Theories of Everything with Curt Jaimungal - The Genius Who Invented Reverse Mathematics

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Starting point is 00:00:22 free of charge. BetMGM operates pursuant to an operating agreement with Eye Gaming Ontario. Pretty outrageous idea. There's all this real number stuff, all this partial and differential equations, even all this set theory stuff, large card moves. It's all fundamentally finite. This is my crazy head. This is Professor Harvey Friedman's first podcast. At 18, he was given not only a PhD, which is outstanding, but the title of Professor at Stanford University for his work in mathematical logic.

Starting point is 00:00:57 The Guinness Book of World Records even listed him as the youngest professor ever. Kurt Gödel, while alive, personally sponsored his last paper for the proceedings of the National Academy of Sciences. And Professor Friedman founded the field of reverse mathematics. Gertl's incompleteness theorems are the most celebrated results in modern logic. The textbook examples are recondite, self-referential curiosities, that no working mathematician tends to meet in practice. However, Friedman says they're pointing at the wrong target. The question is, can ordinary, finite math be trusted? His theorems suggest otherwise.

Starting point is 00:01:36 So now it's harder for the mathematical community to ignore foundations. On this channel, I, Kurtzai Mungle, interview researchers regarding their theories of reality with rigor and technical depth and probe at the foundations. Today, we discuss tree three, reverse mathematics, and the divine consistency proof, where... An angel is a weak form of God. This is such a wide-ranging podcast, and I'm so excited for you to watch it. I hope you continue all the way until the end, especially as the professor and I bond. It was and is such an honor, and we close with why the professor thinks the foundations of math are totally up in the air.

Starting point is 00:02:17 Professor, what misinterpretation of girdles and completeness theorems bother you the most? Well, one is that there really are two separate theorems. and they really are quite different, and most people aren't fully aware of the difference. So one is that in any sufficiently strong system, there are always statements that can't be proved or refuted. That's so-called girdle's first incompleteness there. And the second one is that in any sufficiently strong system,

Starting point is 00:02:53 the system cannot prove that it's without controversy. It cannot prove that it's okay. And these are quite different things. I've heard some people interpret girdle's theorem, first theorem at least, in the sense that we can't know things for sure. That's not quite what it says. It says there are always statements in any sufficiently strong system that can't be resolved within that system. Okay, so what's then the difference between we can't know things for sure and then whatever Gerdel's first incompleteness theorem actually says?

Starting point is 00:03:31 Well, Gertil's first incompleteness theorem, the most famous one, merely says that there are some things, given any particular logical framework, there's going to be some things that that system doesn't handle, that the system doesn't know whether it's true or false, meaning that the system does not prove or refute the statement. However, many statements will be. be provable and many statements will be refutable. Okay, now this is a great opportunity for you to walk us through concrete incompleteness. So what's the punchline beyond what GERL already showed? All right, in a nutshell, GERDL showed that there were statements in the first ink of term showed that there were statements that can't be proved or refuted in, for example, the gold

Starting point is 00:04:21 standard for foundations of math called ZFC. However, those statements are very, very, you know, far removed from what map petitions actually like to think about. And that can be made more precise, but difficult to make completely precise, because what mathematicians like to think about is a little bit up in the air, of course, and changes over time. But basically, the original statements of girdle are very far removed from the kind of mathematics that mathematicians very generally worry about and care about. So it's very different. What happened after the incompleteness, or the first incompleteness, what happened is that mathematicians wanted to know whether, how far would this reach into the kind of mathematics that they care about. And that's been a, there's been a long

Starting point is 00:05:18 development of that, to give you a framework, the ZFC system was pretty much settled by 1930 as to what's in there. And by 1940, Girdle had already shown that a famous problem in set theory, a very famous fundamental problem in set theory, couldn't be refuted in ZFC. And then in the early 1960s, Paul Cohen showed that it couldn't be proved. So the two of them together showed that the continuous hypothesis, famous continuum of policies, is not provable or refutable with the ZFC axioms. However, the continuum of policies is generally regarded as extremely important in general set theory, but is regarded as extremely several steps removed. in abstraction from what math petitions normally do.

Starting point is 00:06:18 And nowadays, we can run searches and change normally into quantity. In other words, so many papers. I mean, nobody's, I don't know that anybody has actually carried this out, but you can actually start putting some numbers on these things. So I think, will that help your understanding of what concrete and completeness is? The idea is that the continual policies involves arbitrary sets of, real numbers. Real numbers are fine in all through mathematics, and the commentatorial method is probably want to go more concrete. But real numbers are fine, but if you talk about an

Starting point is 00:06:59 arbitrary collection of real numbers with no patterns and no ways of generating it, just the notion of completely arbitrary set of real numbers, that's what the continuum of policies is about. And that's something that really is strikingly different than the normal kind of mathematics that mathematicians want to do. Most ordinary mathematicians would just say that the continuum hypothesis is so far removed. It's almost like in physics, there are some counter examples to what people ordinarily think of. So John Norton has this idea of a Norton's dome where you can have indeterminacy in Newtonian physics, but then a counter to Norton, an ordinary physicist,

Starting point is 00:07:41 let's just put that in quotations, would say, that's highly contrived that scenario, and also it's unphysical. And then Norton obviously has retorts like, yeah, sure, but tabletop physics with a rectangle or a square has already has the condition that my dome has. And then we can also say that black holes were thought of as unphysical before and a contrived solution.

Starting point is 00:08:04 Einstein thought so. So what are the similarities and what are the differences here? I think it has some similarities, but there's some differences. If you're talking about abstract set theory, which was a subject of great interest at one time, less so now, but there are still experts doing it, if you're going to do that, then the continual policy is nowhere, is not even remotely contrived. It's absolutely fundamental. We give it that. It's not contrived. However, the notion of arbitrary sets of real numbers where the continual policy starts to gain traction, for arbitrary sets of real numbers, the most general kind of set of real numbers is not commonly understood, well, I wouldn't say commonly appreciated as what mathematics is all about.

Starting point is 00:09:03 in mathematics one is generally concerned with the finite of course which is which is very concrete the finite and one is concerned with what one might be called sequential processes like real numbers real numbers are sequentially understood as an infinite series or infinite decimals so decimal expansions So the heart of mathematics somehow has a certain amount of geometric and combinatorial flavor to it that an arbitrary set of real numbers does not. And for more specific sets of real numbers that are more commonly dealt with in mathematics, like ones based on sequential limit processes, the continuum policies is well known to be proved. it's not independent. It's proved, for instance, for a so-called Burrell sets of real numbers,

Starting point is 00:10:05 the continuum hypothesis is a classic theorem that the continual hypothesis holds there. So let me ask you, Professor, why do you care about the continuum hypothesis so much? If many of your colleagues will say, look, yes, it's

Starting point is 00:10:21 not contrived in the sense that you mentioned, but they don't care about it. Why do you care about it? Well, first of all, I don't particularly, I do care about the consumer policies in a way, but that's not, I don't really care about it for concrete incompleteness, or so I call it tangible incompleteness now. In fact, it's totally irrelevant for tangible incompleteness. So I don't, I don't care about it for that purpose.

Starting point is 00:10:49 Historically, it's very important. I actually got interested somewhat. I wrote an unpublished piece on why I believe the contiguous policy. can be argued better to be false than true. And I'm not the only one who believes that. Many people said there and believe that. I did get a little bit interested in it, but I think that's not really the topic at hand here,

Starting point is 00:11:13 which is the tangible incompleteness. The Borell measurable sets of real numbers. My first big advance in contorts concrete incompleteness was finding statements in the Borell universe or the Borell sets of real numbers. where you need more than ZFC, or at least large hunks of GFC. So my first ventures into concrete and incompleteness

Starting point is 00:11:40 passed through the Borrell sets in the 1980s. When I'm wrestling with a guest's argument about, say, the hard problem of consciousness or quantum foundations, I refuse to let even a scintilla of confusion remain unexamined. Claude is my thinking partner here. Actually, they just released something major, which is Claude Opus 4.6, a state-of-the-art model.

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Starting point is 00:12:40 synthesis, all without producing slovenly reasoning. The responses are decorous, precise, well-structured, never sycophantic, unlike some other models, and it doesn't just hand me the answers. The way that I prompted it is that it helps me think through problems. Ready to tackle bigger problems? Get started with Claude today at clod.AI slash theories of everything. That's Claude.a.I. slash theories of everything and check out Claude Pro, which includes access to all of the features mentioned in today's episode. Where do you contrast with Hugh Wooden on the continuum hypothesis?

Starting point is 00:13:18 Well, Hugh Wooden is a card-carrying believer in abstract set theory, and it's interesting. importance. He knows that mathematicians are not necessarily as enthusiastic about this area as he is, but he has a point of view that the control of policy is false. He wants to create a theory of how the entire universe of sets works. If you think the arbitrary sets of real numbers is, is, uh, is abstract away from the concrete, which it is, the entire set theoretic universe is incomparably more so. It sets of sets of sets of sets of sets of real numbers, for example, and much, much, much more than that. So he's part of the community that is really very much into abstract set theory. He has taken the view that

Starting point is 00:14:24 the Ketuala policies should have a definite answer, even if ZFC doesn't touch it, which ZFC does not touch it. It ought to have a definite answer. And the way that he and colleagues, associates, work, think about this is, it's their job to uncover the truth, the truth. and although it is clear that the ZFC axioms have an intuitive compelling appeal are even visual and very much related to even finite intuitions in some sense it is unclear at this point whether there's any kind of compelling clear visuality to something higher than ZF that would settle the continuum of policy. I think he's more or less thinks of exploring it as a scientist saying what's more plausible than what and what can we find compelling in some way, even if it's not compelling like ZFCX. Now, I've been dubious about this, and we went separate directions. We've known each other for a very long time. So he took this high abstract road,

Starting point is 00:15:53 and I took the point of view that the future of foundations of math is how well it connects up with ordinary mathematical objects and ordinary mathematical intuitions, not extraordinary intuitions that only a few people have. So he and I are kind of the opposite directions. When you're doing math, professor, do you get the sense that you are also probing the truth? Yes, but it's more philosophical.

Starting point is 00:16:27 It's more foundational. I regard what the ultimate view of truth in mathematics is is a totally open question that's been made more mysterious by my efforts. It's been made more mysterious, maybe not clarified, but made more mysterious. Because what I'm actually doing is I'm actually,

Starting point is 00:16:56 although Wooden and company have long since attacked ZFC because it doesn't have some of their abstract pets, like measurable cardinals and other normally viewed as wild things. So they're comfortable with an attack on ZFC that it is, it doesn't handle, it doesn't allow for the kind of very abstract objects that they believe exist. I attack it differently. I say it isn't even good enough to do finitary things that we, that we care about. Now, when I say we care about, this is very much up in the air.

Starting point is 00:17:36 I spent almost my entire life since 1967 trying to uncover uncontrived mathematical contexts in which you can't, the ZFC actions are nowhere near enough. I started off with things that were grotesquely contrived and spent, what, 50 or 60 years, trying to make them less contrived and connecting them up naturally with things that everybody that almost every map petition can relate to immediately. That's been the program for 60 years. And that's a different kind of attack on ZFC.

Starting point is 00:18:27 Okay, let me attack this question in a different manner. So the way that people first learn about math when they're prior to university. Maybe they learn the Pythagorean theorem. Then they're told in university, you're not supposed to have picture proofs. Okay, cool. Maybe they learn the quadratic formula.

Starting point is 00:18:45 Maybe they learn how to multiply matrices or what an integral derivatives are. And then they get to university, and if they take a math course or if they're studying, if they're specializing in math, you start to learn about axioms, and then from those axioms

Starting point is 00:18:58 you derive everything that comes upward. So maybe they have the idea in their head that math is about these seeds on the ground that grow trees, and the theorems are somehow the leaves on these trees. Is that the view that you have of math? Basically, yes. In fact, it's been historically pretty much adhered to,

Starting point is 00:19:22 that there's a purely logical part of math that has nothing to do, pure logic, that has nothing to do particularly with mathematical objects. It could be anything. And that goes back to Aristotle with its syllogism. And the big thing that Aristotle didn't have was the idea of binary relations, relations between things. He had a monadic things like being a man or being mortal, which is unary, so to speak, whereas less than between numbers is binary.

Starting point is 00:19:59 There's a purely logical framework, and this, it took to about 1900 for this to settle down, completely as what's called the first order predicate calculus, or for short, predicate calculus. There's something called second order predicates, but it's a confusion. First order predicate calculus, and that's the purity logical part of the EFC. And then when you apply pure logic, first order logic, you have proper axioms, which you get to pick. And then you study what can be proved upwards from there. So there's the. upwards from the purely logical axioms and logical rules of inference, which are fixed. Then you have throw in proper axioms, and that's the framework that ZFC fits in.

Starting point is 00:20:48 That goes up. How do category theorists view math? Category theorists have a different kind of point of view. Some of it is practical. That's the non-problematic part, the practical part, that, uh, it's good to think of categories because it facilitates a lot of mathematical work in topology at other places. Also, the more extreme ones have a point of view that logic itself is a special case of category theory, that there's something more fundamental or more different than logic playing around with for all and exists and and or.

Starting point is 00:21:32 Well, that's just operations in category. Whereas the logician says, Oh, no, no, no. Category theory is a special case of everything that we do, which is pure logic, and it happens to be a thing where we have optics called categories, and we have operations on category, just like any, just like ZN.

Starting point is 00:21:55 There's nothing special about it. It's just another thing. So we both think we're more fundamental than each other. I will point out that Saunders-McLean, who I knew, he's a different generation than me, But he wrote a book called Categories for the Working Map petition. In the beginning, he defines a category as a set together with blah, blah, blah, blah. So he actually defines category theory in terms of set theory.

Starting point is 00:22:21 So in a way, he wasn't one of the extremists. Half of my days spent in conversations and research, the other half is spent trying to remember what was said and read. That's where Plod comes in, PLA-U-D. I have far too many thoughts. I always tell myself that I'm going to remember them and then I almost never do. It's incredibly frustrating. But I've been using Plod for many months now and it's a huge game changer. It's actually ridiculous.

Starting point is 00:22:50 I've noticed that this channel's output theories of everything. If you've also noticed that it's increased in quality and clarity over the past few months, if it's improved at all, Plot is responsible for a non-trivial chunk of that. I would say something like a 5% gain in thinking and that workflow compounds fast. If you're in knowledge work, you know how valuable, even incremental, small, 1%, 2% changes are. This trounce is that. What I like is that it removes friction. I'll use the note pin S when I'm walking, moving around, or just thinking out loud. It's hands free. I don't have to look at a screen and I could stay present with the world. I don't have to worry that I'm going to drop my phone.

Starting point is 00:23:25 Then I use note pro in more structured settings when I actually want to capture and organize conversations more deliberately. So instead of ideas slipping away, everything that's captured, and then turn back into something usable. It's basically a system for extending one's working memory into the real world, and if you know anything about intelligence, intelligence is highly correlated with working memory. So if your work depends on conversations, research, business, medicine, law, this kind of thing matters far more than you'd expect. They're also serious about privacy and security,

Starting point is 00:23:57 SLC2, HIPAA, and GDPR compliant. If you want to check it out, go to plod.a.ai slash C-U-R-T and use code Kurt. Okay, now how do these extreme category theorists, your words, not mine, I assume you mean top-post theorists. How do they justify that logic comes from them and not the other way around? Well, there's no question that they can do sensible mathematical work like this, which then they want to interpret it. it that way, whereas I don't interpret it that way.

Starting point is 00:24:38 In other words, so the question is, the biggest rival to ZFC being the gold standard for foundations of math is some of these things, the category theorist Bush, Topos theory, for example, as the real foundations. And I should point out, although they try to have a defense, that the big events in foundations,

Starting point is 00:25:04 of mathematics that most people hear about were not done by this point of view. It was done by our point of view. My point, I would say, I don't want to say my point of view. That's too extreme. But, you know, but the point of view that I like, I mean, the actual people, the actual results, after the fact they did do some reworking, but never first. And in particular, at a concrete and completeness, this is on our side. We don't have category theories doing this, or developing the techniques for it. So if you want to just put on blinders and look at what's happening in foundations of mathematics that can readily be described neutrally in some sense, because you can take my concrete incompleteness and say it's,

Starting point is 00:26:06 also not provable in the category systems either. It's equivalent, more of less. If you just put on blinders, you find out that the people and the events and the language of presentation for the big events has been all on, quote, my side. How is the foundations of mathematics viewed by other mathematicians? As pretty marginal. Sorry, marginalize and unimportant or marginalizing... Right, unimportant and irrelevant is the most common reaction. However, there are these periods of history where some blockbuster comes out, comes to surface, girdle especially.

Starting point is 00:26:51 Right. And then some of the leading mathematicians in the world, even, suddenly take notice and make some comments about it. people start to say, well, I don't know anything about this, but maybe it is important, and now I'll go back to my own work. So you do have occasional things like that, but the feeling is that it doesn't really impact their work enough for them to take it too seriously. That's similar in physics as well. However, there's no doubt that what I am trying to do, is to change that radically. How well I succeed is not quite clear.

Starting point is 00:27:41 I've definitely made some inroads. See, there's a thing called, there are these weaker forms of incomplete. It's where you're not incomplete of all of ZFC, but you're incomplete of big, of serious portions of it. That's important.

Starting point is 00:27:56 One famous serious portion of it is called Zamelo set theory. instead of Zamelo-Frankel. This is without Frankl's replacement axiom, so to speak. There's an axiom called replacement in the Z-Efti. Zamello's set theory is already an enormously strong system, far, far more than any mathematicians normally used. But I'll give you an example.

Starting point is 00:28:20 There was a fairly well-known problem in what's called infinite game theory, and it's called Brel determinacy. This is the problem. And Donald Martin proved Beryl and Terminacy using way more than ZFC. And that created some impression in certain parts of the mathematical community. No question. But nobody had any confidence that this had anything to do with foundations. It was just overkill.

Starting point is 00:28:57 It was just using something probably overkill. then I proved that there is no way to prove Rale determinacy without using infinitely many uncountable cardinals. In fact, uncountably many uncountable card. That Zermello said there he was not good enough to prove Rural determinacy and that you needed at least a big hunk, an unusual hunk of ZFC to do it.

Starting point is 00:29:24 And then Tony Martins looked at my argument and said, oh my gosh, this gives me a hint as to how I can prove Borrell determinancy in ZFC. I would have never, he said, I never would have thought of using these strong methods

Starting point is 00:29:42 that Freiband proved were needed. I never would have even thought along those lines, and now I see how to do this. So he published the famous paper proof of Burrell determinancy, using exactly the outer limit of what I said

Starting point is 00:29:58 was needed, just beyond what I said is it was not sufficient. So that's an interesting story. By the way, since then, I came up with a lot of statements that are not involving infinite game theory, another subject that has some interest but not huge, just involving Borrell measurable sets and this kind of descriptive set theory or analysis, which also requires uncountably many uncountable carders to prove. So there was an interaction. This is already 1980s and earlier.

Starting point is 00:30:34 So I'm laughing because I'm reminded of, I was watching a talk of yours, and you were saying that there was some reporter who gave three titles to something of yours. One was that the man who blew up infinity or something like that. And the second was the man who wants to rescue infinity. And you said that one was discarded.

Starting point is 00:30:52 It was just some internal title. And then the one that you said was more accurate, maybe not still entirely accurate, but more accurate. Harvey Friedman is about to bring incompleteness and infinity out of quarantine. Right. That's a nice one. Okay.

Starting point is 00:31:05 So I want to ask you about the foundations of math. That's the theme that unifies this entire podcast between us. So the foundations of math, suppose someone was to make a similarly sensational title. They could say, Harvey Friedman says the foundations of math are what? Like, are wrong, are broken, are not. known, are not investigated enough, or what? Are totally up in the air? Or are more mysterious than ever.

Starting point is 00:31:39 Okay, interesting. Okay, but that would be a correct, that would be a correct encapsulation? That's my view, yeah. Okay, so then obviously someone's like, so what? Like, wait. Let me give you the analogy, because the analogy is, look, in physics, the people study the foundations of physics, and I'm terribly interested in that as well. But then the counter is, okay, yes, go you little, you few people, go about and study your foundations. I'm here

Starting point is 00:32:07 building rockets. I'm here using quantum mechanics in the lab, phones work, et cetera, et cetera. So I'm sure that's the sort of mentality that you see that you encounter. So what, so what, Harvey? Yeah. All right. So I'm trying to deal with that directly. But it took me 60 years to get get to get to the place I'm at. So important, new, attractive mathematical subjects appear, depending upon what size you're talking about, appear regularly. People get excited about some,

Starting point is 00:32:50 there's some surprising result and stuff they're familiar with, and then they want to probe more deeply and get a complete analysis of it, whatever. There are two concepts among the many fundamental concepts in mathematics, two of them are embedding and maximality. Both of them appear in undergraduate math and advanced math, you know, in many things, in many situations. And what I do now is do what I call embedded maximality. I say that these two concepts can be combined into new striking theorems.

Starting point is 00:33:37 And I call the subject embedded maximality. This is the title of the book that I'm trying to finish. This is novel in the sense that two very basic concepts, which are generally not mixed, are suddenly mixed with some surprise, not just mixed. The context in which I do this is not arbitrary sets of real numbers. context is the rational numbers with the ordering less them. So embedded maximality is the title of this book, and the context is among the most concrete situations in the whole of mathematics, which is simply the ordering of rational numbers without even addition or multiplication.

Starting point is 00:34:25 And I show that there's a finite theory of this where I look at all the finite embeddings that can be used for the for the maximality principle. And I analyze that first. And then I say, well, let's look at some very simple infinite maps, not just finite maps, but infinite map. Maps within infinite domain. Okay. In the rationales, we're talking about, you know, on an interval, it might be the identity

Starting point is 00:34:59 on an interval, is already infinite. All right. So, and then all of a sudden it completely explodes into beyond ZFC. Right there. So the plan is to define the subject very gently called embedded maximality, nice and friendly, simple definitions, half page to know what the subject is. state the finite theory, 30 or 40 pages of somewhat complicated stuff, completely analyzing the finite case, attractive commentatorial mathematics of the kind that people know and love, nothing funny, and then say, well, I want to look at these finite maps on the rationales, I want to extend them by the identity map on, on,

Starting point is 00:35:54 the outer parts. In other words, if the finite function lives in here, let's extend it to be the identity above and below the function. That's a very simple kind of function, right? A very simple kind of infinite function. It's the identity almost everywhere, right? All of a sudden, you ask the same question. Is this, can you use these embeddings for the maximality principle? Can you use these embeddings? I solved it for the finite case completely. And the answer is it's independent of ZFC. That's the point of this book.

Starting point is 00:36:33 So now it's harder for the mathematical community to ignore foundations. See, there's now this question of whether these theorems, there's a question of whether these theorems are, in fact, of the outer extension usability theorem, OEU, the outer extension

Starting point is 00:36:56 usability theorem. Is this a legitimate theorem of mathematics or not? Is this been proved? Well, we have a proof using some monster cardinals that go way beyond ZFC and we know ZFC is nowhere and near strong enough to prove it. So is this

Starting point is 00:37:12 math legitimate? Has it been proven? It's like the old classic issue about is it okay to use functions that aren't given by expressions. You know, in classical mathematics, everything, you know, is a formula or something, you know. Yes.

Starting point is 00:37:28 The notion of arbitrary function came up late, relatively late, as you know, and people were concerned about it. Or even definition by cases originally, you know, because you have a discontinuity that's artificially put on, right? So, I never heard of the definition by cases. What's that?

Starting point is 00:37:48 Definition by cases. F of X is zero if X is negative. F of X is five if X is positive. Yeah, something like that. Yeah, you see. That doesn't come from the old idea of formula. All right. So anyways, if, in fact, the mathematical community falls in love with embedded

Starting point is 00:38:12 maximality, because I figure out how to say it and motivate it. And connect it, by the way, with Stanford. Mathematics, which I have a chapter on that, how to connect it up with semi-algebraic functions and piecewise linear function. You know, the stuff that beef, the bread and butter. I do connect it up with that, though when that connection is for background, I don't get independence from ZFC, but that it's so tightly motivated and clear in the finite part is so interesting, allegedly, then one is compelled to.

Starting point is 00:38:50 take seriously the outer extension embedding theorem, which has been only proved using much more than ZFC and cannot be proved in ZFC. So this subject embedded maximality is drenched with ZFC incompletely. How the mathematical community will view this is not clear. I have to roll it out. And where are you with regard to that book? We wrote it four times or something. I rewrote it because it got so much more convincing. But are you publishing slivers of it, or is it all going to come out at once? I have a, there's a website that maintains maybe 30 or 40, one-hour lectures on it. So I have a complete record of that, and there's some manuscripts and unpublished stuff.

Starting point is 00:39:43 Now, a lot of this is, some of this is wrong. So I, you know, I believe keeping the historical record is good. this. Some of these lectures, I overstated some stuff and I take it back, so forth. So it's not a I don't have any problem with that. It's in fits and starts, but it's pretty, it's well on its way. In fact, I have to go to talk on tomorrow on Zoom to a very small audience of experts. But it has to be done just right for something like this, because the skepticism, that foundations, is really important is so strong. This video is sponsored by Shortform.

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Starting point is 00:42:55 That's economist.com slash T-O-E for 35% off. Give a sense of how monstrously large the number tree three is. Okay, well, I think the best way to talk about tree three, there's some technical definitions that require some, expertise in a subject called recursion theory. But one way I look at it is this. Tree of three is so big that you can't even prove it exists, that you can't prove it exists even in a system like,

Starting point is 00:43:36 strong system like piano, orthantic. If that were true, that would indeed be mind-blowing, because any finite number can be shown to exist, depending on your definition of exists. I'm sorry, I'm sorry, I'm sorry. I've misstated this. You can't prove it exists without using two to the one trillion

Starting point is 00:43:55 pieces of paper. That doesn't mean that it's bigger than two to the trillion. It's incredibly bigger than that. I'm saying even with two trillion pieces of paper that makes sense, you're not even proving it exists. Right.

Starting point is 00:44:13 I don't know how that, that helps a lot. And also a considerably stronger system than piano rhythm take. The statement I made is correct. How does it compare to Graham's number? Well, if I remember

Starting point is 00:44:28 Grabs number, I think it's kind of not noticeable. Yes. It's minuscule in comparison. It's an epsilon. Now, you can make artificial, you can play this artificial game in many ways. But this is a particularly

Starting point is 00:44:41 elegant one that is mathematically very friendly, if you remember the definition. Yeah, what is the definition of that number? Yeah. Okay, you look at finite trees. Okay, first there's an infinite fact that's behind it. And then you think of this as a finite form, so speak. And the infinite approximation is you have an infinite sequence of, it goes back to J.B. Kraskel. You have an infinite sequence of finite trees with three colors, if you want. I do it with three colors, with three colors on the vertices. Each vertex has one of the three colors. You have an infinite sequence of finite trees. Then one of the trees is a part of a later tree and part of means there's a color preserving embedding that's infreserving. It's a slightly

Starting point is 00:45:45 technical notion, but it's the obvious notion of homeomorphic embedding. There's a whole comatoric that was well known of finite trees and how they fit together and what does it mean, but, you know, isomorphism and all that. Okay, so once again, given any infinite sequence of three colored trees, finite trees, sorry, finite, finite, one of the trees in the list, is embedded in one of the later ones in the list. This is a famous theorem of J.B. Creskel. And I noticed, or maybe I wasn't the first to notice, but I made it the first to do something with it,

Starting point is 00:46:23 I noticed that this proof involved looking at all possible infinite sequences of trees, uncommonly many, and this proof was so far from being local, you might expect, You just locally figure out it in this piece. It was so unusual this J.B. Crusco proof. Yes.

Starting point is 00:46:45 And I proved why it was so crazy, so strange. I proved that Cresco's theorem can't be proved in systems you'd expect it to be proved. Now, this is still nothing like ZFC. ZFC incompleteness is something entirely different, a different level. But this is a micro-incompleteness, so to speak. And so I kind of prove that you need uncountable sets to prove this cruscal thing. Okay? Now, how do you approximate the cruscal thing?

Starting point is 00:47:21 Because after all, it's about infinite sequences. Well, it turns out that from the things like the compactus theorem, well-known in mathematics, you can show that if you put a bound, if you say that the Ith tree has it most, eye vertices. The ice tree has at most eye vertices. So you put a bound on the growth rate of this trees. Then you can find one of the trees is embedded in a later one, but you can even find a stopping place. You can even put a bound on how far you have to go up in order to know that. You follow me? In other words, you have to go a long way. So how far do you have to go up to get this? That's called tree of three. Tree of three says how long do you have to go on like this so that one of them is

Starting point is 00:48:12 embeddable in a later one? As long as the I tree is, as long as the I tree has the most eye vertices and we have three colors. That's what tree of three is. How long do you have to go up there? Now from general mathematics, you can prove from cross-quist there, which is about the infinite sequences, you can prove that there is a bounding place. There is a place. so high that you can get there. So if you know Kraskel's theorem, you actually know tree of three. Okay. Now, what does this have to do with the relationship between infinity and the finite? Well, this says that there's really a very close connection

Starting point is 00:48:53 between the really big finite, the outrageous finite, and the small infinite. The smallest infinity is just omega, the size of the natural numbers. that's the smallest infinity. You know, Cantor has higher infinities, like the real number line and all that. Yes, yes. I have a video about that.

Starting point is 00:49:15 Yeah. So the smallest infinity is very closely associated and approximated by the outrageously finite. Arrighteously large finite. And we see this pattern over and over again. Now, I actually took this even more crazily. Put on my crazy edge. Okay.

Starting point is 00:49:38 I said, suppose I'm a finitist. I'd say, okay, only finite things, maybe big things, maybe not even, I don't even like big things. If you're an applied computer scientist, you don't care about gigantic numbers. It's too impractical, right? Too impractical. You only care about things like a quadrillion or something is about the most you can stand for. So what if I put my finitist hat on? And I say, okay, all of mathematics I'm going to make finite.

Starting point is 00:50:12 I know it's not finite directly, but I'm going to find finite approximations. And thesis, all of mathematical ideas can be, are already represented in the finite. Pretty outrageous idea. In other words, all this real number stuff, all this partial and infertional equations, even all this set theory stuff, large cardinals. it's all fundamentally finite. But this is my crazy hat. Yes.

Starting point is 00:50:45 A finitist hat. And in fact, the book, Embedded Maximality, has a section on purely finite forms, which shows, in a sense, that all these large cardinals beyond ZFC can be thought of in finite terms. Now let me ask you this. Yeah.

Starting point is 00:51:08 What about ultra-finite terms? Okay. You're talking about ultra-finiteism? Well, no, that's the point. That's the idea that you shouldn't go past very small numbers. That already two to the 100 is absurd. That's ultra-fiant. Yes, that's part of the crazy act.

Starting point is 00:51:31 Interesting. If you really want to push this really hard, all mathematical adventures can be properly imitated or realized in a computer screen, that level of detail. In other words, pixels with colors, everything there is is just pictures. That's already more than enough. Now, we kind of know that this is true in some sense, because the human mind is thinking about things. I'm looking at this screen and I'm looking at everything.

Starting point is 00:52:04 everything I think about, my work, my pace, this is all well within the computer screen's size of information, which is only X bits where X is, you know, you can make thousands of whatever, right? So this is actually the most radical. And now the question is all these fancy mathematical ideas we have, including large cardinals and ZFC, how do we actually say that we've actually properly represented them in a picture. I believe this can be done. I think what's in my book is nowhere near that, that powerful, but it does suggest some things. What are the large cardinals? Yeah, so this is a whole subject in and of itself.

Starting point is 00:52:57 The smallest large cardinal, okay, from the point of view of a working map petition, there's some cardinals that are tiny from the set theorist point of view that are already pretty damn large. Now, if you really want to talk about hard-nosed mappitians, they usually live in the countable, but they do are aware of the real line. So there's this cardinal called two to the olive knot. Yes. So called two to the out of knot.

Starting point is 00:53:24 So that's not really a large cardinal. These map petitions aren't going to call that a large cardinal. If you're a real finite commentatorist, you might think of it. It's pretty large. Of course, yes. Okay. Then there's two to the two to the alibi. And then there's two to the two to the alibi.

Starting point is 00:53:40 See, Cantor, I mean, this is maybe not exactly his original notation, but the idea is that given any cardinal, you can go to a higher cardinal. And so if you take the first omega cardinals, if you know what I mean. In the words, do it once, exponentiate it once, exquisite it again, and esplan again, and so forth. You get a tower. Sorry, Professor. Just a moment. To interrupt, just briefly, for the layman, can they substitute infinity for the cardinal, for the word cardinal here?

Starting point is 00:54:14 But if you take any infinite. Yes, we're talking about different levels of infinity. Great. Yeah. Cardinal came up, canter to find these. Cardinal is an abstraction on, is the way we talk about. about levels of infinity. So I can talk about levels of infinity.

Starting point is 00:54:32 The lowest level of infinity is the integers, represented by the integers. The next level of infinity that people know about, a lot about is the real number line. However, there's something in between called omega-1, and there's a hierarchy, omega-0, omega-1, omega-2, omega-3. And the continuum of policies ask whether the,

Starting point is 00:54:57 real numbers are the second largest, second smallest infinity. Is the set of real numbers the second largest infinity? And that's what's called the continuum of omnibuses. The divine consistency proof. Walk me through how God proves math is consistent. What does that mean? Okay. There has been an idea that goes back to ancient,

Starting point is 00:55:27 theology. And Gertl took it quite seriously, and then I took it seriously because he did, that attributes or properties, properties of things, being a human, whatever, properties are either good or bad, or either positive or negative, a classification of arbitrary attributes into positive and negative. This is very high-browed idea. Now, if you ask me practically, you know, what does this mean, you know, for real actual attributes that people think about who are not abstract philosophers and logicians, it gets a little bit strange. So we won't go quite there. We have this abstract idea that we're going to classify attributes as positive or negative.

Starting point is 00:56:20 And there was this idea that God is the unique entity that's in the positive ones and none of the negative ones. And in all of the positive ones are just only? No, absolutely all. God is perfect. Now, Gertl took this in a different direction than I took it. He wanted to take this idea and actually prove, he actually proves that it's possibly necessary. So he has some sort of edge that there is a perfect being. He wanted to come up with some abstract philosophical principles

Starting point is 00:56:59 that there's got to be something that has only all and only the positive properties. And therefore, he's proving in some sense that God exists or he's proving that actually he states a weaker thing, that necessarily it's possible or some what's called modological hedge. Yes, please talk about your divine consistency. this idea of classifying things into positive and negative it treat me a lot and there is something in mathematics that is very much like this it's called an ultra filter yes it's a mathematical thing maybe you've heard this an ultra filter on a on a set is you divide the subsets of the set into

Starting point is 00:57:49 the big and the little and everything is either big or little and the intersection of two bigs is a big and so forth, you can, I guess. Anything big has got to be infinite. Okay, or at least two elements. There's many ways to say this. Sure. And ultra filters are kind of useful things in a lot of different kind of mathematics. It was exploited by setters.

Starting point is 00:58:15 If you have an ultra filter that has a very strong property, namely, if you have an, if you have a sequence of biggies, then it's intersection is big. And that's not a finite sequence, that's an infinite sequence of biggies, then the intersection is big. And that turns out to be independent whether there is such a thing.

Starting point is 00:58:43 And that turns out to be equivalent, more or less, to some large cardinal called measurable cardinals. So there is something bigger than ZFC there going on if you have a very strong kind of ultra-filter. So I started with this knowledge, right? Now, it turns out that we don't want to consider the ultra-filter.

Starting point is 00:59:06 Okay, so there's an ultra-filter of all the big properties, of all the big ensembles. Well, the positive, sorry, positive and negative, remember. If you take all the positive ensembles or the positive classes, positive classes, that forms an ultra-filter. That's what GERLs started with. If God is in there, it's all sucked up by God, right? So it's just the ultra filter of all things that contain just this one special point called God.

Starting point is 00:59:35 That's not very interesting. That's called a non-principle or trivial ultra-filter. Mathematically, that doesn't get you anywhere. Mathematically, though, you want to consider a situation where you don't have a lump like that. You know, like God is a lump right there, right? You don't have a lump. Then I define what an angel is. an angel is a weak form of God.

Starting point is 00:59:58 An angel is something which isn't in all the positives. That would be God. But it's in all the definable, all the explicitly definable positives. In other words, it's a member of all the good properties, the positive sets that you can

Starting point is 01:00:23 that you can name, all the nameable ones. See, this is weaker. Yes. But that's a pretty good approximation of being a godlike, right? You're only in the good ones that are defined. All the other defined ones you're not in. And I call that an angel.

Starting point is 01:00:39 Yes. This is just a theory in which you postulate that there is an angel, and you add the standard trappings you would like for, for a system. Some mathematical infrastructure called the choice operator.

Starting point is 01:00:57 And you get a system in which you can prove that ZFC is consistent. And the key axiom that drives it is the axiom that angel there exists at least one angel. Now, this system

Starting point is 01:01:10 could be worthless because it could be inconsistent, right? It could be, I could just be blowing smoke, right? But I can prove that this system is consistent using a measure cardinal.

Starting point is 01:01:22 And a measurable cardinal is something that the set there is swear up and down is consistent, is okay, even though they can't prove it. So we've got ZFC proof consistent using angels. And we think that the angels are okay, that logically it's okay to have angels. Now, speaking of putting on the crazy hat. Yeah. So what do your colleagues think when you tell them about this divine consistency, especially with the attributions of the monocers?

Starting point is 01:01:57 Okay, well, one of them said, Friedman, you're just trying to get grant money from the Templeton Foundation, which is known to be very friendly to theology. So that was one reaction. And this was actually from a very credible known set theorist, who should love it, actually. Because, you know, this is a set theory. I use all this set theory to do this, right? Some others are actually, I actually, they, I think they think that I've, that this is another reason to ignore me or something. But it's actually a pretty serious thing.

Starting point is 01:02:37 It was refereed and accepted. It was actually, they found a referee that, I don't know who he is, it was an obvious strong set there is. He asked me to, why don't you go back and prove something? something better or something. No, I mean, he was being positive about it, but he liked it a lot. So I know that at least one, obviously very skill set there is kind of bought it, kind of liked it.

Starting point is 01:03:01 But they'll like it more. You see, there were all these concepts in theology, omnipotence, God has all these properties, omnipresence, omnipathomable, whatever, all this stuff. And every one of those, I believe, has a serious set theoretic analog, interesting. Or analog like this.

Starting point is 01:03:27 I had the ambition to actually go through all of these properties of God and develop all of these connections with mathematics. But, you know, my eyes are bigger than my stomach, you know. Yes. So then did your math inform your theology or your theology inform your math or neither. Well, this was a specific thing. I went to, speak of the depth, I went to the big famous Templeton meeting. There was a famous Templeton meeting, Congress. I was invited speaker, and there were other ones. Gary Kasparov, the world champion chess player, was there

Starting point is 01:04:07 to talk about artificial chess players, right? But it was basically a foundations of math meeting. and one of the speakers talked about Girdle's ontological argument and song, but I remember positive and negative properties, you know, and I said, well, wait a minute, now this sounds like ultra filters, and I think that there's something here of a different kind.

Starting point is 01:04:35 So that's how I got into it. But I already knew I had a lot of the, I mean, the math was fully under my belt to do this. I just had to figure out what kind of math we're going to make this work. And how do I make the axioms? So I had to come up with the notion of angels to make this work.

Starting point is 01:04:53 So speaking of thinking of a theorem or a result and then going to the axioms, reverse mathematics. Oh, yeah. Tell me some of the biggest surprises. That goes back to my real view. I started to think along these lines in the late 60s. So I was fully familiar back then. soon after getting my degree,

Starting point is 01:05:17 the mathematicians didn't really like foundation much. And they were suspicious of these things called formal systems that are bread and butter. And so I got interested in trying to show that these formal systems are really essentially purely mathematical, in sense. There was some sort of mathematical. in the axiens,

Starting point is 01:05:49 that the formal systems were kind of forced by the mathematics that even these people, even these logic haters, like and do. So I then wanted to prove the axioms from the theorems.

Starting point is 01:06:09 This is something that came out of being criticized by people who didn't like what I was doing. This is a thing. was at Stanford. I had an office in math. I was a professor of philosophy. I had an office in math. So therefore, there was a tease at the regular teas in the bath department. And so I used to go to these teas and talk about, probably convinced them that logic and formal systems are important, right? So that's how that came about. Tell me about how you think about math. What is your style of doing math?

Starting point is 01:06:44 Well, nothing is coming to mind immediately here. I can tell you what 95% of my mathematical thinking is about. I got some contrived situation that I know is connected with. I've got some contrived pseudo-mathematical situation that I know is somehow, connected to both math, although it's contrived and kind of ugly, and somehow I can squeeze out some heavy-duty set theory out of it, like what's going on in ZF. And if it knows too much about ZF, of course, then it's going to be independent of ZF, see, that's the idea. See, because if you can prove the consistency of ZF,

Starting point is 01:07:44 that you know you can't be in ZFC, because ZFC doesn't prove its own consistency. I use that all the time. That's called Gerdle's seconding completeist there, not the first. Second. So I'm forever trying to make something mathematical that isn't. In other words, I've tried to take things that are pseudo-mathematical and then redoing them and rethinking them to make them purely mathematical. So that's the kind of thing I'm doing all the time. There's always a ha in there of the kind that you may be asking me about,

Starting point is 01:08:19 but it's rather specialized. And I did this for 60 years. Uh-huh. Would you say that that's something that separates you from your colleagues or from the broader community? Is that a skill of yours that you seem to have that others don't? I, right. I am able to come up with attractive and perhaps

Starting point is 01:08:42 unassailably attractive mathematical forms of things that are artificial. Well, seemingly artificial, or at least they start that way, because then you come back to concreteness. Right. They start artificial, and then they're somehow reworked

Starting point is 01:09:04 and connected with things that are not so that they finally are solidly equivalent. I've done a lot of that. In fact, I do that all the time. I do that. I'm doing that in this book as we speak. Yes. Better and better.

Starting point is 01:09:20 The book, it's got to a certain high level, but I'm pushing it as high as I could possibly tolerate as high as possible for me. So I'm doing a lot of that kind of thing. And you asked the question, things like this have been done, by others. It's not frequent. The most famous case of this is Leo Harrington. There's something called the Paris Harrington theorem, which is a version of the finite Ramsey theorem, which is tweaked, and this time very successfully, to be independent of piano arithmetic. You know,

Starting point is 01:10:07 you can easily get an account of this on a search. Paris hyphen Harrington. Jeffrey Parris did this, but it was comparatively clumsy. Harrington did this and turned it into gold. This is the kind of thing. I mean, he was able to do it. It was kind of interesting with Paris. No question about it.

Starting point is 01:10:29 And Harrington gives him full credit for the initial step. But Harrington actually perfected it. So that's the 1970s. I'm sorry, I mean, I had the date you exactly right. This is quite a while ago. Sure, sure, sure. And I did this, I did this around the same time I did this at something that wasn't finitary.

Starting point is 01:10:52 I did this with Cantor's Diagnal Theorem. I could tell you what I did with Cantor's Diagnal Theorem. This was the beginning of my really getting into the concrete or tangible incompleteness. So canter prove that I have an infinite sequence of real numbers, you can find a real number that's missing. The real line is uncountable. So you're given an infinite sequence of real numbers that you can find a real number that's missing. Now, in modern terms, you can ask, can you really find it? And it turns out that there is a way to find it.

Starting point is 01:11:24 It's not continuous, though, but there is a way to find it that's brilliant. in. You know, it's, it's, it's one of these sequential constructions. But there is a mapping from infinite sequences of real numbers to a real number that's off the sequence. This you could do. I mean, Canter didn't think of this particular issue, but this is common in something called descriptive set theories. So there's a way of going from an infinite sequence of real numbers to a real number off sequence. So I had this crazy idea. If you look at it, how this is done, the real number that you get depends on the order of the real numbers you're given. Not only does it depend on the real numbers you're given, of course, but it depends on the

Starting point is 01:12:12 order in which you're given, given that. In other words, the decent way of doing this will give you a different answer according to which order the real numbers you're given is in, like who's first, who's second, who's third, who's fourth. Right. So I asked, can you do this? independently of the order in which it's given. Can you do this by a Borrell function, you know, a sequentially defined function. A Borrell function in such a way that it doesn't depend on the order

Starting point is 01:12:42 in which the sequence is given. And I prove that you couldn't. This is impossible. It's called the Borrell Diagnization Theory. It's actually the Borrell anti-diagnization theory because it says you can't, right? And I prove this using uncountable sets. I prove this using some serious portion

Starting point is 01:12:59 of ZFs. see that one normally doesn't use. So that's how I got, it's an early event in this story. What are your thoughts on constructive math and intuitionism? On constructive math. You know, I did write a number of papers in this area. Maybe I should start with my success in this area. I can state very easily.

Starting point is 01:13:22 Sure. You know that there is one of the key successes of construction. mathematical thinking and constructive logic is the following property of the good constructive systems. If you can prove A or B, then either you could prove A or you could prove B. And this is considered to be a very strong success of the point of view, right? Because that's not true in the classical, meaning the non-constructive world. You can prove A or B in a in CFC or anything, classical, but without being able to prove A or B.

Starting point is 01:14:07 You may not be able to prove A or prove B. You can prove the continuum of policies or the negation of the continual policies, right? A or not be, right? But you can't prove either one of them, right? So, but on a constructive system, if you can prove A, one of the fundamental properties and the fundamental successes is that if you can prove A or B,

Starting point is 01:14:27 you can prove A or prove B. There's another success that's stronger. if you can prove there's an integer N with a certain property, with a certain concrete property, certain effective property, then there's always a number of N so you can prove that that works. You know, if you can prove there is an N, then there's a particular end that proves it.

Starting point is 01:14:53 If there is an N says that, if you can prove there is an N something, then there's an answer to you can prove that that works. That's even stronger. Okay. So I prove, so everybody's, you know, disjunction property is the first one, and the numerical existence property is the second one, right? So everybody's sort of writing papers. Well, he proved the disruption property. Now we can, we can also prove the numerical existence property by a similar argument or whatever.

Starting point is 01:15:22 So I came in and I proved that for any reasonable system, if it has a disjuncted property, it automatically has a numerical existence property. It's automatic. By crazy proof, I don't even understand today. A totally weird diagonalization argument that's crazy. Kurt Gertel liked this. He published it.

Starting point is 01:15:44 He sponsored it in the Proceedings of the National Academy of Sciences. So I have communicated by K. Gurnal on this paper, the disjunction of property applies in America Existence Property. And I'm very proud to say this is the last paper he sponsored for the procedure. of the National Academy.

Starting point is 01:16:02 Other papers he sponsored, like Paul Cohen's, independence of the continual policies, and acts of Detroit. I mean, there's some real blockbusters that he sponsored. So I kind of honored that he actually put that in there. And then I visited him. He wanted me to come over. So I met the great man.

Starting point is 01:16:24 That was an experience. What advice do you have to a 20-year-old student? Well, I think tangible incompleteness and embedded maximality is a real rich subject, is one of the things and foundations they can really look at. I think that if they're philosophically inclined, at least, thinking about the serious foundational issues is going to be fruitful. But this is not so easy. And I think that with the embedded maximality stuff, I may have made it much easier because here's a real live subject with an enormous number of open questions that's brand new.

Starting point is 01:17:19 Now, I'm not saying that I think they first should learn all the standard stuff, of course. Yes, yes. And of course, I want to tell you that I am biased about my own book. Of course, right. How is it that you go about attacking a new problem? Well, I want to tell you, for quite a number of years, certainly very lately, my new problems are somewhat focused, you know, surrounding this, things like embedded maximality. So I can speak, I mean, I can speak more generally. Of course, you have to decide whether you want to invest more time on whether it's true or whether it's false.

Starting point is 01:18:09 let's say it's specific problem. Let's say you've got it down to specific problems. Okay, so there's several different kind of things you could be talking about. You're talking about when the problem is actually fixed. When the problem is actually fixed, you really have to decide like I've played around with

Starting point is 01:18:27 problems. I played around the P equals NP problem a bit. You're familiar with that of course, that's super famous. But I didn't really work on it hard. But you have to decide whether you believe P equals or not, right? That's interesting.

Starting point is 01:18:43 Okay. So you have to have a position on the problem? Well, you don't necessarily have to take a position, but you have to take a position where your time of Portshund is going to be. Interesting. Because you know if you pick the wrong side and you spend all your time on that, you could wind up with nothing. You still might wind up with something.

Starting point is 01:19:04 You might refute a strong form of it, right, or something. You know, you still could wind up with something, but it's good to develop some intuition for which side you believe more in and why. So that's one thing. Then you have to decide, is this really going to be something that needs a totally new theory? How do you—okay, so what are the partial results going to be? I like a situation where you claw at it getting very weak forms and then try to build it up. These are, you know, lots of people's methods, not just fine. What do you do when you're stuck on a problem then?

Starting point is 01:19:45 Do you switch problems? Do you still, do you attack it continually? Do you start playing music? What do you do? Well, I am something of a musician. I don't know if you do this. I know. I watched your YouTube channel.

Starting point is 01:20:01 I do my research, professor. And I've got a lot better at that over the years. better than anything you've seen. And I also run a chess club. I'm not that strong a chess part. I didn't put the time in. I did play correspondence chess when it was okay, before the computers came in and ruined it.

Starting point is 01:20:26 But I also started a book on the mathematics of chess. So I don't know if you want to go there. No, I just want to know about what is it that you do, what occupies your time, how much time do you spend on math? What does your desk look like? How are you productive? What part of the day?

Starting point is 01:20:43 Well, right. Okay, so I retired in 19, that right, 2012. I'm sorry, getting old here. 2012, I retired, yeah. And I'm single now. I have cleared off and made my life extremely simple in a kind of an apartment house which takes care of everything, including food.

Starting point is 01:21:09 So I have an enormous amount of flex time. I don't write grant proposals anymore. You know, I did some of that, even after I was retired, but I rapidly quit that. But basically, I think it's some money from Temple to Codendish. I should say that.

Starting point is 01:21:28 But basically, I'm spending an enormous amount of time on math now. maybe 12 hours a day. And music is going through my head all the time. That's why I don't have to practice too much and I can still play. So I do some of that. And I run this chess club for 10 hours a week where I mainly coaching. There aren't strong players in the building, although sometimes they make me think. Yes.

Starting point is 01:21:58 But on the chess side, I got very interested in the possibility of a very serious subject right in the midpoint between math and chess. So I'm hoping to write a book on that, but I have to finish this book than I'm talking about. So I'm basically spending a lot of time, a lot of its exposition, perfecting the rollout of embedded maximality. And that involves lots of new concepts, lots of new math, refining concepts, running up into really hard problems I know I don't want to do now because the books will never finish. There's a lot of hard problems, you know,

Starting point is 01:22:42 compiling them and putting the open problems in the book. So I'm really focused on this embedded maximality a lot. I also would like to write a book, write something serious about piano performance. How do professionals actually make it sound so smooth, and emotional, what's going on there. This is very difficult to write about, but interesting. So I have a lot of plans.

Starting point is 01:23:09 So I guess I'm just starting a serious part of my life now, yeah. What are your religious views? Religious, religion. Well, of course, there is the theology I told you about, but that's not what you're driving as. Yes, by positive and negative. But I find that, look, until we really understand how we came to be, i.e. origin of life, what is consciousness, all these things that make what I do look totally trivial.

Starting point is 01:23:49 Everything's up in the air. Any conceptions of God make sense, big some sense, until we understand a lot more. everything's on the table. That's the way I look at it. I also kind of attracted. I used to understand a little bit about what Einstein was saying, about laws of the universe and some sort of non-standard notions of God that he had,

Starting point is 01:24:18 although I don't remember exactly what he said. So I'm not part of any seriously organized religion, although I do maintain connections with a local rabbi that's quite important, actually. We're talking about some possible programs for gifted youth that we might do together. So that's about as far as I think I can go with this. You were one of the youngest professors ever in America. Yeah, I was a professor, assistant professor of philosophy, at Stanford just before 19.

Starting point is 01:24:59 I was actually 18. And I also had my PhD from MIT at that point. I wasn't a classic. I was a relatively strong mathematician, but I think I was more of a mathematical philosopher from the beginning. Interesting. Strong mathematically, but even stronger, I think,

Starting point is 01:25:21 in some ways, given the age, the mathematical side of philosophy. I remember telling, seeing a book that my mother left on the table and looking through it. And I think I was about, I don't know, seven or maybe, maybe a little younger, six. I remember looking through this book and telling her this book was totally worthless. And she looked at me. Why is this book worthless?

Starting point is 01:25:46 And I said, look, I looked up one word, and I looked at the definitions, the words in the definition. And then I looked those up. And I looked at it. And then I went, it came back in a circle. Nothing got explained. It was all in cycles. And I said, that's worth us. That's no good.

Starting point is 01:26:07 Doesn't make any sense. It's all circuit. This is, of course, very naive. And she says, well, I'm going to still keep the dictionary. I'm not throwing it away. Right. So you brought me to where I was going to go next. And also to wrap up, speaking about a circle.

Starting point is 01:26:22 I was going to bring up this story about when you were a child and you looked at the dictionary, I was going to connect that now to consciousness and to meaning. I wanted to know if after all these decades, have you come to some other realization about what words are, what grounds us, what are symbols, what does it all mean? Well, that's more in the direction of what's normally called philosophical logic, not mathematical logic and not foundations of mathematics. There are really three different subjects that are really, that's the even the people involved are pretty separate. Philosophical logicians want to go deeper and look into what is and what does and really mean?

Starting point is 01:27:07 What does and or and not really mean? Now, foundations people do that a little bit with the distinction between constructive and or intuitionistic and classical math, but they don't go as deep as some of the philosophers like to with this. I find this very fascinating, but I only have one life. And I think that I could, in fact, get very deep into this. So I don't know. I find it all mysterious.

Starting point is 01:27:36 I'm not somebody who thinks, oh, well, this is uninteresting. I've interested too many things, maybe. I see. Then let me change the question. You said you only have one life. And at the same time earlier, you mentioned that you are attracted to some of the ideas of Einstein, who said that he's recapitulating Spinoza and that the universe is somehow God or what have you.

Starting point is 01:28:00 Do you indeed believe you only have one life? That's very interesting. Well, there's certainly a way that's coming where we definitely have more than one life in a sense. And I had nothing to do with this, of course. And that's to AI. So if you leave a corpus of Internet material, like this interview

Starting point is 01:28:27 and you broaden your internet path the AI knows about all this because it it swallows the internet at least right now it is and into probabilistic stuff that we all know a little bit about

Starting point is 01:28:42 I don't know a lot about it can extrapolate exactly what you would approximate what you would say in any context I could be here after I'm physically dead. You can have an interview with me, and the AI will answer based on their entire knowledge of my output.

Starting point is 01:29:09 Now, this is maybe not the same thing you're thinking of, like the soul getting regenerated into another form of being in the standard sense. But it's an interesting thought, and this could be a very commonly person, I'm talking about in the future, commonly purchased device to have one's loved ones there for Thanksgiving or Christmas, carrying on conversations that are totally realistic. So that's a form of immortality. Just through one's internet deposits, one's internet contributions. I'm sure that I'm nowhere near the first person to make this comment.

Starting point is 01:29:53 In fact, I think in a way, you know, Pat Supies was my mentor in a way at Stanford. Patrick Supes. He's an amazing person. He died at the age of 92, some, you know, maybe 20 years ago or something. And he was a real polymath. He was a professor of four departments at Stanford. He hired me. He was the one who had this hairbrain idea.

Starting point is 01:30:18 He was chairman of the philosophy department at Stanford. He has a very large archive. maintained by Stanford, you can just look at his collective works. And I remember the last video he made had hints of just exactly what I was saying. This is before the AI exploded. Absolutely, before the AI exploded,

Starting point is 01:30:38 he had said things just like this. It's kind of eerie. And his lecture is speculating on this, I believe, is in that archive. If you just look at his very, very late lectures, you know, because I think everything is there. Professor, thank you for spending so much time with me. Listen, I love this.

Starting point is 01:31:00 I'm thrilled with this, and I think this is terrific. I enjoy myself completely, and thank you for inviting me. At some points, they were very challenging, indeed. So, I mean, you're a big shot in this. You know, you've got this big podcast with these very well-known people spread all over the place. So it's new for me. If you think I'm good at it, maybe I should do more of it.

Starting point is 01:31:26 I don't know. I think you should do more. So just so you know about my interest, I'm interested in foundations in general. So foundations of physics, foundations of math, foundations of philosophy. Philosophy is almost always just about foundations anyhow. And it's best, by the way.

Starting point is 01:31:42 Yes, yes. And also even the foundations of biology. Okay. I have said, and you could, I have said it, I think you can find it. I don't know where I said it. But I said that I want to write it. My ultimate book is called The Foundational Life. And I do talk seriously.

Starting point is 01:32:04 I mean, I've given talks in more formal settings where I'm asked. Like I gave a talk, the Russian Academy of Science people, some of them who are logicians, asked me to give a, presentation about my intellectual life and they asked me a lot of they asked me some good questions and I had to present a whole bunch of stuff about what I did but I yeah I believe there is such a thing in fact I had this idea when I was even before I got out of school which is like 16 and 17 I had this idea of of general foundations that I wanted to do foundations of math foundations of physics foundations of law, foundations of... Interesting.

Starting point is 01:32:49 Of economics. I had a whole list of order of stuff I would do. And I would start with foundations of math, because I thought that was the most well-developed one already, and so much to lever off it. And, of course, what happened is I've had a failed life. I didn't get to the others much. Just a little bit, I got to the others.

Starting point is 01:33:13 You know, you know what I mean? sense that I had all this ambition to do these other ones. But I found the foundations of mathematics around when I was just after, just before going finishing school, I said, look, the key thing is to show that ZFC matters and that ZFC may not be sufficient for things that are really completely attractive mathematically, that are just totally mainstream, that are totally transparent, that are totally natural mathematically. And people thought this was then too ambitious. Well, they were right in a way.

Starting point is 01:33:51 But I may have succeeded. Well, we'll see how well I succeeded. But the point is that I always had this general foundation's idea. I mean, even explicitly back when I was a kid. So when you said this, you and I are soulmates, you know that? Yes. Well, let's continue to speak. and I'm sure this won't be the last time.

Starting point is 01:34:13 Well, I hope not. I enjoyed this so much. You know, I didn't know that I could loosen up. I was worried about, you know, accuracy and all this technical stuff, but you got me to loosen up. You made it easy. Thank you, sir. Okay. Thank you.

Starting point is 01:34:30 Hi there. Kurt here. If you'd like more content from theories of everything and the very best listening experience, then be sure to check out my substack at Kurt Jymongel. Some of the top perks are that every week you get brand new episodes ahead of time. You also get bonus written content exclusively for our members. That's c-U-R-T-J-A-I-M-U-N-G-A-L.org.

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Theories of Everything with Curt Jaimungal - The Genius Who Invented Reverse Mathematics

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