Unification & explanation, algebraic geometry / discussion

0:00 And so I thought that this would be a good venue for just sort of raising the issue of science. And that's what I'd like to do today. So let me start by giving you a sort of an analytic table of content. So, again, I apologize. The talk is going to be not very organized, informal. I'm going to read all of them. There's some very things that are written. But really, the point is to try to get the discussion going. but here's what I'm going to do. So, to get some focus on some things to talk about, I'm going to discuss some particular philosophical puzzles, as I call them in the philosophy of mathematics. So the mystery of multiple proofs is something like a good murder mystery or something like that, but it's nothing of the sort. So it's just on the traditional view, the role of a proof is to tell you that the resulting theorem is true. It kind of tells you what makes the theorem correct or what the correct rules of inferences are. And you can argue about how we have knowledge of the accidents and so on. But really, that entire theory is designed to account for the correctness and the ground for knowledge that the resulting theorem is true. But it's a salient and remarkable fact that in many cases, there are many, many proofs of the given theorem. And even after you have a number of proofs, you're still happy to find a new one. And so there's something that needs to be explained there. And so I'll come back to that. The other problem I'm going to say a little bit about is the problem of talking about conceptual advances, or the differences between conceptual and logical strength. So this happens, for example, when you want to discuss historical developments. I mean, it's common to say that thus and such was an important conceptual advance in the history of mathematics. So to take one example, let's think about the prime numbers theorem. So this was, you know, noticed and conjectured by Gauss. The phenomena was served and conjectured by Gauss and Legendre around the turn of the 19th century.

2:30 A lot of very bright people worked very hard on it, but there was very little progress until the 1850s when Chebyshev made some progress on proving this theorem. and then again things were silent well, there was a lot of work but there was very little progress until the 1890s and the theorem was finally approved by in 1897 and in doing so they made use of complex analysis in a very central and essential way so they made use in particular of Riemann's data function so they're following up on a suggestion by Riemann at this paper that he would do not. And so you want to say something like, well, you took that long because you really needed certain conceptual resources that Chebyshev didn't have, and Gauss didn't have. So you want to say, you know, what it is that people later in the century had once they had complex analysis that people didn't have before. The thing that doesn't work is to say, well, they got new assets. And it's not, it's not, so if you want to make sense of say that it couldn't be done before, and it could be done with these conceptual resources. This could is not a logical could. It's not that they didn't have the logical methods to prove the theorem, but rather something more amorphous. In both cases, the issues of understanding is lurking nearby. I mean, you want to say something like, different proofs of the theorem improve your understanding, or help you understand different aspects of the theory of the theorem, or developments in complex analysis gave you another way of understanding the number theoretic problem. And so you find yourself thinking about mathematical understanding. So then the question is, well, how do you start talking about this understanding? And I fixated on the word method to try to do so. So the point is that one should think of mathematics not as being a body of facts, but more as a way of doing it. So understanding mathematics doesn't mean knowing lots of statements, knowing lots of theorems, But it means knowing what to do in various contexts. It means knowing how to behave appropriately in certain situations.

5:00 It means knowing how to respond appropriately and act appropriately in certain contexts. Put differently, mathematics do not have a body of knowledge of facts, but a body of competencies or faculties or abilities. In which case, I take the philosophical task, at least in parts, to describe these concepts or these faculties, to explain, you know, to understand what they are and how they relate to you, and so on. You know, what they do for you, what they're meant to accomplish. And so the first thing you need is sort of a language to discuss this. And so I will try to give some ideas along those lines in part two. But furthermore, developing this will require, so look, as I've described the comments, it's just incredibly broad and vague. The other thing you need to do is really focus your efforts and find more particular questions to work on. And also focus on the data that you're trying to explain. So you should be aware of what it is you're trying to explain. And so that's, as time allows, I'll talk about that in the third part. Okay, so for the first one, so for the mystery of multiple truths, I'm going to read a snippet from a paper written called Mathematical Method and Proof, which should appear in Synthes sooner or later. There's a copy of it on my webpage. I'm just going to read a little bit from the very beginning. is generally acknowledged that at least one goal of mathematics is to provide correct fruits of true theorems. Traditional approaches to the philosophy of mathematics have therefore, quite reasonably, tried to clarify standards of correctness and ground emotion of truth. But even an informal survey of mathematical practice shows that a much broader range of terms is employed in the evaluation of mathematical development. Concepts can be truthful, questions natural, solutions elegant, is powerful, theorems deep, proofs insightful, research programs promising. In so far as judgments like these channel the efforts in resources we devote to the practice, it is both a philosophical and pragmatic challenge to clarify the meaning of the term. Value judgments applied to mathematical proofs provide particularly interesting examples. For, on a traditional view, the role of a proof

7:30 to demonstrate that a theorem is proved. But it is very often the case that new proofs are valued, a fact that is rendered utterly mysterious by the standard characterization. Salient examples of the phenomenon are Dedeck and Weber's algebraic proofs of the Riemann-Roch theorem, the Selberg-Erdic proofs of the Adhemard-De-Levelet percent-time number here, or the 150 or so proofs of the law of quadratic reciprocity that have been published in Schauss's distribution as Earth meditate. But the phenomenon is ubiquitous, from the most elementary proofs to the most complex. So this number, 150, you know, for proofs of quadratic reciprocity, it's a number that's been kicked around. And so there are actually papers appearing, you know, the 151st proof of quadratic reciprocity. But in a very, very scholarly work, Franz Lundermeyer has written a book on reciprocity laws. And in the appendix, he actually documents 196 published which proves the law of quadratic reciprocity. And so, you know, just to put it simply, I mean, if you're trying to tell the history of quadratic reciprocity, well, you know, in 1796, oh, we're, you know, here from 1796 to 1808, what Gauss proved it. In fact, he proves it six times. In 1829, Kaussi proves it. 1830, Jacobi, 1835, Schley. 1844 to 1845, Einstein proves it, again, Well, I mean, this is sort of a funny historical story. What's going on here? I mean, look, so presumably, after Gauss says it, we know that it's true. And after Kaushin and Joko, we really know that it's true. And so how do you explain this? This is just a very stony phenomenon that calls out for philosophical explanation. And I think we have a lot of intuitions, by the way. I'm not saying that it's a hard problem. I think that we have a lot of intuitions as to what's a savior. But that's a good thing. But, you know, we have a philosophical question that has a shot at having an answer, having some questions. Let's see. So simply, the challenge is to explain what can be gained from a proof beyond knowledge that the result of the theorem is true. Of course, one sense in which a proof may be viewed as constituting an advance is that it may actually establish a stronger or more general statement from which the original theorem easily follows it. But even in cases like that, we need to account

10:00 that the proof can also augment our understanding of the original theorem itself, providing a better sense of why the theorem is true. Such proofs are sometimes called explanatory in the philosophical literature, and there is a small but growing body of work on the notion of explanation in mathematics. I would use the term here only gentrally for two reasons. So I'm really cautious about using the term explanatory for these reasons. So first, the term is not so very often used discourse. So certainly, you can sometimes see that this gives you a better explanation. There are various cognates and similar words, but it's not. Well, certainly not. So second, it's certainly not the only term which is used to voice positive judgments among truths. So here, I prefer to remain agnostic as to whether there's a single overarching concept that accounts for all set positive judgments, or rather a consolation of related notions. And also, as to whether the particular virtues that I consider in this paper are best labeled explanatory. A further difficulty with respect to obtaining a satisfactory theory is that judgments often bear as to the relative merits of different truths. So that's why it's common to find a dozen specialists in the subject writing 13 introductory textbooks. So the best we can therefore hope for is a theory that clarifies the factors that underlie such judgment and helps and claiming the differences, for example, ascribing them to differences of content, purpose, or emphasis. OK, so let me set that aside now. So that was the first thing on my list, the mystery of multiple proofs. So as to conceptual versus logical strength, I'm going to push and read a snippet from another paper called Number Theory in Elementary Ehrlichism. And there I focus on the proof theorists' understanding that, in fact, a lot of mathematics can be carried out, or even, quote, unquote, most mathematics can be carried out in very, very weak, logically weak theories, so very, very weak fragments of arithmetic. So in the paper, you know, I talk about the, you know, the philosopher's focus on the axioms of Zermelo-Pranck, So set theory and stronger theories. But then I know that proof theorists have traditionally studied and been interested in much weaker theories than full set theories. And the general feeling is that most, quote, unquote, ordinary mathematics can be carried out formally without

12:30 using the full strength of CFC. So I'm reading from the paper now. So I should qualify these remarks. And in fact, I will do so in many ways throughout the course of this essay. One issue that arises has to do with the choice of formal of the informal bodies of mathematics and of your education. So you have to remember, you're talking about formal theories and formal representations, but you're trying to study mathematics informally, or informal mathematics. So when dealing with weaker theories, linguistic and axiomatic restrictions force one to pay careful attention to the way in which the relevant mathematical notions are defined. And there's the unsettling fact that the outcome of the analysis at one's choice of definition. So when you're working with very weak theories of arithmetic, you have to be really careful how you formalize the notion. Because depending on your formalization, you might get, well, it's provable to formalize it this way. It's not provable to express it that way. So for example, the question as to whether the mean value theorem of undergraduate calculus can be derived from a certain restricted theory may well depend on one's definition of the real number or of a continuous function. When it comes to finitary objects like numbers, finite sets, and finite sequences, however, issues of representation seem less problematic, because it usually turns out that various natural definitions are easily shown to be equivalent, or at least have equivalent properties on the basis of minimal systems of absence. So when you talk about formalizing theorems of number theory in common torts. There, any reasonable formalization of the statement of number theory should pretty much be the same. Various formalizations will be provably equivalent. So with this in mind, it makes sense to limit the playing field by asking which binatory theorems of number theory in common torts can be derived in a particular formal theory, restricting one's attention in this way, thereby of the analysis. OK, so it's useful to focus your attention on just finite, explicit, finite, commentatorial number theoretic statements and ask, what theorems can you prove where? Now, in the hierarchy of formal foundations for mathematics, Zermelo-Frankel's theory is stronger than Zermelo's theory, which is, in turn, stronger than higher-order arithmetic, second-order arithmetic, first-order arithmetic, and then primitive recursive arithmetic, in order of degrees and strength.

15:00 So in the paper, I say, in the next section, I will describe a first-order theory known as elementary arithmetic, EA, or sometimes Harvey Friedman calls it elementary function arithmetic. In the British tradition, it's sometimes called I, delta, and oct of X. So it goes under a number of names, but it's a very, very, it's a fragment of arithmetic that's even weaker than primitive recursive arithmetic. It's so weak that it cannot prove the totality of an iterated exponential function. So there's a function 2 to the 2 to the 2 to the 2 to the 2, x times. It can't prove that that's a function. And certainly, any mathematician can prove that that's a function. It just knows that that's a function. So it's a very weak theory of arithmetic. it. So from the point of view of any set theorist, EA is almost laughably weak when considered as a foundation for mathematicians. And just to flash some axioms up at you, so there, so here it is. Here are the axioms. I don't want to dwell on them, but just to point it So the language, as a symbol for zero, successor, so it's a theory of the natural numbers here. Zero, successor, addition, plus, times, and exponentiation, and less than. And most of the axioms are just quantified and redefining equations for the basic symbols. So zero is not the successor of any number. Successor is inductive. There's the recursive defining equations for addition, multiplication, and exponentiation. Less than is just defined, but x is less than y So these are just, I mean, there's nothing going on here. And then the only real axiom is the scheme of induction. So you have induction, but you have induction, a very, very restricted form of induction. You have the induction principle only from bounded formulas. So a formula is said to be bounded if every quantifier is bounded explicitly by some time. So roughly, any bounded formula is extracted properly that's checkable just by running through all the possibilities. And so you have the principle in the document 5 holds 0, and for every x, if it holds an x, it holds a successor of x, then it holds for every x. But only for bounded formulas. And so there are a number of senses in which this theory

17:30 is very weak. So I already mentioned one, that you can't prove the totality of an iterated x-dimensional. In fact, roughly any for all exist statement you can prove will be found by some term in the language. You have a finite stack of exponentials to balance the rate of growth of any pi-t statement. Another sense in which it's weak is that it has a finiteary consistency group. So this does, I mean, so we know that there's a limit to how much you can get with finiteary methods. But you can do the consistency of this in primitive recursive reading. It has a finite consistency group. So if you can do, you know, as far as you can do mathematics of its program has succeeded in that respect. So, but I said that, to the point of view of any set theories, this is an incredibly weak foundation for mathematical reasoning. But the proof theorist begs us to consider whether there is more TEA than BCI. From the point of view of binary number theory and combinatorics, elementary arithmetic turns out to be surprisingly robust. It can prove a lot more in this theory than one might think. And so much so that Harvey Friedman has made the following, so this is a very Friedman-esque type thing. So Harvey Friedman has made the following grand conjecture. So I put, you know, you'll see, I mean, it makes sense to put conjecture in quotes. But you have to imagine, this is, I took this from, you know, Harvey's expression. You have to try to imagine Harvey, those of you who know Harvey, you have to try to imagine him saying this. He sounds like a used car salesman. But so here is Friedman's grand conjecture. The conjecture is that every theorem, every theorem, published in the Annals of Mathematics, whose statement involves only fine experiment mathematical objects. So in other words, what logicians can call an arithmetical statement. So essentially, any theorem that can be expressed in the language of the theory I just described can be proved in elementary words. All of them. Every single theory. Every single theory. Now, presumably there are some theorems in the analysis of mathematics that are false or incorrect, so you have to read this with a grain of salt. It doesn't change the conjecture. Yeah, I guess it does. I wrote a conjecture conjecture. So Friedman's conjecture is a clear and pointed manifestation of the proof-theoretic attitude and alluded to above, namely that a lot of mathematics

20:00 can be carried out in this theory. Now, unlike most mathematical conjectures, this one may be spectacularly true, spectacularly false, or somewhere in between. Since the conjecture was posted to the Foundations of Mathematics discussion group on April 16, 1999, it covers as a special case the specific conjecture that Fermat's last theorem is derogable in elementary arithmetic. Now, we're a long way from settling Making real progress towards that end will require finding a deep understanding of some of the most advanced methods of model number theory, with the proof theorist's penchant for developing mathematics and distribution theories. Originally, I had used the word fetish, proof theorist fetish, for saying, in mathematics. And Robert Thomas, the editor, said, well, penchant is a better word, so I didn't. So, but the conjectures are at least interesting because many proof theorists consider them plausible, whereas I suspect most mathematicians would lay long odds against them. So this is really the point I wanted to make. I don't want to, so in this paper I go on and on discussing the quote unquote evidence, or sort of the proof theoretical understanding that makes it not unreasonable to say this. And I don't want to spend the time developing that now. But I'll ask you to sort of take it on faith that if you were to read the paper, you'd come away convinced that this is at least not impossible. I mean, if you told me that there was somebody in the next room who had the answer in an envelope, I'd give you two to one odds, you know, in a reasonable, you know, I'd bet in favor that he did his performance from Albus Deinemann's performance in the last But again, so, but the point then becomes, well, so why is Hermann Glass Theorem hard? Well, it's not that it requires long axioms. I mean, if this is true, and for all we know it might be true, it's not that you need big axioms, axioms above the infinity. I mean, in principle, you may well be able to prove it in this group's theory. So then the question is, well, you know, what makes it a theorem? In particular, by the way, it is a theorem of logic. I mean, it has been shown that the prime number theorem, the etymological level A, the prime number theorem, can be proved in elementary references. So, again, if you just focus on that particular example, trying to make sense of the conceptual resources given to you

22:30 by analytic number theory and complex analysis and so on. logic is, I mean, this just shows you that logic is not going to do it. So coming back to, so those are two, the two philosophical puzzles. So now, so again, in both cases you want to say that somehow you need, different proofs give you different understandings, and somehow to prove things, again, you need conceptual resources that give you an appropriate understanding. So now let me try to give you some suggestions as how one can start talking about the topic. So now I'm going to go back to the first paper, Mathematical Method and Proof, and pick up roughly where I left off. And, yeah, so there, remember I talked about the problem of explaining why, what it is that you get from different proofs, why it's useful to have different proofs of the same theorem. So we do have some fairly good intuitions as to some of the reasons that one may appreciate a particular proof. For example, we often value a proof when it exhibits methods that are powerful and informative. That is, we value methods that are generally and uniformly applicable, methods that make it easy to follow a complex chain of information, or provide useful information beyond the truth of the theorem that is being proved. As a philosophical thesis, however, this claim is lacking. So I just said that we value proof when it exhibits powerful methods. So for one thing, it's vague. I have not said what it means for a proof to, quote, exhibit, quote, a method, let alone what it means for a method to be general and uniformly applicable. Nor have I said anything about how methods can render proof intelligible or the types of information they can convey. The first objection is that it's vague. A second objection is that the claim is rather toothless. few would deny that the attributes indicated they're generally sireful. If I'm going around saying, look, I'm going to prove to the world that mathematical methods, powerful methods, are useful, there's not much of a claim there. My goal here is to suggest that the first objection, so the one of vagueness, can be reasonably

25:00 addressed. In other words, it's possible to develop an analytic theory of proof and method that can do philosophical work, and in particular auditory terms. To that end, I will discuss a model proof that is currently used in the field of automated deductions, an attempt to list a corresponding terminology and framework for more conceptual analysis. So I try to, in this paper, try to tell a story of at least methods, so I focus on the problem of methods of understanding proofs, reading proofs. And the account is very much inspired by the understanding of mathematical proof interactive theorem for this, as was described in the talk yesterday. Now, if this is successful, the second objection, so that was the one that is obvious, or not a very strong claim, the second objection noted above will instead become an asset, which is to say, insofar as the terms can be made sense of, the result will be a philosophical claim that stands a good chance of being correct. I mean, it's clear that methods are powerful and we benefit from proofs by learning new methods. So if you can develop a good theory of methods, then you'll just give me this. There's got to be right. And that would be the case. Okay, so let me set this aside now. Again, so I'm not going to tell you the story that's developed in that paper. but as I mentioned it was very much built on the model of proof that you get from an interactive theorem prover and roughly the idea is so when you work with a theorem you say I want to prove this theorem you declare the theorem you want to prove and the computer says okay, go ahead and what you do is you have to repeat it you say well okay, you do this and do that, so you give it a recipe and we should view this as a recipe formal proof of correctness. So you give it instructions, you say, well, do this, and do that, and the computer then goes off and actually constructs the formal axiomatic proof at the bottom. And so the way to think about this is when you declare the theorem, you have a certain epistemic statement, how can you meet this goal? And as you apply different, quote unquote, methods, you then change the epistemic requirements. So in other words, after applying the first rule of

27:30 interest, you have to show this and that, and you sort of you've met all the requirements of who's in theory is true. So it's very much based on the notion of you have a certain state and you're operating on that state. And so this is a kind of model that's commonly used in artificial intelligence. And in that, one of the two referees for this paper said, well, this sounds, I think he or she meant this as a criticism, this sounds just like strong AI. And I don't want to get into the sense in which, I mean, I don't want to try to make sense of what exactly the names of strong artificial intelligence are. So, in revising the paper, you know, I had to respond to the comments. I hear the referee a little pert, you know, response. But I am sensitive to this. I mean, the question of, is sort of providing this sort of fairly computational model, Is this going to give you the type of philosophical explanation, explanations that you want? And so before coming here, I thought about this, I wrote something that I'd like to try out on you. So this is the topic, so why understanding is like having an algorithm. And, yeah, so, okay, so let's just do it. So if this works, this might make it eventually into a paper, but let's just try it out. So it turns out happily with a function that you're not familiar with, or a variable function that you're not very familiar with. So this is an exercise in the theory of computability. So I start by saying, there's a cautionary exercise that's sometimes given to undergraduate students of taking a first course in the abstract theory of computation. It goes like this. So you tell the student that the Goldbach conjecture asserts that every even number greater than 2 can be written as the sum of two primes. To date, this conjecture has been neither proved nor reputed. Then you say, let f be the function defined as follows. So for every axis, f of x is 1 if the Goldbach conjecture and zero otherwise, and then you say, is this a computable function? Really, you have to believe me that I had no idea that this would come up in the discussion yesterday, and the uses that I'll put this example to are entirely different. There are three responses that are common. Two of them are reasonable,

30:00 whereas the third is not. The first response is the one that seems correct by modern mathematical standards. Either the Goldbach projection is true or it's false. If it's true, then after the constant function value 1, okay. And if it's false, then f is the constant function of value 0. In each case, f can be computed by a trivial algorithm. So f is computable, even though we do not as yet know which of the two algorithms we do. So one answer is yes, it's computable because it's a constant function. It's either the constant 1 function or the constant 0 function. The second response is to call the presuppositions of the problem into question and claim that the quote-unquote definition of that is false. One may feel that any reasonable definition of a function from the natural numbers to natural number should enable us to determine, in particular, whether or not f of 0 is equal to 0. The fact that the description above does not do so, therefore, disqualifies it as a meaningful definition. Indeed, this is the response that is correct out of construction to mathematics. examples like this are commonly presented as a critique of non-constructivism, showing that modern mathematics sanctions certain quote-unquote definitions that are pathfully unsatisfying. So thanks to yesterday's developments, I mean, this is important, because in that divided extent, constructivists present this as an example of what's wrong with classical reasoning. But I don't want to spend all the time talking about the two, you know, reasonable responses and trying to communicate between them. More into the third response, which does make sense. So the third response is that F is not computable because we do not have an algorithm that tells us whether or not the Golovkin conjecture is true. Students often say this. You often get the assignment back. It's not computable because we don't have an algorithm to decide whether or not that here is true. This is the response that, I'm supposed to inspection, does not make me much sense. Or, if the conjecture is true, the simple algorithm that says, yes, tells us that fact. And we have an algorithm that tells us it's true, you know, print yes. Now, and if it is false, an equally simple algorithm is saying no. Now, granted, we do not currently know whether the conjecture is true, but it seems odd to express this in terms of not having the algorithm. We usually think of an algorithm as something that provides responses plus a range of inputs.

32:30 What can it possibly mean to have an algorithm to answer a single yes, no no question? This is something that, you know, you have to set students straight, and this is something that... You want to get, of course, the point that when you talk about having a problem being computable, it's always a class of problems that you're computing, or deciding, or that's being computable or not computable. It is, nonetheless, the third response that I wish to dwell on here. On the surface, it sounds like a reasonable thing to say, and in fact, it's not hard to give it a more charitable reading. The Goldbach conjecture is one of a class of number theoretic statements that makes a universal claim, that is, an assertion to the fact that every natural number has some finitely checkable time. It says that every natural number can do with it. So given a natural number, you just check all the times left that in, check all the sums, to determine whether or not that particular number is going to hit. There is a general procedure for reputing assertions of this form when they are, in fact, reputable. You just search for a counter-exception. The problem, however, is that in case of what the assertion is true, the algorithm never falls. And indeed, the unsolvability of the halting problem tells us that there is no general algorithm for answering questions of the sort directly in every instance. So if we view this class of problems as being salient and implicit when the problem is posed. You give the students a problem to the students. Well, it's clear that this is a universal assertion. And if you sort of keep that in mind, you can reasonably interpret the students' statements as follows. So the statement that there is no algorithm for deciding whether the Bobak conjecture is true can be interpreted as follows. The Bobak conjecture is an instance of a universal number theoretic assertion. And there is no algorithm that decides in general whether such assertions are true. So again, to claim that you can't decide for this thing means that this is an instant, and what we really mean is this is an instant of a more general class of problems, and there's no algorithm for deciding more general classes. This type of locution is not uncommon. When we say that a schoolchild knows how to multiply 17 by 23, we do not mean simply that the schoolchild is capable of uttering 391. We mean rather that the school child knows how to multiply, or at least that he or she knows how to multiply decimal numbers of a moderate size and can carry out the procedure in this particular instance.

35:00 Think about it. When you say that, you know, that this child can multiply 17 by 23, I mean, you mean something more general. You mean that they can multiply. Similarly, when we say that someone knows how to bisect a particular angle or solve a particular word problem. In fact, the student can solve this problem in the book. We typically mean that he or she knows how to bisect angles in general, or solve a class of similar problems. So again, a simple observation is that very often when we attribute ability to do something particular in mathematics, we mean the ability to do something more general, of which the quoted example is a particular instance. Now, up to this point, I have not said anything about I'm just talking about just, you know, calculation or computation algorithms. But that's, you know, what I want to talk about here. Now, although it's common to speak of understanding a mathematical problem, solution, theorem, proof, or concept, there has been remarkably little work in the philosophy of mathematics that can help us make sense of such thought. Since the topic is vast, in this essay I will focus specifically on types of understanding that are usually associated with mathematical proof. But the reader should keep in mind that much of what I say should apply to other types of understanding as well. In particular, I would like to explore what it means to understand a proof, and what it means to say that a proof helps us understand the resulting theory. By the way, am I looking for a 9th proof? So let's explore what it means to say that you understand the proof, and what it means to say that the proof helps us understand the results of the correctness. The first type of vocution may be taken to include, in particular, knowing or recognizing that the proof is correct. When you say that you understand the proof, in particular, it means that you recognize that it's correct. Which is to say that it relies on true assumptions and background knowledge, and that its inferences are valid. The second type of locution, so namely saying that the proof helps you understand why the theorem is true, may similarly be taken to include knowing that the resultant of the theorem is true. But it seems clear that both locutions go beyond the associated knowledge. So it's not just knowing that the proof is correct and knowing that the theorem is true. For example, it is not uncommon for a mathematician to say, I have followed each line of the proof and you can see that it's correct, but you don't really

37:30 want to understand what's going on. It's very common to hear mathematicians. You can say, I can understand it, but I can verify it line by line, but I really don't understand it. Or similarly, after seeing a student recite a proof from memory, so just by rote, you know, imagine the student just saying, oh, then you do this, and so on, we may reasonably feel that the student does not adequately understand the proof. As far as the second type of research, namely that a proof enables us to understand and solve the theorem. It's very common in mathematics to see a new proof of a theorem valued, even when many proofs of the same theorem are already available. So that was the first problem I mentioned. Very often you have a number of different proofs of a theorem and you value each one. And so saying that you understand, that the proof helps you understand the theorem, it's not just that it helps you that the theorem is true. So when it comes to mathematical proof, it seems that understanding consists of something more than knowledge that the proof is correct and that the resulting theorem is true. We may therefore look to a philosophical theory to tell us what that something is. Attempts to make sense of mathematical understanding often fixate on the common experience of an ah-ah moment. For example, the powerful feeling we experience when we suddenly grasp the proof or see why the result is true. So we've all had that feeling when we're reading the proof and all of a sudden, ah, you get it, and it's a very satisfying feeling. It's a very potent feeling. And so it's common to say, ah, that's the type of thing that we want to describe. But surely these states of mind are not what the philosophy of mathematics seeks to explain. Any more than economics aims to explain our pleasure in finding a $20 bill lined in the street. I should say here, a 20-euro bill, which is even better, lined in the street. Economic theories tend to describe agents, preferences, utilities, and commodities in more abstract terms. An economic theory will do something like that. Though when combined with social, political, psychological, biological considerations, these theories might perfectly adequately explain the subjective appeal of cold, hard catch. In a similar way, we should expect a philosophical theory to characterize mathematical understanding in more outstanding terms, in a manner that's consistent with, but independent of our subjective experience.

40:00 So here's the key point. The claim I would like to defend here is that in at least one respect, understanding of proof is like having an algorithm to carry out the calculation, in at least one respect. when we speak of understanding a particular proof or theorem, often what we mean is that we have developed or now possess more general faculties that are instantiated in the case of that. For example, understanding a proof may entail, among other things, being able to follow certain types of complex high-level infancies that are commonly used in the associated sentence. So when you say that you understand a particular proof, one thing, later I'll give you a long list of things that we may associate with understanding of proof. But one thing you may mean is that, look, not only can I follow this proof, but I can follow a certain type of pattern of inference that's common across a range of proofs in the subject. Similarly, when we say that a proof provides a better understanding of a particular theorem, we may mean, in part, that the proof exhibits methods that enable us to prove similar theorems So our answer types of questions do a copy of the app. And so now, I'm trying to think how to jump around. OK, let me skip a little bit. So what I'm saying is that, again, when you say that you understand a proof, it's analogous. I'm not trying to conflate the notions, but it's analogous to having an algorithm that you're saying that you've got some faculties that are more generally applicable and in particular of my case. This analysis entails that understanding only becomes manifest in an agent's behavior across a range of concepts. So if you want to determine whether or not somebody understands the proof, you're not supposed to look into their eyes and see if you can see a glow inside. You're not supposed to measure their brain state or whatever. What you want to do is you want to see how they behave in a range of contexts that bear on the subject. So it may be tempting to identify understanding with the range of relative behaviors. Such a dispositional analysis of understanding has famously been put forth by Gilbert Ryle as a part of a more general philosophy of mind. Since Ryle's approach is commonly viewed as having failed, it's worth reviewing some of your criticisms to see what bearing they have on the more specific issues

42:30 addressed here. in his book, The Concept of Mind, you know, famously tries to develop a dispositional account of ascriptions of mental states. So he's trying to explain things, for example, when you say that a chess player is playing chess skillfully. I mean, the question is, what distinguishes it from either a machine that's just doing something mechanically or a novice who just happens to be thinking the right moves? And what Weil is trying to argue against is the picture that, When we describe a chess player playing chess skillfully, there are really two things going on. There's the physical act of moving the pieces. And then there are these mental acts of thinking and cogitating and moving the pieces and intending for something to happen. So the problem is that you have two stories of the action playing chess. There's a physical one and a mental one. And then there are all the notorious typical reasons of what the connection between the two. So Raoul refers to that as the Cartesian fallacy. And he tries to argue against it, saying, no, no, there's only one act. It's only the physical act. And so when you say that the chess player is playing skillfully, what you really mean is that you're making a disposition of the scripture. You're saying that you expect that in a range of contexts this chess player would do things that exist in understanding. And he does this, again, broadly in the philosophy of mind, for not ascriptions of belief and volition and so on and so forth. So it's a very behavioristic account and it's one that's taken to have failed and so what I'm proposing sounds somewhat like that I should at least respond to that. But I'm going to skip over my response to it. So the short answer is that, so first of all I review the critique of Braille's analysis some that are commonly put forth, and argue that they aren't as problematic in the particular case here. I mean, the difference is Weil is trying to give a general theory of Marx, and I'm trying to give a much more focused theory of mathematical understanding. So a lot of the problems that come up aren't relevant here. But even so, so let me just pick up and finish up in just a few minutes here. So perhaps the most compelling criticism of Weil's dispositional account. So I've skipped a number of paragraphs. criticism of Ryle's dispositional account is that even if we could

45:00 characterize the behaviors that have correlated with various mental states, identifying the mental states with the associated behaviors simply tells the wrong kind of story. We expect a philosophical theory to provide some sort of causal explanation that tells us how intelligent and intentional behavior is brought about. So it seems unsatisfying to identify knowing how to to play the piano with successful performance. When what one really wants is an account of the mental activity that makes such performance possible. So while we say knowing how to play the piano just means that when you sit with somebody in front of the piano, they play it. But to say that knowing how to piano just means ability to play the piano in the right context. It just doesn't sound like the right type of theory. You really want some explanation of what it is that brings about the performance. In the case at hand, so namely, with respect of theory to mathematical understanding, we would similarly like a theory that explains how proper understanding enables one to function mathematically. This insistence is not only an intuitive appeal, but also a pragmatic one. For example, in so far as our theory is to be relevant to mathematical lecture history and pedagogy, we would expect it not only to characterize the outward signs of mathematical understanding, but provides some hints as to how they can be encoded from the plot. A theory of understanding should not just tell you how you can measure whether or not a theory of understanding, but give you some clues as to how you can foster an understanding. Similarly, I take it that a theory of mathematical understanding should be a service to computer scientists trying to write software that exhibits various types of competent mathematical behavior. Even if we set aside the question as to whether, under any circumstances, it is appropriate to attribute understanding to a computer, we might expect a good philosophical theory not just to clarify and characterize the desired behaviors, but also to provide some guidance for bringing them to that. For that reason, I expect that a satisfactory theory of understanding will transcend behavioral description. Nonetheless, I think it is reasonably safe to say that developing a theory of mathematical understanding, will at least require it to come to terms with the relevant behaviors and the starting point. Even this we claim, however, has strong implications for the philosophy of mathematics. So it's a three-line argument. You just go over it and throw up your hands when you have an objection. So line one, it would be nice to have a theory of mathematical understanding, or theories of

47:30 of mathematical understanding. Claim two is that, insofar as the theory of mathematical theory of understanding can be successful, we need to at least come to terms with the conditions under which we typically ascribe understanding. That's it, and then apply multiple. It's something that we really need to understand, the conditions under which we say that somebody understands something. But even this reclaim has strong implications for the philosophy of mathematics. When we pause to reflect on what it means to understand mathematical proof, a rowdy and insistent co-forge of quick-tiered proofs. So now I have a long list, and this will be the last information you've done, so I don't have time to do it. Read this list and then stop. But here are lists of things just off the top of my head as I was writing this, various things that understanding the proofs made it together. So one, the ability to respond to challenges as to the correct answer to the truth. I'm still in details of justifying this as I skipped a few points. So if you say you understand the proof, and I say, well, And for example, I don't see why this follows. If you understand it, you should be able to fill in the justification. You should be able to say to me, well, this is from the definition, or this is invoking this lemma, or this is, you should be able to fill in the details that are left out and justify anything that I asked you to. Second, the ability to give a high-level outline or an overview of the proof. So if you understand the proof, you should roughly tell me what the idea is. How does the proof show? The ability to cast the proof in different terms, say, terminology. So maybe it's algebraic, but not essentially algebraic. You can sort of translate out the very uses of algebraic. So I might ask you to put it in slightly different terms. Well, Nick gave a good example of things like that. If you understand a proof, you should be able to relate it to a picture. That's another one might be trying to do. The ability to indicate key or novel points of jargon, Let's separate them from the steps of the three-folds. You should be able to look at it and say, here's the point and point, and here's the point and point, and this is just a three-fold argument. The ability to quote unquote motivate, that is to explain why certain steps are natural or to be expected. So very often in hearing a mathematical talk, some people will be, you know, driving a little bit, let me, before this next step, let me motivate it. And then what you get is some kind of story as to what you would expect to do or what why this next step seems natural.

50:00 So that's motivating the definition, and motivating this definition. Again, it's something we associate with that. The ability to indicate where and approve certain of this theorem's hypotheses are needed, and perhaps to provide counterexamples that show what goes wrong with various hypotheses. So assuming A, B, C, and D, you get E. Well, why are all these things necessary? Well, this one is used here, this one is used here. If you leave this off, here's a counterexample. assume that it's complete, then, you know, you get this kind of thing. The ability to view the proof in terms of a parallel development, e.g., for example, as a generalization, adaptation of a well-known proof of a similar thing. Oh, this is just like, doesn't such, or this is a generalization of this particular thing. The ability to offer generalizations, for example, to suggest an interesting weakening of the conclusion that can be obtained with a corresponding weakening of the hypothesis. Or another example that's usually associated with understanding is the ability to calculate a particular quantity or provide an explicit description of an object whose existence is guaranteed by the truth. So, for example, if you prove that a matrix with non-zero determinants is very known, and once you understand the proof, you should be able to calculate it. Or, you know, the theorem of sorts, space with certain properties, well, you should be able to describe such a way, give an explicit one. And so on. So this is just a rudimentary list, and it's likely that you have plenty of other suggestions. But the philosophical challenge is to characterize these abilities with clarity and precision, and then try to fit them into a structure to make something. Thanks. Thank you very much for the school. I just wanted to ask you what is your present and some better, if I may, your statement that I'm telling you about hiding and hiding. Because I look at... Only in that one particular respect. I'm not trying to say better from the point of view that it means to have an idol.

52:30 The reason why I start from this time is that for me, having an algorithm, there is not any place that you understand. And now the question is that you have to understand algorithm to understand who, I suppose. So I need to understand better how you relate, having an algorithm to understand the truth. Because it seems to me that the wrong algorithm perhaps should be quantified, or the way in which you have an algorithm should be quantified, and for us to understand better how it needs to understand the truth. Yeah. So in fact, I just left out one paragraph as I was reading that. That just, you know, tried to clarify this. So let me make it clear that I don't mean to identify the notions or conflate them. I'm only using it as an analogy. And so let me spell out carefully that I use my analogy. So just as when one says that one has an algorithm to do something to solve a particular problem, what you really mean is you have an algorithm to solve a more general type of problem, So now, when you say that somebody knows how to multiply 13 by 27, you really mean that they can do something more general, and in particular they can do that one in the student. In a similar way, what I'm suggesting is when you say you understand the proof, what What you really mean is you've got an ability or faculty to do, or multiple, to do a broader range of things, of which, in this particular case, you're doing particular institutions. So for example, I just sort of read off the list. So when you understand a proof, in a particular proof, then you say you have a way of recognizing the correctness of the example, in particular, recognizing the correctness of the infancy. And it's a general method of recognizing correctness that works in related proofs and other proofs. You're saying that, when you say you understand the proof, I have a general method of recognizing correctness of infancy that, in particular, I'm applying it to the situation.

55:00 There's only the analogy in the picture. So that means that you refer to a particular way of describing the analogy, which is to describe a general algorithm in the framework of a pantheon. Right. Exactly. So, but there are other ways of providing the analogy, but perhaps... Yeah, so I don't want to make the analogy a piece from it. Let me say that there's also, I mean, it's just an analogy, there's one, there's at least one selling respect in the analogy, in which the analogy is not a very good one. Which is to say, either you have an algorithm, either you can calculate it, you can't. You know an algorithm to do something, or you don't. It's a boolean, it's a yes-to-do. But on the other hand, we commonly think that those are degrees of understanding. So it's not, I mean, the analogy is flawed in the respect. So it's just that one feature, being able to do something in general, I think that's important to make you something of understanding. Two points. First, the mystery of multiple groups. Any constructively mystery at all, because when you convince yourself of the truth of a proposition, you actually are building a proof object for the proposition in question. particular algorithm, and of course we are interested in different algorithms for solving the problem. Right, so that's part of the story. There are a lot of things that's one thing. So one thing is different constructive proofs are associated or give you different algorithms. Or comparing constructive and non-constructive proofs. Constructive proofs give you algorithms. So you may have non-constructive proofs, and then you find a constructive point, and you Aha! Now I have a constructive proof and estimation. That's one of many things that... I mean, so, when you're referring to the paper for that and many other types of things, I hope that's very good. Well, the second point is concerning your example with a go-to conjecture. You said that the first two answers were okay, that you were very happy with the students getting the paradigm. Well, I'll say yes, that's okay. To my mind, it's perfectly all right. suppose that you have a group of G or not G, where G is go by conjecture, then you can define the function F from G,

57:30 but it has two arguments, it depends on P and it depends on X, and we don't get the function on X only unless we have a group of G or not G. So what I said is that, and I think the idea makes sense, but at least the expression is a little bit odd to say that you don't, I mean, because it's just a particular, I mean, it's just a term. It's not a function, but it's a particular term. So what, again, we don't have an algorithm to decide whether or not the theorem is true. That is the proof of theory or not theory which we have to happen. Then why would you call it an algorithm? We don't have a proof of theory or not theory. Right, and such a group, what is it for such a group which terminates either with a yes and a group of B, or no and a group of non-B. Okay. So you seem to have to stop this. Okay. So I'll just, okay. But again, as I say, so, all right, so, so, so perhaps that, okay, so I'll have to revisit that. But still, the point that they've talked about having an algorithm is, again, to provide responses to the main issue. It's a little bit unusual to talk about having an algorithm to solve one problem, one yes-no question. Do you have a question? one more questions at least, so I'm asking a good question, do you understand? Okay. It's sort of a common question I share. I'll digress. This has walked his mind when I decided I wanted to focus just on complication, so I talked to my students. You're reading a student's other books on the Turing test. Anyway, this made me think about all of these issues, because it seems like in a way which you're trying to do, it could sort of come up with

1:00:00 you know, sort of behaviors Right. You're doing this in the mathematical Right. So I'm trying to distance myself from, so, again, I'm leaving out things that I don't mean. Let me talk to you afterwards about that. I like to distance myself from that a little bit. And I'd be interested in elsewhere, you know, to see that. But the second thing, so that's just a little comment, there's a little connection between sort of, you know, behavior as part of the debate. Right, but I'm trying to stay away from this, there's a breadth of the plot, because I think I focus on it. I'm sorry, I make you happy. But the second thing is, look, absolutely sympathetic to all of this, but it seems like some of this issue would say, well, really, there are some more foundational requirements one of them seems to be, in my view, in this world of Jenny's opinion, the term, think about the story, that this notion of high-level, you know, being able to recognize and point out sort of high-level inference patterns, something like that. If you want to be able to, if you want to get away from the notion of an algorithm when you're saying because you're not looking for a mechanical low-level procedure, you're looking for a high-level inference. And I'm wondering, again with a big question, to what extent do you think some of these other fine abilities would quite rest on that one? So if you think that one might be a foundation of science. I'm sorry, which point? The ability to recognize and point out some sort of high-level inference patterns. Yeah, so there are two issues here. I mean, one is you don't want to be tied to an algorithmic language, you don't want mathematical and the other issue is, again, as I've put the issues, it's just so much to be brought. In the particular paper about mathematical methods, I focus very narrowly on methods of making relatively small incremental steps. But as I mentioned in the introduction, there are, I think, methods of looking for a proof in general, attacking, you know, Riemann hypothesis, solving, and there's a, so I better not say much more, because I don't know, but I think really what we need to do is focus on the

1:02:30 and be sensitive to the issues we face. But I think we're going to do that. We have so much to do. Thank you. Yeah, I wanted to ask you about the comment you made at the end, about the fourth end, about having to fill in the test and how that communication of understanding. That seems to be a rather complex word and a rather context sensitive, and actually a rather subtle manner of things. So let me give you a couple of examples. Suppose that you have the proof and it's fully formalized, but then you get the skeptic and the guy who's a classical with Dr. Young. Can you explain that to me? Or a stand elimination. I don't quite see that. Somehow, I don't hope we have to demand that the person be able to explain that much. Oh, right. But this is something that I do. The mathematical method of proof is 55 journal. So this I do go on at length. And again, I'll try to just say something brief here. But again, I could go on and on for as long as you wanted to listen. Okay, so I think that back when I was in graduate school, I can understand a lot, I understood reasonably well, a lot of the proofs in Rogers, both on computability, even though I probably couldn't, especially these fields to interpret people, right, even though I couldn't fill in, I couldn't actually come up with a sorting machine because I'm not very good at that sort of thing, but you know, we have to recognize it. And I'm going to say, oh, yes, it's recursive. Now, you could just say, OK, well, in that case, Church is decent to supremacist. But you don't really have to do it that way. You're just doing this as a sort of abbreviation, right? But this is one case where I just couldn't believe it. At least that was a fact. But at least you had the impression that you were good at recognizing whether or not something was computable. No, no, no. It just be shoveled with the arm. That's right. I could recognize it's computable, and then I would conclude that it was recursive. Right. OK, so recognize it's computable in the intuitive sense Yeah, and then I would conclude that it was recursive. Right. Again, I'm not claiming, you know, it's a great problem here. Right. No, we all do this. Right. But it might be that in the 50s, or if I were taking this course in the 50s or 60s, then you really could say, no, we're just understanding the difference, because you can't actually write the sewing machine. Right.

1:05:00 Right. So, but it sounds simply that there are two facets here. One is recognizing intuitively. I mean, there are general features that you use to recognize that something intuitively There's another faculty for, you know, writing out either descriptions of the or views, and you can do one and not the other. But I mean, it seems that there's room in the theater for accounts and things like that. I mean, there's a faculty that you can't be able to do. Right. I mean, the general point is the same as the other one, that it's just a subtle matter as to what exactly is challenged, or would you be required to put it? Yeah, so I'm not saying that, so I'm not looking for a pre-definition of understanding of group discussion. So a theory of understanding, I think, shouldn't, needn't necessarily a specific, crisp definition of understanding. But it should be at least a characterization of the various aspects of understanding. You want to fill in details, but at least ultimately you want to fill in details like this one. You know, like how much detail do you fill in and what circumstances? Yeah. Okay. So, yeah. I guess I'm not sure I understand what you're saying when you make this comparison to understanding something and having an algorithm. But it seems to me that there are two signal aspects in which understanding is not going to happen. I just think there are two key things, two kinds of reflections that are essential to understand that are not typical of what it is. that's the primary reflection and then there's another reflection on that you have to you have to have knowledge with respect to that description of what you're doing that is appropriate Okay, now this notion of appropriate is a complex line that, you know, includes things like, you know, no wasted effort, stuff like that. But it seems to me that that's kind of the logic of understanding, is that it's not just doing something, it's reflecting. It's having a description of what's being done, and then it's having a reflection on that, devaluing the description of what's being done.

1:07:30 So first, again, let me emphasize that, again, the analogy with having an algorithm is just bringing up a sense of general and specific. So I'm saying that one aspect of understanding is being able to behave appropriately in a range of contexts. You're saying that it seems that there's more understanding than that. I'm not convinced. I'm not convinced. I mean, but I'm not unconvincible. So I'm not convinced. It seems that one can't have understanding without being able to vocalize what you're doing. That didn't mean to say that you could vocalize what you're doing, but you have to have some representation of yourself, to yourself of what you're doing. Whether you can to express that in the words of politics in the same circle. And that's an interesting thing about music. People talk about it. Thank you, sir. So, who is Kashina Pink? And who is the next? So, we're going to... Oh shit. Thank you very much. So, my talk is on Mathematical Explanation, and as you by now probably saw, Mathematical Explanation is an area that overlaps, to some extent, with some of the topics that were discussed here today on visualization, and obviously with Mathematical Understanding, as presented by Jeremy.

1:10:00 Now, I've published other papers before on this topic, and so I'm not going to present the topic as it were, motivating some ecological choices, arguments of why mathematical science is a good subject to science. This is going to be a contribution on the topic, so it would be a more specific, focused topic. The paper is a joint paper with Johannes Athler, who Okay, so there are, here is the plan of the talk. What we aim to do is to present teacher's theory of explanation, and that will be part of the challenge. It's not an easy theory, so we're really trying to give you a better outline so that you really understand what's going on in theory of explanation. We will give you some of the formal details of the model. And after doing that, I will introduce some basic concepts from semi-algebraic geometry. We'll look at the simple theorem and a variety of proofs. And we'll use this kind of space to show what, in our opinion, is really wrong with thinking that explanation is unification as a general thing, which is, in fact, interesting. So, there are at the moment two theories of mathematical explanation on offer. The first is due to Steiner, published in an article from 1978, and it's been extensively discussed, in fact, among others like me and Johannes in another paper just came out this year in a volume on visualization and explanation. Now, the second theory is due to Philip Kitcher. Kitcher is a well-known defender of an account of scientific explanation as theoretical nullification. Kitcher sees as one of the virtues of his account that it can also be applied to explanation

1:12:30 in mathematics, unlike other theories of scientific explanation whose central concepts, say causality or laws of nature, do not seem relevant to mathematics. Keecher has not devoted any single article to mathematical explanation, and that his position can only be gathered from what he says about mathematics in his major articles on scientific explanation. In And his later work, such as Keecher, 1989, is a paper called Explanation and the Cocoa Structure of the World. It uses unification as the overarching model for explanation both in science and mathematics. And it says, the fact that the unification approach provides an account of explanation and explanatory symmetry in mathematics stands to its credit. And like Steiner's model of mathematical explanation, Kitcher's account of mathematical explanation has not been extensively discussed. There is some discussion by Stafford and Buck on the overall idea of communication and the specificity of the model. Now, our aim here is pretty cold. We will present an Now, by doing this, we will give teachers' advice from another paper from 1981 called Scientific Explanation, where it says, quite evidently, I've only sketched an account of precise analysis of the notions I've introduced, the basic approach to explanation offered here must be refined against concrete examples of mathematical practice. What needs to be done is to look closely at the argument patterns favored by scientists and attempt to understand what characteristics they share. So let's see a teacher's account of explanation as unification. Now, we will follow here the account of explanation given by Kitscher in this article I mentioned called Explanatory Unification and the Photos Structures, the World from 1989. Kitscher explained that behind the account of explanation

1:15:00 given by EMPL, EMPL's Covenant Law model, there was an unofficial model which saw explanation as unification. What should one expect from an account of explanation? Well, in his paper from 1981, he points out two things. First, a theorem of explanation should account for how science advances our understanding of the world. And secondly, it should help us in evaluating or arbitrating disputes in science. He claims that the covering law model fails on both counts, and he proposes that its unification account fares much better. So what's the basic intuition? Let's try to get that straight. Well, teachers found inspiration in an article by Friedman from 1974 called Explanation and Scientific Understanding, where Friedman put forward the idea that understanding of the world is achieved by science, by reducing the number of facts which they pass through. And this is what Friedman said in 1974, this is the essence of scientific explanation. Science increases our understanding of the world by reducing the total number of independent phenomena that we have to accept as ultimately or given. a world with fewer independent phenomena is other things equal, more comprehensible than one with more. Now, already Friedman had tried to make this intuition more precise by substituting for the notion of phenomena and laws linguistic descriptions of such, so that the model could be treated in fact formally. The teacher disagrees with the specific details of Friedman's proposal, but thinks that the general intuition is correct. And it modifies Friedman's proposal by emphasizing that what lies behind unification is the reduction of the number of argument patterns used in providing explanations, while being as comprehensive as possible in the number of phenomena explained. So it says, understanding the phenomena is not simply a matter of reducing the fundamental incompatible communities, but of seeing connections, common patterns, in what initially appeared

1:17:30 to be different situations. Here, the switching conception from premise-conclusion pairs to derivations proves vital. Science advances our understanding of nature by showing us how to derive descriptions of using the same patterns of derivation, again and again, and it demonstrates how to reduce the number of types of patterns that we have to accept as follows. So the criterion of signification I would like to articulate, which is able to get that EK is the type of derivation that makes the best trade-off between minimizing the number of patterns of derivation employed and maximizing the number of conclusions generated. I'm going to now define this term, but the key thesis here is really the flat one, that this EK, what we call the explanatory store over K, is going to be what is a set of derivation that makes the best trade-off between minimizing the number of patterns you're going to use. And maximizing the non-strangling. And as I say, we will come back to the distinction between arguments and derivations and to the clarification of what is below. Let me make a distinction that might be useful to clarify certain general alternative positions in the philosophy of explanation by distinguishing local versus global notions of explanation. of explanation, including the Empyrean one, explanations are arguments, pairs of premises conclusions. Arguments are identified with these pairs of premises conclusions and can be assessed individually with respect to explanatoriums. Following Friedman, we would like to say that whether an argument is an explanation, then in this case it is a local property, You need only look at the specific argument to decide whether you can explanation or not. And it doesn't depend on more global constraints. Teachers rejects both the identification of explanations as arguments conceived as pairs of premises conclusions, and the local characterization of explanations. The informal idea is that explanations qualify as such because they belong to the best systematization of our beings, and that's a global constraint.

1:20:00 Moreover, explanations are not pairs of Genesis conclusions, as in Hempel, but rather derivations, and here is what he said from the systematization account, Now, an argument is considered as a derivation, as a sequence of statements, whose status as a premise for us, following from previous members in accordance with a specified rule, is clearly specified. An ideal explanation does not simply miss the premises, but shows how the premises yield the conclusion. So the way from which you go from the premises to the conclusion makes a difference. Okay, let's try to understand a bit better, because this is the same for our later material, how the formal details of this account work. Let us start with a set of beliefs, which are assumed to be consistent and deductively closed. Informally, you can think of this as a set of statements endorsed by an ideal scientific community at a certain specific moment in time. Then, a systematization of k is any set of arguments which derives some sentences in k from other sentences in k. So you have a set of arguments which gives you these derivations starting points are in k, ending points are in k. That's the systematization of K. The explanatory star over K, EK, is the best systematization of K. Of course, we have to say, according to what criteria. And now, Kitcher makes the idealized claim that, in fact, the EK is unique. That there is a best systematization. Now, Now corresponding to different systematization then, we have different degrees of unification. The highest degree of unification is that given by EK, the explanatory score over K. But according to what criteria, and that's what I'm going to try to explain now more forward, according to what criteria can a systematization be judged to be the best?

1:22:30 Now, Keechard lists two criteria, sorry, three criteria for judging the quality of a systematization case, although only two of them will be relevant to our discussion, and I'll make a claim taken on faith that the third one will not affect the details of what we're going to look at for our research. Now, in order to introduce them, we need to clarify three notions that are central to a teacher's model. The notion of argument pattern, the notion of a generating set for sigma, and a conclusion set for sigma. So what's an argument pattern? Kietcher derives the notion from specific examples taken from the natural sciences, for instance, Newtonian mechanics and Darwin's theory of evolution. But we can exemplify it, in fact, with an example taken from mathematics. Let's begin with the notion of a schematic sentence right here. A schematic sentence is simply an expression obtained by replacing some or all of the non-logical expressions in a sentence by down in letters. You might have an argument. One of the premises might be Socrates is Greek. you might decide to replace Greek with the variable for French. That gives you a schematic sentence at that point. So you can replace with that X now any other phrase that we might want to have. Eschematic argument is, by the way, notice that, sorry, in a schematic sentence you might replace some or all. It's up to you to decide what you want to pluck out of the sentence. A set of filling instructions tells us how the dandelions in a schematic sentence are to be replaced. So, for instance, if you take off Greek from Socrates' Greek, I might give you filling instructions by, say, only put properties in there that can be applied to humans. Or I can give you other instructions. Any property whatsoever can be filled for that. Now, a classification for a schematic argument is a set of sentences which tells us exactly what role each sentence in a schematic argument is playing, i.e. is it a premise? which sentences are inferred from it, and according to what it is, and so on and so forth.

1:25:00 So basically tells you the inferential role if you want that that sentence is working in that sentence. So a general argument pattern then, SFC, SFC, becomes a triple consisting of a schematic argument S, a set F of sets of filling instructions, and a classification C for S. Okay, so let me very quickly flash an example, because this is all very abstract, very general. As I said, the kind of examples key tricks are taken from Newtonian mechanics and Darwin's variology evolutionary theory. we have concocted an example from mathematics that has been tailored exactly to reflect the kind of examples that teacher gives, and to give you an idea of what he really thinks about this sort of argument patterns, what they look like. So let's start with a very specific problem. Determine the equation of the line tangent to the curve y equals 2x squared plus 3x plus 1 at 0.16. Okay, now, to make also the example more interesting from the historical point of view, you know that up to the 17th century, solution of tangent problems used to be done case by case. You had a single curve, you studied a tangent problem to that curve. You didn't have a general pattern. So this is to bring you into what kitchen is really thinking about here. You didn't have a pattern. Every single problem had to be addressed individually. Well, suppose we allow ourselves all the power of differentiation. Well, at that point, we can very quickly find out the solution using derivatives. We take first premise. The derivative of this equation is 4x plus 3. Replace for x the point 1. We're trying to find the tangent at point 1, 6. And we get 7. Then we use the fact that the tangent line to this curve, in fact, can be written by this list equation for the line x minus 1 times 7 equals y minus this. Now of course, in this argument, we have been using various facts about the calculus, there are premises, mathematical premises we're using, and so forth.

1:27:30 But this is the schematic argument, in a way, how we can, in fact, write it as a schematic argument immediately afterwards. to determine the tangent line to a differentiable curve f of x at point x0, y0, can be obtained as follows. First premise, f of x derivative equals g of x. g of x evaluated at x0 equals c, that the tangent line is that equation. Now, what are the filling instructions? Well, of course I have to say that in place of x, just put the description of this function here in place of gx, put the description of the derivative of x of x. In place of c, put the value of g of x at x0. Classification, 1 and 2 are premises, 3 follows from 1 and 2, and we say here, back out here. So that's the example that I think should give you a good idea of what it meant by an argument pattern in this, in this convo. So, as in, um, what Kietcher says then about argument patterns is that whereas logicians in general are concerned to display all the schematic premises which are employed and sorry, are included to specify exactly which rules of inference are used, our example allows for the use of primacies, mathematical assumptions, which do not occur as terms of the schematic arguments. And it does give a complete description of the way in which the rules from 1 and 2 to 3 is to go. I mean, there's a lot of stuff you need to know to work on the previous example. So, you know, already at the start, of course, some of the very hard problems might be examined completely the derivative of the functions they're offered. We have an algorithm for polynomials, you know, transcendental curves might give us more problems, but that doesn't affect that you have a general structure that allows you to say, well, if you have that particular information, you can go on. But down here, you know, there might be, depending on the complexity of the functions, different different ways to set up things and so on, to the equivocation. Okay, so, we're getting almost there to the definition of what's going on here.

1:30:00 Now, as we define the notion of the general argument pattern, we now need to capture formally the notion of nx plan to the star over k. Remember k is that set of sentences given from the star which are trying to systemize. E of K, which informally will turn out to be, I'll repeat, the set of derivations that makes the best trade-off between minimizing the number of patterns of derivation employed and maximizing the number of concluding values. Okay, now, a set of derivations, a few more definitions, please bear with me that there There is no other way to do this pattern to look at what it's done. A set of derivations is acceptable relative to k, just in case every set in the derivation is deductively valued and each primus of each derivation belongs to k. Okay, so it's acceptable relative to k if you stay in fact with k. Now, a generating set for a set of derivation sigma is a set of argument patterns phi such that each derivation in the set sigma instantiates some patterns in the generating set phi. So, for instance, if you go back to the previous example, and suppose you are, in fact, within the sphere of calculus, where you are not only solving columns to draw in tangents, but say, you also want to discover areas under curves. Well, you would have a general pattern for the tangent column. You might have another general pattern for the integration column of calculating areas. These are general patterns in the sense that all the particular examples here in your theory are instances inside of these patterns. So a generating set, then, for a full cycle derivation that you have, will be made up of this set of patterns that are enough to generate what you want. And now, a generating set pi for sigma is complete with respect to k, if and only if every derivation which is acceptable relative to k and which instantiates a pattern in pi belongs to sigma. Okay, now we can give the recipe for how do you go about computing the best systematization.

1:32:30 First of all, you select among all possible systematizations of K, only the acceptable systematizations of K. Okay? That is the set of those derivations, those sets of derivations that are acceptable relative to K. You might have system accusations, for instance, that are not acceptable. Two, to each acceptable system accusation selected in one, associate the collection of the generated cells for that system accusation that are complete, relative, with respect to K. So here the problem is this, that you have all this possible systematization. Once you move at the level of looking at what possible argument patterns might be the framework that generated all of the relations. Well, if you're silly enough, you might come up with two main argument patterns. For instance, suppose I distinguished in my case from the polynomials. polynomials have ended up with a coefficient in two, and other polynomials. I might have ended up giving two different argument patterns, maybe, for the tangent problem. And that's still a, you know, it's still a complete study, not necessarily that's one in one in the Okay, three, for each acceptable axiomatic basis, where a basis is an element of the collection of the generating sets for that systematization, which ranks best according to the criteria of falsity of atoms. Okay? That's the key point. You have all your systematization, then you look at all the possible generating sets for those systematizations, pick the one that has the least number of argument-anities in it. That's one of the criteria. Sorry, what about uniqueness? It doesn't, yeah, uniqueness is just assumed. It doesn't really deal with that. You know, if you think in terms of numbers, right, if we could count them, then I guess that every two, every two, sorry, every two, what's the word I want, Every two basis with the same number of patterns would count just as the same.

1:35:00 Okay, finally there is one more criteria, so this was the first criteria, pausity of patterns of the system attribution. The second one has to do with how much can you generate from this pattern. And this is what is going to allow us to move to step four. We've run all bases of the acceptable system applications according to the unified power. So we have one factor already, the positive fact, and the other one has to do with the conclusion, and that will allow us to go to step four. Now, the conclusion set, um, the conclusion set of a set of arguments sigma, C sigma, is the set of sentences which occur as conclusions of some arguments in sigma. And so, at this point, the degree of the implication of the systematization is directly proportional to the size of the consequences we can get from the systematization, and inversely proportional to the numbers of actions. So we go back really to the intuition I stated at the outset, what the system, the best systematization, the explanatory score of the case, is what brings the best combination of falsity of factors, right, minimizing the assumptions, the pattern, the structure patterns, and maximizing the components we got. Now, one question one needs to ask immediately is how well that teacher's model fits the concrete case? And teacher doesn't, as I said, present this model of explanation with specific mathematical examples in mind, but in various applications, he has made his claims referring

1:37:30 the big things can be, in fact, accounted for by each model. So, for instance, in Chapter 9 of this book on the nature of mathematical knowledge, it discusses what we call patterns of mathematical change. And one of the patterns it discusses is called systematization, and it refers explicitly to this model. In particular, one type of systematization is called systematization by conceptualization. And in this case, he mentions, for instance, the improvement obtained by Viet's algebra in treating systematically the chances of equations of the same degree, say degree 3, in contraposition to the case-by-case analysis which is typical of Cardano's R. S. N. N. Similarly, Lagrange, in his analysis of resolving equations and permutations of equations represents a huge step forward in our understanding of why the solution of certain equations can be reduced to the solution of equations of lower degree. But here is an important point. In fact, in both these cases, one adopts new language which allows for the replacement of a disparate set of questions and accepted solutions with a single questions and a single pattern of reasons, which subsume the prior questions and solutions. Generally, system accusations like conceptualization consist in no-defining language to enable statements, questions, and reasoning, which were formally treated separately to be drawn together under a common formulation. The new language enables us to perceive the common threads, which runs through our old problem solutions, thereby increasing our insight into why those solutions work. This is especially apparent in the case of Lagrange, where antecedently there seems to be neither run nor reason for the choice of substitutions, and that's a genuine explanatory problem. Okay, so the problem we'd like to point out here is the following. Kitcher is very explicit about the fact that the new unification provided by Lagrange or even Galois, with Galois theory, in the theory of equations, must use a richer language. New concepts and new properties of this concept. And we're all familiar with the fact of the mathematics. In fact, that's when you get the more interesting stuff going on.

1:40:00 But then, where earlier on we had to account for a set K of sentences formulated in a language LK, we now have a set K-star of sentences formulated in a language LK-star, right? I mean, the language we need to do the theory of substitutions, or even Galois theory, is much richer than what Viac had at his disposal or what Tardin had at his disposal. we would still not like to say that we have a better explanation of the original problems, right, than we had before. But Kitscher's model seems to force us to compare only systematizations of K and not only each other, and such a systematization as to appeal only to sentences from K. that in order to make teacher's system more adequate to the actual situation we face when making a valuable judgment of explanatoriness, we modify slightly this model in such a way that such a systematization of K can appeal to a class of science larger than K. And indeed, such a move receives also textual support from teacher himself, who seems to have recognized this in his 1989 article. So these are the last two definitions for this model. exactly the same as what you saw before, except that now, a systematization allows you to conclude things in case, but can have its premises reaching outside of case. In other words, you can dispose of this larger theory to explain the old problem that's what we're trying to account for. And that's really what teachers would like to be able to do, to say Why is it that Lagrange's theory of equations or Galois theory is a better explanation of those phenomena that already carved out and yet we're looking at? Of course, the new theory might explain or give new results and new stuff, but we want to be able to do at least that to say in what sense would we have explained an increase in expansory power with respect to the old stuff. As I said, there is nothing really new here, except for the fact that now your premises can come from K-star, a larger set, but your conclusions have to come from K.

1:42:30 Okay, now we move to a test case from real algebraic geometry. In its monograph on partially ordered readings and semi-algebraic geometry from 1979, Brownfield contrasts different methods for proving theorems about real-closed fields. One of them relies on a decision procedure for a particular axiomatization of the theory of real-closed fields. By this method, one can find elementary proofs of sentences formulated in a language of that here. At least in principle, since Brantley himself remarks, it certainly might be very tedious, if not physically impossible, to work out this elementary proof. Now, another method of proof, another proof method in fact, consists in using a so-called transfer principle, which allows to infer the truth of a sentence in all real-closed field from its being true in one specific real close field, for instance, and the real numbers. Despite the fact that the transfer principle is a very efficient tool, Brantfield does not make any use of this. And it's very clear about this, and this is where It says, in this book, we absolutely and unequivocally refuse to give proofs of the second part. Every result is proved uniformly for all real-taught ground fields. Our philosophical objection to transcendental proofs is that they might logically prove the result, but they do not explain it, except for the special case of real numbers. And this is exactly that sort of tension, of course, that my project deals in, you know, this recognition that some of these are better than others with respect to certain extremist virtues. One of them is explanation. Going back to Jeremy's talk, I'm not saying that all of the new proofs we introduced as this explanatory intent. But I think that there is a class of things that can be truly characterized under the time explanation. What exactly that is, then, is part of the project to find out.

1:45:00 Now, Bradfield prefers et cetera proof, which aims at giving non-transcendental proofs of purely algebraic results. This does not mean that he restricts himself to merely elemental matters. He does use stronger tools, but it is crucial that they apply uniformly to all real closed fields. It is also clear from the context that Bramfield does not consider proofs obtained from applying the decision procedure for RCL, real closed fields, the theory of RCL, as explanatory. So, there are two types of proofs with each other. The one obtained by using the decision procedures for SDS, and this one that means transcendental method. Now, this notion of transcendental is very interesting for, actually, for what will come out in the next few days on purity of method. The distinction algebraic genomics it uses in this particular case is that between algebraic and transcendental, they know what you're talking about. I will not try to make it concise, but they have a very definite feeling about where the partition line falls on that one. But to give you a sense, a purely algebraic proof would be one that, in fact, relies on things that are uniformly true for real closed fields. A transcendental proof is one, for instance, uses specific properties of R to then get the result that can be lifted to all real closed fields. You're using maybe integration on R, compactness, things which do not hold across the board, but because of its great transfer result, you can use that transcendentally, get the statement, elementary statement, out of that, and then transfer it all across the board, even if the original proof does not go through in the other real closed fields. Okay, so let me try to at least give a very quick outline of semi-algebraic geometry, just a mere backbone, to at least see in a specific instance what this distinction comes to be. Okay, so what's a real closed field? It's a field which admits a unique ordering, a total ordering, such that every positive element has a square root, and every polynomial of odd degree has a root.

1:47:30 That's the definition. I can give it more formally. Here you have just a summary of, this one tells you that every positive element is a square root. The last one tells you that every polynomial of all degrees has at least one root in the field. Now, the theory of real-close fields, RCF, is the redactive closure of the above axioms. An important result about RCF is that there is a decision procedure used to parse to ensalender, i.e., given any sentence phi in the language of RCF, the algorithm outputs 1 in RCF proofs phi, and it outputs zero if RCF proves not one. This decision procedure works uniformly for every sentence found in the language of RCF. Now, examples of real close fields include the set of real numbers, R, and the set of real algebraic numbers, R algebraic. That is the set of groups of some non-zero polynomials with rational coefficients. Now the latter field is not complete. So the algebraic numbers are not complete in the way the real numbers are. I.e. not every Cauchy's sequence converges. There are lots of real close fields between the algebraic numbers and the real masses. In fact, there are two TDIF nodes. For real closed field R, we denote the ring of polynomials in N variables by R, X1, Xn. A subset of Rn, and now we're talking about an arbitrary real closed field, okay? So whenever we mean R as the real numbers, it's bold, as opposed to just normal R, which A may be obvious. A subset of Rn, the upon n-space over R, is called semi-algebra, if it belongs to the smallest family of subsets of Rn containing all types of form. And basically, this R-text says, on which a polynomial is formed. the semi-algebraic side, and which is closed, inside the closing conditions, and we're taking

1:50:00 finite intersections, finite unions, and commons. So the semi-algebraic subset of a real closed field, now we're talking in one dimension, are exactly the finite unions of points and bounded or unbounded. And that's the algebra characterization of what these sets look like. I'll say it again, are exactly the finite union of points and open intervals either bounded or unbounded. Let A and B be subsets in RM and RM respectively, and be semi-algebraic, and mapping F from A to B. It's semi-algebraic, if it's graphed, is in fact semi-algebraic in the sum, in the space which has the sum of the two spaces. And finally, you can endow, this is very important, an arbitrary real close field with a Euclidean topology coming from the ordering on the real close field. You exploit that fact. And so you can define the norm as the square root, and then you can define the open ball around the point. So you have to make an approach on that. Okay, finally, let's look at the simple theorem and a variety of proofs to bring in this value judgment that Brownfield sort of gave on us, on what's explanatory and what's not in semi-algebraic millimeters. Let's look at a very simple theorem. A polynomial fx1 extent assumes a maximum value on any bounded closed semi-algebraic of course, if you think of this in R, if you think of this in R, that's an obvious statement. And, of course, the tool requires very specific tools. In this scenario, you need to be able to balance these various things that are very specific of the system's work.

1:52:30 OK. Well, how could we have come up with a terminology? Let's put it this way. One possibility is that within RCS, we might have run the algorithm that we know we have. Right? And that one, when we put what inside? Well, here's the problem. On RCS, we can't put this statement because that's not the statement of RCS. But, we can put a parallel chart version of the world, where we work with a specific polynomial, and we work with specific semi-algebraic graphs. So, what we're saying is that in our theory, you could have gotten instances of this theorem, but then you run the algorithm. You get, you know, you run it on x cubed, on the zero-one interval, the algorithm goes on for a while, now it's one. then you say, it's a theorem. But you trust the algorithm. So we could have arrived at the result in this way. And every single instance, in fact, of this theorem could have been obtained in this way. By the way, there is no loss of generality here in proving all the instances as opposed to proving the universal claim. numerably many instances, as a universal plane, it's about all polynomials, and those are the numerable two. So, in other words, they're not used to match by looking at specific instances as opposed to a human agent or plane in a particular situation. Let's look at proof number two, second strategy for the truth. Right, here is, of course, a proof, the first one, not the one-fold practice. It's going to be relevant to what we say later. Proof number two. The following proof strategy draws on transfer mental methods. This approach is based not so much on the task of studying your decision procedure, but rather on one of its consequences, namely that RCS is a complete theory. This gives rise to the following transfer principle. If a sentence 5 in the language R-C-F can be shown to hold in one

1:55:00 concrete real close field, say the real numbers R, then it must be true for any real close field because of the completeness of R-C-F. Now, the theorem at hand on the maximum of an arbitrary polynomial can of course be established for the real numbers. But what we need is, both handled by extra theorem that says that every bounded sequence has a converging sub-sequence and the leaves that are bound principally. These basic properties of R do not hold in general a real close field. For instance, they both fail on the algebraic numbers. However, once the theorem has been proved for R, by whatever means, we can conclude by a field of transfer principle that equals for all re-objectments. Third type of truth, which is the one preferred by ground truth. Another way of establishing the theorem relies on purely algebraic means, exploiting the fact that if A is in our end and is a closed and bound semi-algebraic state, and G and into RP, and it's a continuous semi-algebraic mapping, then G of A is a closed and bounded semi-algebraic set. Now, this proof is very delicate. You'd be tempted at this point to prove this result exactly by appealing to the standard techniques that we use when we do this theorem on the real numbers. But you can't, because once again, what you would end up using will not be allowed in this particular context. That's why the proof becomes sophisticated. And so I'm not giving it here. I'm just saying this is a result that you will find in any textbook on the algebraic geometry. A very good exposition, if you're interested, the book by Volchner, Kost and Rose called Semi-Algebraic Geometry, published by Springer in 1998. Now, since a polynomial is a continuous semi-algebraic mapping, and we assume that we are given a closed and bounded semi-algebraic set S in our end, it follows that F of S, assuming the theorem I just gave, is closed and a bounded semi-algebraic set.

1:57:30 But now, f is enough. And we know by the characterization of semi-algebraic sets that it is a five union of points and closed and bounded interval. Right? It's a five union of points and closed and bounded interval. So you can go there and see the last one, right, the last element of what has the right most interval you have using the ordinary. And so, f as a maximum. Okay, to conclude now the assessment of each of the model in this particular example. The above three proof strategies exemplify different approaches to systematization, to systematizing knowledge about the of closed fields. More precisely, we will in the following be concerned with systematization of the theory RCL, So we intend the theory now as the deductive closure of the field, which is the consistent and deductively closed, in fact, complete set of elementary sentences true in real-world field or equivalently what follows from our theory. To be in line with teachers' terminology, let's call this K, right? So K, the set of statements through, of elementary statements through, for real close fields, is what we want to systematize. What's the best systematization? That's teachers' project. Well, as was pointed out about, a systematization of sentence S might have to go beyond S premises from S-star, i.e., a consistent superset of S. That's what necessitated the slight modification of the teacher model. And the different systematizations connected with the above three proofs are cases and forms. The first proof strategy uses some meta-theoretical machinery in addition to K. For instance, it uses the decision algorithm for sentences of K, which is not in itself a theorem of the theory RCF. It's part of the meta-theory. So let us call K1 star, K together with some meta-theories for K corresponding to this

2:00:00 decision algorithm, we leave it often, to some extent, how much metatheory there is in K star 1. For instance, certainly the decision algorithm, perhaps more, in action might be needed to actually establish the main properties of the algorithm. It really doesn't matter. Because the systematization of K, now provided by K1 star, is given by a very simple argument pattern, OK? pattern is you take the function from the decision algorithm, you take the decision algorithm, you write it as a term function from that. So now you take an arbitrary standard of R3a and you say f of 5 equals 1 infer 5. That's the pattern. So, all our maxima problems here are systematized by that. You put x cubed, you put the interval 0, 1, or whatever you choose, does it give 1 at the decision level of that premise, then you infer the same. That's the fact. The second proof strategy employs, besides the transfer principle, transcendental methods, like the use of completeness and compartments, which alter the real numbers, but not for real cross-field. As in this case, the K-star-2, the system accusation from which you can draw parameters, as it were, includes some of the theory of the classical real numbers, and a formulation of the Tarski-Sylumber transfer principle, which is based on the computing theorem called k. Because this requires a use of model-theoretic concepts and methods, in addition to syntactical ones, in order to talk about truth, for instance, in the specific real-close field, which is the real numbers, the meta-theoretic part of the second systematization, k2 star, is more encompassing than the meta-theoretic part that we used in the previous example. Again, however, we do not need to specify the necessary meta-tree, 3 star, and can be the alternate, as two examples in there.

2:02:30 The third proof belongs to a systematization of K relative to a wider framework of real algebraic geometry. Let's call this K3 star. Brantley stresses that K3 star is not confined to elementary methods. For instance, he said, we can use dead between cuts, we can use total orders, we can use a lot of stuff that is beyond the elementary level when we do this work. But the point is that, in the form we use this concept, they are, and I'm quoting, they apply uniformly to all the across fields. One advantage of developing such techniques is precisely that one is not tied down to elementary sentences, and that was a quote. work in an elementary theory, doesn't mean you can't use stuff that applies uniformly to all real false fields. Stuff about ordering, stuff about dedicating counts, and so on, which go well beyond the expressive power of the elementary theory. Okay, now we can ask the question, what is the best systematization of theory? Okay? So, at this point, Kitcher's model should predict that one of these three is going to be the best. Well, what does this mean? If we can find a systematization that makes use of only one argument pattern to generate all cases, then any other systematization which uses a greater number of patterns is inferior. Well, it turns out, then, that the best systematization of cases is the one provided by the Tau C-side on the decision procedure for our solution. The one is identified by the first two strategies. And why is that? It's not only because it solves uniformly all the problems about the specific polynomials and intervals. It's because you can use it for all of these. Any other problem you look at, right, will have exactly the same strategy. You plug in the problem. If s, the algorithm, tells you 1, then you insert s. across the board, you have one argument pattern that decides everything on that particular structure. The uniformity of the procedure shows that we have, in principle, a single argument pattern which we can use to generate all of K. Comparing the situation with those in which we try to prove

2:05:00 arbitrary elementary sentences of K by means of proofs such as those given in 2 and 3, it is obvious that we cannot get by with this simple argument pattern. Meaning, we were able to get one argument pattern in the other types of proof for this specific problem of finding maximal polynomials. But that particular argument pattern is going to be useless, say, when you try to apply it to the semi-isobrite form of Brouwer's fixed-point theorem. Well, then you would need a completely different type of argument pattern to solve that. But now, in the first case, where again, you can use the general algorithm given by Charleston Salinger, plug in the Brouwer 6-point theorem, and again, that's just one single argument. Well, similarly, in the case of proof 5-2, in this case, we need to determine that Thank you.