Transcription
pretty outrageous idea. In other words, it's all this real number stuff, all this partial differential equations, even all this set theory stuff, large cards, it's all fundamentally finite. This is my crazy head. This is Professor Harvey Freriedman'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. The Guinness Book of World Records even listed him as the youngest professor ever. Kurt Girdle, while alive, personally sponsored his last paper for the proceedings of the National Academy of Sciences, and Professor Freriedman founded the field of reverse mathematics. Girdle's incompleteness theorems are the most celebrated results in modern logic. The textbook examples are recendite, self-referential curiosities that no working mathematician tends to meet in practice. However, Freriedman says they're pointing at the wrong target. The question is, can ordinary finite math be trusted? His theorems suggest otherwise. So now it's harder for the mathematical community to ignore foundations.
>> On this channel, I kung interview researchers regarding their theories of reality with rigor and technical depth and probe at the foundations. Today we discuss tree 3 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. Professor, what misinterpretation of girdles incompleteness theorems bother you the most?
>> Well, one is that there really are two separate theorems and they really uh uh are quite different and most people aren't fully aware of the difference. So, one is that uh in any sufficiently strong system there are always statements that can't be proved or refuted. That's so-called Girdle's first incompleteness theorem. And the second one is that in any sufficiently strong system, the system cannot prove that it's without contradiction. It cannot prove that it's okay. And these are these are quite different things. I I've heard some people interpret girls theorem, first theorem at least, in the sense that we can't know uh things for sure. That's not quite what it says.
>> It says there are 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 Girdle's first incompleteness theorem actually says. Well, Girdle uh Girdle's first incompleteness theorem, the most famous one, merely says that there are some things given any particular logical framework, there's always going to be some things that that system doesn't handle that that the system doesn't know whether it's true or false. Meaning that that the system does not prove or refute the statement. However, many statements will 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 Girdle already showed?
>> All right. In a nutshell, uh Girdle showed that there were statements in the first incompleteness theorem showed that there were statements that can't be proved or refuted in in in for example the gold standard for foundations of math called ZFC. However, those statements are very far removed from what mathematicians 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, it changes over time. But basically the original statements of girdle are very far removed from the kind of mathematics that mathematicians very generally uh worry about and care about. So it's very different. uh what happened after the incompleteness or the first incompleteness what happened is that mathematicians wanted to know whether how how far would this reach into the kind of mathematics that they care about and that's been a there's been a long development of that to give you a framework the the ZFC system was pretty much settled by 1930 as to what it what's in there and um 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 in ZFC. And then in 19 in the early 1960s, Paul Cohen showed that it couldn't be uh proved. So the two of them together showed that the continual hypothesis famous continuum hypothesis uh is uh uh is not provable or refutable with the ZFC axioms. However, the continuum hypothesis is generally regarded as extremely important in general set theory but is regarded as extremely several steps removed in abstraction from what mathematicians normally do. And nowadays we can run searches and and change normally into quantity. In other words, so many papers I mean nobody's I don't know that anybody's actually carried this out but you can actually start putting some numbers on these things. Uh so I think uh uh will that help your uh understanding of what concrete and completed? The idea is that the continual pauses involves arbitrary sets of real numbers. Real numbers are fine in all three mathematics. Uh and uh uh the cominatorial mathematicians probably want to go more concrete. But real numbers are fine. But if you talk about an 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 continuous policies is about. And that's something that really is strikingly different uh than the normal kind of mathemat 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 Nort 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 let's just put that in quotations would say that's highly contrived that that scenario and also it's unphysical and then Norton obviously has retorts like yeah sure but tabletop physics with a with a like a rectangle or a square is 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. Einstein thought so. So what are the similarities and what are the differences here?
>> Uh I think it's 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 uh less so now but but but uh but there are still experts doing it. Uh if you're going to do that then the cater is 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. uh ar for arbitrary sets of real numbers the the most general kind of set of real numbers is not commonly uh understood well I wouldn't say commonly uh is not commonly appreciated as what mathematics is all about in mathematics one is generally concerned with the finite of course which is which is very concrete the finite uh and one is concerned with what what might be called sequential processes.
>> Mhm.
>> Like real numbers. Real numbers are sequentially understood as infinite uh uh series or infinite uh decimals. So uh 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 for uh more specific sets of real numbers that are more commonly uh dealt with in mathematics like ones based on sequential limit processes, the continuum hypothesis is well known to be proved. It's not independent. It's proved for instance for so-called burell sets of real numbers. The continuum hypothesis is is is a it's a classic theorem uh uh that the cumulus 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 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 I do care about the hypothesis 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 I don't care about it for that purpose. Historically, it's very important. I actually got interested somewhat. I I wrote a uh an unpublished piece on uh why I believe that continuous can be argued better to be false than true. And I'm not the only one who believes that that many people uh sectors believe that. uh I did get a little bit interested in it but I think that's not really the topic at hand here uh which is uh the tangible incompleteness the burell measurable sets of real numbers my first big advance in towards concrete incompleteness was finding statements in in the burell universe or the burell sense of real numbers where u you need um more than zfc or at least large hunks of gf zfc Uh, so my first ventures into concrete incompleteness passed through the burell sets in the 1980s.
Where do you contrast with Hugh Wooden on the continuum hypothesis?
>> Well, Wood Hugh Wooden is a is a card carrying believer in abstract set theory and its intrinsic importance. uh he knows that mathematicians are not necessarily as enthusiastic about this area as he is but but he has a point of view that the policies is false. He uh wants to create a theory of how the entire universe of sets works. If you think the arbitrary sets of real numbers is is abstract away from the concrete which it is uh the entire set theoretic universe is incomparably more so it's sets of sets of sets of sets of real numbers for example and much much much more than that. uh so he's part of the community that is really very much into abstract set theory. uh he has taken the view uh that the couture policy should have a definite answer even if ZFC doesn't touch it which ZFC does not touch it uh it ought to have a definite uh answer and the way that he and colleagues associates uh work think about this is it's their job to uncover the truth the truth and um although it is clear that the ZFC axioms have a an intuitive compelling appeal or even visual and very much related to even finite intuitions in some sense uh it is unclear at this point whether there's any kind of compelling clear visuality to something higher than ZF that would settle continental pauses. uh I think he more more or less thinks of exploring it as a scientist saying what's more plausible than what and what can we find uh uh uh compelling in some way even if it's not compelling like ZFC access. Now I've been dubious about this uh and I we went separate directions. We've known each other for a very long time. So he took uh uh this high uh abstract road 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 if 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?
>> Uh yes, but it's more philosophical. It's more foundational. Um, I regard what the ultimate view of truth in mathematics is as a totally open question that's been uh uh made more mysterious by 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 although Wooden and company are 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 they're comfortable with with an attack on ZFC that it is uh it doesn't handle it doesn't allow for the kind of very abstract objects that they uh believe exist. I attack it differently. I say it isn't even good enough to do finite things that we that we care about. Now when I say we care about, this is very much up in the air. I spent uh almost my entire life since 1967 trying to uncover uh uncontrived mathematical contexts in which you can't the ZFC axins are nowhere near enough. I started off with things that were grotesqually contrived uh and spent what 50 or 60 years uh trying to make them less contrived and connecting them up naturally with thing with with things that everybody that almost every mathematician can relate to immediately. That was that's been the program for 60 years and that's a different kind of attack on ZFC.
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. 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 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?
>> Um basically yes. In fact, it's uh it's been historically uh uh uh pretty much adhered to that there's a purely logical part of math that has nothing to do uh uh pure logic that has nothing to do particularly with mathematical objects. It could be anything. And that goes back to Aristotle with his syllogisms. And the big thing that Aristotle didn't have was the idea of binary relations, relations between things. He had a manatic things like uh being a man or being mortal which is unary so to speak. Whereas less than between numbers is binary.
>> Mhm.
There's there's a purely logical framework and this this uh 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 pic calculus but that's a confusion of first order predicate calculus and that's the purity logical part of ZFC and then one uh and then when you apply pure logic uh 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 rules of and logical rules of inference which are fixed. Then you have throw in proper axioms and that's the framework that ZFC fits in that goes up.
How do category theorists view math?
Category theorists uh have a different kind of point of view. Some of it is practical. That's the non-p problematic part, the practical part
>> that uh it's good to think of categories because it makes it facilitates a lot of mathematical work in topology and other places.
>> They 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 all. Well, that's just u operations on in category theory. Whereas the logician says, "Oh, no, no, no. Category theory is a special case of everything everything that we do >> which is pure logic and it happens to be a a thing where we have objects called categories and we have operations on category just like any just like Zen. There's nothing special about it. It's just another thing. So, we both think we we're more fundamental than each other. I will point out that Saunders Mlan who I knew he's a different generation than me but but he wrote a book called categories for the working mathematician. In the beginning he defines a category as a set together with blah blah blah blah blah. So he he actually he actually defines category theory in terms of set theory. Yes. So in a way he wasn't one of the extremists.
>> Half of my day is spent in conversations and research. The other half is spent trying to remember what was said and read. That's where Plaude comes in. P L A 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 Plaude for many months now and it's a huge gamecher. It's actually ridiculous. 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 it's something like a 5% gain in thinking and that workflow compounds fast. If you're a knowledge work, you know how valuable even incremental small 1% 2% changes are. This trounces that. What I like is that it removes friction. I'll use the notepin 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 can stay present with the world. I don't have to worry that I'm going to drop my phone. 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 gets captured, processed, and then turned 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, SLC2, HIPAA, and GDPR compliant. If you want to check it out, go to plaud.ai/cur and use code kurt.
Okay. Okay. Now, how do these extreme category theorists, your words, not mine, I assume you mean topos 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 that way, whereas I don't interpret that way. In other words, are 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 push uh Toppo's theory for example uh uh as the real foundations and uh I should point out although they try to have a defense that the big events in foundations 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 not I would say my I don't want to say my point of view that's too extreme but you know but the the point of view that I like I mean the the actual people the actual results after the fact they did do some reworking but never first and in particular concrete incompleteness this is this is uh on our side we don't have category theorist 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 also not provable in the category systems either. You can it's equivalent more or less. Yeah. Uh 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 u quote my side.
How is the foundations of mathematics viewed by other mathematicians
>> uh as pretty marginal? Sorry. Marginal as in unimportant or marginal as in
>> uh right unimportant and irrelevant is the most common reaction. However, there are these periods of history where some blockbuster comes out comes comes to surface. Girdle especially,
>> right?
>> Uh and then some of the leading mathematicians in the world even uh suddenly take notice and make some comments about it. people start to uh 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 uh things like that, but the feeling is that it doesn't really impact their work enough for them to uh uh 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. Uh how well I succeed is not quite clear. Uh I've definitely made some inroads. See, there's a thing called there are these weaker forms of incompleteness where you're not incomplete of all of ZFC, but you're incomplete of big of serious portions of it. That's important. One famous serious portion of it is called Zumelo set theory instead of Zlo Frankle.
>> This is without Frankle's replacement action so to speak. There's an axm called replacement in the ZFC. U Zlo set theory is already an enormously strong system far far more than any mathematicians normally use. But I'll give you an example. There was a there was a a fairly well-known problem in what's called infinite game theory and it's called burell determinacy. This is this is the problem and Donald Martin proved burell determinacy using way more than ZFC and that created some impression in certain parts of the mathematical community. No question. And um but nobody had any confidence that this had anything to do with foundations. It it was just overkill. It was just using something probably overkill. Then I proved that there is no way to prove real determiny without using infinitely many uncountable cardinals. In fact, uncountably many uncountable car that Zerto set theory was not good enough to prove real determiny. and that you needed at least a big hunk, an unusual hunk of ZFC to do it. And then Tony Martin looked at my argument and said, "Oh my gosh, this gives me a hint as to how I can prove determiny in ZFC. I would have never," he said, "I never would have thought of using these strong methods that Freeman proved were needed. I never would have uh even thought along those lines and now I I see how to do this. So he published a famous paper proof of burell determiny using exactly the outer limit of what I said 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 u just uh involving burell measurable sets and and and and uh this kind of descriptive set theory or analysis which also requires uncountably many uncountable cards to prove there. So there was an interaction. This is already 1980s and earlier.
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. It was just some internal title. And then the the one that you said was more accurate, maybe not still entirely accurate, but more accurate. Harvey Freriedman is about to bring incompleteness and infinity out of quarantine. Right.
>> That's a nice one. Okay. 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 Freriedman says the foundations of math are what? Like are wrong, are broken, are not known, are not investigated enough, are what? I'm totally up in the air or are more more mysterious than ever.
>> 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.
>> Okay. So
>> let me give you let me give you the analogy cuz 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 building rockets. I'm here using quantum mechanics in the lab. Phones work etc etc. So I'm sure that's the sort of mentality that that that you see that you encounter. So the so what so what Harvey? Yeah. All right. So I'm trying to deal with that directly. Um but it took me 60 years to get get to uh get to the place I'm at. Um so important new attractive mathematical subjects appear uh depending upon what size you're talking about appear regularly. People get excited about some 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 them among the many fundamental concepts of mathematics two of them are embedding and maximality. Both of them appear in undergraduate math and in advanced math. Uh it you know in in many things in many situations and um what I uh uh do now is do what I call embedded maximality. I say that these two concepts can be combined into new striking theorems and I call this subject embedded ma maximality. This is the title of the book that I'm trying to finish. Um, 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. The context is the rational numbers with the ordering less than. So embedded maximality is the title of this book and the context is among the most concrete uh situations in the whole of mathematics which is simply the ordering of rational numbers without even addition or multiplication. And I show that there's a finite theory of this where I look at all the finite embeddings that can be used for for the maximality uh principle and I analyze that first and then I I I say well let's look at some very simple infinite uh maps not just finite maps but infinite map maps with infinite domain.
>> Okay. um uh uh uh in the rationals we're talking about you know uh on an interval it might be the identity on an interval is already infinite. All right so uh 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 in uh 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 combinatorial mathematics of the kind that people have uh uh uh know and love nothing funny and then say well I want to look at finite these finite maps on the rationals I want to extend them by the identity map on on the uh outer parts in other words if the finite function lives in here let's make let's extend it to be the identity above and below 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 uh uh uh 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. So now it's harder for the mathematical community to ignore foundations. See there's now this question of whether these theorems uh there's a question of whether these theorems uh are in fact u the outer extension usability theorem o eu the outer extension 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 that go way beyond ZFC and we know ZFC is nowhere near uh strong enough to prove it. So, is this is this math legitimate? Has it been proven? It's like the old classic uh issue about is it okay to use functions that aren't given by expressions? You know, classical mathematics, everything you know is is a formula or something, you know.
>> Yes.
the notion of arbitrary function came up late, relatively late as you know and and and people were concerned about it uh or even definition by cases originally you know because you have a a discontinuity that's artificially put on right so
>> I never heard of the definition by cases what's that
>> definition by cases oh um f ofx is zero if x is negative f of x is
>> is five if x is positive. Yeah, you know, something like that.
>> Yeah,
>> you see that doesn't come from a the old idea of formula. All right. So, so anyways, um if in fact the mathematical community falls in love with embedded maximality because I figure out how to say it and motivate it >> and connect it by the way with standard mathematics which I have a chapter on that how to connect it up with semi-algebraic functions and peace-wise linear function you know the stuff that the bread and butter I do connect it up with that though when that connection is for background proud. I don't get independence from ZFC, but that it's so tightly motivated and clear and the um finite part is so interesting allegedly then one is compelled to 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 incomplete. 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?
>> Rewrote it four times or something. Uh I rewrote it because it got so such much more convincing. But are you publishing slivers of it or is it all you can I I have a there's a website that maintains uh maybe 30 or 40 onehour lectures on it. So I have a complete record of that and there's some manuscripts and unpublished stuff. Now a lot of this is some of this is wrong. So I keep I you know I believe keeping the historical record is good for this. Some of these lectures I overstated some stuff and I take it back and so forth. So it's not a pro I don't have any problem with that. It's in fits and starts but it's pretty uh it's well on it. It's well on its way. In fact I the I have to give a talk on it tomorrow uh on Zoom uh to a very small audience of experts. But um it has to be done just right for something like this because the skepticism uh that foundations is really important is so strong.
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Give a sense of how monstrously large the number tree is.
Okay. Well, I think the the best way to talk about tree three and there's some technical definitions that require some expertise in a subject called recursion theory. But one way I I look at it is this tree of three is so big that you can't even prove it exists uh that you can't prove it exists even in a system like strong system like piano arithmetic. If that were true, that would indeed be mindblowing because any finite number can be shown to exist depending on your definition of exist. I'm sorry. I'm sorry. I'm sorry. I misstated this. You can't prove it exists without using two to the one trillion 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, uh, that makes sense. You can't poss you're not even proving it exists,
>> right?
I I don't know how that if that helps a lot, but and also a a considerably stronger system than piano rhythmic. The statement I made is correct.
How does it compare to Graham's number?
>> Well, if I remember Graham's number, I think it's kind of not noticeable.
>> Yes.
It's minuscule in comparison epsilon.
>> Now you can make artificial you can play this artificial game in many ways. Uh but this is a particularly elegant one that is mathematically very uh uh friendly if you remember the definition.
>> Yeah. What is the definition of that number?
>> Yeah. Um okay you look at 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 approximate a finite form so to speak and the infinite approximation is you have an infinite sequence of it goes back to JB Kros uh you have an infinite sequence of finite trees with uh three colors if you want I do it with three colors uh with three colors on the vertices uh each vertex has one of the tree colors. You have an infinite sequence of 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 if preserving. It's a it's a slightly technical notion but it's a but it's the obvious notion of homeomorphic embedding. There's a there's a whole comm that was well known of of fi of finite trees and how they fit together and what does it mean by you know isomorphism and all that. Okay. So once again and given any infinite sequence of three colored trees finite trees sorry finite one of the trees in the list is a is embedded in one of the later ones in the list. This is a famous theorem of JB Kresccoll and I noticed or maybe I wasn't the first to notice but I did so I made the first to do something with it. Uh, I noticed that this proof involved looking at all possible infinite sequences of trees, uncommonly many. And and this proof was so far from being local, you might expect you just locally figure out this piece is so it was so unusual. This JB Kroskll proof.
>> Yes.
>> And I proved why it was so crazy, so strange. I proved that that cros theorem can't be proved in systems you'd expect it to be proved. Now this is still nothing like ZFC in completeness is something entirely different a different level but but but this is a micro incompleteness so to speak and uh I so I kind of proved that you need uncountable sets to prove this crusal thing. Okay. Now how do you approximate the cross thing because after all it's it's about infinite sequences. Well it turns out that from the things like the compactus theorem and well known in mathematics.
>> Yeah.
>> You can show that if you put a bound on if you say that the E tree has at most I vertices. The E tree has at most I vertices. So you put a bound on the grow 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 trio3. Trio3 says, how long do you have to go on like this so that one of them is embeddible in a later one? As long as they're the eye tree as long as the i tree has the most eye vertices and we have three colas that's what tree of three is how how long do you have to go up there now from general mathematics you can prove from cross theorem 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 krescos's theorem you actually know tree of three hm H. 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 between the really big finite, the outrageous finite, and the small infinite. The smallest infinity is just omega. The, you know, the the size of the natural numbers. That's the smallest infinity. You know, caner has higher infinities like the real number line and all that.
>> Yes. Yes. I have a video about that.
>> Yeah. So, so the smallest infinity is very closely associated and approximated by the outrageously finite outrageously 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? I said suppose I'm a finiteist. I I'd say okay only finite things maybe big things maybe maybe not even I don't even like big things you know 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 is all about the most you could stand for so uh so what if I put my finiteist hat on and I say okay all of mathematics I'm going to make finite 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 differential equations, even all this set theory stuff, large cardinals, it's all fundamentally finite. But uh this is my crazy hat.
>> Yes,
>> a finiteist hat. And in fact uh the book embedded maximality has a section on purely finite forms which shows in a sense that all these large cardinals beyond ZFC uh can be thought of in finite terms.
>> Now let me ask you this.
>> Yeah.
>> What about ultra finite terms?
>> Okay you talking about ultrainatism? Well, that no, that's the point. That's the that's the idea that the that you shouldn't go past uh very small numbers that already two to the 100 is absurd.
>> That's ultrafunded. Yes, that's part of the crazy ad.
>> Interesting.
>> Uh 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 of of detail. In other words, pixels with colors. Everything there is 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 at at this screen and I'm looking at everything. Everything I think about my my my work my ps this is all well within the computer screen's size of information which is only x bits where x is you know from the thousand >> 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 uh properly represented them in a picture. I believe this can be done and I think the what's in my book is is nowhere near that that powerful but it does suggest some things.
What are the large cardinals?
Yeah. So that this is a whole subject in and of itself. Um uh the smallest large cardinal. Okay. from the point of view of a working mathematician, there's some there's some cardinals that are uh that are tiny from the set theorist's point of view that are already pretty damn large.
Now, if you really want to talk about hard-nosed mathematicians, they usually live in the countable, but they do are aware of the real line. So, there's this cardinal called two of the olive not.
>> Yes.
>> So-called two of the olive not. So, that's not really a large cardinal. The mathematicians aren't going to call that a large cardinal. If you're a real finite commentatorist, you might think of it as pretty large, of course. Yes.
>> Okay. Then there's two to the two of the olive nod and then there's two to the two to the two to the hour of nod. see Caner.
>> Mhm.
>> Um I mean this is uh maybe not exactly his original notation but but but uh the idea is that you 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 other words do it once exponentiated once exponated again and expon so forth you get a tower.
Sorry professor just a moment to to interrupt just briefly for the layman can they substitute infinity for the cardinal for the word cardinal here but if you take any
>> we're talking about different levels of infinity
>> great
>> yeah cardinal came up caner defined these cardinal is an abstraction uh on uh is the way we talk about levels of infinity so I can talk about levels of infinity the lowest level of infinity is the integer uh 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 continus asks whether the real numbers are the second largest second smallest infinity is the is the real set of real numbers the second largest infinity and that's what's called the continuous the divine consistency proof walk me through how God proves math is consistent. What does that mean?
>> Okay. Uh there has been an idea that goes back to ancient theology and Girdle took it quite seriously and then I took it seriously because he did. um 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 hybr. Uh now if you ask me practically you know what does this mean you know for real actual attributes that people think about uh who are not abstract philosophers and logicians it gets a little bit uh uh strange so we won't go quite there we have this abstract idea that we're going to classify uh attributes as positive or negative 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 or just
>> no absolutely all God is perfect. Uh now Girdle took this in a different in a different direction than I took it. Uh he wanted to take this idea and actually prove he actually proves that it's possibly necessary. So he has some sort of a hedge that there is a perfect being. He wanted to come up with some abstract princip philosophical principles that there's got to be something that has only all and only the positive properties. And therefore he's proving he's proving in some sense that God exists or he's proving that actually he states a weaker thing uh that necessarily it's possible or or some mo what's called modalological uh uh hedge.
>> Yes, please talk about your divine consistency. uh this idea of classifying
Things into positive and negative, uh, intrigued me a lot. And there is something in mathematics that, uh, is very much like this. It's called an ultrafilter.
Yes, it's a mathematical thing. Maybe you've heard of this. An ultrafilter on a set is you divide the subsets of the set into the big and the little, and everything is either big or little. And the intersection of two bigs is a big, and, uh, and so forth. You can guess now, anything big has got to be infinite, okay? Uh, or at least two elements. There are many ways to say this.
Sure. And ultrafilters are, uh, are a kind of useful thing in a lot of different kinds of mathematics. Uh, it was exploited by set theorists. If you have an ultrafilter that has a very strong property, namely, if you have a sequence of biggies, then its intersection is big. And that's not a finite sequence. That's an infinite sequence of biggies. Then the, then the intersection is big. And that turns out to be independent whether there is such a thing. And that turns out to be equivalent, more or less, to some large cardinal called a measurable cardinal. So there is something bigger than ZFC there going on if you have a very strong kind of ultrafilter.
So I started with this knowledge. Right now, it turns out that we don't want to consider, uh, the ultrafilter. Okay. So there's an ultrafilter of all the big properties of all the big ensembles of all the positive, sorry, positive and negative. Remember, there's the, if you take all the positive ensembles or the positive classes, positive classes, yes, uh, that forms an ultrafilter. That's what Gödel started with. But if God is in there, it's all sucked up by God, right? So, so it's just the, uh, uh, ultrafilter of all things that contain just this one special point called God. That's not very interesting. That's called a non-principal or trivial ultrafilter. 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 in there, right? You don't have a lump. Then I define what an angel is. An angel is a weak form of God. Uh, an angel is some, is something which isn't in all the positives. That would be God. But it, it's in all the definable, all the explicitly definable, uh, positives. In other words, it's, it's a member of all the good properties, the good, the positive sets that you can, that you can name, all the nameable ones. See, this is weaker.
Yes. But that's a pretty good approximation of being godlike, right? You, you're only in the good ones that, that are defined. All the other defined ones, you're not in. And I call that an angel.
Yes. This is just a theory in which you postulate that there is an angel and you add, uh, the standard trappings you would, you would like for, uh, for a system, some mathematical infrastructure called the choice operator, 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 exists" and "there exists at least one angel."
Now, this system 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 measurable cardinal. And a measurable cardinal is something that the set theorists swear up and down is almost, is, is consistent, is okay, even though they can't prove it. So, we've got ZFC proved consistent using, uh, angels, and we think that the angels are okay, that logically it's okay to have angels.
So, 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 monikers?
Okay. Well, one of them, one of them said, "Freeman, you're just trying to get grant money from the Templeton Foundation, which is known to be very friendly to theology." So, that was my reaction. And this was actually from a very credible, known set theorist who should love it, actually, because, you know, this is set theory. I use all this set theory to do this, right? Some others are actually, uh, 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. It was refereed and and accepted. Uh, it was actually, they found a referee that I don't know who he is. He was an obvious strong set theorist. He asked me to, "Why don't you go back and prove 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 skilled set theorist kind of bought it, kind of liked it. But they'll like it more. You see, there are all these, um, concepts in theology: omnipotence, God has all these properties, omnipresence, omnipotence, uh, unfathomable, whatever, all this stuff. And every one of those, I believe, has a serious set theoretic, uh, analog.
Interesting. Or analog like this. Uh, I had the ambition to actually go through all of these properties of God and develop all of these connections with mathematics. Uh, but, you know, I have my eyes are bigger than my stomach, you know.
So.
So then did your math inform your theology or your theology inform your math, or neither?
Well, this was a specific thing. I, I, I went to speak at the, uh, the big famous, uh, Templeton meeting. There was a famous Templeton meeting, Congress. I was an invited speaker, and there were other ones. Gary Kasparov, the world champion chess player, was there to talk about artificial chess players, right? But it was basically a, a foundations of math meeting. And, uh, one of the speakers talked about Burle's ontological argument, and some, and, uh, but I remember positive and negative properties, you know.
And I said, "Well, wait a minute now. This sounds like ultrafilters." And I think that there's something here, uh, of a different kind. So that's how I got into it. Uh, 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 how, what kind of math will really 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.
So, speaking of thinking of a theorem and then, 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 you. I started to think along these lines in the late '60s. So I was fully familiar back then, soon after getting my degree, that mathematicians didn't really like foundations much, and they were suspicious of, um, 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 a sense that there was there, there was some sort of mathematics in the, in the axioms, that the formal systems were kind of forced by the mathematics that even these people, even these logic haters, uh, uh, uh, uh, like and do. So I want, so I then wanted to prove the axioms from the theorems. This is something that came out of being criticized by people who didn't like what I was doing. This was at Stanford in the, I had an office in math. I was a professor of philosophy. I had an office in math.
Mhm.
So, therefore, there was a tease at the, uh, regular teas in the, uh, math department. And so I used to go to these teas and talk about, try to convince 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?
Well, I, nothing is coming to mind of me here. Uh, I can tell you what 99, 95% of my mathematical, my, my, uh, thinking, uh, is about. I got some contrived situation that I know is connected with, I 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, like, like, what's, what's going on in ZF. And if it knows too much about ZF, of course, then it's going to be independent of ZFC. See, that's the idea. Uh, see, because if you can prove the consistency of ZFC, then 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 Gödel's second incompleteness, not the first, second. So I'm forever trying to make something mathematical that isn't. In other words, it's, I'm trying to take things that are pseudo-mathematical and and then redoing them and rethinking them to make them purely mathematical. So that's, that's the kind of thing I'm doing all the time. There's always an "aha" in there of the kind that you may be asking me about, but it's rather specialized and, and, 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?
Right. I am able to come up with attractive and perhaps unassailably attractive mathematical forms of things that are artificial, uh, well, seemingly artificial, or at least they start that way, but because then you come back to concreteness. They start artificial and then they're somehow reworked and connected with things that are not so that they finally are solid, 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, better it with the book. It's got to a certain high level, but it's, uh, I'm pushing it as as high as I could possibly stand, tolerate, I, as high as possible for me. Uh, 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, the most famous case of this, uh, is Leo Harrington. There's something called the Paris-Harrington theorem, uh, which is a version of the finite Ramsey theorem which is tweaked and this time very successfully to be independent of Peano arithmetic. You know, you can easily get, uh, an account of this on a search Paris-Harrington. Jeffrey Paris did this, but it was comparatively clumsy. Harrington did, 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. And Harrington gives him full credit for for the initial step, but Harrington actually perfected it. So that's the 1970s.
Uh-huh.
I'm sorry. I may not have the date exactly right. This is quite a while ago.
Sure. Sure. Sure.
And, um, I did this. I, I did this around the same time. I did this. It's something that wasn't finite. Uh, I did this with Cantor's diagonal theorem. I could tell you what I did with, with Cantor's diagonal theorem. This was the beginning of my, of really getting into the, uh, concrete or tangible incompleteness.
Uh, so Cantor proved that if you have an infinite sequence of real numbers, you can find a real number that's missing. Right? 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. It's not continuous, though, but there is a way to find it. That's brilliant. You know, it's, it's, it's one of these sequential, uh, constructions. But there is a mapping from infinite sequences of real numbers to a real number that's off the sequence. This you can do. I mean, Cantor didn't think of this particular issue, but, uh, this is common in something called descriptive set theory. 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 how, 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 order in which you're given them.
Uh-huh.
In other words, the decent way of doing this will, will give you a different answer according to, uh, 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 Borel function, you know, a sequentially defined function, a Borel function, in such a way that that it doesn't depend on the order in which the sequence is given?" And I proved that you could. This is impossible. It's called the Borel diagonalization theorem. It's actually the Borel anti-diagonalization theorem, because it says you can't, right? And I prove this using uncountable sets. I, I prove this using some serious portion of ZFC that one normally doesn't use. So that's how I got it. It's an early event in this, in this story.
What are your thoughts on constructive math and intuitionism?
Oh, on constructive math, you know, I did write a number of papers in this area. Uh, maybe I should start with my success in this area. I could state very easily.
Sure. Uh, you know that there is one of the, uh, one of the key, uh, successes of constructive mathematical thinking and constructive logic is the following property of the good constructive systems: If you could prove A or B, then you can prove, 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, uh, in the classical meaning, the non-constructive world. You can prove A or not A or B in, in, in ZFC or anything, uh, classical, uh, but without being able to prove A or B. You may not be able to prove A or prove B. You could prove the continuum hypothesis or the negation of the continuum hypothesis, right? A or not B, 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, 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, uh, concrete property, certain effective property, then there's always a number n. So you can prove that that works. In other words, if you can prove there is an n, then there's a particular n that you can, that that proves it. If there is an answer, such as if you can prove there is an n, something, then you can, then there's an answer such that you can prove that that works. That's even stronger. Okay. So I proved, so that everybody's, you know, disjunction property is the first one, and the existence numerical existence property is the second one, right? So everybody's sort of writing papers, "Well, he proved the disjunction property. Now we can, we can also prove the numerical existence property by a similar argument or whatever." So I came in and I proved that for any reasonable system, if it has a disjunction property, it automatically has a numerical existence property. It's automatic by a crazy proof I don't even understand today, a totally weird diagonalization argument that's crazy. Kurt Gödel liked this. He published it. He sponsored it in the Proceedings of the National Academy of Sciences. So, so I have communicated by K. Gödel on this paper, the disjunction property applies the numerical existence property, and I'm very proud to say this is the last paper he sponsored for the Proceedings of the National Academy. Other papers he sponsored, like Paul Cohen's, independence of the continuum hypothesis, an accident. I mean, there's some real blockbusters he sponsored. So I'm kind of honored that he actually, he actually, uh, put that in there. And then I visited, then I visited him. He, he wanted me to come over. So I met, I met the great man. That was an experience.
What advice do you have to a 20-year-old?
Well, I, I think tangible incompleteness and embedded maximality is a real rich subject. It's one of the things and foundations they can really look at. I, I think that, um, uh, if they're philosophically inclined, at least thinking about the serious foundational issues is going to be, uh, 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, with an enormous number of open questions. That's brand new. Now, I'm not saying that I think they first should learn all the standard stuff. Of course.
Yes. Yes.
But, and of course, I want to tell you that I am biased about my own book.
Oh, 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 these things like embedded maximality. Um, so I, I, I can speak, I mean, I can speak more generally. Uh, of course, you have to decide whether you want to invest more time on whether it's true or whether it's false. Let's say it's, it's a specific pro. Let's say you've got it down to a specific problem. Okay. So, there are several different kinds 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, um, like I've played around with problems. I played around with the P equals NP problem a bit. You're familiar with that, of course. It's super famous. Uh, but I didn't really work on it hard, but you have to decide whether you believe P equals NP or not, right?
That's interesting. Okay. So you have to have a position on the problem.
Well, you have, you don't necessarily have to take a position, but you have to take a position where, where your time apportionment 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. You might refute, we might refute a strong form of it, right, or something, you know. You know, you still could wind up with something. But it's good to develop some intuition for for which side you believe more in and, uh, 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, I like a situation where you claw at it, getting very weak forms, and then try to build it up. These are so, these are, you know, lots of people's methods, not just mine.
What do you do when you're stuck on a problem, Ben? Do you switch problems? Do you, 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 should, I don't know if you knew this.
I know. I watched your YouTube channel. I do my research, professor.
Uh, and I've got a lot better at that over the years. Uh, better than anything you've seen, actually.
Um, and I also run a chess club.
Ah.
Uh, I'm not that strong a chess player. I didn't put the time in. I, I did play correspondence chess when it was, um, okay, before the computers came in and ruined it. Uh, but I, I also started a book on the mathematics of chess. So I don't know if you want to go there, but.
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?
Well, right. Okay. So, I retired in 19, uh, 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, uh, a kind of an apartment house which takes care of everything, including food. So I have an enormous amount of flex time. I don't write grant proposals anymore. You know, I, I did some of that, uh, uh, even after I was retired, but I got, I rapidly quit that. But, but basically, I, uh, I did get some money from the Templeton Foundation. I should re, should say that. But, but basically, um, I'm spending an enormous amount of time on math now. Uh, maybe 12 hours a day.
And music is going through my head all the time. That's why I, I don't have to practice too much and I can still play. Uh, so I do some of that. And, um, I run this chess club for 10 hours a week. Uh, where I mainly coaching. There aren't strong players in the building, although they sometimes they make me think.
Yes.
Um, uh, but I, but on the chess side, I got very interested in, in the possibility of a very, a very serious subject right at the border, right in the midpoint between math and chess. So I, I, I, I'm hoping to write write a book on that, but I have to finish this book that I'm talking about. So I'm basically spending a lot of time, a lot of its exposition, perfecting the rollout of, um, 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 and, you know, compiling them and putting the open problems in the book. So I'm really focused on 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, uh, smooth and emotional? What's going on there? You know, this is very difficult to write about, but interesting. Uh, so I have a lot of plans. So I, I guess I'm just starting my serious part of my life now. Yeah.
What are your religious views?
I, I, religious religion. Um, well, of course, there is the theology I told you about, but that, that's not meant to be what you're driving at. Um, yes. By positive and negative. Yeah. But, uh, I find that 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. Um, everything's up in the air. Uh, any conceptions of God make sense, make some sense. Until we understand a lot more, everything's on the table. That's why, or 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, although I don't remember exactly, uh, what he said. Uh, so I, I'm not part of any, uh, seriously organized religion, although I, I do maintain connections with, uh, local rabbi. That's, that's quite important, actually. We're talking about some possible programs for gifted youth that we might, uh, do together. So I, 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, um, I was a professor, assistant professor of philosophy at Stanford just before 19. I was actually 18, and I also had my PhD from MIT at that at that point. I wasn't a classic, I was, I was a, you know, relatively strong mathematician, but I think I was more of a mathematical philosopher from the beginning.
Mhm. Interesting.
Uh, strong mathematically, but, but, but even stronger, I think, in some ways, given the age, the mathematical side of philosophy. I remember, uh, telling, seeing a book that my mother left on on the table and looking through it, and I think I was about, I don't know, seven, or maybe a little younger, six. I remember looking through this book and telling her, "This book was totally worthless." And she looked at me and she said, "Why is this book worthless?" And I said, "Look, I looked up one word, and I looked at the, and I looked at the definitions that the words in the definition, and then I looked those up, and I looked at, and then I went, it came back in a circle. Nothing got explained. It was all in cycles." And I said, "That's worthless. That's that's no good. Doesn't make any sense. It's all circular." Now, 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. 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, that's more in the direction of what what's normally called philosophical logic, not mathematical logic, and not foundations of mathematics. There are really three different subjects that are really, that even the people involved are pretty separated. Uh, philosophical logicians want to go deeper and and look into what does "and" really mean. What does "and" and "or" and "not" really mean. Now, foundations people do that a little bit with 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. Um, I find this very fascinating, but I, I, 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. I'm not somebody who thinks, "Oh, well, this is uninteresting." I'm interested in too many things, maybe.
I see.
You know.
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. Do you indeed believe you only have one life?
Oh, okay. Ah, that's very interesting. Um, 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 through AI. So, if you leave a corpus of internet material like this interview, and you broaden your your internet path, the, uh, AI knows about all this because it, it swallows the internet, at least right now it is. Uh, and it, to probabilistic stuff that we all, you know, know a little bit about, I don't know a lot about it, can extrapolate exactly what you would approximate, exactly approximate what you would say in any context. Uh, I could, I could be here after I'm physically dead, and you can have an interview with me, and the AI will answer based on their entire knowledge of my output. 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, a commonly purchased, I'm talking about in the future, commonly purchased device to have one's loved ones, uh, there for Thanksgiving or Christmas, carrying on conversations that are totally realistic. So that's a form of immortality. One's internet, uh, uh, deposits, one's internet contributions. I, I'm, I'm sure that I'm, I'm nowhere near the first person to make this comment. In fact, I think in a way, you know, Pat Suppes was a, was my mentor in a way at Stanford. Patrick Suppes. He's an amazing person. He died at the age of 92, some, you know, maybe 20 years ago or something. Um, 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 hairbrained idea. He was chairman of the philosophy department at Stanford. He has a very large archive, uh, 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, he had said things just like this. It's kind of eerie. And, and his, his lecturer speculating on this, I believe, is, uh, in that archive if you just look at his very, very late lectures, you know, uh, because I think everything is there.
Professor, thank you for spending so much time with me.
Listen, I love this. I, I, I'm thrilled with this, and I, uh, I think this is terrific. I, I enjoyed myself completely, and, uh, thank you for inviting me. So, at some points, they were very challenging indeed. So, I mean, you're a, a big shot in this. You know, you've got this big, 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. 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, but philosophy is almost always just about foundations anyhow.
At its best, by the way, sometimes.
Yes. Yes. And also even the foundations of biology.
Okay. I have said, and you could, I have said, I think you could find it. I don't know where I said it, but I, I, I said that, um, I want to write, my ultimate book is called "The Foundational Life." And I do talk seriously. I mean, I've given talks in more formal settings where I'm asked, like, I gave a talk at the Russian Academy of Sciences. 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, I believe there is such a thing. In fact, I had this idea when I was, uh, even before I, uh, 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, of, of law, foundations of,
Interesting.
Of economics. I had, I had a, 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. And of course, what happened is, uh, I've had a failed life. I didn't get to the others much, just a little bit. I got to the others, you know, you know what I mean, in a sense that I had all this ambition to do these other ones, but I found the foundations of mathematics around, 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, uh, attractive mathematically, that are just totally mainstream, that are totally transparent, that are that, that, that are totally natural mathematically." And people thought this was too, too ambitious. Well, they were right in a way, but, uh, I may have succeeded. Well, we'll see how well I succeeded. But the point is that, uh, I always had this general foundations 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, sir. And I'm sure this won't be the last time.
Well, I hope not. I, I enjoyed this so much, you know. I didn't, I, I didn't know that I could loosen up. I was worried about, you know, accuracy and all this technical stuff, but, but you got me to loosen up.
You made it easy. Thank you, sir.
Okay. Thank you. 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 kurtjimong.org. 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 k u r t a i m u n g a l.org. You can also just search my name and the word Substack on Google. Since I started that Substack, it somehow already became number two in the science category. Now, Substack, for those who are unfamiliar, is like a newsletter, one that's beautifully formatted. There's zero spam. This is the best place to follow the content of this channel that isn't anywhere else. It's not on YouTube. It's not on Patreon. It's exclusive to the Substack. It's free. There are ways for you to support me on Substack if you want, and you'll get special bonuses if you do. Several people ask me like, "Hey Kurt, you've spoken to so many people in the field of theoretical physics, of philosophy, of consciousness. What are your thoughts, man?" Well, while I remain impartial in interviews, this Substack is a way to peer into my present deliberations on these topics, and it's the perfect way to support me directly. Kurtjongle.org or or search Kurtjungle Substack on Google. Oh, and I've received several messages, emails, and comments from professors and researchers saying that they recommend Theories of Everything to their students. That's fantastic. If you're a professor or a lecturer or what have you, and there's a particular standout episode that students can benefit from or your friends, please do share. And of course, a huge thank you to our advertising sponsor, The Economist. Visit economist.com/toe to get a massive discount on their annual subscription. I subscribe to The Economist, and you'll love it as well. Toe is actually the only podcast that they currently partner with, so it's a huge honor for me. And for you, you're getting an exclusive discount. That's economist.com/toe. And finally, you should know this podcast is on iTunes, it's on Spotify, it's on all the audio platforms. All you have to do is type in Theories of Everything, and you'll find it. I know my last name is complicated, so maybe you don't want to type in Jiongle, but you can type in Theories of Everything, and you'll find it. Personally, I gain from re-watching lectures and podcasts. I also read in the comments that Toe listeners also gain from replaying. So, how about instead you relisten on one of those platforms like iTunes, Spotify, Google Podcasts, whatever podcast catcher you use, I'm there with you. Thank you for listening.