is working with his school in Goettingen in the 1920s. And he knows that the foundations of mathematics is a mess because we know that they've revised the contradictions. So, he says, we want to make sure it's not going to happen again. So, what we're going to do is we're going to axiomatise mathematics, give a complete axiomatisation, and then prove that it's consistent so we can't be subject to these things again. So, that's his programme called Hilbert's programme. It was destroyed by Goettingen, but that was his programme. Okay, if you want to prove mathematics consistent, what tools are you going to use to do it? Well, if it's going to be a mathematical proof, you've got to use mathematical tools. But the mathematical tools are the very things you're worrying about. So he says, well, okay, so we're going to have to, in this consistency proof, apply a very, very restricted kind of mathematical reasoning which is entirely unproblematic and he called this finite tree. Okay. So he said, look at finite tree mathematics, that's kind of contentful. You can do it with matchsticks. And all the other junk, they're kind of super real on top of that with a lot of vice-over and for that things. This is just kind of playing games with symbols. So, you can think of that as just worrying about the formalism. So, he didn't think all mathematics was a game with formal systems. He thought there's this really contentful part of mathematics called finite tree mathematics. And then the other stuff was kind of just using symbols in the appropriate fashion. When you say finite tree mathematics, do you mean, does that like an explicit term? So, we're talking about the mathematical structures that are built on an assumption that there are no infinities. Is that what he's talking about or does that term mean something else? He never actually gave a very precise definition of finite tree. But it's the sort of things which you can do with finite tree combinatorics. So, think of symbol manipulation. Yeah. So, if you want to add three to two, you've got two matchsticks and you've got three matchsticks and you put them together and you've got five matchsticks. So, that's sort of the combinatorial reasoning of mathematical objects. Is that kind of thing? So, how much of this development then of higher mathematics is driven by the assumption of the existence of, I guess you could say, of infinite sets of the work of Cantor and the idea of the natural numbers being more than just the things you can count on your fingers that you're dealing with like a higher level of mathematics here that includes infinities? The theory of infinity is completely standard in modern mathematics. I mean, Cantor opened up this whole new realm of investigation for mathematics. And it turned out to have problems of the kind that Russell's paradox says. So, again, the paradox of self-reference come in. But, okay, and there's another bit of history. What happened to Cantor's theory after the failure of logicism and Fray the Well? What happens is you get mathematicians in about 1905, a mathematician called Zomilo who says, you know, never mind these down foundational programs. Let's just write down what we think holds about mathematics and we'll make it strong enough to do the kind of set theoretic construction if we want to, including the infinite, but not strong enough to get us into these problems of self-reference. So, Zomilo produced an axiom system for set theory called Zomilo set theory. And it was sort of strengthened a bit later by some other people. But essentially, his axiom system for set theory has now become kind of orthodox. So, most mathematicians will assume that when they do set theory, that they're doing it in something like Zomilo set theory or its extensions Zomilo-Frankl set theory. And I'm not going to tell you anything about the nature of mathematical objects. It's just a bunch of axioms for dealing with certain kind of mathematical objects, namely sets. So, in those mathematical structures, how many of the, I don't know, rephrased it. So, you said, Cantor opened up this whole new world of investigation for mathematicians. Of those worlds, how many of those do you think are immediately applicable? We could say, ah, this theory about the uncountability of the reels is applicable to how the engineers solve this problem here. Or is that just kind of the area of pure math? Like, is that maybe what Goethe was talking about, that you have the finitary mathematics, which is maybe what the engineers are using and this whole, this pure mathematics stuff, which the mathematicians are talking about. Is that not as applicable to the world? No, no, no. Look, the engineering mathematics is based on calculus. Calculus concerns the real numbers. So, you're certainly not dealing with finite free objects. You're dealing with infinite objects big time. Because even to construct these things, you need several degrees of infinity. So, standard mathematical theory that gets applied in engineering is already dealing with what are infinities for a set theory. The engineers themselves, they won't be dealing with infinities, right? They're going to be dealing with kind of finite mathematical calculations that there's a story told about the calculations about how they presuppose infinities, but they themselves won't be encountering any infinities, right? Well, they're dealing with real numbers and the solutions to differential equations. And how do you solve a differential equation? Well, you look to the pure mathematicians and the pure mathematicians tell you how you solve differential equations by appealing to certain properties of real numbers. How do you know what properties those real numbers have? Well, you look to your theory of real numbers. And theory of real numbers. Real numbers are already larger than small to infinity. So, of course, you know, engineers just want the damn building knocked over. Mathematics they're using deals with infinite sizes, which are larger than the smallest size of infinity. Isn't that, though, kind of the theoretical story that's put on top of what the actual engineers are doing? So, I mean, you could definitely do calculus without any completion of infinite calculations, right? Because no actual calculator, no actual engineer is dealing with real infinities. It's just kind of ensumed infinities. We talk about like the epsilon delta definition of a limit. We're not like dealing with actual infinitesimals or anything like that. You're still dealing with infinite totalities. And on the standard understanding of a real number, a real number is itself an infinite totality. I mean, engineers, of course, aren't interested in the foundations or what they're doing. They just want the private everything. But mathematicians are. And mathematicians will tell them how to do these things. Now, mathematicians will often use infinite tree methods to tell engineers how to solve the equations they want. So, would you say that kind of putting this all together, come a full circle here, going back to the self-reference and contradiction, that there is modern mathematics points to the idea that there is some inescapable contradictions in any logical system, like any formal mathematical system is definitely going to run into problems of self-reference or even contradiction, explicit contradiction. If it tries to be like a complete mathematical theory, like the only way you can avoid problems is by restricting what you can do with your mathematics. Well, most mathematicians don't buy it. I'm not interested in inconsistent theories because they think that once you've got an inconsistent theory, you can prove everything. If you use classical logic, that's true. So, they certainly hope that their theories are consistent. And as far as we know, things like arithmetic and some of the Frankl set theory are consistent. But there's this kind of, there's a saw that's there has never really been diffused because what Zamelo did was get rid of inconsistencies, except at what Kantor called the absolute. So, this is what you get when you increase the size of things and go as far as you can go, however far that is. And that's a hard question mathematically and philosophically. And a lot of these problems actually arise at the level of the absolute infinite. And Zamelo solves the problem by certain assumptions about the absolute. But once you start to look at them, they're a bit problematic. And so they're all kind of clungence, which mathematicians do at the level of absolute infinity. Most of them realize that there's sort of something a bit funny going on. Well, that's a wonderful note to end the conversation on. It's always interesting to me, like prior to investigating anything in mathematics, I had a default assumption that, you know, all of mathematics is sorted out. It's crystal clear, like logically precise. There's there cannot possibly be any contradictions. It's all I guess that would be something like a logistic by default that math is just an extension of logic. And we actually look into it, at least historically speaking, there was that might have been an acceptable view in 1820. But as you get into the end of the 1800s in the first part of the 20th century, you have such radical revisions taking place at the foundations of mathematics that the kind of common sense notions about surely mathematics is consistent and rigorous as precise as we like, those start to get a little bit fuzzier. Well, mathematics is much more rigorous than anything before the 19th century. But it's true that, you know, mathematical methods and techniques develop over time, and they're still developing. We're witnessing the development of computer generated proofs and things. The status of those is somewhat an issue. And we now know that there can be branches of mathematics based on intuitionistic logic or param consistent logic. So perhaps mathematics, which use different kinds of logic from classical logic. So mathematics has always been and still is in this process of evolution. So mathematics, you know, is not fixed once and for isn't it interesting that the idea that the mathematical you said the math now is more rigorous than it has been before. Then that means the confidence that prior mathematicians had in their proofs and their assumptions about the world were misplaced confidence. I think of something like Euclidean geometry that something that seems self evident and indubitable turned out to be there's a little bit more room for skepticism here. Yeah, we know a lot of the things that they assumed were sort of problematic or at least require very careful handling in a way that these guys can realize. All right, thanks so much, Dr. Priest. This has been a fantastic conversation. I appreciate your time. You're more than welcome. All right, that was my conversation with Dr. Graham Priest. I hope you guys enjoyed it. Lots, lots, lots more to be said. As Dr. Priest mentioned, there are three different schools, roughly speaking, trying to refound mathematics after the crumbling of the certainty of Euclidean geometry, the intuitionists, the formalists and the logisticists. And I think each one of them gets a little piece of the puzzle correctly. I like the metaphysics of the intuitionists. Mathematics is mental constructions. But I like the logical grounding of the logisticist. What the logisticist got wrong, I think, was metaphysics. So if they got their metaphysics right, they didn't try to build it on set theory. I think the logisticist actually would have had a better job of it of it. But anyway, those are conversations for another time, and I'll talk to you guys next week.