 Well, thank you for the opportunity. Some of you might know, this is a talk, a version of a talk I gave at the philosophy department here at the door, so if you've sought, if you were at that, it's not that much different so you can feel free to leave. But otherwise, I will proceed. Anyway, as I indicated in the flyer that went around, when people ask what do you teach, and I say philosophy and mathematics, they say what? What's the connection? Well, I want to make the case that it's a huge connection between these two fields, historically and even today. So, when I saw the flyer, Robert had asked me to send me a picture of myself for the flyer. And when I saw this, I thought, let's see, oh, that must be, again, that guy must be Robert. And this is a famous picture called the school of Athens with my rafial, it's fresco in the Vatican, and it really nicely made together serendipitously because I didn't send him a picture. But it brings together the theme of my interest, which is mathematics, philosophy, and the fact that it's a mnemonic, and another image of why this is religion and theology, given my current point, but that's a fact of an interfaith institute. We'll come back to this image at the end as kind of a summary, because in a way, it nicely summarizes what I want to say today. We're going to start actually with Pythagoras, a little before we go, because when I asked, it seems to be 495 was the most famous theorem, and I'll be saying the Pythagorean theorem. And this is a statue on the island of Samos for Pythagoras lived, and obviously it depicts the right triangle and his theorem, Pythagoras theorem. But Bertrand Roswell said at Pythagoras that he was intellectually one of the most important men that ever lived, because the whole conception of an external world revealed to the intellect, but not that the census is derived from him. So, Roswell looks back to Pythagoras as the touch point for this issue, between can we have knowledge that is not necessarily from the census, but is from our rational view. Plato was famous for having said the knowledge which geometry aims is the knowledge of the eternal, and not of the perishing or the transient from his Republic books out. And then that brings us quickly to Euclid, who was a student in Plato's Academy, approximately 300 before the Commodera. And Euclid was, as I indicated, a student there. He went on to establish the School of Mathematics in Alexandria, wrote the elements, and is probably most famous for today, especially for the axiomatic development of geometry. With five process, five common notions, he was able to prove 465 theorems. Now, this is gonna be old hat to the math people here, but this is what I had to tell the philosophers to remind them. He starts with common notions. Things equal the same thing, we equal each other. Equals added equals or equals, attracted or equal, things which coincide or equal, and the whole is greater than a part. Nothing really profound, right? Pretty common sense of all kind of notions. Then he added to that the posthumous, two points to turn the straight line. You have to be clear, straight lines can be extended. A circle can be drawn given any center point and radius, and all right angles are equal to each other. Well, a fairly efficient set of axioms so far that he was able to develop so much out of these things. There was one kicker, of course, and that's the famous fifth postulate, which in Euclid's elements reads, if a straight line following on two straight lines makes the interior angles on the same side less than two right angles, but two straight lines if produced and definitely meet on that side on which the angles are less than two right angles. In the character of this postulate, right from the very beginning, it's so different than the common notions and the others, and of course, it caused mathematicians for centuries, a lot of problems. Why should something that is so rich and not in content, and is far from intuitively out of this where the others seem to be, why is that a postulate and why can't we prove that? But anyway, that takes us down the story as it's in its relationship to philosophy as well. Of course, that postulate was rewritten in pre-fair in the 1700s and the one that we usually learned now in high school geometry. Given a line and the point not on the line, there is one on only one line through the point parallel to the given line. Out of this, as I indicated, Duke was able to prove 465 theorems, including the sum angles of the triangle equal to right angles, the pathetic green theorem, and brewed into similar triangles, prime numbers are infinite, volumes of cones, the cylinders, the spheres, and the ekenesis book 13 with the theorem about regular solves, which I want to discuss in the next. Regular solves are polyhedra where each face is a regular polygon, all the sides are equal and the angles are equal, and each vertex of the polyhedron has the same number of faces and edges. He proved that there are exactly five such regular polyhedra, and they were called a poitonic solves. Now, in the math group, you probably don't need to see the proof, but for the philosophers I did, who run through the proof real quickly, we'll start with what the five that were known in the time of Plato and Euclid, the tetrahedron, the four-sided, the cube, or hexahedron, the octahedron, the dodecahedron, and the acrosahedron, those 20 hexahedron sides. Now, Plato, they're called poitonic solves, because Plato refers to these in his famous biolog, the Tamaeus. And the reason he refers to it is that way back at the time of Plato, there was this desire to combine science with mathematics, or to understand the scientific knowledge that we have in mathematical terms, and that's still a very driving force in science today. If once you can get it, explain something in mathematical terms, you've explained it, or as Newton famously said, I don't know what grammar it is, but here's how it works, here's the formula that works. So that's kind of started back in Plato's time. If you can reduce your science to your mathematics, then you've pretty much done the job that you're at. And the science of Plato's day was a periodic chart of four elements, fire, earth, air, and water. We think that's pretty primitive, but in his day, there were people who argued that everything is water. And their argument was that, well, we see water in a solid form, ice, we see it in a liquid form, we see it in a gaseous form, so it covers all the states of, everything must be made out of this water. And there's other sort of intuitive reasons. When a fire burns, you see the air escaping from the smoke, you see the fire that's there, you see the crackling, which is the water that is leaving, and then what's left over the ashes are the earth. So wood and what it's burned, you see it's been made up of those four elements and you can decompose it. Anyway, Plato said that the fire was actually little atoms of tetrahedra. And earth, because the tetrahedra, by the way, because they're very pointy, it said the fire breaks things up. The cube represented earth because it's nice and stable. Air was the octahedron, water because it flows and it's almost round was the icosahedron. And of course that left the dodecahedron, the 12 sided, and he says less the dome that covers the universe. Now, just to take this a little bit further with Plato's Timaeus, and this is a quote from the dialogue, to the earth, then, which is the most stable of the bodies and the most easily modeled of them may be assigned the form of the cube. And the remaining forms to the other elements, to fire, pyramid, to air, the octahedron, to water, the icosahedron, according to their degrees of lightness or heaviness or powers of penetration. The single particles of any of the elements are not seen by reason of their smallness. Well, you got that right. You can all see how it was very well. They only become visible when collected. The ratios of their motions, numbers, and other properties are ordered by the God who harmonizes them as far as necessity permitted. And there's a fifth figure, which is made out of the 12 pentagons, the dodecahedron, this God used as a model for the 12-fold division of the zodiac. Now, he didn't just lay this out, he also did what we would call today in chemistry, he balanced equations. For example, water, which, remember, is the icosahedron as 20 faces. If you break it up, you get fire, which is the tetrahedron, four faces, plus two airs, which are the octahedron. So two times eight plus four is your 20. So there's got the equation balance there. A volumetric air divided in two becomes two fire. Two and a half parts condense, and a very condense in the one of water. An equation like that, we would like to make those whole numbers, so we would probably say five air, which is 40 faces, equal two waters, which is also 40 faces. So he had a whole concept in his science of balancing the equations as we do now in chemistry class. The point I want to make is that the science in Plato's day has been replaced many, many times over, and we kind of all smile when we hear about Plato's science. But Euclid's proof is still valid today. The mathematics that was proven during the Greek era is still good mathematics today. Mathematics is the best and most certain script in reality is the idea that people have for... Again, this is for the philosophers, but I'll just quickly run through it, so we can just quickly see how Euclid's proof, quote, was written. He says, if we're going to look at polyhedron made out of triangles, and we're going to argue there's only five, let's look at what we're dealing with. He starts with, and I'm putting this, of course, in degrees instead of right triangles, but we start with 360 degrees in a circle and 180 degrees in a right triangle. And so in a equilateral triangle, which we're going to build our polyhedron out, each angle would be 60 degrees. The tetrahedron has three vertices on each, I mean, three faces on each vertex, and the sum of the angles then would be 180 degrees. So that's consistent, though, that works. And the octahedron has four faces on each of the vertices, and the sum of those angles are 240. And then a cosahedron has five triangles on each, and you take five times 60 degrees, and you see we get 300 degrees, that's still possible. But if we try to make something out of six triangles, the sum of them at the vertex would be 360, and so it would be flat. They could not form a three-dimensional form. And then he does the same thing with the cube. Of course, now we're dealing with 90 degrees, and the sum of three of them at each vertex would be 270. But if we try to make one out of four squares, it would be 360, and again, our figure would be flat, and it would not have three dimensions. Then we go on to the dodecahedron, which is made up of pentagons, and each vertex has three pentagons. And the angle of each pentagon, if you don't know of hand, we just break it into three triangles, and add those up, and divide by five, and we see that each angle is gonna have 108 degrees. And when we have three of those, we end up with 324 degrees, which is still possible. But if we tried to put four together, it would exceed 360, so we're not gonna be able to be a convex volleyball. And then we go on and say, what about hexagons? And we thought of it as equal animal triangles, and you can break that up into equal animal triangles, so each of those angles would be 120, and of course, three of those give us one 360. And anything larger than that cannot be a building block for a polyhedron because it would have too many degrees at the vertex. Anyway, that's just a real quick run of Euclid's proof, and that concept still holds today. That's still good after 2,000 some years. So, they knew at the time there were five, and the issue for Euclid was that they were to spill ballot, like the science is not. And the impact that this had on mathematics and our knowledge in general is this idea that mathematics gives us certain knowledge. It's not knowledge that will change. Once we've proven something, we now enter the realm of what Plato called the eternal. We have eternal truths that are not subject to change. There can be no better description of reality than what mathematics provides. Mathematics was seen as the perfect science. It was seen as a science in the end. At the time, mathematics was not as we see it today, a model building, but it was a actual description of reality. It was the queen of the sciences. Gal, this continued on into the Renaissance. Galileo said, the great book of nature can be read only by those who know the language in which it is written, and this language is mathematics. Kepler said, the laws of nature are but the mathematical thoughts of God. Kepler, by the way, was still intrigued with these Platonic solids to the point that his first attempt to provide the laws for the orbits of the planets around the sun, he attempted to embed circumscribe around the Platonic solids these spheres, and then around that one, another Platonic solid, and he found that the ratios between them was fairly intriguingly close to what the empirical measurements were, but it didn't work very well. Fortunately, he gave up on what's interesting to me is that here, you know, 1500 years later, more than that, they were still trying to make sense out of the Platonic solids as a principle of explanation for science, for all we were discovering. Even as Plato did, Kepler was trying to do it. He did find it like that, and said, it's not circles, but the Greeks thought that the circle was the perfect figure, and so the circle must be the way the orbits would go. He gives up the circles and says, let's go for ellipses instead, with a sign out of the foci, and he gives up constant speed. Another principle in the Greek ideal was not only the perfect figures, but also constant speed. And because that speed is not constant, the planets go faster as they get closer to the sun, but this constant is the area that is mapped out at any given time frame. And if you think about this conception, why in the world would area of a two-dimensional ellipse have anything to do with the speed of planets going around? I mean, the conceptual break that he did with his laws of planetary motion is profound when you think of the thousand years plus of Greek ideas of constant speed and constant radius, or the circle is the perfect figure. It's the area that's constant. Descartes goes on as a philosopher, we know him as a mathematician, that the philosophers like to claim him, too. And what's interesting about the connection here is that for Descartes, mathematics is a model for the best kind of knowledge. Descartes was living at the time of renaissance, reformation in Europe. A lot of things that were believed to be eternally true were kind of falling apart. He traveled around Europe and elsewhere to try to get him broadened his education. Finally realized that there was really nothing he could put his stakes down for that would be certain, except mathematics. He found mathematics was the model. In his discourse on methods, of which, by the way, the appendix was called a lot of geometry, or alec geometry is actually just an appendix to this philosophical work, he says it's only the mathematicians who've been able to find some proofs, that is to say some certain and evident reasons. I was especially delighted with mathematics on the account of their servitude and the evidence of their reasons. And so he set forth his four points on method. These are what they are, and as mathematicians we will recognize them. Not to accept anything is true, which did not clearly know to be true. You know, there's euclids based in common notions and in prostitutes. Divide each difficulty into as many parts as possible. In an orderly way, begin with the simplest objects, gradually climb right up to the knowledge of the most complex. And for it, make my calculations throughout so complete and my review so general that I would be confident of not omitting everything. I'll put a set of guidelines for any math students. Those long chains of reasoning, all simple and easy, which geometries have obitually used to reach their most difficult proofs, gave me occasion to imagine to myself that everything which could fall under human knowledge would fall in the same point. Any kind of, oh man. As a solution of S, as his method for philosophy. Even you know, the great skeptic who questioned everything, including causality and everything else said, though there never were a circle or a triangle in nature, you need to understand that these were abstract things that we never received at all because it's two dimensional. Anyway, the truths demonstrated by Euclid would forever retain their certainty and evidence. So even a great skeptic, like human, sees mathematics as that certain truth. Fainless mathematician Kant said, all probably mathematical judgments are synthetic a priori. For example, if you review your introduction to philosophy, he divided things which were analytic and synthetic. And analytic were things like mathematical statements are all, all bachelors are varied, things that are true by the very nature of the words. Synthetic are things that are true about nature. They're not true just because of the way the words, the meanings of the words. And a priori, those things that we know before we've experimented, a posteriori are things that we learn from experience. And Kant famously said that mathematics is synthetic. It's not true just because of the meaning of the words because there's nothing in all triangles that have 180 degrees. There's nothing in the meaning of those words that would make it true. It is something that's true apart from the meaning. And yet it is not, and yet it is not something that we find through measurement evidence. We don't go out of nature triangles and then come to that conclusion. So his whole idea of mathematics is synthetic and a priori. And that was an important part of his philosophy. In every specific natural science, there can only be found so much science proper as there is mathematics presented in it. Now some of this is still true in science today. No people still. When you've got it down in the mathematics, you pretty well got it done. The science of mathematics presents the most brilliant example of the extension of the sphere of pure reason without the aid of experience. And as this debates through the history of philosophy between does our knowledge all come through the senses, the empirical senses, what we can measure and experiment on, or is there knowledge that comes through reason that is not dependent on the senses? And this is the, you can describe the whole history of philosophy as a debate between these two. And mathematics plays that pivotal role always in the middle of that debate. Is mathematics empirical? You say no, it's logical. And does that mean therefore, is that in an argument in the case of the rationalists and the rationalists that argue, of course, as did Kant and Descartes and many others. And the empiricists would say, well, you know, there's other ways of looking at it. They have to account for mathematics, which is sometimes a tough job. How does mathematics function if everything has to be empirical? Anyway, I'll come back to the summary a little bit later, but it's really clear here in Kant and some of these others appearing this time period. Of course, the thing that really changed the whole world happened after Kant and Descartes and the early scientists, and that is the development of 9th Napoleon John. Going back to the five postulates of Eulogy, we see that that fifth postulate on the parallel given a line and a point. The line on the line is one of our one line through the point parallel to the line. That was the one that had been questioned in many, many attempts throughout history to explain or to prove that based on the other, on the other axioms. Or to take other axioms in and prove this, for example, if you've wanted your axioms, is that all right, all triangles, angles add up to two right angles. If you have that as an axiom, you can prove the parallel posh. So there were lots of alternatives to the parallel posh, but nobody could get mild with just using the four posh lists and prove this. But anyway, so that then became a question that I won't go into all of the historical details as to how that came about with the word of psychoacup. But the big three for the first time in Clinton Java tree were Baus in Germany, Bolia in Hungary, and Lvovchesky in Russia, all working about the same time. There's a lot of Wolfgangs effort. God's sake, please give a theoretical lesson of sensual passion, because it too may take up all your time and deprive you of your health, peace of mind, and happiness of life. Another old historical footnote here is that Wolfgang Bolia sent to Gauss his son's work, and Gauss said, I would like to praise it, but if I did, I'd only be praising myself, because I had actually worked that out years before. You know what, what a rejection matter. God. Anyway. Then Riemann came up with the alternative from us. Assume there's more than one parallel line, and you can get a consistent geometry. What if you assume there are no parallel lines? And of course, as you all know, in the Lvovcheskyan or Bolia non-inclidean geometry, triangles have less than 180 degrees, and in Riemann's geometry, which is modeled by the steer, triangles have more than 180 degrees. Now, Morris Klein, in his mathematics, Western culture, in his big, huge history book, says the importance of non-inclidean geometry in the general history of thought cannot be exaggerated. Like a Permacus heliocentric theory, Newton's law of gravitation, and Darwin's theory of evolution, non-inclidean geometry has radically affected science, philosophy, and religion. It is fair to say that no more cataclysmic event has ever taken place in the history of all thought. But in the math theory, we know that non-inclidean geometry was a very creative point which a whole lot of new kinds of mathematics was developed. And we celebrate that. But Klein is arguing that it was such a shock, not it was the mathematics or famous mathematicians who refused to be considered non-inclidean geometry. But he's saying that it also affected science and philosophy and religion. And the big thing here is why didn't it have that huge effect? Because for 2000 years, there was this concept of eternal truths. And if you question whether there's eternal truths, just go check with your local mathematician and he'll explain it to you. Here's the eternal truth and how that's clear. Okay, now let's go on. So when you have more than one geometry and you no longer can claim that you've taken geometry, it is the actual description of reality. Not just the consistent system or a model, but it is the actual description of reality. If you could no longer claim that, then what can you claim? Where is absolute truth? What do we do with this? Geometry is the knowledge of the eternal. What happens then? And the influence of that over 2000 years was profound and client recognized it. It led to a rethinking of mathematics. The non-inclidean geometry took place during the 19th century. And at the end of the 19th century were the great foundations of mathematics debates. And that's a whole lecture itself, so I won't go through the details, except just to highlight who some of the major thinkers were. Logicism was developed by Bertrand Russell. Intuitionism by Brauer. And Formalism by David Hilbert. If you want to talk about those, we can, but like I said, that's a whole other lecture. But I want to jump now to Bertrand Russell because my talk is from Plano to Russell. And in particular, I've subtitled, I said that this is the 100th anniversary of the publication of the second volume that is famous for keeping in mathematics. Well, I did this for the philosophy grant, but it's the 100th anniversary of the volume one. Well, I'm keeping in mathematics. The timing's been fine here. Anyway, Bertrand Russell, I want to tell his story, but I'm going to shift gears a little bit and move into a comic book form. It's nice to get a little change of pace. And if you're not familiar with this book on Mojic comics, it's a really fun book that tells the whole story of Russell, the white head, and I'll hold it all in some of the problems I've dealt with. I'll give you a little excerpt from that comic form. Here's Bertrand Russell's first wife saying, oh, Master Bertie is also displeased with philosophy. At least mathematicians try not to contradict one another. Not so philosophers. They are all great and all in total disagreement. They call this transphosphory. I want to find my way to reality, man. I want a method to acquire a certain knowledge. See that goal that was there from Plato, Nucleon, and it's still with Bertrand Russell. He wants a certain knowledge. If we unite the healthy parts of mathematics and the conceptual sophistication of the new logic, we can launch a powerful attack. You really want to build a foundation, not only for mathematics, but for philosophy. Well, he's sitting in on Alfred North Whitehead's class and Whitehead says to achieve any kind of certainty in mathematics when you must re-examine its basic assumptions, we must begin at the beginning and Bertie sits back and there, here, here. So he meets with Whitehead and says, sure it's kind of a certain way since Aristotle, but is it strong enough to deal with mathematics? Whitehead says, let's join forces. And Whitehead was famous for his book on universal algebra. And so Russell says, you can write a second volume of universal algebra. He says, no, write together a brand new book. So they work on what becomes pre-Kippian Mathematica, which is kind of an interesting bit of overconfidence, maybe. Pre-Kippian Mathematica was what the great Newtonian decided to call theirs the same thing. And he gets like, Russell gets excited, Whitehead, I've got it this time, I've finally done it, what the damn thing. And he says, real, here, you got it. What is it? It's proven, one plus five is two. On page 362, it says, and therefore, one plus one equals two. And the kid says, I don't get it, when you write 362 pages, let me play with you. It was such an enormous rethinking of logic and symbolic logic and the whole basis that it took that long for him to prove. Russell then starts, as things develop, he starts losing confidence in his own program. And says, to understand my predicament, remember that my profound underlying aid ain't that never changed, to acquire certain knowledge about the world. Same idea, somehow mathematics should tell to give us information about the world. Knowledge, which should only come from science. But science depended upon mathematics, which was a total mess, plagued by unproved assumptions and circular definitions. To repair it, a powerful logic was needed, but there wasn't one, and so it came to an end fast. He finally said, mathematics is a subject where we don't know what we're talking about, whether what we're saying is true. And this was actually an insight is important for all of mathematics, because Euclind tried, one of the problems with Euclind's elements is that he's trying to define everything. And he defines a point as location without extension. And that helps. The line is, breadth is length without breadth. And as we often point out to our students, if you try to define everything, you get into circular reasoning. One example I like, I think it's Oxford Extreme, that defines a dog as a quadruped of the genus Cana. You end up using more complicated words to describe simple words. And even if you have simple words, if you look at the meaning of those, it'll eventually cycle around. So definitions aren't necessarily circular. And Russell recognized that. He also recognized that our axioms are not always intuitively true. They are beginning ones. So we can't prove the axioms and we can't ultimately define them. So we don't know what we're talking about. And no matter what we're saying is true. In his portraits from memory as autobiography, he says, I wanted certainty, again this key certainty, in the kind of way in which people on religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. After some 20 years of practice trial, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge and do it all. The splendid certainty which I had always hoped for in mathematics was lost in the bewildering maze. So 100 years ago, they published their three-volume pre-Kipley Mathematica. He died in 98 in 1970. 42 years ago now approximately. But the question of mathematics in reality and the search that Russell, along with Euclid Descartes and many others, is still a question that we often honor. Does mathematics describe reality? Is mathematics just a game? As some of the formulas bring. Is mathematics meaningless? Are mathematical objects real? Are numbers real? Are points in line as real? Reality is one of the issues philosophers deal with. It's called ontology. Ontology, what is real, what is true? And so these are weird questions that a philosopher would want to publish to a mathematician. If mathematics is not real, it's... I mean we have this much certainty about mathematics and everything else is less certain. Mathematics isn't real. What's going on here? Questions of, is mathematics discovered or invented? Question of certainty in mathematics, realism. The status of logics is a formless intuition. These are kind of the questions we deal with in the philosophy of mathematics. And then finally the question of structural realism. In the last 20 years or so, there's been a new movement in that philosophy of mathematics called structuralism. And that's another level talk, which I don't have time to do today, but it's one that I have presented at some international leagues. And it's a very intriguing way of making sense out of this whole question of whether mathematics is real. And it really comes back to the picture that Robert picked for that notice. And this is, like I said, from Raphael's famous School of Ethics. It's a good way to summarize the part of this for you, I want to say. The person on the left is supposed to be Plato and notice that he is pointing upward. He is pointing to the rational truth, to the certainty that transcends our experience. And Aristotle is saying, no, it's here on earth, it's empiricism, it's what we can see, it's what we can touch, it's what we can feel, it's what we can hear, it's what we can measure. It comes from the sciences. And Plato says, oh, there is knowledge that comes from outside, it comes through our reason. And so this is a summary, and in a way, of what the role of mathematics has been in history of philosophy. Is this tension between can there be knowledge that is not empirical? Is there knowledge, even better knowledge, that comes through our reasoning and through our rationality that is not something to empiricism and to our senses, tell us. But since I have a little bit of time, I guess I want to race through this pretty fast, realizing I was under time constraint, I'm going to take the opportunity to do a personal post-scratch. And I want to talk about three famous philosophers, Wittgenstein, Russell, and Karl Popper. And Russell, of course, we've talked about, became not only a famous mathematician, but a famous philosopher. Karl Popper was one of the famous philosophers of science. And Wittgenstein was the philosopher who in many ways caused the revolution towards analytic philosophy, and then a whole new way of looking at philosophy in his philosophical investigations. Probably most people would argue is perhaps the most influential philosopher of the 20th century. A lot of interesting stories about Wittgenstein, too. But the story I want to tell is when Popper was invited to the Cambridge University of Moral Science as well. This is the talk that the greats had belonged to Russell, Witte, and lots of Wittgenstein. This was sort of the elite club in Cambridge, and you had to be nominated to be, and approved to be a member of the club. In October, 1946, Karl Popper was invited for the first time, he was not a member, but he was invited as an outside lecturer to give a talk, aren't there philosophical problems? Wittgenstein was famous for saying there really aren't any philosophical problems. It's just problems of language. We confuse ourselves with the way in which we talk about things. We talk about, I have keys in my pocket. Therefore, I have ideas in my head. Because the grammar is similar, we think that ideas are things like keys. If they're in your mind, then it's like keys in your pocket. And he says, his approach, in my philosophy, was to analyze things from a grammar and understand why they weren't really philosophical problems. And so, Popper, on the other hand, as a philosopher of science, was committed to the idea that science does, in fact, describe reality, and there's a pathology for it, falsification was the key term that Popper used. And he wanted to argue that there are philosophical problems. This whole story is recorded in a book that was published about 10 years ago called Wittgenstein's Poker. Because the seminar took place in Brathwaite, another famous philosopher, in his library, in Cambridge, if you've ever been there, the faculty, once you meet a professor, it's pretty nice. You have an office, you have your own, a joining library, fireplace to keep warm. And so, the seminar was actually taking place in Brathwaite's library. And at one point in the debate, in the discussion after Popper made his claim, Wittgenstein got very, very agitated. Argued against Popper. And said, finally, and to make his point, he grabbed the poker from the fireplace and was waving it, according to that. By the way, this story is very fascinating because there's eight different versions of what happened, depending on if you're a pro-Popperian, or a pro-Viccentian, as to who did what, and what was exactly said, but they turned out, you know, unpacked the whole thing. And finally Wittgenstein says, give me an example of a moral truth. And Popper said, one shouldn't threaten a guest speaker with a poker. Anyway, so anyway, this one's what happened. Now in Brathwaite's library, there was a set of the pre-Kipfium Mathematica. The three volumes, and I happened to have the three volumes of the pre-Kipfium Mathematica. If you want to come up and take a look at it afterwards, I've even put a little sticky on page 362, where it says, from this proposition, it will follow, when the Arithmetical Edition has been defined, that one plus one equals two. But the other interesting part of this is that if you look in the front page, this was our Bureau at Brathwaite's personal copy of the pre-Kipfium Mathematica. And when I left the dean's office in 2004, was it? The faculty, some of you were still here, collected some funds and bought me a bunch of philosophy books because I was moving into my faculty role. And one of the treasured things were the three volumes of pre-Kipfium Mathematica. There aren't very many original editions of this around, I don't think anymore. And not only have one, thanks to my faculty colleagues back then, but to have Brathwaite's when it was part of the story of Vickensites Poker, is kind of a special little personal thing for me. And so anyway, it for me, that kind of pulls together my own personal history, as well as my exploration over the decades in philosophy and mathematics and how these things sit together. So I think with that, any questions, is that? No, you can do it. Any questions for our speaker? Yes, I'm just trying to think, it seems to us in this presidency that the last three mathematics are further in place. Is that because of the strength of science and the strong of it, and the long lines of the last four to five years? Well, certainly the success of science has had a big impact. But I would argue that that's sort of a positive force on why we tend to see things scientifically rather than mathematically. But I would argue that the other part of it is the shift in the understanding of mathematics is no longer a science. It is not description of the empirical reality. It is something else. And whether it's a game or whether it's meaningless or whether it's, you know, what it is, we still don't have a good handle on it. But no one, very few people would argue that it is a description of empirical reality. And maybe a description of another part of reality. If one argues that there is more to the universe than matter, and science is primarily looking at matter, but there are other things like structure and structural philosophy, and all these things are things that mathematicians also like to talk about. If you say that their reality includes those kind of things, not just material things, then, you know, there may be a place for that. But it doesn't have the same role as it had. And I think that the case I would like to make is that really the turning point of that again as more clients would get with the That's when the triangle hold on reality in mathematics. So there's a real push and have been for quite a while that in the teaching I said K-12 mathematics, that it's real life. I mean, do you see that as part of the issue as well? It's real life? Yeah, that, you know, the question often becomes is, as you're teaching this particular lesson, how does it relate to real life? Well, that's me. I mean, I got, I might want to think about a little bit more, but my first reaction would be that's a pedagogical question. And I think pedagogically, of course, we want to teach things that are relevant to real life. And so, is mathematics relevant? Absolutely. In fact, if science is more dependent upon mathematics now than it's ever been. And so, does mathematics inherit its reality via science? You know, maybe. That's one way you could argue the case. Or is science reach a dead end when it puts things in mathematical terms? And it finally says, we can't go any further because we have now put it in terms that are no longer in parallel, but are rational. And that's as far as we can go. Science does not proceed. The reason, of course, when it doesn't proceed with an exhumatic base that is apart from the person's. And as I find it really interesting that people like Dawkins and others say that there is no reality other than what the scientists can discover. And yet the scientists use mathematics all the time. And they have discovered that through their interval method which is supposed to be the sum of all knowledge. You know, when they finally get to mathematics that's as far as they think they're done. They think, hey, we've got a better name. So anyway, your question is interesting but I would tend to think of it more in pedagogical terms. We want our students to relate to these things in real life terms. And I try to do it in college classes too. Do I have time for my foundations lecture? If there's any more questions, let's thank Prisik and Jay again. Thank you. If you're interested in the print hippie or those in that kind of panel, it's something a lot of all students