Varieties of mathematical explanation

2:30 A joint work with William James was faced with a task of writing in an encompassing way on religion.

5:00 He emphasized the variety of phenomena under the topic and warned against the dangers of oversimplification. It's not quite usual to start a talk on philosophy. Besides, most books on the philosophy of religion try to, with a precise definition of what its essence consists of, Some of these would-be definitions might possibly come before us in later portions of the course, and I shall not be pedantic enough to enumerate any of them to you now. Meanwhile, the very fact that there are so many and so different from one another is enough to prove that the word religion cannot stand for any principle or essence, but is rather a collective name. The theorizing environment tends always to the oversimplification of its materials. Let us not fall immediately into a one-sided view of our subject, but let us rather a bit freely at the outset that we might very likely find not one essence, but many characters which might, alternately, be very important to religion. If we substitute explanation in the above broad, the result captures our point of view about ecological explanation. Contemporary work in scientific explanation has pursued, to a great extent, the project of a single, unified account of the nature of explanation. I'm talking here about the philosophy of science. There's a huge literature on explanation. If you take a look at the book by Wesley Salmon called Four Years of Scientific Explanation, there is a full-page bibliography in terms of the quantity of work that has been done there. The drive towards unification has also ignored an important number of phenomena. In particular, many theories of scientific explanation do not address mathematical explanations, either because they rule mathematical explanations out of court from the outset, or because they hold that their account of explanation automatically takes care of mathematical explanations. Most of the time, mathematical explanation is not an explanation. This is a symptom following dreams of the dangers of the theorizing mind and like him we propose to begin by addressing ourselves directly to the concrete facts.

7:30 Now of course it is not our intention to downplay the importance of the work that has been pursued in the area of scientific explanation and which has yielded many remarkable insights. Do not even pass judgment on whether a more careful analysis of concrete scientific case studies of explanatory activity in the empirical sciences might have been beneficial for this subject as a whole. However, indeed, in the case of mathematics, we cannot rely, as people do in the natural sciences, on well-entrenched concern in paradigmatic examples. So in this paper, we will begin with some general methodological remarks. You're wondering right now, what are these things? I would have to motivate, in fact, the whole project to start with and then go more into detail as we progress. Now this realization, so we will then point out that attention to mathematical practice reveals the presence of a great variety of mathematics. This realization affects two important aspects of the discussion of the nature of mathematical explanation. First of all, most of the traditional debates. have focused on the opposition between explanatory and non-explanatory proofs. There are mathematical explanations that do not come under the form of proof. And this has in fact been recognized by several scholars and I've also contributed. However, there are mathematical explanations. Second, sorry, the variety of mathematical explanations challenges the current philosophical accounts of mathematical explanations, i.e. those of Kircher and Steiner. So there are two accounts of mathematical explanations. One is due to Kircher, the other one is due to Spiner. And what we are claiming is that because both accounts suffer of this over-theorizing simplification, right, one model is going to work for everything, that mathematical practice, in fact, has bearing on evaluating whether these models can really be considered good.

10:00 Now, as detailed discussion of case studies is necessary to see the limitations of such accounts, in the second part of the paper, we restrict our focus to Steiner's theory and to the discussion of an example of an explanatory proof taken from real analysis, which we claim Steiner's theory. I should say that in general, this type of approach, mathematical explanatory, well, it doesn't go without saying that there is a lot of philosophy in mathematics that can be done by restricting observations about mathematics. Certain other problems require a completely different one, indicating as a new selection. Section 2 is called Mathematical Explanation or Explanation in Mathematics. In the above, we've been freely using the expression mathematical explanation. The use was intentionally ambiguous, and we should now clarify the source of the ambiguity. In the first case, we intend to refer to explanatory practices that take place in the realm of mathematics. In the second case, we would include, among other things, mathematical explanations of physics, which clearly do not. Now, the second kind of explanation is part of a large problem area concerning mathematics. Shapiro, well, here I have a little spiel about a recent book by Shapiro where he points out the general problem, if you want to account. For the explanatory role of mathematical science, you better have it. But that's an aside, because fortunately, we won't have to tell. Our goal is to talk about explanatory activity within pure mathematics.

12:30 Explanation is a notoriously ambiguous word, and this ambiguity shows up in mathematics just as much as it does in mathematics. We can explain the rules of a certain calculus, the meaning of a symbol, how to carry out a construction, how to fix or set up a proof. These are all instructions on how to master which should be no more puzzling than the equivalent. While doing physics, we might ask for an explanation of a certain notation or of how to describe a certain phenomenon by means of the new formalism. These uses of explanation are of a different category from that involved in explaining, for instance, why salt dissolves in water. So now section 3, the search for explanation within mathematics. In addition to explanation, mathematicians and physicists use a cluster of expressions to refer to this phenomenon. Here is an illustrative sample of expressions we found in the mathematical and philosophical literature. In which the search for explanations is sometimes characterized as a search for, and here I give a list, A, the deep reasons, B, an understanding of the answer, C, a better understanding, D, a satisfying, E, the reason why, F, the true reason, and then the account of the fact that causes of. This is all ordinary. These are ordinary ways of talking you find in mathematical articles that people use very often. Now, of course, we are not claiming that the above expressions have the same intention. However, we maintain that the cluster of notions we indicated is not accidentally related. That mathematicians seek explanations in their ordinary practice and cherish different types of explanations is for us, after working on this topic for so long, so obvious.

15:00 This has to require almost no proof. However, some of the philosophical literature on the topic has denied that there are explanations in mathematics, and thus it will be useful here to provide some examples of explanatory talk in mathematics. Let me just, before going on with this example, point out that I think there are two methodological points of importance here, and they relate very centrally to the theme. If we are able to establish that explanation cast across the two devices, then that already will show that there are interesting issues that in fact can be posed for both realms, and it would be the same methodological issues. Of course, the explanations we look for in both areas might be different, right? You're not going to look for causal explanations and that transcendental number of times is obviously different. A difference in the nature of explanation, perhaps, but that's the first thing. There are methodological problems. And the second one is basically internal to fields. The foundational literature has basically privileged the point of view of either certainty. What we're saying here is, look, mathematical practice, mathematicians pass by your judgments for certainty over others. This happens in many different ways. We're not saying this is the only way in which it happens. This, we think, is a very interesting actual value judgment that is passed very often. Another one would be related to issues of visualizability, which is also another topic that would be interesting. Let's look then at some examples here of ways in which mathematicians often refer. And so, in conclusion, for the second point then, this would explain if we managed to bring home the point

17:30 If these were all there is, it just would not explain why after every... Okay, so, let's see the first quote. These are all taken from recent articles in mathematics. You don't need to know the mathematics to really appreciate them. So, one of the oldest and still one of the most interesting applications of group analysis is the study of the transformation of an ordinary differential, a case in point, which serves to explain and to exemplify the use of... This is actually a standard case where... A number of mathematical phenomena are perceived as too complete, a desire to bring order in the realm of fact. Equations are given to us, but somehow we want more, we want some way of ordering, of giving it. Leads one to look for a deeper explanation or an explanation of what's going on. Best cases, in my opinion, come from the desire to explain resemblances, which are either mysterious or remarkable. that leave us puzzled and usually being puzzled is what forces us to look for explanation. Let's look just at the second case here. The aim of this paper is to begin exploring a new algebra or a geometric approach to the study of the geometry. Which establishes a bridge between the arithmetic approaches. It might help explain some of the mysterious analogies.

20:00 And why is already the ending. Number three is interesting because mathematicians very often modify explanations. By using a modifier that tells you what kind of explanation you're looking for. So, for instance, we found that explanations can be analytic, algebraic, group theoretical, combinatorial, categorical, geometric, and so on and so forth. The quote I have here is quite interesting because three of them appear at the same time. So, this is about an article in Android here. We know two strange dualities. The duality of 14 exceptional unimodular singularities and the duality of 14. The second is to give explanation for the second duality from various viewpoints. In section 5, we give a number theoretic explanation. Then in section 6, we provide a geometric explanation. Finally, the last category, I won't even flash the slide, is basically a neutral one. These quotes were basically analyzed and explained why the area of a Pythagorean triangle can never be a square. So this concludes some mythological concepts. It should be obvious from the above that mathematicians in what form do these two possibilities follow two alternative approaches, top-down or bottom-up.

22:30 In the former approach, one starts with a general model of explanation, perhaps because of its success in the natural sciences, In the latter approach, one begins by avoiding as much as possible any commitment to any particular theory. We favor the second approach for the following reasons. As a rule, contemporary accounts of explanations have been developed within the philosophy of natural science without addressing the specificity of mathematical explanations, as the conceptual resources of those accounts, for instance the notion of causal connections or laws of nature, which are also involved, seem inappropriate for capturing what is going on in mathematical or in explanation mathematics. Furthermore, even if some more abstract features of those accounts, for instance, construing the general form of explanations as answers to why questions, could perhaps be adopted for... Thinking in this way would mean forcing the evidence from mathematical practice into a predefined probability predisposition. The same holds for the few philosophical accounts These are all examples of mathematical explanations found in the literature. So I'm referring here to the account given by Spiner in 1978 in an article called Mathematical Explanations in Journal of Philosophy. And the other one is Kicher. Kicher has never written actually just singly one thing, but if you look at all his books and articles you will see, well the book on mathematical knowledge from 1984 has sections on mathematical explanations, but also his articles on scientific explanations. Usually, which define the unification theory, we usually say, and the fact that these models also account for explanation method, right? Speaks in their face. You solve the problem and immediately you also get it. What we say is that making either theoretical unification a la feature or deformatism is the hallmark of this.

25:00 And that's the imposition of a defining characteristic feature or on what ought to be counted. Theories, methods, etc. which do not satisfy and are then disregarded or discounted, regardless whether they are taken to be by working mathematics. Thus, in our mind, the fruitful approach would consist in giving, first of all, a taxonomy of recurrent types of explanations, and then trying to see whether these types are heterogeneous, neither giving the taxonomy nor arguing for the previous single case study. This can be seen, as it were, as a case study of how to show that the variety of math is Steiner's theorem and discuss a counterexample in Witte's argument from 1978. Keeping his own account of explanatory proofs in mathematics, Mark Steiner discusses and rejects a number of initially plausible criteria for explanation. So, you might think at first, well, what is it to be more explanatory? Oh, it's just to be more abstract. And then you try that criteria. Or you could say, well, an explanatory proof is a visualizable one, and you can run some examples with, say, Gauss sampling the first 100 integers, and you'll see that certain proofs, in fact, seem to explain that. Or you could say that an explanatory proof is the one that tells you how you could have discovered the result. In other words, it displays the genes that create the typical objection to Archimedes' theory in the 17th century.

27:30 Well, Steiner rejects all these criteria and takes up the idea from the essence in order to avoid the notorious difficulty in defining the concept of essence or essential or necessary property, which, moreover, do not seem to be very useful in mathematics because, anyway, Steiner introduces the concept of a characterizing property, okay, so this is one of the keys, a property unique to a given entity or structure. Within a family of such entities, again, a characterizing form is a property which is unique to a given entity or structure within a family or domain of such entities. I'm soon going to give an example. By this he means, and first of all, the notion of family is here taken as undefined. For Agon, I should say something. This paper is 40 pages long. There are 20 pages later, which I will not present, where we really take it apart. We take this model apart. Because there are so many undesigned terms that, in a way, even our discussion as I will present it, leaves a lot of leeway for Steiner to make comebacks for the simple reason that the model is being described in a vague way. But we don't give him a lot of leeway. In other words, we pursue every possible trend we can think of in pushing his undefined notions in such a way that he could... If you feel a certain sense of unsatisfaction, the model I grant you from the other is characterizing properly.

30:00 Hence, what distinguishes an explanatory proof from a non-explanatory proof is that, in Steiner's words, and I quote again, an explanatory proof makes reference to a characterisation of an entity or it is evident that the result depends on that property. There is another, is that proof must be generalisable. By an array of the formations of the original proof, the status arrives at two criteria. Dependence on a characteristic and generalizability through varying of that proof. The following proof, just to give now a concrete example of what you might have had in mind. Consider a proof of the irrationality of square root of 2 given as follows. There is another proof which is just as simple which goes as follows. You say you square both sides and you get 2 equals a squared over b squared and then you get 2b squared and now you observe the following. You observe that a squared is even and in fact not only is it even because it's a square

32:30 The prime power expansion of A squared will contain an even number of powers of 2. Similarly, that B squared is also even. But now there is this extra 2. So that B squared has an even number of prime power expansion of 2. When you multiply by 2, it becomes odd. So now you have the same number as an odd and an even number of powers of 2. It appeals to the fact that every single natural number has a unique, and in fact, not only that, within the family of natural numbers, each single number has its own unique prime power expansion, has its own unique prime power expansion. It's only true of two that it has that very specific. That's the theory somehow, is that, look, in this case, what have you got? You've got a proof that goes to a correct order. However, can you generalize this proof? It's quite simple to actually extend this proof to square roots, not just of two, but of any. And you can also generalize it. The article they wrote is explanatory proofs. So what I mean by that is, apart from those appeal to make it evident and deep,

35:00 which means that in the realm of all possible mathematics, proofs that are explanatory, our job is simply to find the criteria that we shall have to make important. For instance, there is no appeal to context, with the small exception of this make-evident 2, but that's not really what he's insisting on. He's insisting on these very, very low properties that they deny. There is a conclusion. Simply do not believe that there is such a divine in reality. And so they deny an objectivist, but more concretely, they also challenge Steiner's account by proposing counter-examples, i.e. a proof that that is not accepted, a claim that there are proofs. Namely, a certain group of data to qualify as explanatory, but to begin with, what justifications are put forward by Riesling and Kushner for the classification of their examples as indeed explanatory, besides simply claiming that these groups, and I quote, would seem to qualify as explanatory if any need be. It is contended with some Rather vague reference to mathematical logical practice, that Engels' proof, for instance, I quote again, is generally regarded as really showing what goes on in the computer sphere.

37:30 And the proof idea is being used again and again in obtaining results of the logical system. And with respect to the proof of the intermediate value theorem, they also say, and I quote the quote, they found it hard to see how anyone could understand this proof and yet ask why the theorem is true or what makes it true. This is the nature of their counterexample to science, first showing that these theorems do not. Yet we are not given any hint as to what exactly the explanatory features of these proofs are supposed. They appeal to simply, they are just evidence. Well, we think that for counter-examples to Steiner's theory to come a real way, they would have to come to be much more closely connected to mathematics. Contrary to what Resnick and Kuchner, mathematicians often discard themselves and their judgments concerning explanatory versus non-explanatory has to figure as the basic evidence. It seems to be a fact that certain proofs are explanatory come, in our mind, from within mathematics, not from philosophers. Working mathematicians are furthermore precisely identifiable, and the case for explanatoriness will be even stronger if a certain proof is put forward explicitly with the aim to explain a mathematical phenomena which has been acknowledged for a long time to be mysterious or puzzling by a subgroup of the mathematics. And it is such a test case from the work of Prinsine in the theory of infinite series that we want to present.

40:00 So far I have given you basically the general methodological picture, one of the most available for talking about mathematics. Now we're going to get into some more technical material. Of course, what's the mathematical practice that we can appeal to? Well, let's see their interviews. So we're going to look at an historical case of explanatory activity. This has to do with Kummer's convergence test. That's where it all starts, our case, in 1835 and will go up to 1916. With some preliminary observations concerning infinite series. There are pretty trivial observations for anyone. Because we are using Trinsheim's work, that's our end in the line of analysis, we will be dealing with infinite series, where you have summations from n to 1, n equals 1 to infinity, of positive terms. So the a n's, the coefficients appearing for the c n's and so on, are always positive terms, greater than 0. It's different convergence and divergence s. Now, of fundamental importance are the following comparison tests. First of all, and look at another trick, when we have a convergent series, we use as a coefficient cn. When the series will be divergent, or we intend to refer to a divergent series, we have as a coefficient dn.

42:30 If we want to say that we don't know which one, you know, whether it's divergent or convergent, then we use generic an or bn. Such that the terms of an order series, now with a n's, satisfy that every a n is bounded by the respective c n on some value of n. Well, if c n is convergent, then of course... Same thing about mirror image, as it were, for the divergence series. If you have a divergence series and you show that you have a series that majorizes a coordinate series, Now, it turns out that comparison tests of this sort are often easier to work with in practice when they are stated in slightly different forms. In order to simplify the exposition, cn stands for convergence, dn for divergence. What's the simpler form in which you can state some of these results? You can let cn will stand for 1 over and capital dn is 1 over small dn. I can state some other tests, which, however, I just realized I didn't need, but this will be useful to have for later on. Okay, so let's just stay with the first two. Now, the first two tests, together with other tests of a similar type, are called comparison tests of the first common. But in practice, it's often much more useful to work with ratios of subsequent terms in the series.

45:00 And you study that. Those are called, the tests you get out of that are called comparison tests of the second kind. Here are two examples. If for all n greater than or equal to n, you have that a n plus 1 over a n is majorized by c n plus 1 over c n, then the a n's converge. Of course, the c n's here. And conversely, reciprocally, for... If you have that for all n greater than or equal to m, a n plus 1 over a n majorizes d n plus 1 over a n, then the series of a. Now let's assume we have defined capital C n equal 1 over 1 over, we're going to simplify just the, if you do that and you do a few algebraic transformations, which I won't waste your time doing, okay, so here you have it, the convention C n and capital C n and capital V n. When you do a few algebraic transformations, you get this new criteria that that complex term cn times an over an plus one minus cn plus one is greater than or equal to zero. If that condition is accurate, then the an number ten, the limits exist. And then you have that if the limit, then the an. The interesting thing about these two tests, the last two, because that's the one we're focusing on right now.

47:30 As I said, they are obtained by simple algebraic manipulation on the previous ones I gave you. It's just a question of replacing capital CN by one order and so on. But what's important here is to focus on the role of these capital CN and DNs. See, the capital CNs originate from a series that we know to be convergent. That's the data on which we are building. We know that that series of lowercase c n's is convergent. And so what we are doing there is exploiting this fact to get a convergence criteria. In the second one, we know that the capital D n's originate from a series which is divergent, and we exploit that fact to get a divergence criteria. So it would seem to be essential that anyway you start from either a convergent or a divergent series. That says that if you have the limit of n going to infinity of bn times an over n, then the summation of the an... But what's surprising about it is that, in other words, they don't originate from either... You don't know from what they originate. It's just simply any sequence whatsoever. That's a surprising thing because up there you feel like you're basically home. Well, of course, I've got the information about the convergent sequence to start with, or I've got the information about the divergent sequence. Here, no. Here it's just any old sequence will do the job. And this is a rather striking text, exactly because... Sorry? Anything. Yeah. And of course, anything, any sequence you give me of uppercase BNs can be also seen as 1 over BN, right? You can take the reciprocal, but... Yeah, the condition is to be satisfied, of course, right? You must have the antecedent of the error.

50:00 So if you have that the antecedent of the error holds, we're not saying that for any sequence whatsoever you're going to get the antecedent. If the antecedent works and is convergent and is greater than zero, in other words, there is a limit which is greater than zero, then, so yeah, it's slightly different. What is rather striking about The extreme generality or arbitrariness of this sequence behind O'Connor. Whereas the tests 11 and 12 at all required the use of sequences which derived respectively from O'Connor, we see in any old sequence. String science calls it a most remarkable criterion in its lectures on mathematics, of indeed surprising generality that studies in it was by no means exceptional. Many mathematicians must have been similarly puzzled and left unsatisfied by Cougar's original approval. As Knoop notes, it wasn't until 1885 that Otto Stolz gave an extreme, by means of which the criterion was first rendered even after another 30 years. This criterion was still viewed as an anomaly of sorts, defined smooth integration. Prinsheim noticed that Kummer's criterion appeared as totally erratic in other accounts. It is seemingly lacking any analog among the convergence criteria, and so it does aim at presenting it, gives two different proofs, and one which proves the correctness of the criteria a posteriori in a simpler way.

52:30 Let's begin with the latter. It's basically Stoltz's proof. Trying to bring home now the idea that there are now two proofs that for this mathematician have obviously opposed epistemic values. One is explanatory, the other one is not. The first proof that we are looking at is the one that Stoltz gave. It's supposed to be, right, according to what he says, to basically show the result and still leave you pretty puzzled about how does this work. Okay, so this is the one that proves the correctness of the criteria posteriorly, according to Stoltz. By the way, just to lighten up the talk. An interesting mathematician from the 20th century, very important person in Munich at that time, and his daughter, in fact, was Katia Mann, was the wife of, so he had a very strong connection. In fact, I think that at some point Thomas Mann even threatened to shoot him with a gun for some debate they had in the family. Let's go back to the proof batch thoughts. Okay, so remember... Sorry? Is the proof by Scholes presented by Pringson, but it's basically the old proof, yeah. So remember that it's an if-then statement, so we assume the truth of the antecedent and we're going to try to prove the conclusion. So we're given that the limit of n going to infinity of bn, where bn is arbitrary, times an over an plus one minus bn plus one is greater than zero. Well, if we know that... Then there must be a positive rho such that from some stage n on, the bn times an over an plus 1 minus bn plus 1 are greater than or equal to rho, which is greater than z plus just out of the definition.

55:00 Now, if that's the case, if it is true that from some n onward you have that inequality, look at what's happening. Multiply on both sides now by an plus 1. You get bn times an minus bn plus 1 times an plus 1 greater than or equal to rho times an plus 1. Remember, all these are positive quantities. But if that's the case, bn times an minus bn plus 1 times an plus 1, it's always a positive number. What that means is that you have an infinite decrease. In other words, each new term, an plus 1 times bn plus 1, is going to be smaller than the previous one. If it is an infinite sequence of positive numbers, it will converge. Let's call this limit alpha. Alpha is greater than or equal to zero. Let's look at those terms, bn times an minus bn plus one times. And let's start summing them up from a certain point m onward to a certain point k. Let's just look at a finite segment from n up to m plus k. We'll write it up, and what we get is this expression, bn times an minus bn plus one times an. All the intermediate terms here can get deleted because they are the negative ones, and so by going the sum, term by term, we are reduced basically to Let's let k vary to infinity. What do we get? Well, we knew that bn times an, if n is varying to infinity, was going to alpha. So that's basically what you get here. We are making this thing vary to infinity, so we get the limit alpha.

57:30 And here what do we get? Well, we get rho times this infinite summation from n to infinity. This is positive. This is positive. It's greater than or equal to a positive number times the summation. So this thing must convert because that's the whole point of the claim made by Prinsan. Even if you don't feel passive, make our claim a proof. Now I want to finish simply by looking at the other proof. Yes. No, not very different at all. No, it's very similar. Right, so this is this is Stoltz's proof, basically the original one. Now we are looking at Prinsheim's proof, and it's a time to say, look, I have a proof that will show you why things are going really the way they're going. And now the proof is not simpler in the sense that it requires less steps. On the contrary, it requires a representation theorem that we do in the appendix That takes time, but it's not anything that changes anything, but the representation theorem needs to be established for what is going on here. Okay, so Prinsan proof goes this way. For any summation of Cn's, okay, so the Cn's are convergent now, pick a strictly increasing, sorry, this is a result, result number 15. Which satisfy that they diverge, in other words, they limit, and is such that Cn equals Mn minus Mn minus 1 over Mn times Mn minus 1. No, I'm not claiming you should give, this is the representation, as I said, it's proved in the appendix, but I'm just putting it there otherwise we never...

1:00:00 So, conversely though, every sequence of terms defined in this way turns out to be convergent. In the case of divergent theories, we also have a representation theorem of this kind that says that we can find a sequence Mn, like the one described above, which is such that the dn's are now Mn-Mn-1 over Mn-1. In this case there is no limit of the Mn? It's the same, no, same conditions. I say that in fact in the slide. As above, with the same conditions. And conversely, every sequence of terms defined in this way turns out to be a mistake. It's divergent. Okay, now, let's assume, again, now we're proving the theorem, if we have this too fast. Again, let's get the antecedent of the condition. Let's assume, in fact, the limit of those bn's and so on is greater than or equal to zero. Now we argue by cases, and we say, well, look. The sum of the bn's, remember those lowercase bn's originate, right, from the uppercase bn. I mean they're one the reciprocal of the other, so that's how they're related. So you can argue this way. Either the summation of the bn's, lowercase bn's, converges, and then the an's simply converge because of our criterion 11, which I've already given, which is they diverge. And so bn turns out to be of type dn because it takes origin, it originates from a divergent series, so it is of type dn. So we can reformulate our assumption now as the following assumption, this one here, that the limit for n going to infinity with those dn's now in place of the arbitrary dn's is greater than 0. We reason as before and we claim that there is a role, a positive role, which is such that

1:02:30 A little algebraic transformation down there, dividing by rho on both sides, gives us the equation. If the summation of the bn's is divergent, so will be multiplying termwise by rho, a positive number. We get another divergent series. Hence the terms rho times bn can be expressed by means of a sequence m over mn. Right, now we're using the representation theorem I mentioned before. So we have a sequence of numbers, positive real numbers, mn, such that each rho times vn, in fact, has that peculiar form, mn minus mn plus 1, sorry, mn minus 1 over m sub n minus 1. Okay, the next step is purely algebraic. And now, well, I'm sorry that this is hard for you to see because it's on two different slides, but there is a substitution in line 18, from line 18 as you see here. So, if we substitute broad times dn in 18, we get this very complex expression. We transform it algebraically again, I know you can't take it in in jazz, but the fact of the matter is that at the end of the game, we end up with This particular expression where you have a certain term times a n over a n plus one minus another term which is greater than or equal to z. And these are all simple algebraic transformations. Again, we appeal to the criterion, to the representability, and we see that that particular statement can be rewritten as c n times a n over a n plus one minus c n plus one greater than or equal to z.

1:05:00 And so we're back to one of the old forms we had from equation number nine. It was one of the old criterion. We have found these are the C n's which come from convergent, from particular convergent. And now once we have that assumption, we know that in fact the series of A n's converge. So that's the end of it. Okay, as I say, I realized that Looking at this machinery will just leave you a bit out. There is nothing technically very complicated. It's basically algebraic. The hard part is, in fact, coming up with a representation, the two representation criteria for convergence. One way to look at what this has done is to think in the following terms. Look, you argued by case and you said, take an arbitrary bn. Either the sequence they gave us to it was convergent or it was divergent. If it was convergent, we're back to the first criterion we knew. How about if it is divergent? Well, don't worry, we have a representation criteria. If you do all the algebraic transformations, you end up with this equation that brings you back to a criterion you already knew. A perfect example of reduction to the familiar, explaining as reduction to the familiar. And that's exactly what, he doesn't put it this way, of course, Prinsheim, but that's exactly the point of his claim that goes well beyond the proof given by Stoltz and that really explains what's going on. Okay, let me conclude because I went on for too long. And here I apologize, I realize that this is too long a paper to present, but at least I think you got some of the methodology that we want to use in addressing these very thorny issues. Intuition that is going to carry everyone on board, because people talk about this phenomenon, but putting everyone on board on the same example is not going to work.

1:07:30 It doesn't really affect what we're doing, because we claim that if a certain mathematician or a certain subgroup of mathematicians, that is a thing we can explain, even if we or other mathematicians do not consider those proofs to be explanatory. In other words, for us, an explanation is a context-dependent phenomenon. So we don't believe, like Steiner, that there are these things in the heavens, some are explanatory, some are not. We'll bring in context also, and that's part of the general outline of the project. Now, you might be asking the last question, and maybe we can address it in the question and answer. How is this related, then, to Steiner? Why is this a counter-example? Let me just say two things. As I said, we spent about 20 pages arguing back and forth. But it's basically two things. Remember how the proof of square root of 2 worked. You needed to identify one property which was typical of one element, like 2, and use that as the characterizer. Well here, from the very outside, you are at the most extreme level of generality. Your theorem is about all sequences. There is no characterizing property. There is a problem of the characterizing property. There is also a problem of dependence in order to be explanatory according to Steiner. We'll have to show how this property, sorry let me just step back. So let me say this, besides liking characterizing property, Prinsheim's proof also fails to meet Steiner's second criteria for unexplorative proof, namely generalizability.

1:10:00 We illustrated this phenomenon, this criterion, by recalling the proof of the rational hyperscope. In analogy with this example, one could imagine the formulation of convergent tests for infinite series involving a concrete series. Whose proof could then be generalized, and thus count as explanatory, and to establish similar convention tests involving other concrete sequences, and eventually to establish a more general test, referring only to a generic sequence. However, Prinsheim's proof doesn't fit this pattern. Comer's test is already the most general with respect to the number sequence figuring in it, and you cannot generalize any further.

1:12:30 A number is irrational if and only if it is proved that for any then misconstrued rational expression is periodic and therefore make it the most explanatory proof. Good, good. But not at all, because it reduces by a very simple algebraic trick the problem to the basic fact of real analysis that any decreasing sequence of positive numbers converges as if. A very simple and very precise observation reserve is interesting only in so far as once you are represented then you prove to be constricted. Well, yeah, okay, so two things. Thanks for both points. For one thing, they immediately show the problems I was addressing from the very beginning. Just as much as you make your claim about the infinite being more explanatory than other groups, you find other methodologies. And so that's a fact we have to face. You can take two ways to go on a basic realization. You can either say, look, this is just like a state, not something you can account for. So the old phenomena, like in the old times, can be taken and put back in the dustbin of problems that can be solved. That's, I think, what many people might want to do. Of course, given that we have invested in this project, we are saying you can't do this. In other words, let's see how far we can go. But we have to take as evidence that there are wild differences in...

1:15:00 Now, Steiner cannot get away from it, see, because Steiner, having given his theory as being about an objective, as soon as someone comes around and says, I don't find that explanatory, he basically has to say, you're not a good mathematician. You don't see far enough in the realm of truth and so on. We have a different way out. We have a way out which is the following. We think that still new judgments, explanatoriness, can be accounted for. And can be accounted for in such a way that we can rationally allow for people to have different judgments about the explanatoriness of certain results. Let me give an example that maybe we'll talk to. Another typical example where this type of place is, people work in number theory. Some people will tell you that the only explanatory is one that is rooted in the elementary facts about it. So, a number theoretic statement. You know why we invent complex analysis? It's exactly to put things in context and to show you why certain things hold and we explain.

1:17:30 That is, if we don't want to throw the whole game out of the room, right? There's always the first option. Someone might be skeptical and say, we want to say, well, let's look at it first, see where we can go. Since we're so sensitive to what happens in the world, that people diverge wildly on their intuition. You had a second point, though, that we can do that, right? Both of my questions, what I said was about the use of representations. Oh, they, they, right. Yes, again, it's the same problem, but I take it seriously that if a mathematician like Prinsheim, who, by the way, doesn't just do it for this particular theorem, Prinsheim is known for having given a complete reformulation of complex analysis along the Weyerstrassian lines, but with a different central concept. That he claimed to have an advantage over both Riemannian analysis and Cauchy, and over various stress-to-reach approaches, exactly non-explanatory. What I want to say again, I can very well grant you that you might find this proof non-explanatory. The fact remains that here I present a particular case of one mathematician motivated by explanatory concerns, who in fact goes on to make this rational. And I think you can also then give examples for general groups, i.e. sub-communities of the mathematical community, maybe you don't want to say, you know, when you talk to category theorists, they will tell you that they don't understand any result unless it's stated in category theoretic terms. I don't know how much truth there is to that. I'm sure you can find cases, of course we're just starting with some examples, but where you have segments of the mathematical community involved on example issues.

1:20:00 The comment is that the notion of explanatory power, in my opinion, is not a law. So I think the issue is not to say we have explanatory proof. Now, extraordinary to me means more the question of organising, organising your conference, and organising your conference. So it's not just, it is the whole market. It seems that from this example that one way to make things intelligent is more or less to come back to Descartes. And I was surprised too that my colleagues did not mention Descartes. Descartes has much to say about that. About what makes the things inevitable, what makes the events interesting, what do you have to do to convince. Okay, so that, to me, something becomes explanatory when you are able to split the whole body into a number of subunits. Each one having a well-defined meaning, a well-defined meaning. So, positive, double, positive. That's one step. Then, we have another difference, what you mentioned, one of the comparisons. So, the point is that we can arrange the whole thing by splitting it into each one, and I think that we move on. One of the methods that people, mathematicians, know to construct a computation, you first make an algebra, and you say,

1:22:30 I will first rely on this general principle over this, and then I will write it down. So, organizing complicated groups to split the program into many certain programs. Each one has a well-defined name. Each one has a deserved name. And then, so, you want to do an outline. And then you want, say, like, something from one store to another one. And then you need the area to be what you need to be. That means each step has to be just enough. Maybe by subdividing it into sorts. And so on and so on. But what you are looking at is what you are looking at is to arrange all these concepts in a very simple way so that what is not simple and explicit is weighed on the implicit knowledge, I mean on the know-how of the people. I mean if you say, right, and that's the way you do it, you write it down. Here I've written such a general principle about it. And then individualize it, the simple algebraic proof may be transformed.