Towards a theory of indistinguishables

0:00 I put on first the fact, usually to say a few words about Parker-Rhodes combinatorial hierarchy, so I thought maybe it was time for a change this year, it reminded me of a cartoon I remember many years ago in some Cambridge magazine, I can't remember what it was, of two aged dons leaving Mill Lane lecture rooms and one of them says The tenth time. Surely the young gentlemen know it by now. It's not that I've been neglecting the combinatorial hierarchy in the past year. Ted and I have been trying for a cast on a philosophical basis and I'm not ready to talk about that yet. But in any case, if you really wanted to start off with that, you've only to wait until Sunday when I shall begin with a quick look at the hierarchy. In my spare time with Ted in the past year, I've been replying to letters from George Bickman in Sweden, who some of you I think know, about the other major effort of Frederick's, his theory of indistinguishables. I undertook to work on it with George, and later John Amson made some contributions, but what I'm saying here is essentially my view of the matter. They've both been a great help, but must not be held responsible for any defects in what follows. Such was the initial unreadability of the book, Frederick's book, for me, that I no longer have a copy. I lent my gift copy from Frederick to someone without wondering too much who it was to get it back. Wasn't it in our letter? No, not before I knew it. But George has helped in a very concrete way, providing me with a Xerox copy. One thing that the three of us are certainly agreed on is that the definite article in Frederick's title should be replaced by the indefinite article,

2:30 for there are, I'm sure, several theories of indistinguishables. I've been a very poor correspondent with George, so poor, I may say, that George has just sent me a paper he's been reading on indistinguishables in Sweden. That is inconsistent with what I'm mostly going to say now, and one result of that is that my carefully prepared Parker Rose lecture has had to be changed considerably at the last moment. Still, I think I have something to say today, even if it's only a program for a year's work ahead. Of course, we're not the first to make this attempt on Frederick's book. I believe Chris Clarke made some progress, but regrettably didn't publish it. Frederick is careful to distinguish his theory and his distinctions from the hierarchy, and in particular from what Ted Bastin and I have done. I think he thought, but he was too polite to say so, that we'd taken the wrong path. As you will see, if I get to the end of what I've prepared, There's an odd sense in which both parties are actually engaged in similar projects. We both provided a substructure that will seek to validate Frederick's original odd piece of hierarchical matrix algebra which produced the 3, 7, 127 and 10 to the 38 elements with, importantly, the bar to go any further. I'll start by surveying some of the difficulties in Frederick's book. A process which may leave me with rather little time to tell you how I intend to go ahead. I'll do the best. I'll try and avoid the following situation. When Bondi was quite a young lecturer here, He gave a part three course on general relativity which a friend of mine attended, a person who knew the subject well but wanted to see what Onn thought of it.

5:00 And he said, it's terrible. People there, certainly our undergraduates, will come away convinced that the theory is entirely worthless because he's only told us what's wrong with it. Well I try not to do that, but it's quite difficult. One thing I will do, I should say straight away, in case you know the book, is to ignore Frederick's Chapter 3. So I'm not going to be involved in discussion about the incoherent plane and those sorts of things. I'm just going to deal with part one of the book, omitting Frederick's so-called physical applications. Now the book was number 150 in Riedel's series called Studies in Epistemology, Logic, Methodology and Philosophy of Science. This must be borne in mind when thinking about it, because although it partakes of all those four headings, it is mainly about logic. What Frederick's doing is not a general theory of indistinguishables, but a formal logic of them. It's helpful to see this, I find, as a total reversal of what Wittgenstein was doing in the Tractatus. As I see it, what he was doing was setting out just what sort of world was being described by Whitehead and Russell in Principia Deo Mathematica and marking this world off. From those parts of the experience world which, though they may be important, you can't speak of in that language. Conversely, Frederick is asking, just what logical structure is needed for a world containing its extinguishers? And is such a consistent logic possible? He believed the answer to be yes, and that this had consequences for physics. My present view is that it's still an open question. And I say this because even if we suppose that Frederick's formulation is satisfactory, or can be made so, we still do not have a proof that the whole system is a consistent one.

7:30 The matter is complicated by the fact that not all the axioms of the system come at the beginning, but at various stages in the development. Is it all right? Yes, it's okay. Clive, do you know if Frederick had to rush the volume? Was he late for the copy date? Well, I don't know. As a writer of books, I've never yet not been late for the copy date. But that's an interesting point because there is evidence that it hasn't been as carefully... Well, of course, it would be a bit hard to ask any copy editor to improve the matter, but there are signs that might be the objective. I don't know. The restriction of logic is one of the difficulties. Another is the question, what are indistinguishables and why did Frederick worry about them? One might suppose that he was thinking about elementary particles. Those are Einstein's statistics and all that. I don't know if he started from there, but he's at pains to make a distinction. Already in page 4 of the book, he talks of electrons as being equivalents, but not genuinely distinguished. And says about them, they can consistently be treated in one context as identical and in another as distinct. Now his reason for rejecting this as a definition of indistinguishables is that equivalence in this sense is, he says, a technique and not a theory. And also, he says, because it's essentially only an epistemic matter. He's asking rather whether one can have A certain class of things which are fundamentally indistinguishable, fundamentally indistinguishable, and he glosses fundamentally by saying that for equivalence he went to the basic level only by throwing away information whereas his hope Is to see whether information could emerge somehow in the process of ascending from the lower level of indistinguishables to the higher one of distinct things.

10:00 You see, there are new people here who perhaps don't know the definition of the statement, wouldn't it be helpful to just give this definition? No, because this is what I'm just coming to. There is no... I mean, the word has clear English meaning and that will have to do on the moment. All right, OK, so it works out. Because it's far from clearing after you've read all Frederick's books, just one of these things, which it was hard, one of them I've dropped it at the end of the day, he gives another way in after thinking about electrons and putting it on one side, he gives an example for ordinary mathematics, but doesn't tell very much that there's He said, look at the expression A plus A plus B for the sin. He says the two occurrences of A and plus are indistinguishable symbols. But the divisors of the A and B here are identical symbols of C, so he's trying to point there at the indistinguishability meaning that they're not the same, these A's, but of course they are nearly the same, but the... They're nearly the same, it's up to the printer to get them exactly the same, but this C is really absolutely the same, but that's his second attempt to find a way in without giving him... But wouldn't that produce a change if you were to say instead of A plus B over C, you were to say A over C plus B over C? Yeah, then the two C's would be indistinguishable, according to Fred. Well, Frederick doesn't give any very explicit definition of an indistinguishable. So you can point at indistinguishables? What Tony's just said says peculiar things about the equals.

12:30 You mean... Yeah, yeah, right. I'll also... A bit discomforted by your saying you can point at them because, of course, another thing Frederick says a lot is that if you've got some, then you can't, as you can't distinguish them, you can't point at them. Anyway, Frederick doesn't give any explicit definition and now turning foot to George Rickman for a moment and my correspondence. He puts a lot of emphasis on one of the things Frederick says, which is the inability of being able to count, meaning two electrons can be differentiated if you do the right sort of observation, but at other times in statistics they need to be counted as well. Frederick, on the other hand, prefers to skate around the difficulty by defining instead what he calls a triparatus mathematics. In which two things can be related in one of three ways. Either you can say that A is equal to B, or you could say, and you usually put it in this form, that A and B are twins, which is a merely short, I mean, not actually, twin isn't a very good word here, because it takes a certain overtone to a part. Or, thirdly, you can say it's neither of those, and I slightly like to say neither. This notation isn't quite Frederick's, it's just simplified for your purposes. So, Frederick then, having done that, has a quick look at negation. He takes a not equal to b to mean either this or that. John Anderson's gone in quite at the deep end over this, like inheritance mathematics, and he prefers a different definition of negation, in which the negation of each of these statements is the one below, just the one below, and then of course we get to the bottom, to the top again. But there isn't a world of difference between these two because you can always

15:00 I'm going to express one of these negations in terms of the other, so it's merely a matter of notation of the fundamental idea. If you were to take the notion of two things being twins as clear, which of course it isn't at this stage, then you could define what an indistinguishable is by saying, well, x has the property of being an indistinguishable if and only if there exists some y such that x and y are twins. Then those would be the indistinguishables that we're talking about. But, see, is this notion of a twin clear? And that would be true that there is implicitly more than one x after the n as well? Any x would be an indistinguishable if there were a y of which it was a twin. Right, right. But it's not as helpful as one would hope because we haven't got the notion of a tween clear. This is a bit cumulonimation in a way as well because it's context-dependent. Of course, yeah. No, no, but I mean, that's the next point. Thank you for bringing that up. Frederick says the notation is of entities which can or cannot be distinguished according to context. And to refer to my correspondence with George Bickman, he goes to tarot context. He tries to give, actually because I prompted him to do this I think, he tries to give two different kinds of context in terms of which one can understand this sort of thing. Which very closely follows the way Frederick starts, is to say, well, suppose you have two objects, two symbols, Frederick, A and B, and there is some collection with some other things, like that, say some collection, we don't want to use such technical phrases like set, because that brings all sorts of...

17:30 People will say, oh, you do mean, you know, someone's had a set here, doesn't he? But there's a collection. We have the time, I think, isn't it? Sorry. In a set, technically speaking, the elements must be distinct. Well, yes. So, indeed, these might not be... But both of them have to be in some collection. And then... The idea would be that if, in this context, the cardinality of this collection was raised by two by putting them in, then the A and B are distinct, but if they're twins, it's only raised by one, and if they're equal, it's only raised by one. And that's one of George's things, and then the other idea of context which he raises is that if A and B are related by a set of functions or operators or something which allow for a direct comparison. I'll have to look at his letter a bit further and do more study, but I'll return now to my description of Frederick. The whole notion of context gives the mathematician a real shock, because after all, the painful course of mathematical history since 1850, if not earlier, has been a struggle to be context-free, and the same goes for the whole of formal logic. And with this in mind, this important notion of context will need a lot of formula. Pritik doesn't do this. He mentions the idea of taking context as a set, but that soon takes a back seat. Because, perhaps because, if you put A and B into different contexts, suppose you put A into S and B into T, and so distinguish them because they're in different contexts, you have to say, but then at that very same time, they are in a union at S and T, and so they're then in a context in which they wouldn't be distinguished.

20:00 Maybe one can sort that problem out, but it means that even before we can get any explicit definition of context, we're in a certain amount of trouble. There's another difficulty which has been troubling me for the last few months. I think I know what I want to say about it now, but it's quite different from what George said, and that is over the question of whether this twinship relation is transitive. It seems to me that it's not transitive, but Frederick, at some points in the book, as George pointed out to me, uses the transitivity in his proofs. So, there is a problem there if you agree with these aspects. However you're going to define it, it seems clear that it's possible, because if A and B are unable to be distinguished in one context, but they can in others, for example, and then the same for B and C, well, there might not be an appropriate context for A and C. So it doesn't seem to be any possibility that these things should be transcribed. So you would agree that twinship within a given context should be transcribed? Yeah, yeah. Well, it would be as long as we had a clear understanding of what within a given context is, and that's one of the problems. This non-transitive character of twinship takes me back about 800 years to one of the heresies which brought Peter Abelard into trouble about the Trinity because he argued that a difficulty which they felt about the Trinity at the time could be overcome I modernised the notation but in effect he felt that there was a twinship between the Father and the Son and between the Father and the Holy Ghost.

22:30 But these two, because of the non-plastic character, these two weren't twins, and that apparently helped his thinking over a problem that worried him at the time, but we may not feel this quite so early. However, as I said a moment ago, the non-transitive character of Queenship is important because at least in one point in the book, on page 64, it plays an important part in one of his proofs, and if one reads the book very carefully, there may be others as well. This is also a major difference from what George has been producing and sending to me because he decided After some discussion we'll be doing this. Sorry? I'm sorry. What about the reflexivity? Isn't there indistinguishable itself? I certainly take it to be reflexive, yes. Well, let's just say I take it to be... No, I don't know about reflexivity, sorry. That's rather just a matter of notation, because if you say it reflects it, then you have to allow that it subsumes equality as a special case. And if you want to do that, you can, but if you don't... Perhaps I'm asking whether A is indistinguishable from Z. Well, I'm saying if you have, I'm finding it difficult to get a real meaning of this discussion. Sorry, I'm saying it, but it's because if we are dealing with equations, obviously, but if we are dealing with this logic where there could be situations that Are these things distinguishable from each other? Do you have any suggestions?

25:00 Well, nail my colours to the mast on this one and say, no, it's not reflexive. There are these two things, there's A equals B, so I should use the same symbol for it. And twinship is different. What I wish you'd asked is is it symmetric? It is symmetric. I mean if A is a twin thing then B is a twin thing. So that's alright. But I feel that the reflexivity is a matter of how one wants to use language rather. Do you want to let equality be a special case of this or not? And personally I don't want to. Does that have implications for the formal situation? I don't think it does actually, but I wouldn't commit myself to class with him. If we're talking about distinctions, we're talking about sort of properties, attributes, I think it doesn't have to be objects at all. Maybe they're not necessarily objects. No. Some concepts within something. Oh yeah, sure, sure, quite again. Now in the last two sections, I've made things easy for you by translating Frederick's notation into one with which I'm familiar. The notation of the book is a major obstacle to understanding, and this obstacle has two separate parts to it. The first part is that Frederick uses reverse Polish notation and, as far as possible, uses only symbols that he would type. I don't mind Polish notation. I'm not at one with the Oxford logician who professed to be unable to read logic except in Polish. But it's less than helpful to see a set inclusion relation in the form, this is Frederick, to mean that S is a subset of T. One can disentangle it, but it's not helpful to have everything what I think of as the wrong way round, because at least you could type in collusion, which is what happens to C, actually.

27:30 I made it a bit easier by putting in two. Turning to the keyboard of Frederick Starlight, which of course I haven't seen, but one can tell that it is by reading the book. It's evidently that of a linguist because it's got a circumflex accent, which he didn't really use, and a German umlaut inserted. One could imagine, and I did initially imagine this myself, that there's a bread and butter task of simply transcribing the whole book into a notation that everyone could read. But once you start doing this, it turns out not to be what is really wanted. Because you soon find proofs that don't seem to go the way that Frederick says. Another matter of notation is that Frederick, writing when category theory was young and exciting, likes to write in terms of functors. And I've got personal difficulties with his use of these functors over the question of Pairs, ordered and unordered pairs of things. I think I've got time just to show you the kind of difficulty that Frederick lands me in here. You're familiar, I expect, with the orthodox way of defining the unordered and the ordered pairs being used. The unordered pair, A and B. Conventional mathematics uses something which, if anything, belongs to this set, then it's either A or it's B. And then you have the ordered pair. And one way, quite a common way of defining the ordered pair, is to say when you take an unordered pair of either A or A, and take that as the definition of the ordered pair. Now, this one would have tripped. Because, as he said, you've got this expression, you've got to ask whether these two A's are the same A or merely indistinguishable.

30:00 And so you will land in the North Pole. Exactly. So you need another way of doing it. And he prefers to do it in terms of two pair functions. One of them, which is a circumflex, and the other one which is the omelette. And the way that, well, slightly simplifying matters, his statement about these two functors, I'll just put this in to show you how hard I've been working to show you his notation. At this stage, this is the definition of the circumflex and this is the definition of the omelette functor. See the reverse Polish notation makes it not immediately transparent. The top line in fact could be written in a more orthodox way by saying, oh I should have talked about the IN which is a, I've underlined it as Clarendon type, meaning the information in something is, and these two are all information. We'll come back to what that means in a moment. And this top line then translated out of the Frederick thing is that the union of the information in X and the information in Y is equal to the information in Y. And then the circumflex of xy. So that makes circumflex xy essentially the unordered pair and then this lower one would come out with an inequality sign and then that would say that the information in the information in x And the information in Y taken together don't provide enough information for umlaut XY, so the umlaut XY is the ordered pair because the amount of information contained in that is both the X and the Y together with the order. So, the notational problem which arises from that is trouble. But there's more trouble rather, for me, In this in the way that it seems to me there's a great deal less in this than meets the eye but you have to ask me you have to ask the question well what is this um it's all now it's uh essentially we're cutting out all the

32:30 So, essentially you're saying that the umlaut function applied to x and y is the ordered pair x, y. But then you've got to ask yourself, how do you actually operate with this thing, this umlaut? Well, you've got to operate on a certain pair of quantities, x and y, and naturally you've got to take them in appropriate order. So it seems to me that before you can start doing this, you've got to have the ordered pair already. And so this all-out functor, which Frederick puts rather a lot of emphasis on, seems then to be the identity to me. Not to be despised on that account, of course, but not worth making a lot of fuss about it. You'll also notice over there that a lot of emphasis is put on the operator i.n. applied to these things, which in Frederick's case, since it's logic he's doing, are streams of symbols. And unfortunately Frederick doesn't give a very clear idea of what this information content is. Moreover, this is not from negligence. But it's because, I quote him, the elements structured in a mathematical formula carry the dictionary information. The latter is a major problem in linguistics, but for the mathematician is simply disposed of by referring to the stated definition of every term. So, in other words, as far as this idea of the information in a string of symbols is concerned, he feels there's no problem for him. There was when he was being a linguist, but as a mathematician it's all straightforward. It doesn't seem at all straightforward to me. If I pass on, though, to the second part of the notation obstacle, I'm also passing, you'll be glad to know, to a more positive attitude to Frederick's book.

35:00 The second part of the obstacle deals with his different levels of notation. He begins by claiming that in ordinary mathematics we can distinguish two levels, a substantive one, He says, contains expressions denoting relations within the object theory. He takes as example his elementary arithmetic, who might bias A is less than B. And then he claims a theory would need inferences to be drawn. So that in order to do those, you'd have to, you'd want to say something like, if A is less than B, then, B minus A is greater than 0. And as well as these symbols being all in the original substantive notation, you also need this arrow, which is part of the inferential notation. And then he goes on to say that he needs two substantive notations, an objective one, To express the theory, and a semantic one to work out the rules for context finding, and then he needs the inferential ones before, and he denotes these three by T, U, and V, in certain lots, sometimes just like that, and sometimes in curvy letters, for reasons which I won't go into. I applaud the subtlety of the distinctions that Frederick's drawn here, but I must say I think it's misplaced. In his original example, I could easily alter the notations in such a way so that the whole inference is put into one particular notation, but the distinctions that Frederick's drawing seem to be artificial ones. And I have to ask, well, why is he doing that? I think there are two parts to the answer to that question. One part, I think, is what I would call, perhaps wrongly, traditional Cambridge logic.

37:30 The distinction is a bit like the distinction which Whitehead and Russell draw in Contemporary Mathematica, and which people in Cambridge drew for quite a long while afterwards. They draw a distinction between The proposition P, and asserting the proposition P, for which they use the Frege's symbols, Frege's assertion. Well, I say, like most people, Frege's. Naturally, Frege didn't use it quite that way at all, but they took it over, they took the sign over from it. Fairly soon after the world, I think I'm talking about perhaps in the 30s of the last century, People in Oxford and the United States found that they didn't really have any reason to write a composition of P unless they were asserting it, and so the assertion symbol dropped out of use. But we shall see later on that Frederick does need an equivalent of this inside his system in order to express what he really wants to say, and this alone Shows that, unlike his claim, his calculus is essentially different from classical first-order logic. But the second part, that's one part of the answer about his distinction between different notations, and the second part of the answer is that what Frederick's doing is trying to fence in those parts of his system which have non-classical aspects because of the possibility of indistinguishables, so that he can deal with the part outside the fence more easily. I want to Try and just get along with one notation. Leave Frederick's distinctions out and conflate all three. When Frederick's talking about these three notations, he says that the third one, the V, which is the inferential one, rests on the axioms of classical logic. Later, he says, Perhaps one could have the added complication of using, say, intuitionistic logic, but it's bad enough as it is without that.

40:00 But before you breathe a sigh of relief, you have to think what he says later on when he's talking about sorts, which are his equivalent of sets, on page 66 of the book. It was, by the way, George Rickman who impressed on me the importance of getting on with the book as far as Frederick's sort theory. Frederick's definition of sorts, and just to have another bit of fun with his notation, I've written it down here for you. This is definition of belonging to S, heavy S, which is collection of sorts. I've translated that as best I can into an orthodox form. It says that if a class C is a sort, if and only if, not for all x and y that belong to C, it's false that x and y are twins. These double negations are important. Then Frederick. So it contains no twins. Is that right? It is not the case that it contains no twins. Now, that's my first, my ordinary first order translation of Frederick. But actually I don't think I quite do him justice because in explaining the reverse poach In his book he says, you see, it asserts that a class is a sort if it is not claimed that every pair of members is not a pair of twins. Note that it may be claimed that some pairs are so, and it may even be the case that all are, provided that this has not been proved or asserted. And so you see there's really quite a strong, oh, well, before I say that, that in other words, my version of what Frederick's saying won't do.

42:30 I think we do need to say it is not the case that we have the assertion that there are... So is it like an extension of a concept of... Something which could be said, but also something which is more general, in a sense, including in this thing, is it something like that? The idea is, because as you, I think, said a while ago, And sets, the members of a set have to be distinct. Is it substantially different from the multi-set? Yes, yes. Because multi-sets have cut them out. Yeah, they have. Fredericks was aware of multi-sets and is explicit that that's not it. Now, if you think of what Frederick said, it may be claimed that some pairs are so, and it may even be the case that all are, provided this has not been proved or asserted. It seems to me that that reading has got a strong time dependence about it, because you can, you know, tomorrow someone will prove or assert this. And so there's quite a whiff of Brouwer coming in there, even though he's been eschewed earlier. In fact, what the conclusion I've come to is, That Frederick's logic is not really classical as he claims, even in his fenced-off part, and I'm inclined to accuse him of trying to get the advantages of a Brauerian logic without the disadvantages by just altering it by incorporating the assertion symbol inside the system instead of just talking about the system. Before I finish on notation, I'll say a few words about quantifiers. More problems here.

45:00 Frederick accepts, I think a bit too readily, that existential quantification is a problem. Impossible in general, he says. Though his argument for this seems to me a bit odd. He says, well, if you take this example, you take the formula that for all n there exists an x such that x is greater than n, you say, well, this presupposes, he says, some sort of testing, say, by examining the class of integers one by one until a bigger one than n turns up. And there's of course a problem if some members of a class are indistinguishable, this will not work because indistinguishables cannot be labelled so as to know which have been tested and which have not. He, I should say, there's an exception he makes to this in the case of a definition. If you had a definition with an existential quantifier, you'd say, well, that does not so much assert that something exists as command, but it shall. At the very least, I must say that I think this account of how existential quantification is introduced into classical logic, where there aren't any indexing variables, is not the usual one. And in Frederick's spirit, I suppose what he really has in mind, not there explicitly, is that he wants to distinguish between there exists an x with some property 5, And he wants to distinguish between that and asserting that there exists an expert across the entire and he would say well you could only do this if you've got some testing procedure or a proof of some kind whereas this is just a statement. I think that's the distinction. And maybe the second one would only be justified.

47:30 I'm afraid I would be very glumping about that and say, well, if you're, why are you making the statement if you have known us? I sympathise with that approach, the sort of plain man's approach. If I say something, it's because it's true, not... Not rhetorical. Quite. Well, there are difficulties anyway with that little bit of rhetoric and one would have expected if you have difficulties as he claims to have with the existential quantifier, then you must have some with the universal one as well since, I mean, it's commonplace that for all x, phi of x ought to be the same as that it's false. That's where exists an x for which 5x is called. So, if the existential one finds a problem, there should be an equal problem with the universal one. Well, yes, there is to some extent, but he manages, with quite a lot of hard work, to propose something acceptably similar, he says, to universal quantification, which he calls conditional. Universal quantification. Essentially, what that means is that you have a predicate P which you want to talk about. The idea is to have something which, if it's not allowable to say for all x, P of x, because nothing distinguishes it from someone, Tama proposed something which would Do instead of that in some cases, and he seems to be able to justify saying this, that for all x, qx, arrow, px, now this p is the original predicate, and this q is a predicate of a special kind which belongs to the class of decidable ones.

50:00 So if you've got this class of decidable predicates, then, according to Fredling, you can actually introduce this weaker form of... Decidable, it means that he's saying that if we could decide something about X, then it is a proper... Yes. And X need not be distinguishable in that the cube is aimed for deciding. Well I won't go on with my increasingly pedantic criticisms of Frederick's book. I'll just use the remaining 20 minutes to try and sketch out a positive program for constructing a theory of indistinguishables which may perhaps be different from his, but I hope will eventually be able to reproduce his applications to physics. A three-pronged attack, and I don't know which prong is going to pierce the membrane or whatever, and they're not a very good metaphor. The first prong is, as George Lickman's been doing, to try to found the theory on the notion of contents. Frederick seems, as I said, to Take context simply to mean as member of a set. And then, of course, if you did that, it wouldn't be very difficult. In fact, you could jump in at the deep end and define twins straight away by saying, well, x and y are twins. It would mean that there were sets S and T. Of course, a bit of us first on the logic of this, that there's an S and a T such that the union of X and S is the same as the union of Y and S, but the union of X and T is not the same. And then of course there's a bit more needs adding because S would be trivially true if you had X and Y already in those two sets, so you have to have a lot more stuff down here to say that S and T don't contain X and Y. But that's not the problem as far as the English words are concerned.

52:30 So, if you add X and Y to one of the sets, it makes no difference which you do, but if you add it to the other set, it makes a difference. That's the meaning of that, that they never belong to the same set. No, no. Like the definition, you know, that they distinguish if they never belong to the same set. If they never belong... I'm not quite sure what's going on here. I don't understand the use. I tried to give a meaning to that. Whether it would mean the same. This was... No, yeah, do come in. I mean, don't worry. Okay, so don't worry about it. Just go on. The second quote is just to make sure they're not equal. Yeah, yeah, yeah. That's exactly what I was going to say. And then when you define twinship, you could build everything on that and the idea of indistinguishable, as we said before, would simply be anything which had a twin was indistinguishable. Then, according to this approach, the twinship wouldn't be transitive, so some of Frederick's proofs would need attention. I don't really see any technical snags in this approach, but the problem about it which I must What I want to point out to you is that it's a doubtful assumption. It's doubtful whether mere context dependence captures the whole of the notion of indistinguishability that Frederick's trying to get. I suppose I sort of feel this all the more strongly because for many years I thought it was. I thought, oh, yes, he's worrying about context and how awful.

55:00 Let's name the book to somebody else. And it has dawned on me that what he's trying to get at, if there is something there that he's trying to get at, which is, you know, it's not an illusion, then it's more than merely context. As you'll see shortly, Frederick does actually consider kinds of indistinguishables which don't seem to depend on context at all. So this approach, this first approach, would only capture, I succeeded in it, it would only capture part of Friedman's work. Of course, it might be the whole consistent part of it. And if the approach just using sets like that were to fail, I could imagine fighting a rearguard action, saying, oh, well, maybe just using sets is a bit too general for them to do what I want. Perhaps the context sort of thing like this would have to be formulated with something cleverer, so something... Not wholly unlike that of neighborhood in topology or something of that sort, but I'm not going down that line until failure forces me to. It's worth saying at this point, if I had gone after the other methods, something about the physics, which Frederick marginalizes rather in his part one, that is my motivation. Of course, if we go back to the despised equivalent, say, the indistinguishability of electrons, that's only a modern version of a problem that already troubled Leibniz. He took an object to be defined as a bundle of properties, but whereas he was happy about their color and size and shape as those properties, he was equivocal about position. In some ways, he wanted to include it along with the others, but in others he didn't because he wanted to formulate a theory of space, so he got into a muddle over there. In a way, position is context, isn't it? Yes, that's right.

57:30 But the way, this doesn't worry physicists, not because of their brutality, but because the distinction there is between a set of properties, charge and spin and so on, which are all quantized, and in another set which are position and momentum, of course, Leibniz didn't think about momentum. Which aren't monetized, so that the two are of different kinds and the problem of the non-transitivity of twinship doesn't really arise. I mean, you can see that the way the physicists go about it is they have a set of properties like charge and spin and so on. Let's just call them E for the moment. And then they have another set which are position and momentum, or we just call those Qs. And then if you have two objects, A and B, which are such that, well, these are various possibilities. If they have the same E and they have the same Q, Then you'd say the two were equal, and if they had different, you'd say they were different, and then, for the physicists, you'd say that they were twins if they had the same B, but different Q. So all electrons were then twins? Yes, they had that property. They could be, if you... I have always had the problem with this word equal because in elementary arithmetic equal has a context of arithmetic that is not the same as. It doesn't mean quite the same thing as is the same as.

1:00:00 Because you have to evaluate the sides. And this twin idea is, again, different from is the same as. In my mind, that phrase is much more like is indistinguishable from. But you see, if you think in pictures, which I don't, The problem would be how can I say that something is indistinguishable from something else? It has to be in the same place and has all the same attributes. I can't see it. I think you're always saying indistinguishability of objects, not indistinguishable of properties within objects, which I think is really what you can use this for better than you can use it for objects. Well, yes, yes. As soon as you go into that abstraction, it seems demeaning. You have to, because it doesn't make any sense otherwise. Right. You're just saying something's identical. But that is the distinction between, pardon the pun, physicists... And the logicians, the physicists are talking about things they believe are separate in some way. I'm not quite sure I'm on with you Tony, but I just want to say here that these E and Q refer to arithmetical quantities, don't they? Yes, they do. And the arithmetical quantities place them in a spatial context and physicists can point to them. Can I just ask one question? Is there a possibility of coming up with a concept of anti-indistinguishability as opposed to mere, make not distinguishable, indistinguishable, something more, much more definite, because if you did, if you did, you mean very distinguishable, as indistinguishable, as not indistinguishable as could conceivably be. Because if you did, then you put indistinguishability and anti-indistinguishability together in two concepts, then you have a perfect definition of symmetry.

1:02:30 That's precisely what symmetry is. I really need notice of that question, I think. Well, I mean, it's just struck me that it's exactly what symmetry is. It's indistinguishability plus anti-indistinguishability. Well, I mean, it really looks... The main response on the spot is that we're not dealing with a very symmetric situation. I can't really see what the opposite of indistinguishable would be. Well, I think I could in a physical context. Yeah, I know. But that's the problem. Because we're developing an object of this. I'm not very clear, I think, about what I'm doing, whether I'm developing a logic of it or a general verbal theory of it, something that we could try and work on. I could give a good trivial example of what I think, an indistinguishable example of what one comes across it. For example, if I write something in my word process, a word, In times and times re-roll, it looks exactly the same. If I then transfer that material onto PowerPoint, it will default the fonts separately, so the font that's in times will look completely different. This is a practical problem I actually came across. In fact, I got distorted fonts. But if I look at it in the word processor, it's exactly the same, indistinguishable. So in that context, if you've got an indistinguishable, what I do is times it, times it wrong. But as soon as I put it into PowerPoint, it will default the times it wrong to something else. The times it to something else. So that's it to me, it's a fair changeover. Yes, and so I think what you're getting at in that example is that there are some things you put in that come out right, you know, so they're, yeah, they're very, yeah. But I mean you could take the practical view that the thing is simply telling you lies. Well it depends though, if all I want to use is words, then it doesn't matter which one. It doesn't matter. But if I then use PowerPoint, it does matter, it matters a great deal. It has a practical problem though. I think I'll just stay on for another five minutes to tell you the... I won't say very much about the second prong of my program, that is that one could alternatively try and, since a lot of Frederick depends on this information theme, which he doesn't make at all clear, one could try and...

1:05:00 Axiomatise is information which is in things, and then when you've got a clear axiomatisation of that, you could see how much you could turn the handle and do some of Frederick's things. I don't think that would be all that hard, but before one expends too much effort on it, you'd have to see whether it was worthwhile because of the amount of use that Frederick actually makes of it. And to a large extent, it only comes up over questions like defining pairs and so on. So I'm leaving that for the time being as not being the thing that's going to pay off best. So I come on to the third prong, which was essentially George Whitman's suggestion to me, which I pooh-poohed at the time, but I've come to realize is the most promising one. And that is to crash on ahead as far as you can with more or less classical theory to get as many of Frederick's later results as may be, and in particular, as George emphasized to me, the crucial test for this approach, which is the importance of two of Frederick's innovative notions of sorts, which we were talking about, and the idea of a blur. Now, say a word or two about blurs, because they are something that one can get one's hands on fairly easily and see what sort of things come out of it. I'm inclined to translate Frederick's definition of blur into this form. I'd say that Z is the blur of X and Y. I'm using you, actually. Frederick uses a little, although he uses various things which I'll need to talk about. Z is the blur of X and Y. I would say it means that either Z equals X or Z equals Y. You could question, I tend to question myself, whether I've really captured Frederick's idea, which is to, his idea is to emphasise the unspecified character here in the blurb.

1:07:30 You don't know which of these is true. Say that his words are good ones, blurb. But it's not the same thing as Hussie. Because. No, no, not fuzzy in the sense of. No, no, that's not. A fusion of. A fusion of. A fusion of. A fusion of difficulties. Well, the reason that I want to put the emphasis on this idea of preference is because this blur is indistinguishable in his sense. He says it is, and there's no question of context here, it is just, and why I don't feel that it's quite captured what he's trying to say is because he makes an important part of his definition is that, if I have any predicate, any predicate please, Then, if that predicate holds for the blur of x and y, then it holds for x or, of course, for y, so that, because it holds for this unspecified either x or y... Because you don't know which is which. Yes, since you don't know which, then it must hold for x. And equally it must go with the Y. So this blur, the unspecified character of the blur comes in in that way and the other thing is that X, says Frederick, and I agree with him, X is a twin of the blur of X and Y. And so is Y of course. And incidentally this shows that George must be wrong in thinking that twinship could be transitive because if you have an X and Y which are definitely different and you form their blur and then X is a twin of X of the blur and Y is a twin of the blur and X and Y are different, you can't separate them.

1:10:00 So I think one's got to accept that twinship isn't transitive. The left blur of the signal is the point of vector. Ah, well, yes. Then it does depend on how you look at it. Your context comes up again because the blur is rather like a variable, ranging over the two. And if you put it in that context, then it shifts. Yeah, it doesn't range, it switches. You don't know which way the switch is going to go. I see what Lou's saying. I'm not sure. I'm not sure whether to accept it or not, actually. The old use of variables, just like that. Yes, the old use, yes, that is true. Now, I just want to push on. I wouldn't like to not show the good stuff on the last page. The idea of blurs comes up more explicitly in Frederick's theory of sorts. Which, as we said above, are things that would be sex, in which there's no assurance that twins are absent. It turns out, in pregnant sort theory, that they're very different animals from sex. For one thing, the members of a sort, if it's got any, must be sorts themselves. It certainly has a hole in it in Frederick's book, but I think it's probably a hole which could be filled in, covered over or something. So I'm assuming that result of this is right, although I haven't been able to quite pottage-proof up enough to do it. The fact that there aren't any members of sorts which aren't themselves sorts, This means that it's no good going down and down and down until you get to individuals, because there aren't any individuals. But there is a provision that Frederick makes instead of individuals. He has things called elementary sorts, and they're defined in terms of perfect sorts.

1:12:30 Without going into all the details, elementary sorts are the ones that are members of themselves. Incidentally, Frederick takes an extraordinarily cavalier line about things like self-membership and so on. He simply says, well, there are problems in classical logic and paradox and so on. They don't arise here. So, I hope he's right. Now, just to conclude then, He, I must go back to his chapter three for a moment, which I said I wasn't going to talk about, there he argues that in physical observation, this is a bit like the distinction of people having quantity, I mean it's sort of a co-behaving kind of distinction, in physical observations you can only find bioparitous entities, you can't find twins and so on. And so his aim, he says, is to construct, for as many sorts as possible, a bicaritas representation, which is an injection and a homomorphism. Actually, what homomorphism means in this context is a bit more complicated than it normally would be, but he puts in some details, which... Because of the idea that some of the things may be twin. That's right. So you've got to put a bit more in, but it's alright. And then he says And then he concludes part one by rather triumphantly enumerating those perfect sorts, that's a technical term of his, which had got by apparatus representations and this is where the details are rather interesting because he said well the simplest such sort Here's if you have two members, and he's got a rather awful notation for this, he has two members, this is the sort which he calls D2, subject 0. It has two members, I and J.

1:15:00 Then he looks at what he calls the closure, which consists of I and J together with their blur. So the closure of the most animated one has got three elements, and in fact any two of these is the blur of the other. So you've got a situation which has quite a resemblance to some of the things we're doing in quite a different way in the hierarchy. Yeah, that's right. And then he goes up to the case where we have three of the major and the k, and the blur, all the possible blurs, so that the closure there then has seven elements, so that three, two, produces three, three produces seven, and then the next one up, and this is where it gets quite interesting, where you have four to start with, in the The closure would naturally have 15 elements, but that won't do because although there is a bi-character representation of this and of the closure of the 3-naught, when you get the 4-naught, there is no such representation. So he says, well, for the present purposes, that must be rejected. And then he looks at more complicated things which are in fact. Univariate operations of sorts and he finishes up with how many distinct pair trees you can get and with bipartisan representation and he finishes up with the numbers two, three, which is this one here, and seven.

1:17:30 127, and then this is a familiar set of numbers in this particular audience, and then at that point, even with the alien hunters, things stop because after that there aren't any vicarious representations. I'm not holding it correctly, I think the B, Casar, Vortigai, and so on, and which finishes up with the same mysterious numbers which his early algebraic calculations did. So he did have the algebraic ones earlier? Yeah, yeah, sure, yes, I mean those came out just by algebra, but he managed to write a whole book to justify getting the numbers. Well... I think he would have said getting the numbers quite crudely like this, but I think he would have said we've got the numbers in a different way. I'd say that he's provided a different justification. Any questions? Did you say that the blur of J and the blur of I and J is I? Yes, yes, because it's quite general you see, not only in that context, well if the blur of A and B means that, if Z is the blur of A and B, then Z equals A or Z equals B. And so if If you look for the blur of A and the blur of A and B, well then W is equal to that, means either that W equals A, or it means that W is the blur, in other words it's dead here, and so another W equals A or W equals B.

1:20:00 Sorry, that's of course, if my rendering of blur is what it really means, if it meant something more subtle, then your question is more difficult. Sorry, is the W equal to the level of A and B? Yeah, sure, I mean W is... Now, that W is the blur of A and the blur of A and B. Yeah, yeah. Now, yes, our letter is worrying about what is the blur of A by itself. No, no, no. I think that the second line looks like W is the blur of A and B. Because it's just a repetition of w equals a, w equals a, which you may write as w equals a, or w equals p, which comes to put that together. I cannot prove what I've claimed to have happened. Yes, that's right. So what he claimed he found may be more true. This is true. I would like to reject, to sort of pass over this as a technical detail. I was perfectly satisfied when I saw what Frederick did in the book at this point. I think it's all right. I don't think I can do it out of my head at the moment. Maybe I'll make an announcement after lunch or something. It would be nice to see that report. So if there is no more questions, let's go and play again. I have a comment really, it's just that I did in fact buy Parker-Rowe's book and it cost me £210. You should play it on Amazon. Yes, yes, yes, I would. I wouldn't want to cut it out of print and use it. Yes, yes, I'd hang on to anybody that's got copies. I'd hang on to them, don't I? Well, I'll see you once in the settlement.

1:22:30 All right, so after the lunch, people is, oh, unless you want to ask a question, no? It's not being closed after lunch. Yeah, you are, I'm sorry. Let me change it then. Ah, no, I'm sorry. Yeah, it's, yeah. Incredible. Does he have a term for what happens in notation all the time, where you have, you regard the A as indistinguishable from the A over here, and it's really only distinguished in terms of its position. Does he have some notion of formalizing that? No. I think that's something that's missing from the book, which struck me at the time. It doesn't have means of externalizing things. That's what I get out of it, and you gave an example of an externalization that seems to work. I'm not, I don't understand. My problem with it is that I can imagine something as being indistinguishable from something else, either a concept or an object, but if I want to externalize that imagination as objects that I can play with in the external world, then I find it very difficult. I mean I think it's worse than what Peter said, well you can't just use pictures all the time. And this is one of the things that makes my mind boggle even before I try to read the book. It's now 12.43, that was the delivery by Clive Kilnister. It's Friday the 5th of August 12.44 now.