 Hello, my friends, and welcome to the 97th episode of Patterson in Pursuit. I got an excellent interview this week with another mathematical heretic. But unlike yours truly, this mathematical heretic is in fact a distinguished professor of mathematics at Rutgers University. I spoke with Dr. Doren Zeilberger, who was both a very accomplished professional mathematician as well as a strong dissident who disagrees with many of the foundational axioms of modern mathematics. I recently discovered his work in my own research, and as you can imagine, I was extremely excited to be able to interview him. In my opinion, there just aren't that many mathematical minds who are, first of all, interested in foundational questions and more philosophical questions in mathematics, much less interested and radically independent thinkers. Dr. Zeilberger is both interested and a radically independent thinker. In his professional career, he's also won a bunch of awards. He's won the Lester Ford Award for his work in mathematics. He's the recipient of the Leroy Steel Prize for his co-development of WZ Theory. In 2004, he was awarded the Euler Medal. In 2016, he received the David Robbins Prize of the American Mathematical Society. So this is a man with impeccable credentials and yet disagrees with 99% of his colleagues about the fundamental ideas in their discipline. We cover a huge range of topics in this interview. We talk a lot about infinity, and since Dr. Zeilberger is an ultra-finitist, he rejects the idea of the completed infinity. We also talk about the notion of mathematical proof and how it might be a bit overrated, especially in the modern mathematical world. He's got a ton of great articles. I also recommend people check out if they're interested. Like one entitled Real Analysis is a degenerate case of discrete analysis, which is a hysterical title. He's got articles talking about mathematics as a religion. He's got a ton of lectures out there on YouTube talking about his unorthodox views of mathematics and new ways to think about mathematics. What I found also interesting that we covered is the future, the predicted future of the math profession, where Dr. Zeilberger thinks there's just going to be a paradigm shift in the math profession. That people might look back on 20th century mathematics, kind of laugh at lots of the superstitious thinking that found its way nestled into the foundations of mathematics. So if you've been following my work with my written work or Patterson in pursuit for a while, you know, this is like the holy grail of interviews. I'll give you no more introduction. I hope you guys enjoy my conversation with Dr. Doren Zeilberger, a distinguished professor of mathematics at Rutgers University. All right, Dr. Doren Zeilberger, thanks so much for coming on Patterson in pursuit. I'm really excited to get to talk to you today. Yeah, I'm very excited too. So I've got a ton of questions for you because just in the last several years, I've been researching mathematics more from a philosophical perspective. And my opinion about math has radically changed in a way that I didn't even know was possible. I had the assumption that a lot of people have that math as a discipline is kind of the most rigorous of all areas of thought, that there's not really room for disagreement in mathematics, the kind of the story of mathematics is one proof that's absolutely logically certain that's built on top of another. And that mathematicians can't really disagree with one another because everything is so crystal clear, logical and laid out and precise. Yeah, exactly. This is the dogma of the prevailing religion that unfortunately is still the dominant religion, but hopefully this will change eventually. So I would say that you are an unorthodox mathematician. And even putting those two words together, I think would be surprising to people like there is such a thing as an unorthodox mathematician. What does that mean? Yeah, so what is your perspective on this on mathematics as a general discipline? Do you think the story that we're told is just completely wrong? Is there a lot of proofs that maybe aren't so certain? What are your thoughts? Oh, that's a good point. This is, of course, even the most pure mathematicians concede that nothing is certain, 100% certain, because there's always a possibility that there was an error. So even the most orthodox mathematician would concede that this is an approximate ideal that mathematics is completely certain. It tries to be certain, but I'm sure that even if you look at the most religious fanatical mathematician, if he or she would be pressed, they would admit that it is a tiny probability that, for example, the Pythagorean theorem's proof is flawed because there were quite a few examples of proofs that were believed to be correct. And later, they were found errors, but it was a disinitization like a perfect vacuum. So, of course, ideally people think that indeed, mathematics is absolutely true and certain, but this is, of course, a false. Now, it's true that many, for most of the established theorems, for example, definitely the Pythagorean theorem, it's absolutely certain, but my take is the reason it's absolutely certain or almost not absolutely, that's too strong, but with very high probability, it's a correct proof and correct statement, because all this human-generated theorems were really kind of shallow. If a human can do it, it cannot be very deep. I like that perspective. So what about on this idea of absolute certainty? What about even the more elementary claims? Let's say of arithmetic. So if we're not talking about necessarily the higher order proofs or anything, do you think arithmetic can be something we can be relatively certain about? Absolutely certain about. Well, absolutely with this disclaimer, yeah, it's absolutely certain, and you can be as long as you stick to finite statements. Everything that has infinite in it is already a priori completely meaningless. It's not even wrong. It's utterly meaningless. Okay, can you go into that a little bit, because I want to ask you some questions from the perspective of somebody that wouldn't agree with you. I already agree with you, and this is why I'm so excited to talk with you, because so few people say that, but from the outside, it seems like the notion of infinities is absolutely rooted in a bunch of central conceptions of modern mathematics. So when you say the notion of infinity might be dubious or ridiculous or not even wrong, it sounds like you're saying something that's like very, very radical. So can you explain your conception of infinity and where you think we went wrong and maybe what ideas have to be revised based on these ideas? No, I'm not a professional philosopher. And in a way, it's embarrassingly simple. So, you know, for a long time, God was the big dogma. And then they had other movements and reformists, but they never doubted the existence of God. Martin Luther never ever doubted that God exists or that Jesus exists. He said that had some minor, or maybe not so minor to him, tweaks on the old dogma. So similarly, with infinity, it's not a rooted thing. But from a point of view of an atheist, all these scrubbers about religion are completely at best culturally interesting. But you cannot relate to it. So in a way, I'm not a specialist because for me, it's so obvious that the whole notion of infinity is completely bogus. So I don't even see the point of talking about it and no offense to. And for example, it was very interesting interview with a philosopher, Michael Hummer, and he wrote a whole book about it. So I'm not going to write a whole book about something to me. It's so obvious. For example, if an atheist write a book, God doesn't exist. He doesn't need 300 pages to prove it. It's just obvious it is not God. Period. So in a book, it would be only one sentence. That God does not exist. So my book would be infinity doesn't exist. Period. OK, so that sounds, I mean, that sounds very provocative. And I agree with the conclusion. But doesn't that imply that you've like you've just broken mathematics? Like if we if we say infinity doesn't exist. Well, how can we rescue things like calculus? And what are your thoughts? I mean, I mean, it's a very, you know, once upon a time, there was an emperor called Alexander. And there was also a very, very challenging note. The golden note that nobody could untangle. And people tried very, very hard. Then came Alexander the great and with one swoop of his sword. Unrevealed it. So this for my solution is also similar. There is a canonical way of rescuing 90 percent of current mathematics without ever mentioning infinity. Without all this mental gymnastics of the into into East and East and constructivist, they still are like Lutherans. They still believe in the potential infinity. And they go to lots of a mental, intellectual acrobatics to resolve, quote unquote, all the paradoxes, quote unquote. So for me, it's extremely simple. And so once again, also, I want to issue with Michael Homo in what he mentioned, that there's no largest and largest integer. You can always add one. That's another stupid thing. How does it know? There's no proof. It's just a stupid axiom that you can take, take it or leave it. And I. So this is still in this respect, I'm an agnostic. I famously once said that there is a largest number which nobody knows. But if you add one to it, you go back to zero. So it's like a cycle when you go around the planet, you go back. If you go the same direction all the time, you head west. Eventually you wind up where you are. So this is a little bit tongue in cheek. But this is still an open question that you can redo mathematics without using the so-called piano axiom. You can redo everything also calculus. You can redo everything by completely finetistic means. OK, what about me? So obvious, I don't even bother to write it down. So so when I talk to some people about this, because I come at this from a philosophical perspective, not really a mathematical one. A lot of times people will bring up real analysis like the whole subject of real analysis is analysis of real numbers, which seem to have this connection with infinity. So so what are your thoughts on real analysis? There's not a number of theology behind this theology. And and then you had famously a Newton and Leibniz who developed calculus in a quote unquote and rigorous way. Then people had all this paradoxes and then came the 19th century. Mr. Matisse and Squashie and Weierstrasse and allegedly made it rigorous with the epsilon and data and all that. And it was in a way, it's also finetistic. It's an algorithm. So in a way, they didn't know it, but they didn't really have to use infinity. They in a way, they rescued it by making it an algorithm. So to prove that something. A function is continuous. For example, X square is continuous. It's really an algorithm for every epsilon, blah, blah, blah. I have a data and you have a explicit algorithm that the computer can be implemented on the computer. So in a way, they made the way step in the right direction. But there is even a simpler way. Make everything discrete analysis. Yes, no, no more limits. So you have, like in physics, you have a plank distance, this belief to be the shortest distance possible in the physical universe. So you have something analogous in calculus that we have no idea what it is. And once again, it's an agnostic whether it actually exists or not. That's a moot question, that's an irrelevant question. You can redo the whole calculus. I can rewrite the calculus textbook in much shorter way. By replacing derivatives by final differences. OK, so I have three things to say. First of all, I would love I know you had a quip there where you said it's so obvious. I don't even want to write it down. For the sake of future mathematicians, please do write it down. Because I feel like that would be very, very important. I kind of have a nine page paper that you could look at my website. Real analysis is a degenerate case of discrete analysis. I saw that I actually downloaded that and I just love the title. I thought that was hysterical. Yeah, so the outline of the thing is everything is outlined there. And the rest is just details. OK, another thing I want to say, I want to kind of run this idea by you. So when there's a question of, you know, what might be the largest integer B or what might this I like to call the idea of a fundamental like a mathematical atom, I like to call it a base unit. It's like, what could the base unit be of mathematics if everything is discrete? And I think this is kind of a philosophical question. But perhaps this is a resolution. The reason that there that it's kind of confusing or or misleading to talk about the largest integer, oh, you could just add one to it. Therefore, there must be an actualized infinity. What if numbers themselves are just constructions of our own mind? And if that's the case, then at any given time, there was a largest number that has been conceived or constructed. And yeah, you can add one to it. But in the process of adding one to it, you've created then a new largest number, could that work? No, I don't give you this. Well, what? So that's another aspect of my redoing the semantics because I'm very big on symbolic computations. For example, I was Maple, but Mathematica and as a free software sage is symbolic. So if you so, you know, it's a little bit like no nominalism. So if n is a symbol, n plus one is a symbol. So that's one way to rescue mathematics by redoing the piano axiom. If something is true for n, it's also true for n plus one. But n is not a concrete number. If you plug in one of the finite numbers, the only thing is finding many numbers. Then you get a correct statement because of keep n as a symbol. And then it's just true symbolically. So to give you a simple example, if A and B are two integers or numbers, no matter A plus B equals B plus A, this is a fact. Now, the conventional wisdom is it's true for every number A and B, support or an integer for any two integers A and B, A plus B equals B plus A. But for me, this is already meaningless because when you say and not even wrong, if A plus B equals B plus A for every two integer A and B, you implicitly assume that you have an infinite supply of them. So the correct statement is A plus B equals B plus A for any finite integers. That's the only integer that exists. And also A plus B equals B plus A for symbolic integers. So once you have the phrase symbolic in front, then everything can be rescued. OK, so do you think in your kind of conceiving of what numbers are, do you think then they are kind of platonic objects in the sense that numbers exist separate of human minds and maybe separate of physical space? They're kind of another. My concession to Platonism, a finite numbers do exist. But once again, it's a philosophical issue and who knows what it had to be. So so how what what kind of a role do you think computation, like actual math on computers, should play in the development of mathematics? Because when I again, from the outside, when I think of like the progression of a discipline and I look at the invention of computers, I think to myself, OK, that's kind of the holy grail of mathematics. And it probably should should maybe like the theories and the theorems and the philosophy might need to change around what we can do with with computers now. So what do you think? What do you think about that? How important are computers to develop to the development of mathematical knowledge and like mathematics knowledge in the abstract? Like our understanding of mathematics is kind of a discipline. No computers are already revolutionizing the practice of mathematics and because of computers, the current religion will completely change. And in a hundred years, possibly 50 years, people make make fun of the superstition of 20th century mathematicians with a pathetic belief that after the 70, because computers like already AlphaGo is much better than any Go player and as it says, playing deep, deep blue can beat any. So analogously, mathematics will already and all the theorems, that mathematicians, even, for example, the thermonuclear theorem and the wilds in 50 years, maybe a computer, a computer can find a much better proof all by itself without any human help, that's by doing machine learning. I mean, this isn't a theory in the future, but eventually everything, all the corpus of mathematics so far will be just be done in a very fast form scale of initial computers. Yeah, it's funny. You mentioned earlier that the move towards epsilon delta kind of maybe rescued some ideas and calculus and made them a little more rigorous. I kind of feel the same way about computation in practice. So it's like, if it's the case that calculus can work on computers, it is thus the case that we must have a discrete calculus because computers operate. Yeah, good point. Yeah, so that's the kind of that's it's funny when I've argued with people about this, they talk about, oh, you can't do calculus without infinities. And I'm saying, no, no, no, we are doing calculus without infinities. Like, yeah, exactly. Good point. There's an irony when you actually do numerical computations, we have a partial differential equation. The first thing you do is discretize it and you actually solve a difference equation, a finite difference equation. OK, so let's get into some more ideas then. So we talked about real numbers a little bit. What about transcendental numbers? What about something like pi? This is the three point one four one five nine dot dot dot and so on. Whatever the end so on means. How what is your thoughts on pi? Yeah, here I also like to take issue with your excellent article. Pi is finite. The famous thing of Derrida is saying about what is is what is the what is the construction. So if you turn the question because it doesn't assume that there is something like is. So if the pi is a number, it's like saying, I don't know, a unicorn is an animal. So pi is not even a number. But it doesn't exist the way. So Lineman was wrong when he proved he claimed to have the claim that pi is a transcendental number. This is nonsense. Of course, what he did was very valuable. What he did prove that pi is not a number. So so in your language, then, when you're talking about. Pi like not not saying that it's a number. What is the thing that we're referencing? What is the three point one four one five nine? What's going on there if it's not a number? Yeah. So the way to rescue it is still a useful thing. But and but it's not three point one point one point nine that that that that that this implicit has shown that you have you can go forever. And this is complete nonsense. You cannot go forever. But it's also not finite. It's not even a number. So the way to get around it is I gave a talk about it in pi day. That's pi day. So it's pi is an equivalence class of algorithms. So you have many algorithms to compute approximations to pi. So it's also called when you try to do for some Thomas Hales in his proof of the Kepler conjecture, the second phase when he made it want to make it completely rigorous. Before he used floating points and floating points uses round of errors. And people criticized it for being non rigorous. And they have that a point or so it was very pedantic point. And then put Tom has spent ten years in setting them up by making everything completely rigorous by using interval arithmetic. So the statement that pi lies between three and one tenth and three and one fifth makes sense. It lies in the interval since rational numbers do exist. So and then you can say pi is larger than 31 over 100 and less than 32 over 100 and so on. So for any epsilon, it's like the epsilon things you can find out. You can come up with an interval a, b, where a and b are rational and hence meaningful entities such that pi, quote unquote, is between a and b. So this is the meaningful thing. So is that also the same with irrational numbers in general? Yes, yes. For example, square root of two makes sense as a symbol. You can define square root of two by its property, by its defining property. It's a symbol, not a number, a symbol that has the property that X square equals two. Then it's also true then. Whatever it is, it lies in the interval between 1.4 and 1.41 and so on. This reminds me, I was doing a little research on imaginary numbers. And if I'm not mistaken, the creator of imaginary numbers called the square root of negative one a useful fiction. And he said something along the lines of, you know, this doesn't really make sense, but if we if we get away with it, if we treat it as just an empty symbol, then actually we can solve these math equations. And yeah, yeah, that's a great analogy. Yeah, yeah, I thought to myself, wow, that's actually really that's a smart way of thinking about it. And then that's that seems to have been lost in the in the theorizing afterwards that the actual creator of it thought of as thought of as a useful fiction. And then everybody else or 99% of the mathematicians after the fact seem to think take it as some transcendental truth or entity. Right, right, right, it all gets able to get used to the force of habits. You have a good point. Very good point. OK, so I want to ask about another question that this is more of an area of my active research I'm trying to really sort out carefully is Gertl's incompleteness theorems. So in philosophy, it's also my pet peeve. Thanks for bringing it up. Excellent. In philosophy, they come up a lot. They come up a lot when people are making very bad arguments. And they say, you know, if you're trying to be logically rigorous or something, you go, oh, no, like you can't be fully rigorous because, you know, Gertl's incompleteness theorems. I'm thinking, what, what are you talking about? So it's been a multi year long investigation, really trying to wrap my head around Gertl. The first thing I would say is this is an area where I'm inherently skeptical because what I found in talking with mathematicians and everybody else is almost nobody has actually gone through Gertl's proofs. Apparently very complicated. And they say, oh, well, I I've never gone through them. But there are these really smart mathematicians out there who have gone through and they've checked all of it and it all checks out. And I'm thinking, well, how do you know that? That seems like a lot of trust. So this is the first thing. So that's my impression from the outside. Is that also what's going on in the inside of math? Do you think that do you think that not a lot of people are actually double checking Gertl at this point? And no, no, I'm sure that professional mathematicians went through it and it's nothing wrong with Gertl's metapoof. And the idea behind it is the proofs are very simple. In fact, Alan Turing famously had something analogous that the halting problem, the halting problem is undecidable. This is you can have in half a page. In fact, I have a little half page note that proofs Turing's statement. And philosophically, instead of using Gertl, so go to the details of Gertl's proof of very daunting, but the main idea is basically a takeoff of the Lyos paradox. But he just misunderstood. So what he did, for my point of view, Gertl, a much simpler Turing, the proof something even more profound that in infinity is baloney. It doesn't exist because it leads to this paradox. So there's a statement that exists true, but unprovable statement is completely stupid. OK, so what's the relation then? This is something I'm really trying to unpack here, because I think this would be a very big deal, but it's hard for me to put my finger exactly on it. So what is the relation between Gertl's incompleteness theorems and infinity? Oh, because what Gertl proved that the Principia Mathematica and the Hilbert program, he used infinite sets. It only applies his paradox, so to speak, only applies to infinite mathematics, because his tricks, the idea behind Gertl is extremely simple. Only the details are complicated, and he was not the best rider. But so basically you look at the typical statement in mainstream number theory. So look at it. He has quantifiers for every and they exist. And also he has logical connectives. So you look at such a statement and then he has this beautiful dictionary that converts any mathematical statement into a number theoretical statement from the meta level to the object level. He has this dictionary. And then you also look at the proof, what's the proof that the sequence, a finite combinatorial sequence of statements and you go from one step to the other using one of the so-called axioms that and then by this very clever dictionary, he concocted a very, very complicated statement that for an orthodox mathematician like Hilbert made perfect sense. And what it says that if you translate it back to the meta level, I am unprovable. So the conclusion was that this is a correct statement, but it's unprovable and approved by contradictions like the Lyos paradox. But the flaw in these things, he implicitly assumed that statements that has quantifiers, the range of infinite sets are meaningful. But if you don't accept it, for me, what he proved is all baloney. So what I, what people call undecidable, I call not even a meaningless a posteriori, even a posteriori meaningless. So some statements, every statement that has quantifiers, for example, a very simple statement for every integer a a plus one equals one plus a for every integer a is a priori, meaning less, because it implicitly assumes that there is an infinite supply of integers, stupid baloney. So this is wrong. So the correct statement, a plus one equals one plus a for symbolic a. And now it's a meaningful statement because a is a finite symbol. So this is true. So in this case, a plus one equals one plus a, the statement is a priori, meaning less, but could be easily made a posteriori, meaning full by instead of doing the quantifier for every a for symbolic a. Now, they sent this a mathematical number, theoretical statement that Gertel concocted is a app. It's not even meaning it's also a posteriori meaningless. So this is my dictionary. So to be to be clear, if it's the case that there are only finite sets, then what happens? Does Gertel's proof not work? No, in a way to prove a contradiction. It's another goes to the cloud of two being irrational. So for my the way my take on Gertel's thing, he proved that most mathematical statements, orthodox mathematical statements, they have quantifiers that range of infinite sets are not meaningless, even a posteriori. Because they need to a pair of graphs to rescue them. Yeah, right. And there was no way to rescue them. Unlike one plus eight with eight plus one, which could be resurrected and deeper, deeper statements. For example, like the theorem, every positive into the end can be written as a sum of four squares can be resurrected for my point of view. But all the statements, the statements of Gertel and many others like it, probably most of them are complete gibberish. And there's no way to resurrect them. OK, so so I'm going to give an analogy here and tell me if this is something similar. So yeah, when I'm when I'm having arguments with people who have math education and we're talking about infinities. Yeah. And some of the more remarkable claims of mathematicians that a lot of times I say, OK, well, give me an example of, you know, where this leads to paradoxical reasoning, OK, well, what's an example? It's interesting that you say Gertel is an example. I'm going to come back to that. But one a proof that I like to bring up is the but Banach-Tarski Paradox. Now, because I because from the outside, you see the claims for people who don't know the idea is like you have a sphere and then you can, you know, cut apart the sphere into a finite amount of sets and then reassemble the sets and then you get a you get two spheres. You get a sphere and then another sphere, the identical size. There's another problem, infinities ballooning. Right. That's exactly so. I see that and say, OK, but you've just like there's a lot of people that will say, oh, that just shows the counterintuitiveness of some of these ideas in mathematics. I'm not counterintuitive. I think that's a pretty clear demonstration of a catastrophic flaw. So you think it's the same thing going on with Gertel? It's a kind of demonstration that something's gone awry here. Yeah. Yeah, yeah, yeah, the Banach-Tarski Paradox is another good example of how everything with infinity is bogus. OK, so it could be an interesting mathematical. It could be still a game you can pretend. So it's still like it's like a chess. The queen says the king and queen of chess are not actually king and queens. It's a game. So it's still as a game is an intellectual game that some people may enjoy. I don't it still makes sense, but only as a game and this paradox is it still makes sense in the game. But the ontological validity is not nil. OK, so so for for the rest of the world, especially they have mathematics training, they're going to say this is just there's no way that everybody in the math community could be fundamentally mistaken like this, that there's an assumption. I think part of the mathematics religion is that mathematics is kind of the language of God and when you discover truth in the language of God, there's no way that it could possibly be wrong. Therefore, to step back and try to analyze whether or not the mathematics is correct and conclude that maybe everybody for the last century or two centuries is completely mistaken is such a radical claim. It's like a qualify it. Sure. Most except for the counter set counter and all the ultra ultra esoteric logic, all the mathematics that is used in science and all the things can be easily resurrected and became and become legit by doing this translation. So it's not like I think we'll be lost forever. It's just the way of thinking about it. Yeah, I like to say that, you know, the math might be able to be rescued. The actual problem is the stories that mathematicians tell about the math. Yes, yes. Yes, it's yes. OK, but so are there areas, though, that you think can't be rescued? Do you think that there's there's fundamental assumptions and maybe it's topology or something in real analysis, some actual conclusions or proofs or that just won't the can't be rescued. They're just too far gone. In principle, if you look at it for for the sense for infinity, sensory, for example, what you were doing and the professional satirist or Sarah Chellach, that I admire admire very much as mathematical athletes and like chess players. So it's a game. So for anybody who cares about it, there's nothing wrong with this. It is a philosophical issue, whether it has any but it represents anything. In fact, ironically, Paul Cohen, the patron saint of satirism, he was in a way a finalist, but since he didn't like to argue, he was a very peaceful person. He never overstated it. But if you look at one of his beautiful papers, he says that it's just a empty game and that the game is a theory. So what he allegedly proved as a continuum hypothesis is undecidable. For him, he didn't believe that infinity for so deep inside or even not deep inside. He meant it. So what he meant was that the game is a game of starting with Zermelo-Franco's choice, that he proved that in this game, like he says, some moves are not reasonable. So in this case, the statement, the continuum hypothesis is true, it's not reasonable and the statement, in fact, that's probably good, I already proved it, the statement as a continuum hypothesis is also not reasonable. So once again, here's another proof that it's all ontologically baloney, but still it's still be an intellect, a fun to some people, not to me. So here's the creepy spot, go ahead. He said it's all meaningless, but it's still a fun game. So I think for me, it's not a fun game. It's an OK game, but I have more fun games in this one. Yeah, so so this is where I also think there are unintended consequences outside of mathematics when questions like this aren't taken seriously or the conclusions like, oh, math is just a game. Don't propagate outside of the math community, because all the other disciplines seem to be looking up to mathematics, as I said, as if they're speaking the language of God, and there's this phrase, the math speaks for itself. Yeah, but if that's not true and not even the mathematicians at the highest level. No, I'm talking about the esoteric century. The stuff that is used in physics is very useful and that completely can be resurrected in my approach. And this is probably in a platonic sense, as certain as can be once it's interpreted correctly, and the century is a game, but it has no relevance to applied math. Yes, yes. Yes. And I'm saying even even the esoteric set theory people will appeal to outside of mathematics outside of mathematics. So that's why I'm saying there's I see it because I have these conversations about philosophy all the time and it's amazing how many conversations will ultimately end up coming to the philosophy of mathematics. People make rather poor arguments and then they'll appeal to set theory. And then they not only there's something unique about set theories. It's not just one esoteric area of mathematics. It's treated in the in the storytelling of mathematicians. It's treated as fundamental. It's treated as a dogma, that's a prevailing dogma. But probably in a hundred years, possibly 50 years, people will laugh at it. Most people, of course, there's still some people who believe that the earth is flat or completely. Right. It's like of all of the areas that we would want. We would want to maybe a little more regular than, oh, this is just a game. It seems like set theory would be one of those areas. Given how it's supposed to be foundational. Right. OK, so that's the thing I want to talk to you about is so you say in a hundred years, you think people will look back and kind of laugh. Can you be a conservative? That's what they say, probably won't be alive in a hundred years. So do you think that so what's kind of your expectation for the future of mathematics, do you think that we're going to see like a revolution or our young mathematicians, you do content wise, become as deeper and more interesting. And all the previous human degenerated things will be done very, very fast automatically and there be so much new discoveries done by computer. Most of them will be what I call semi rigorous or not even rigorous. Also, the adherence to the rigorous proof will become very soon obsolete because even computers can probably the really deep CRM will be out of reach. But computers are finite and they cannot do everything. So. So my vision is that mathematics will become more a science that they really want to know what's actually true and the insistence on a completely rigorous proof that the optional is possible. But and there will be something that what I call semi rigorous proof that you prove something with probability 99.99999 and this would be acceptable to them. So do you think when we when we reach that point, the the storytelling of the 20th century is just going to fade away? Or do you think it's going to be like, well, that's because I have a hard time just when I observe from the outside, seeing much progress. I mean, I see you, I see maybe a couple other mathematicians out there in universities that seem to be seeing kind of similar things, but it seems like 99 percent of the profession is just totally sold on some of the the ideas of the 20th century. Yeah, but seeing change so much, even even the computers 20 years ago, many people doubted the validity of the proof, the apple hack and proof of the four color C.O.M. And prove that computers on count in that experiment. And but and now even the most orthodox one would not doubt it. It's to be reluctant to say, ideally, there'd be a nice human proof. But so things are also the attitude that's changing slowly already. I can see. And so when a new generation comes and sinful change, so we have to wait for the old people to die. So how what then have your experience has been in the math community? So so I kind of want to know when you started having a lot of these these heretical thoughts, if this is something that's been over your whole lifetime, or was there a moment in which things changed? And then when you talk to your colleagues, do people just look at you like you're totally crazy or are they like, OK, yeah, actually, you're making some good points. Maybe maybe we do need some fundamental revision. And yeah, to be honest, most mathematicians are so busy doing their own little tiny acre gardening, they don't really give any considerate, they don't really care about foundational issues. So for them, it's like somebody being so so they don't take me seriously. The thing is that here's somebody who wants to get attention. So in addition to my philosophical so-called outspoken thing, I'm also a mainstream regular one of the mere commentaries and I do conventional stuff. So for most most mathematicians and colleagues, many of them probably aren't even aware of it because that's too busy doing their own very, very narrow, often very deep, but narrow part of mathematics. So mathematicians are very philosophically uninterested, the only they have this belief, very naive belief of God, but they don't really think about it too much. It's like like a naive religious person. So we take calls for granted, but it's not a scholar. What about students? Do you find students are more open to what you're saying or interested in? My students are more interested in this than my colleagues. My colleagues couldn't quite care less, most of them. There's nothing I'm an eccentric. I mean, I'm a good, I'm OK. Mathematicians, even probably a good one in my specialty, which is one of maybe two hundred different specialties. So in my little part of mathematics, which is maybe one one percent of all mathematics, I'm OK. Functionally, good citizen. And everything else is as somebody who has crazy ideas about. So the thing is that maybe you want to have a tension and have a tension. Then in fact, that's my wife claims that claims that I would be raising that to get attention. So but it's just I'm more open. So when did this when did this happen? Have you had suspicions all along that some of the fund? Yeah, no, yeah. Of course, it was gradual. I was I was born and raised in the current dogma. But partly it's also personal. The course that I hated, I still got an A, but the course that I hated most in college was real analysis, because it seemed to me so pedantic and so unattractive compared to other to algebra I liked because it had more content. And when you prove something, you prove something is not intuitively obvious. But this analysis is pedantic epsilon delta was just very, very unattractive to me. So that's how the grain started. Yeah, my PhD was doing this analog of something continuous. So my lab fell with this state started in my PhD. Yeah, I've I've so I don't really have any background in mathematics, like I said, I just come in it from a philosophical perspective to know a lot. Well, thank you. It's just been many years of research and you might find it funny. I don't think mathematicians could possibly understand the amount of bad arguments that are made outside of mathematics by appealing to mathematics. I mean, it is just this is the reason I got interested in it is just from so many bad arguments I was here at propaganda. Yeah, it's a lot of it is shallot, shallotanism, for example, economics. Many economics are shallotans. Some of them consciously, some of them unconsciously. They're trying to prove using the medical rigorous things to make the point. Right. Yes. And in even manners of logic of, you know, for example, I've had so I had a conversation recorded, actually, one of the one of the episodes I don't remember off the top of my head with a philosopher. He was a mathematician. I don't remember if he was a philosopher or a mathematical logician, one of the two from Columbia and we were talking about logical contradictions. And he said, well, yeah, he gave two, I'm not exaggerating. He gave two examples of what he thought could be true logical contradictions. One, he said, could be the pope. Is the pope married or not married? And he said, I'm not kidding. He said, well, there's a sense in which the pope is married to the church. And he's unmarried. I thought, what are you, you're a professor. How was that? And the other, he said, was infinite sets. He said, that seems like it's a contradiction. And yet it's this fundamental part of mathematics. And so I was thinking, OK, well, I got to research this. But what I wanted to say is also I've just in some of the articles that I've written, occasionally, this probably will bother you. Occasionally, I've gotten messages from people will email me and they'll say, hey, Steve, you know, I love this article. I used to be really into mathematics. And as I was studying it, I found these ideas about infinity or set theory to be so incorrect, I then assumed I didn't have the knack for math anymore. And I just quit. Oh, my God, you're chasing me. You're like soccer days, you should see. Yeah, well, it's not it's not me, it's them. It's it's a yeah, they're saying that they thought that the the so so so they like my articles, but they in their train, their formal training, they couldn't make sense of set theory and thought this must mean I'm bad at mathematics, therefore I'm not going to pursue mathematics because it doesn't make sense to me. I just see that as a tragedy. Yeah, yeah. So hopefully they can see the light that finite mathematics is still one. Yes, yes. Well, I'll make sure to send them to to your work and others. And I wonder, I wonder, too, this is this is both a question for my podcast and a personal one. Do you collaborate with many people? Like, do you have a network of of heretical mathematicians that are that are working together? Because I would love to talk with more of them and try to be more networked in this community. Well, my most constant collaborator is my own computer. So she has and most of my recent papers are in collaboration with it. It's not him or her. It's a computer. It's it. And as far as. We couldn't call him heretical, but they're doing experimental mathematics. And I don't really think about the foundations. So implicitly they're doing a new kind of and mathematical experimental mathematics. But to be honest, most of them don't care about the philosophical underpinnings and for them it's just a job and they do what they like. So if they are heretical, it's only implicitly without knowing knowing that they are. OK, have you come across the work of Norman Wildberger, who's he's a professor? Yeah, I really, I really admire him. But once again, in a view, whatever he spent so much time doing, can be summarized in one sentence. So it's good that he prosprosalizes. It's good to have people like him. But for me, it's so obvious. So I don't think I think he always does it as far as the time he spent in preaching the world because it's so obvious to me. But sure, I'm sure he does a good purpose. I agree. This is rational. I really like this book, rational trigonometry, that everything can be done by rational numbers. So the last question I want to ask you is again, I know like the current religion, not just in mathematics, the current intellectual religion is is really utter deference to whatever comes out of the math profession. And so I know a lot of people when hearing this, you know, this is this is like hearing that there's no God if you've been a theist your whole life. So the actual like within the academic process, do you have a lot of respect for or trust in the academic process as it is in terms of publishing? So like, have you found obstacles for getting some of your work published? Definitely, definitely. It's a whole that's like a supposed somebody a published in the if you do submit the paper to the to the journal of Jews, the temple, of course, it gets rejected. So yeah, it's very orthodox. So luckily I no longer my paper that I do with myself and my computer, my computer doesn't care about tenure. And I'm not going to go out for tenure for promotion. So my computer kindly agreed. We only posted on the archive. It is non-referred and in our website, it's completely irrelevant. The peer reviews adorn us that are completely completely prejudiced against new ideas. Unfortunately, I have quite a few PhD students and other collaborators who still need for career, practically isn't official peer review publications. And then I try to go to places where I know the editor, who are more broad minded and not just submitted blindly to a random mainstream journal for which the chance of being accepted is near. In fact, what I do, I first I'd email to the editor who sometimes happens to be happens to be a friend, as for the frank opinion, whether he's comfortable considering it. So that's the way around it. Do you think that's something that's also destined to change in the future is the peer review process? Yeah, the peer review things will probably also be absolutely based on eventually computable check the correctness and as far as the interest of significance, quote unquote, the future will decide. So it's also going to be obsolete. Well, OK, I know I said that was the last question, but I have one more question for you. It sounds like so I did. I was able to interview Dr. Wildberger in Australia and we had a great conversation. And I really admire him. Yeah, I do, too. And we were talking about kind of the role of philosophy in mathematics. And he had the perspective that that philosophy shouldn't have much to say in mathematics, which was very funny to me because I thought so much of his criticism and so much of his work comes from a philosophical perspective, but he doesn't he doesn't like labelled as philosophy thinks it's just mathematics. Maybe he probably meant official philosophy in a way, but he died of philosophy. Right, right. So I wonder for you, when you're thinking about kind of the role, the hierarchy of knowledge in a sense, do you think that philosophy, not philosophy proper, not from professional philosophers, but the philosophy of mathematics or maybe you could call it metamathematics is in a sense more fundamental than the actual applied mathematical work? Well, more is that it's too strong, but it's at least as interesting. And and it's much more interesting than many of the details of the object level mathematics that very soon will be done much faster and better with computers. So yes, in some sense, this is still room for humans. And philosophy in some sense will become more prominent because that's something humans still can do probably better than computers for some time. Right. And for mathematics, we won't need humans anymore. Space one. Yeah, that's a that's a fantastic note to end on. I really appreciate this conversation, Dr. Zairebog, Dr. Zairebog, this has been this has been great and inspiring. And I feel like this is really an area where I want to keep doing more research because every once in a while, like, you know, people will tell me, Steve, you're crazy. You got all these crazy ideas and math. And then when I find somebody that, you know, is in the academy and still is saying some very fine, autistic things, I'm like, ah, OK, I've justified. Yeah. And thank you so much for getting me to know your wonderful side. And I'm looking forward to a browsing and even listening to some of the podcasts that you posted in the past and keep up the good work. Thanks very much. I appreciate it and you as well. Thank you. All right. Bye bye.