Conjecture & natural properties

0:00 Okay. So thanks for inviting me. Thanks, Susan, for having me at the seminar. Thank you all for sitting through my talk after four hours of sitting in one place. It requires a kind of higher in the world, sort of discipline, to borrow the words that Jose was using, discipline. certain kind of certain certain kind of sex and i don't know i like just um anyway yeah uh i'm gonna i'm going to this is going to be taught mostly on in philosophy there every now and then a little bit of history will will pop its head up and be chewed away um and uh mostly i'll be trying to argue trying to give a very skeptical or restrained or um trying to give a of defense of a certain view, you've already heard the view described this is one of the views described in David's talk as something so absolutely lunatic that only deranged people would only hold sort of meaning in a psychoanalysis unfortunately David's right here help me out in that regard but I'm going to defend the idea that there is in some way in some interesting way genuinely natural properties in mathematics that is in some sense objective, at least as objective as the presence of natural properties in physics. Now that's an idea about which people are right to skeptical, and I'm skeptical as well, so my main effort will be to try to find some defensible conception of objectivity. And what I'm going to do is appeal to something that both David and Jose have mentioned, quasi-empiricism, the idea that we make conjectures in mathematics and verify them or falsify them, and I'll tie back to the idea of natural properties in the same way that metaphysicians tie making successful predictions in empirical sciences to the idea of properties being natural. And I guess I should also say that, although it might seem to make myself sound exceptionally bold,

2:30 I can describe this as finding a sense of objectivity for the idea of natural properties in mathematics. I guess maybe perhaps a more accurate way of describing what I'm trying to do, is try to show that mathematical properties are at least no less objective when you have, let's say, the naturalness of mathematical properties can be no less objective than physical properties like, say, pressure. Okay, so we're going to have, you know, mathematical properties splashed all over my mind. Okay, so what I want to do is clarify, in particular one thing, not just the naturalness of properties, but also correspondingly, these are related, although not identical, issues, the naturalness of definitions. I want to find a niche for these. And then mostly I want to concentrate on this pattern, that you can regard it as an advance in knowledge to find, and here you can't resist using scare quotes and so on, find the right definition or the right concepts for setting up a problem, or the natural domain for a function. This can be regarded as an advance in knowledge. And so what I want to do is sort of try to get epistemology, which is after all a theory of knowledge could make sense of these advances in knowledge and also one thing that makes this kind of discussion challenging is that if you're trying to fit it into this framework of the discussion in general metaphysics and epistemology, is that it's hard to come up with examples that have the right level of complexity right they have they have to be of a certain strength otherwise they're going to be so trivial that they're easily shrugged off and they don't really illustrate the point you're trying to illustrate and on the other hand if they're too complicated you you know you lose the house right you lose the audience and so for for public relations reasons you want to find examples that you know somebody with a high school education or maybe one or two math courses in

5:00 university, I could at least get the basic idea. So I'm going to discuss two of them, one simpler than the other. One is the Le Genre symbol in number theory, and the other is the definition of prime number. Now, I'll also be staying largely within mathematical practice, taking judgments at face value, and then I'll try to explain, as I say, how we can defend at least a minimal conception Oh yes, yes, yes. Okay, now the talk goes into four parts. I'll no doubt end up having to skip over bits and pieces because I'm thinking I may want to condense it a little bit in concession to the facts of human attention spans. First, the basic data, definitions of mathematical knowledge and practice. Then I want to talk a little bit about the idea generally of natural properties as it's understood in contemporary metaphysics. Then I want to work out my core examples. One example of a stipulative definition, the LeGonard symbol, and then one of the redefinition of an already accepted concept, and that'll be prime numbers. and then I'll talk about conjecture and verification in mathematics as what I call a foothold on objectivity maybe a toehold, a toenail hold on objectivity okay, now this is the basic observation in mathematical practice it can be regarded as a crucial advance in knowledge to find the right, the proper, or the correct or the natural definition of something This can be something that people try to do. They set it as a research objective and champagne corks are popping when it's finally attained. So we find here are two examples. This is common. You'll find lots of stuff like this. These are two of my favorites. This is from a review by Harris, kind of overview of the last hundred years or so of algebraic geometry. And he says, the thesis presented here, that is saying in the paper he's writing, is that the progress of algebraic geometry definition, as in its theorems. It goes on to explain how it is, but things like the definition of scheme in particular were

7:30 absolutely crucial theoretical advances to discovery and formulation and recognition of its importance. And this is a very popular quote it's a quote from the very popular undergraduate textbook Spivak's Calculus on Manifolds, very popular in North America, Spivak's Calculus on Manifolds. It says, Stokes's theorem shares three important attributes with many fully evolved major theorems. A, it's trivial. B, it's trivial because the terms appearing in it have been properly defined. C, it has significant consequences. Right now, in fact, the proof he then goes on to give involves two applications of Koubini's theorem, so it shows that Stokes' conception of what counts as trivial is a little bit different from mine, but nonetheless, we get the idea. The real success here is finding the right definition. And there are two different ideas here. You don't want to run them together. They're related, they're importantly distinct. One is where you're introducing a new concept, a scheme, and then showing how it, if I may put it that way, car's needs to be joints. And then in another case, you're taking an already defined cluster of ideas and redefining them in such a way that the necessary, that the desired logical relations are later clearer. What does Ivanka mean to treat that? I suppose he means that it's... Easy to understand? Yeah, easy to understand. Proof is sort of transparent. I guess, you know, because even though it does involve, the proof involves an application, a couple of applications, Rubini's theorem, in one sense, are sort of high-powered, but on the other sense, they're kind of conceptually easy. Well, we carve it this way. We carve it that way, and there you go. Okay, anyway. But, yeah, easy to see, and... Easy to formulate. Sorry? Easy to formulate. Thanks to all the definitions to define the objects. Like, from the objects that define, which is very similar. I'm not sure. You think easy to formulate? Yeah, that's true, but it's... When you look at the way, how can I do it, it's less formative. It's certainly formative. I think it is better than one.

10:00 This would be simple. It's a trivial thing. It's simple and something more. Well, not necessarily, but at any rate, the formulation is, I mean, not necessarily easy to formulate in the sense that it's easy to discover the formulation, but it's got to be easy to understand the formulation when it's presented to you. It just, well, anyway, all right, this is, you know, let's lead to the scholarship for the next talk. And at any rate, this is one example of something you'll find in any of a number of places, people pointing to the proper definition of something as being an important conceptual advance. Okay, so anyway. I just sort of would like to emphasize that not only is this sort of spoken of as an advance in knowledge, but it really seems to be rightly regarded as an advance in knowledge, in at least some cases, when you recognize that a certain concept should be taken as fundamental. And so now my question is what can, I mean, So this should suggest to you, taking the idea of epistemology more generally, shouldn't we give an account of that, since epistemology is the theory of knowledge, and after all this seems, at least per se, to be a kind of knowledge. But I'll take the question more narrowly. What counts as epistemology, you can also see, is defined by current debates in the literature, so to say, and taking things more narrowly, just to see how far that can get us, you can ask what connection can research on mathematical definition have to current debates. Okay. Now, one thing is that the talking about things as natural does suggest a connection to metaphysical debates on distinctions generally between natural and artificial properties or kinds, right? This question is relevant to the naturalness of mathematical functions and classifications

12:30 overlap with the corresponding questions about properties generally, at least in the really absolutely elementary, simple paradigm cases, right? So you consider the example like Goodman's groove, that it's green if observed up to a certain time and then blue afterwards, and we distinguish that from green. on the grounds that one is natural and the other isn't, and we would also distinguish divisible by two from is pi or a Riemann surface of genus 7 or the stone shack compactification of omega on the same ground, right? One of them is natural, the other one is artificial. So let's say if we are going to talk, if there is a metaphysical debate about the distinction of naturalness and unnaturalness, we should look to see if mathematical properties can be used. Yeah. You know that Goodman's solution of the paradox of a group using the normal simplicity is a lot of discussion. It's not clear that a group is less simple than a group. Well, yeah, but I think Goodman's sort of... Goodman also... Paul sort of thought simplicity in the sort of context. I mean, in the structure of appearance he had the song and dance about simplicity in terms of the number of constants, but at least the robust solution he had was in terms of the entrenchment. I want to say, the first solution is to say, okay, blue is simpler than blue, and this is the boss. So there is something that is there. Oh, yeah, yeah, but I think you can self-realize that that couldn't work. I mean, I think it's... With respect to your talk, the only thing that I want to say, because you are not... Oh, yeah, yeah, okay. Is that the fact that the divisible factor for two is more symbol than P or E? Yeah, no, and in fact, we'll get to that when I talk about the le genre symbol. You know, the thing is, one thing actually more generally people will say also in the general metaphysics is not just generally appeal to simplicity, but specifically talk about disjunctiveness, about the definition, so there's an order there. And we know what else, that's just not a robust feature of the property as the genre symbol will show. Anyway, if you go to David Lewis, if you want to find metaphysics done in the grand old fashion,

15:00 metaphysics done as if Plato had never lived, as if Socrates had never been born, you know, sort of Heraclitean metaphysics. A little struggle. You look to David Lewis and his story about why you should take seriously the natural artificial distinction is very characteristic of Lewis' approach. You know, he says, well, I'm just going to point out, says Lewis, how many ways the natural artificial distinction shows itself in our everyday thinking. suggest that we, says Lewis, you know, people generally are not prepared to give up those ways of thinking, and so to be consistent, we should accept that there is a robust, objective, natural, artificial distinction. So says Lewis. He says, well, for example, there are clear cases where you do want to say some things are natural and some things are artificial, like grew versus green, and the natural artificial distinction underwrites those. There are cases where we do want to make judgments of simplicity and similarity, and far from it being possible to explain naturalness in terms of simplicity and similarity, it seems more promising to take naturalness as basic and explain simplicity judgments and similarity judgments in those terms, to distinguish general truths as law-like or not, to support assignments of content, like in the so-called Krippenstein cases, of a plus and a plus, underwriting the distinction between intrinsic properties and non-intrinsic properties, singling out intended interpretations in cases of Levinas-Golden-Indeterminacy, and distinguishing correct and incorrect inductive predictions in Drew-type examples, and the list goes on. The only thing Lewis does is say, look, don't be aware of how embedded in our ordinary ways of thinking the natural artificial distinction is. So let's say we sort of say that, then I want to point out, even with respect to the examples that Lewis gives, that this is going to be true of the practical properties as well, at least a couple of cases, like the Krippenstein case, which is specifically about the group of both plus versus plus, and in the case of indeterminacy like the Lydman-Skollum argument,

17:30 for Lewis's story to work, we have to take mathematical examples as core instances. But in other cases, it seems like mathematical examples, in mathematical cases, will have to show the faces as well. So, for example, in the distinction between laws of nature and accidentally true generalizations, it's hard to imagine how an account of natural properties could help, unless at least some mathematical properties, functions, and relations are included, since the criteria and the practice for law-likeness and simplicity of laws often pertain to mathematical form. Can the relation be formulated as a partial differential equation? Is it first or second order? Is it linear? the role of natural properties in inductive reasoning might be taken to mark a disanalogy somebody might say well naturalness fits into things like prediction, this is essentially the suggestion Sidney Schubanker although he doesn't pose the issue in justice form and also somebody might suggest, although this is a complicated issue that I just want to It's too complicated for this one. It has to be discussed elsewhere. Somebody might say, well, natural properties support analyses of causal relations, and since it doesn't seem that you have causal interactions between mathematical objects or mathematical statements stand in the types of relations that causal statements do, that's a disanalogy, I don't think. I think that's here. What I want to sort of talk about is actually that the role of the idea of naturalness as interacting with the correctness of patterns of inductive reasoning doesn't represent its analogy. I just want to make one more remark about terminology because there's just a word off a kind of misunderstanding or just to indicate that there's a potential for misunderstanding and I just hope we can live I mean, in English, the word natural has this sort of double meaning, right? It can mean sort of appropriate or correct, you know, have a kind of correctness as when you say this applies to natural math or this interpretation is the natural way

20:00 to understand constant marks in page 17. But it can also mean that it pertains to the physical world, right? When you talk about reducing topical mental properties to topical natural properties, because you mean properties in nature. And it seems most of the times that I'm going to want to talk about natural properties in metaphysics, most of those discussions seem to me best understood as drawing on the correctness or appropriateness meaning, at least in the cases relevant here, like when somebody counts a plus, that's more natural than a plus. So I'm going to speak about natural properties. I'm going to use the word natural in mathematics as well, just because there isn't really a good, better one, another one, which isn't ambiguous in this way. And since I don't have a better word that's going to work in the exact, all the ways that natural is going to work, I'm going to stick with it. You know, I'll just say, you know, until we understand, when we understand the situation better, we can coin a new word. But for now, I'm going to use natural and just alert everybody fallacies of equivocation and so forth, that we have to be alert to. The problem is that when you speak about the metaphysics, the two meanings are not so completely distinct. Because something is correct and it was reflected. It's natural in the sense of correct. Because it's reflected, natural in the sense of nature, it's like the world. So we have a criterion of naturality. Oh, I'm okay with that. Just, you know, then... Groove is not natural as not true, because a decision between blue and red reflects the world in a better way than groove and not groove. Well, okay, I'm happy to say in this case that... But here, when we speak of correction, you have not this parameter, you cannot have this parameter, at least if you are not... Well, I'm sort of, you know, I'm sort of claiming that it's not so obvious. It makes it negative. Pardon me? It makes it negative. Well, I mean, it could be that it turns out that the, so to speak, criteria for success in identifying things, though it's natural or not, are going to be different. but that doesn't mean that the sort of patterns of reasoning that you engage in to identify

22:30 what's natural or not are going to be different and so i'm just i'm happy if i just if it just turns out that there's a uniform set of patterns of reasoning we engage in and and it may turn out you know uh that there's going to be a difference in the way the world has to be in order for them successful, right? there's sort of also, there's one kind of basic fact I want to mention and then set aside that's really not too important. And that's that one of the things that really makes, you know, I was a graduate student around David Lewis for five years and I sort of saw him argue and saw what sort of things tended to work. And as we were ten minutes here, it was hard to deny that there was this prima facie force to the he would point to something like sir you know look are you going to deny that these two electrons A and B are more like one another than they are you know than either of them is like the moon or the Eiffel tower or a moose and he's like no no no when you put it that way it sure seems hard to deny that they're very much like each other and that sort of gives it some intuitive force right and the sort of same feeling you get with plus being natural objectively somehow and plus isn't, right? But I don't think we can really rest very much on those sorts of intuitive judgments. Not just because I think it's inadequate in general to rely just on brute intuition in philosophical argument, but I think because the force of some of the better mathematical examples and prospective natural categories like genius or scheme and some of the better definitions like prime actually require training to appreciate. You sort of have to natural, right? It's not something that you just kind of sit back and feel struck by. You can actually give a rationale for its natural. So if we can make sense of the idea that categories can be objectively mathematically natural in a way that relates to ongoing mathematical investigation, we'll need to make room for the fact that we can discover that a category is natural, even if it seemed to lack intuitive naturalness at the outset. Even in some cases, The prima facie impression, at first, is going to be that the definition is an obvious hack, an obvious disjunctive hack.

25:00 In politics, this is called a gerrymander. Do people know the word gerrymander means? In politics, in the United States, if you have a political district where they carve it up in a completely artificial way, like so you might have say 41 percent of the yeah yeah say you have you know something like you have you have this this largest district and you know you know 48 of the people are uh you know uh vote vote we and uh you know 52 votes no and you want to bring it about or and but but you're choosing sort of candidates right because we know it's the we party and the no party um you're choosing candidates so that the idea is that each district is going to choose a single candidate and if you were to just sort of draw the boundaries in a normal way you know you'd expect that the we people would get you know some chunk of the candidates but if on the other hand you wanted to curve it up in such a way that the known party got all of the candidates right you just sort of that there's a very small advantage to everyone. But you'd have to sometimes sort of be really clever about where you draw the boundary line. And that's a very powerful point. It's called Pimber Constituency Boundaries. The origin expression is actually from Irish politics. Oh, is that from Irish politics? And it doesn't make sense on the basis of single-member constituencies . Which was not . Oh, I see. Yes, yes. You have to understand that each of these districts is choosing the person. Yeah. The basic idea is what you do is you carve up these things artificially so as to get the desire for itself. So that's what a jerrymandered is. And then it is extended metaphorically which is made in that which is made in that from the point of view of the non-partum anyway what I want to do is I'm actually going to be as I sort of indicated I'm going to be neutral about what we're going to judge what view of natural properties

27:30 generally makes sense, right? If it turns out that overall what we're going to see is the idea of natural property only makes sense relative to our classification schemes, something like, you know, kind of a Goodman entrenchment response, then I'm like, you know, I can live with that as long as the story ends up being that mathematical properties and physical properties are of a piece. they're sort of at least the naturalness of mathematical properties is no less objective so I'm not going to be deeply committed to defending any particular story about naturalness of properties generally in part because just by temperament that's not the sort of thing I ever want to find myself defending so I want to I want to give one example of a kind of natural definition and corresponding to a natural function, where it's introduced stipulatively. And this is one I've actually mentioned a couple times in the talks last week, or two days ago and yesterday. Quadratic reciprocity. And just to give you the basic definitions, I assume you know, but just to remind you, radius congruent to B mod C, meaning that there is an N such that A is equal to NC plus B. Later, if we put it in school arithmetic, A divided by C has remainder B. there is an X squared, such that X squared is congruent to P mod Q, we say that P is a quadratic residue mod Q. the law of quadratic reciprocity is if P and Q are odd primes then X squared is congruent to Q mod P is solvable let's say P is quadratic reciprocity exactly when X squared is congruent to Q mod P except when P and Q are congruent to 3 mod 4 in that case P is a quadratic residue mod Q exactly when Q is not a quadratic residue mod P.

30:00 Now this is often used as an example, a nice example, it's in many ways a nice example, of the search for explanation in mathematics. And one of the reasons it's a good example is because it's been proven and reproven and reproven Many times in recent centuries, the last time I checked the list that's maintained by Lemmermeyer, it's 221 distinct groups. Some of those, I'm not sure exactly how much I count them as distinct, but anyway, lots of distinct groups. Gauss started off the trend. He proved it first, and then he reproved it in seven more times in significantly distinct ways. to be that this to be paradigmatic of the value of finding new natural proofs of known results which he took to do of paramount importance so here's a for me and another all right all right well then i gotta i gotta sort of you know i gotta shut the paper down right it's completely obsolete um uh typical expression of his this is Gauss speaking now just because it ended up being too long for the page, I had to break it up into two chunks. It says, it is characteristic of higher arithmetic that many of its most beautiful theorems can be discovered by induction, by that means you know, enumerative induction, not mathematical induction, with the greatest of ease, but have proofs that lie anywhere but near at hand and are often found after many fruitless investigations with the combinations. This significant phenomenon arises from the wonderful concatenation of different teachings of this branch of mathematics. And from this it often happens that many theorems, whose proof for years the sudden thing, are later proved in many different ways. Continuously. As a new result is discovered by induction, one must consider as the first requirement the finding of a proof by any possible means. But after such good fortune, one must not in higher arithmetic consider the investigation closed, view the search for other proofs as a superfluous luxury. For sometimes one does not at first come upon the most beautiful and simplest proof, and then it is just the insight into the wonderful concatenation of truth in higher arithmetic that is the chief attraction for

32:30 study, and often leads to the discovery of new truths. For these reasons, the finding of new proofs for known truths is often at least as important as the discovery itself. And this was a sound instinct, right? The pursuit of general reciprocity proved to be among the richest beings in the last two centuries, right? So nearly 100 years after Goff recognized the richness of quadratic reciprocity. Hilbert ratified the judgment that quadratic reciprocity was an important phenomenon of the study by setting what he called the proof of the most general law of reciprocity in any number field as nine on this list of central problems, and the solution was that there was reciprocity law's views made it landmark. Now, the richness of the facts incorporated in quadratic reciprocity hasn't run out even after two centuries of exploration. As witnessed, the 2002 Fields Medal was awarded for work on the Langlands Program, which is an even more ambitious generalization of quadratic reciprocity. Now, one of the reasons the theorem attracts attention is that it cries out for explanation. It's exceptionally surprising. Even if you've seen several proofs, I still find it surprising. As Harold Edwards put it, he says, the reason the law of quadratic reciprocity has held such fascination for so many great mathematicians should be apparent. On the face of it, there's absolutely no relation between the question is pi a square and what a lambda, is lambda square nought p, and yet here is a theorem which shows they are practically the same question. Surely the most fascinating theorems in mathematics are those in which the premises bear the least obvious relation to the conclusions, and the law of quadratic reciprocity is an example par excellence. Many great mathematicians have taken up the challenge presented by this theorem to find a natural proof, or to find a more comprehensive reciprocity phenomenon of which this theorem is a special case. So in this case, that law is actually complicated. Yeah, I mean, in this case, natural is applied to proof, not to property, but, yeah, natural, illuminating in that case. In the case of proofs, I suppose natural means explicit, or can. I'm merely, what I'm here trying to do is point out, or indicate that there is this sort of cluster of soft, so to speak, ideas. is explanation, understanding, natural property, natural domain, natural proof.

35:00 And they all kind of interact with one another. If you know enough about one, you can shed light on the other. And knowing enough about natural proof, you could probably shed light on explanation. Knowing enough about an explanation, you can shed light on natural property. since explanation has been something that's been relatively well studied recently, I'm going to say I'm going to use that as one orienting point but anyway, here's our this is the example I want to indicate mathematically central and perhaps natural stipulative definitions and I sort of want to ask, the question I'm going to be asking more generally about this? And in particular, what's so good about this in connection with ideas of naturalism? What's so good about this in connection with ideas of explanation? Just to kind of get the root of it. Right, so here is the Lejeune symbol, n over p, where p is an odd prime. So this looks like that's the logarithmic symbol. And n mod p is defined as follows. It's going to be 1 if x squared is congruent to n mod p has a solution and n is not congruent to 0 mod p. Minus 1 if x squared is congruent to n mod p has no solution and N is sorry, has no solution and N is not congruent to 0. And it's 0 if N is congruent to 0. if you have, now that you've defined the Le Genre symbol, we can go back, recall quadratic reciprocity. Oops. Here we have quadratic reciprocity. P and Q are on primes. this holds in one case doesn't hold in the other case now given that with our new definition we get this nice simple single case statement of quadratic reciprocity for p and q on primes q over q times q over p is minus 1 times p minus 1

37:30 q minus 1 over 4 that's the statement of quadratic reciprocity suddenly it sets things clean and simple and short So the first thing you might say is, well, look, it simplifies the statement. That can't be bad, right? But not only that, it simplifies the statement in a way that turns out to be systematic. It also streamlines proofs. For example, Dirichlet reformulated Gauss's first proof of quadratic reciprocity. Gauss's proof used nothing fancier than mathematical induction. Dirichlet pointed out that this economy of machinery was traded for a kind of fragmentation because Gauss had to prove eight separate cases. So, you know, Dirichlet used a minor generalization of the Lejeune symbol to turn the cases down to two. now here you might appeal to a general philosophical theory of explanation as a kind of unification this is a theory associated with the fourth picture and Michael Friedman might say well well well we know that if you one of the ways that you can explain is to unify a bunch of cases into fewer cases and then it doesn't and Dirichlet shows that by introducing the genre is simple, you can do just that. So that might seem like a textbook example. Okay, now, there's a problem, and I'll kind of, you know, support the problem with a bit of autobiography. It's a problem already recognized by Kitcher, in a way that turns out to have an epicycle, which is interesting for our purposes here. that there are always often going to be ways of artificially unifying theories, artificially unifying arguments. Like if you come up with predicates that are gerrymandered in the right way, you might come up with a unification, which is unilluminating and spurious. So let's say you have a theory of milkshakes and a theory of the planet Saturn, and you find some way of kind of grafting them together and shortening the discussions of milkshakes and Saturn

40:00 by introducing a predicate, X is a milkshake or the planet Saturn, you haven't really explained anything. You may have gained a certain shortening, but you haven't had any cognitive success. Now, a piece of autobiography. When I originally was setting out to write this paper, trying to find examples of natural property, It occurred to me that I should also come up with some examples of unnatural problems to serve as a contrast. So I sat down and thought, okay, let's go back to my days as a math student. What was the least natural, ugliest, most revolting property I saw? And then my thoughts immediately went back to fourth year number theory, genre symbols. God, that's ugly, right? What a hack! What an artificial device for slapping together a bunch of cases. So, okay, that's going to be my example. Right. So I started reading up the genre symbols to illustrate how completely artificial and what a hack it was, and how it only gained you a sort of temporary advantage. then I started learning more I think I'm going to have to change this chapter a little bit as it became clear to me as I learned more about the relevant number theoretic facts that in fact the algebra symbol was right at the heart of very general phenomena of very general features of quadratic reciprocity that in fact though the definition made it seem artificial sort of disjunct it, it was actually quite natural in need. But anyway, I'm getting ahead of myself. Give that bit of autobiography to illustrate that the Le Genre symbol is the sort of thing that one could take at first blush to be just an artificial gerrymandered hack that was used to gain a kind of temporary and not particularly natural shortening of a statement, Okay, now, I also want to mention a way in which this fits into the general theory of explanation. In his most evolved form of his account of explanation is unification.

42:30 Phil Kitcher and this is in his paper Explanatory Unification in the Couple Structure of the World he recognizes that it's a limitation on his account of explanation as unification he has to give some account of naturalness of property or some account of non-gerrymanderedness because people then pointed out that if you're allowing any amount of this, you know, if you're allowing any gerrymandered artificial properties, you can unify virtually any two characters. Right? Just as, you know, a matter of clever, you know, syntactic manipulation. Right? So he says, okay, okay, okay, I've got restricted to only natural properties. He doesn't use the word natural, but then he says, you know, to non-artificial properties. And then he says, well, here's what I'm going to do, at least as a temporary expedient. One that, you know, he never really advances beyond it, says I'm going to restrict the concepts to be projectible. As I say, they've got to be capable of being used in systematically successful and durable predictions. Now, in a paper of a few years ago, I criticized Pitcher on this ground and thought that this actually represented a significant limitation on this view, because it meant even though he had originally wanted his account of mathematical explanation to include mathematical explanation, I suggested that this was, that by restricting his, uh, the account to unifications that involve projectable properties, he was actually losing the advantage of being able to incorporate mathematical explanation. But now I think I was too hasty. I'm now going to suggest that this is not a problem for Kitcher because mathematical practices of conjecture and verification afford more of a basis for a distinction like that between inductively projectable and inductively non-projectable than I appreciate it. So let's sort of put the question directly, right? Is will a genre symbol a natural mathematical function or is it an artifice? As I pointed out, it might be given force

45:00 first blush to be obviously artificial, obviously a gerrymandering. as I said, the question, as we anticipated in earlier discussion, this question is especially pressing, because discussions in the metaphysicist properties often take as a criterion for unnaturalness purported disjunctiveness of a predicate, which would be reflectiveness and disjunctiveness the Le Genre symbol as first introduced does indeed have this sort of disjunctive definition by cases. So here we have the thesis and antithesis. Prepare for some Marxist dialects. One way to say it is, look, here's the hard line in one direction, but the Le Genre symbol is a useful stipulation that contributes to mathematical knowledge. It allows for one-line statements of theorems that had required several clauses. It supports streamlined proofs by unifying a variety of cases into two. This supports the verdict that it's mathematically natural. So that's somebody who sort of emphasized things like the irreplace. Using it to cut down the case is going to prove. Then the antithesis would be, look, just look at the thing. It's a hack. It's valuable only for very limited reasons pertaining to accidental facts of human psychology. And you can even see in the very syntax of its definition that it's a hack and not a conceptual event. So how are you going to proceed? One foothold, so to say internal to mathematical practice, is that the Le Jarre symbol is itself the object of mathematical investigation. You can ask as a mathematical question, does the Le Jarre symbol carve, put it provocatively, mathematical reality of the joints? And if you ask it as a mathematical question in terms of practice, the verdict is unequivocally yes, right? Now, there's so much mathematical detail, all I can do is sort of wave my hands about it. But I'll just sort of gesture the story. Now, there's sort of small stuff, right? You can point to the general properties of the LeGonard symbol, right? So, restricted to p relatively prime to the argument on top, it's a function from numbers, one minus one. For fixed

47:30 multiplicative that the under multiplication this means that the you're given given the relevant groups you're going to have the function the function is going to be a surjective homomorphism also given the multiplication on one minus one is commutative that is to say the group is abelian well this is actually another fact which sort of turns out to be the group one minus one under multiplication is abelian and you might say well that's a pretty trivial fact about the group right i mean obviously it's abelian what else could it be but it just turns out in general it's extremely important i mean when in the general case it turns out to be extremely important that that group be abelian so even though it's a trivial fact in this case that the group is abelian it's sort of important so these are some of the simple criteria you can use, right? But there are even more high-powered considerations. You know, to crack Hilbert's ninth problem, you need to properly generalize in many directions. And to do this, you need to reformulate the question of reciprocity, right? You can rethink the issue as the circumstances under which you can factor an equation in a field extending the rationals, which you can then see as a question about the relations between the rationals and a certain field you get by extending it with a root. These field extensions can be sorted by the smallest degree of polynomial from the base field that splits in the extension field. In the case we're concerned with, the degree is going to be two. A basic fact of Galois theory is if K is the splitting field of a polynomial over a field F, you can associate with the extension field and the base field a group called the Galois group encoding key information about K mod F and the polynomial. Okay, so let's say we've generalized, we've restructured the question in this way. We can then generalize it. We can consider not just degree two polynomials but other degrees under Galois groups. numbers, but more general structures sharing the basic property of prior. Considering other fields besides Q, it gives us the need to generalize the idea of the integers in the field.

50:00 Galois group is a group of functions. We can define other useful functions in terms of those functions, and so on. Lots of ways in which you can generalize. So after this and more reformulating and generalizing, lasting nearly 200 years, if the clock starts with Euler, you arrive at the arts and reciprocity law. as quadratic reciprocity, cubic reciprocity, 17-ic reciprocity, as special cases. The core is a function called the Archon symbol fixed by several parameters. The base and extension field and use general integers and generalize prime. Punchline for us is that when you plug in the right values for quadratic reciprocity, so the extension is 2, the general integers are the ordinary integers, prime is an ordinary prime. You plug in all these values, and then the Lejarva symbol falls out as the relevant special case. So it turns out to be a function that's an instance of a far more general fact. okay, but still, you know, the skeptic is going to say, yeah, yeah, yeah. Given any fact, given any function in a special case, you can always jigger up many more general versions with the given function theorem as special cases right most of these general versions are going to be completely uninteresting right so here you know we have to now look to yet another idea for generalizations