Tame Forms of Non-Associativity

0:00 Clive Kilmister, Frederick Carver, on tamed forms of non-associativity. Construction in which one had matrices over the field with two elements and vectors, and then the important thing, a succession of levels, and the various levels you got out certain numbers which we thought were important, and you also had a bar to going too many steps. I'm not going to go into the details of this, you'll be glad to know. You could look at almost any of the previous proceedings, or indeed you could use the excellent book that Ted and I wrote and sold under the name Combinatorial Physics. Now the important... I've also got copies of your recent formulation, which is very different from the foregoing ones. Does anybody want to borrow it? Yeah, but you're speaking, Ted, of... Yeah, I'm going to go on to the later formulation, but at this stage, I'm simply talking about Frederick. Sorry. And, you know, the algebra. OK, but... And I'm not going to talk about it, in fact. Now, the essential... The three essential bits of the construction, which we did understand at the time, but we didn't emphasise as much as some other things. One of these things which the Frederick system and our reformulations of it did was to produce what I call a discrimination system. That is to say, a set of elements with a binary operation, which at this stage I'll just write as multiplication, and with the property that if ever it happens in the system that the result of the operation on two of the elements is to give the square of another one if ever this should happen then the definition of a

2:30 discrimination system says that can only happen when x and y are equal and actually then they're equal I've put it in this form, which is not the form in which we used to talk about it. I'm sorry to interrupt you, it's not digital, that. Too small? It's the wrong kind of material. Oh, that's a beauty old method, yeah. I'm doing it, I'm doing well. Here, the definition of discrimination system, So that if ever xy equals z squared, then that can only happen if x equals y equals z. So that's one essential feature of the Frederick construction. The second one, as I said, is that one goes up in a number of levels. I won't elaborate on that one. And then the third one is that instead of looking at it in this rather general form, Friedrich had a special element that was in fact the zero element, and that element is the one which served as a signal to show when two things were equal. So, he was working in terms of, oh, Rovanoff, but the hint has been taken by Vanessa's. good, yes, successful he was working with over and using addition as the binary operation and so he as it were encapsulated that requirement in the form that x and y add up naught if a node is S equals Y. So you have a special symbol here, naught, which acted as a signal to show that the two things on the left-hand side were actually equal. The thing that Ted and I have been involved on for somewhat more than half a century is how to understand that peculiar construction

5:00 We've argued many times, and in particular, if you want the argument, I thought it was very well put in Ted's paper in the last proceedings, that a process way of understanding it is what is required. And we started on, well, as he says in that paper, there was a certain element of process was implicit from the beginning, although Frederick didn't take it very seriously. But we started to try and improve these matters, and I can't put a year on this, but I had a wonderful revelation on the journey from King's Cross to York. I was an external examiner in the University of York at the time and I received shortly before the journey I mean a day or two before the journey I received a review copy of John Conway's book on numbers and games I thought oh I'll take that and read it on the train and as we went up through the home counties I thought this is just what we're looking for chapter in which he constructs the succession of fields of characteristic two, what he calls, with one of his jokes, the field on two, you have this construction which originally when we used this, we used to call it Conway's trick, but actually we now call it Conway's construction. and was more respectful to him. And I'll show you what this is in case you haven't read that wonderful book, but I won't waste time by showing you how it is in Conway because we had, from the very beginning, to modify it slightly. Now, the modification is really a rather trivial one, but I could put it this way. If you talk to people who, logicians and so on, they refer to the natural numbers. In fact, there are two schools of thought who often don't actually realise that they're different.

7:30 One lot believe the natural numbers include the number zero, and the other lot, to whom I belong actually, believe they don't include zero, but start with one. I mean, I won't go into that but it just seems absurd for anyone to think of zero as a natural number if it were, why didn't they invent it earlier it came quite late historically however, don't want to be led into that particular blind alley now Conway constructs his field including the zero by his construction And we don't... Because zero is special in the Frederick construction, that's not to be one of the things that is constructing we want to put it in at the beginning. So we modified the construction a little bit. Now, what... There are two stages of the Conway construction, and we only need the first one. So this is not actually Conway, but it's Conway modified by Cure. You've got a series of elements which, in Conway's view, they're being generated, or he says they're being born, and he talks about the birth times of the different things. I would prefer here to simply say, well, elements are coming into play in the algebra. The birth business seems to have a lot of overtones which I don't really want to bring in. Anyway, elements come into play and they are naturally ordered by the order in which they come. so there is really no harm in using as a notation for them a try and trust the ordinal notation we use this vertical stroke for the first one and then we use this sort of halfway round thing for the second and so on I'm not just trying to be amusing and failing in the way I put it But I'm just saying these are the funny symbols which we're going to use for them. I would not at this stage bring in any of the cardinalities.

10:00 Because traditionally they have a certain order, you're going to attach that order to the cardinalities. It's just the order, just the order, nothing else. And then you define, or Conway defines the operation, the class operation in this system. way, that A plus B, where A and B are these elements which are coming into play, and they're some of these things, A plus B is the least C such that it's not equal to anything like A dash plus B, where A dash is anything earlier than A in the sequence, and it's also not equal to any A plus B dash. So this construction generates, as you go along, new elements come into play, and when they come into play, this tells you how to operate. I suppose it seems it must seem a bit odd sorry I have missed something I beg your pardon least seen yes least in an ordinal sense earliest shall we say earliest that's one of my normal I mean I tend to think about all these things quite the wrong way and then have to disinfect the stuff after it's come out thank you for that so could C be A or B? ah, yeah thank you, very good that's the Conway version because he's got a nought at the beginning here, that would have answered your question because C can't an earlier A dash could have been naught and so then it couldn't be B but for us because we don't have naught as part of the generated thing

12:30 we ought to have also that C is not equal to A and it's not equal to then we have to put those in in addition in our construction Conway construction, they're not needed. Right, thank you, that's got that clear. Now, what, and indeed, and thank you again, if I may, because that actually does lead into the next thing, which is what I wanted to say. When we turn this around, it's a normal character. Well, no, I mean, these are, these ordinal numbers label the order in which the things have appeared. So it's least in the order. Least? Least in the order, that's the point. Yes. Well, I mean, when I say, yes, just a moment, when I say earliest, are you worried that it sounds as though time is coming in? No. Well, it is, isn't it? Shall I say first? The first seed for which these things are true. What do you mean? No, well, I don't have a choice of what I mean, because I have got nothing except things in a certain order. i can't make sense out of this except by using a little more of conway yeah so i'm going to say that little more of conway and maybe you can tell me that i don't need it okay conway said numbers will be created in tiers starting from nothing and there will be an ordering and at any given stage we create a new number in the following way we take some numbers we take some other are bigger than that, and then this schism of left set, right set, creates a new number, and this number is in between the others, and its value gets precised by these notions of addition. Yeah, I'm not using that. I don't need it, and in fact, well, two things. Because I haven't had completely forgotten that it was in the book. I remember it.

15:00 I understand your definition here. The way in which I'm using the Conway construction is this. New entities come into play. That's the sort of physics part. So there's some way in which new entities come into play, not necessarily the Conway way. Not Conway's way, but, as it were, from outside. And any given entity is in, it's a strictly ordered set, so any given new entity sits either bigger or less than any previously created entity. Right, yeah. And then this defines the operation. So I leave out the Conway thing. But the sequence of entities is distinct. Yes, absolutely distinct, yes. Everything's discrete. It could be like one-third of something in between one and one. Well, you could call, but I wouldn't call it that, because I'm only going to call it these things. The new entity. Yeah, I'm going to apply this to things like bits of information coming in in physics, or something like that, but they're just going to be labeled. Yeah, they could be just called first, second, third, fourth, which is what you're doing. But then two plus three might not be five in the system. No, it'll be one, actually, according to this definition. Because, you know, would it help anybody, perhaps, if I showed how this worked? You have that, you know, if you have one and two, you know it can't be... it mustn't be one in Conway's original version let me do it that way first one and two can't be naught and two that is two it can't be one and naught that is one so it can't be one and it can't be two so the least thing it can be is three so you put down three on the other hand when you do two plus three which is what Lou's where you know that it can't be 1 and 3, which we've already done and turned out to be 2, and it can't be 2 and 0, which is 2, so you know it can't be 2,

17:30 and similarly one can see it can't be 3, but it turns out that there isn't any reason why it shouldn't be 1, so I just can't wait for a second of all. If there's no reason why it shouldn't be 1, then it is 1, and that's the list. that was the actual construction that gave me this revelation in the train to York and we have to modify it because we didn't want Nort to be in now that leads on to the point which we failed to emphasise at that stage for some while afterwards that this very special character of Nort that 1, 2 and 3 construction which we've just done actually generates we used to say before I became very careful we would say this generates the quadratic group 1, 2, 3 the result of doing any two of them gives the third and the nought is the identity element of the group in the usual way now that was a rash thing to say because it doesn't generate the whole of the quadratic group it only generates elements 1, 2 and 3 are generated the 0 is the signal there for telling you when things are equal so we didn't really generate the quadratic group which we've always called S we generated it with 0 removed and then you say well what about if someone wants to what about sums like that A plus 0 and examining the process carefully you see that the only answer is well if you want to you can say that that's all you see this won't arise in the process because the process consists of new elements coming in and being discriminated together but not norm norm is not an element it's a signal about things being equal so that won't ever come in happen but if you want to know if you want to put things down about that then obviously a fairly good choice would be a because naught then as well as acting as a signal is doing the sort of things

20:00 that naught usually does but you don't have to say it's a if you don't want to act there's no reason not to at this stage, but this is a warning for later on. Now, things took a turn for the worse, you might say, some years ago, when having formulated this this way and then gone through and thought, yes, this is just producing what Frederick did, and then Ted and I wrote the book. And within months of its publication, so it's not as bad as Frager and Russell but within months of the publication I noticed that this discrimination operation which Frederick had introduced us to and which was here denoted by class didn't actually have to be a commutative operation and that in fact if one what shall I say that it would be quite a general matter for it not to be although it might the commutativity was not necessary although it was a possibility I made a lot of fuss about this at the time and historically you see the reason for that was it hadn't dawned on me and the book was published without anything in it, well, in particular with a totally fallacious proof that the operation which we were discussing, the Frederick operation, had to be commutated. I forget what page, but you can easily look this up in the book, and a little homework exercise for you would be to tell me this afternoon what the fallacy is. But I know what it is, I just... it's one of my straightforward cheats over a long life I've done it a number of times and I suppose a lot of mathematicians have you know the answers there and you just leave out a necessary step which turns out to be an impossible step so obviously a devoted a devoted Conway convert of my variety

22:30 realised that what had got to be done was to modify the Conway construction a little bit more so as to involve the non-commutativity or the possibility of the non-commutativity. one thing, part of the modification was to put, well, I mean a trivial part of the thing, let's get that out of the way first, to commute then people won't like it if you denote it by plus they'll be offended and so i've lived a whole life trying not to offend people and i mean i because it is in my nature to do this i mean it's in my nature to offend them and so i've had to learn over 80 years how not to do it. So, of course, we won't use plus, we'll use notification because people are happy to accept that that might be non-commutating. So that's the trivial change. And then, of course, if we're using multiplication, the identity element which proves so useful as a signal, which is not an element which can come from outside of it as a signal, that Now, so the Conway construction then will become, will be, if A and B are two elements which have come into play, then we should have to say, using the same notation as before, these ordinals, 1, 2, 3, and so on, so of course, 1 won't be available for the identity, we'll have to call it something else. so as to remind me that when I go back to the original Frederick construction it was zero. So AB is going to be now the, what did I decide on calling it? The first, the first C in the thing which is different from, now what's it going to be different from? Well, it must be different from A and B. And AB. Sorry? AB is going to be the first scene which is different from either A or B,

25:00 or, oh yes, you know, I'm with you, and any earlier A dash B, and any earlier AB dash, And also, if we've had it already, so then the feedings won't come in. If that hasn't come up yet, then we're all right and we can put down what we like. But once we've put the product down in one order, we've got to a minute for it and not do it the next time. But there's a little bit more to the construction than that, because we've also got to feed in this question I said being not necessary although possible so how do you make it possible well you make it possible by saying and there must be a homomorphism of the structure which comes out of this back onto the original structure so as to preserve the possibility that it was common so there must be some whole homomorphism which i call capital h which takes this structure back Of course, if you take it back onto the original one with multiplication, instead of plus, but that's... So, with the whole construction, then, I should say, is that you do it subject to H, meaning that you carry this construction out, and you've got to make sure at each stage that the system you're producing has a homomorphism back to the original system. you're tempted to label such a product in a way which interferes with this homomorphism and that can come up then you say no all that won't do go on to the next label so perhaps taking up more time than I thought still that's the first page. Now, let me tell you some personal things. Once I did this, and then looked at the first level, the first thing I said to myself, and I think actually I said it to audiences here as well, was that the first level, instead of being the quadratic group, is quaternions.

27:30 ah, I thought, here at last I've spent my life dining out on quaternions I even got an invitation to 10 days in Dublin to talk to the Irish about quaternions I'm home, and I also said rashly if the lowest level is quaternions and when you go up through the automorphisms to the higher levels, there'll be all the other Clifford algebras I said this as an empire meeting lamentably the second statement is more untrue than the first statement they couldn't beat Clifford Algebra but at the lowest level that was an overstatement you see if you draw a multiplication table for Caternions as everyone knows we won't put all the details multiplication table, and it's well known, but down the diagonal, oh well, one IJK, right, one IJK, and although you have a plus one there, you have minus ones there. So, you seem to be using two signals instead of one, and the plus one which was your signal for equality only holds in that phase, and these three are minus one. So it's not the sort of system I said it was going to be. But then I have to go back to what I said here, that in fact you don't actually get returns because these elements down the diagonal, which should have been signals, are not going to turn up automatically out of the process of new elements being generated and so when one asks questions like this one only transferred you don't know what the answer is you can make it various things so what you've really got is not quaternions but you've got quaternions taken away well, if you use

30:00 the abbreviated notation for quaternions which have been derived from quaternion algebra so you've just got three elements plus their negatives and plus and minus one, then what you've got is quaternions with that little subgroup removed taken away of course it's a normal subgroup but that's technicalities which play no important right so the what you've got at the lowest level could be quaternions if you filled in the blanks the right way but you don't have to fill them in that way well of course I did fill them in that way to start with I mean not only when I hadn't realised that there was ambiguity and did it naturally but I filled them in that way because I wanted this and then I've got an associative algebra so that's fine except that when you go up to the next level in the way that Frederick has taught us you find horror the system ceases to be a discrimination system that is it no longer has this property the damn thing just for two elements not in general but there are two elements at the next level which when you multiply them together give minus one which you're taking as a signal here for equality whereas they're definitely different elements so the requirement of a discrimination system breaks down at the next level and so that must be wrong because the whole, the real essence of the Frederick construction is that you have at the next level something which was like what you had at the earlier one so you can do the same thing over again. And if you don't have something that looks like you had before, because it's not a discrimination system, then the whole thing collapses. So you think, well, alright, I've got myself into this mess, I've dunked the hole, now I've stopped digging. What I should have had is plus ones down the diagonal.

32:30 So I put those in, that's right, the signal is plus one as it was before. But the algebra I've got is not quaternions any longer, it is quaternian algebra except that you've put ones down the diagonal instead of minus ones and so of course it is no longer associative because, well you can do it, simple exercise to do afterwards once you put those ones down the associativity vanishes of course it's not terribly non-associative, just a little bit like the housemaid's baby, only a little one so it's what you call slightly non-associative but when you go up to the next level you find it is seriously non-associative so it comes in horror if you're not in the University of Edinburgh where they have done this sort of thing for nearly 50 years I think you think this is bad Oh, non-associativity. No, no, terrible. I was rescued from despair by Arletta in two ways. Firstly, she explained a way of making the non-associativity look harmless by her group mates. That is to say, she said that the system that I'm talking about, as long as I don't worry too much about the identity elements and so on, could be got this way. that I've got a set of elements, a set of three elements at the lowest level, and I'm defining a binary operation between them, which I'll denote by a dot now, which is related to the binary operation between them if they had been fraternions, by the fact that the product of those two comes to this in the situation where well she would say where in quaternions a to the minus 1 times b is equal to c

35:00 so this is all normal quaternion operations on the right hand side but you make an inverse I actually prefer at that time to do it the other way around and say b to the minus 1 a but this is only a trivial difference So, the advantage of that is that it sort of fitted better what I was doing. You see, this will certainly produce the sort of discrimination system I want, because if A to the minus one B comes out to one here, or the unit element, then of course a and b are reciprocal a to the minus 1 and b are reciprocals and so a is equal to b so we only have a equal to b here when they come out to a thing on the right hand side which is what was called 1 over here so that is a way of taking the non-associativity and making it tame, as I put it in the title of this talk. Now, the other way that she helped me was by sending me Xeroxes of a book by a lady called Flugfelder, who shows a whole lot of stuff about non-associative systems of various kinds. I would have preferred it if she... She, I mean Arletta, in this case, not Felder. As Arletta says, there seems to be some peculiar reason for non-associativity to be taken up by ladies. It's something to do with the making of babies or something. No, it's to do with the parking of cars. I think that's probably a nasty remark. But I got quite a lot out of that Xerox that she sent me. Now, 20 minutes, just enough time to get on to the real material.

37:30 the end of the context part of it. The virtue of the group that I hear is, you see, that it enables you to tame the thing in the sense that if you want to ask questions like, are there normal subgroups or something like that, you can transfer these questions across to the original thing and ask it there. Now I'm going to be a bit more general. I mean what I'm now going to say doesn't refer to the hierarchy or things like that although the reason I'm doing it is because I think I may need some of these techniques in investigating the higher levels when the thing's non-commutated but it's simply at this stage a piece of mathematics which I think may be useful so I want to look at various ways of taming non-associativity I did want at one time to formulate a definition of taming but I haven't done so and I'll say something about that at the end I think general taming of like this on the right hand side call it G. Any group, but in fact it needs to be a non-Arbelian group, otherwise everything becomes trivial. So the G corresponds to, in the earlier case, a double term. And then I'm going to, as our letter did, define a different binary operation on the elements of the group which I'll denote by a dot actually this makes me slightly uncomfortable but I would prefer to say that I map the elements of this group into another set and then define the binary operation on that set but it doesn't really matter, that just makes me feel more comfortable but she's happy to have two different binary operations going on in opposition one set of elements. Then I say, well, you see, all that's been done here in the group is A to the minus 1. If you consider the mapping A goes for A to the minus 1, you're permuting

40:00 the elements of the group. So let pi be any permutation of the elements. Then I could my new operation here, as by the rule that a dot b equals c, under my new operation, means that some permutation of the group phi operating on a times b equals c. The phi just operating on the first thing. I tried some other things of this kind at first, but some of them didn't work. some permutation on the underlying set of the group not having to do with the problem no, no, no, nothing, that's right what did you mean when you said you tried other things? oh, I tried permuting both of these and I forget what the other ones were there are things you can do which aren't in use but this seems to be quite useful Of course, one trivial difference would be that you could permute the second factor instead of the first one, but that would just generate the same series. Now, when you say you permute it, do you mean that you pick out from the group some element which isn't A? Yeah, because phi A is a single element. Yeah, yeah. Oh, I'm sorry, yes. Yes, phi is a mapping of the group, the set of elements of the group, into its set. Right, so that's a permutation of all the elements. A permutation of the elements of the group. Yes. Now, so what does phi on a single element mean? Well, it means what that element is taken into by this permutation phi. What it becomes. What it becomes. Are you assuming that phi A is not A? Oh, you mean for a particular A? For some A, phi A is A. No, I'm not assuming it's not A. But of course I am excluding the case where it's for every A. No, it could be. Yeah, you could have some of them are the same and others are different. That would be a permutation of some of the

42:30 elements leaving some fixed, which is a permutation for a whole lot. Then what we get by this process is what is called by the people in the trade a quasi-group and I denote it by G suffix phi. that's the first step and indeed it's the main step I haven't got a lot more to say here's a special case of this you might take the permutation to be defined in general terms by its operation on any element A is the result of multiplying in front by some fixed element of the group. So you pick out anything by calling Q, and you say QA will be an element of the group, and all the different A's, the QA's will all be different because it's a group. So pre-multiplication by Q does generate a permutation of group elements. And then this operation here, this would hold when QAB equals C. I call that a Q-form, not a very good word, but anyway, it's what I just called it to myself. Q-forms are often non-associative, but as we'll see in a moment, not always. Well, having done that, I thought, oh, well, now, you know, it's almost in business. So let's have a look at what happens if you do it the other way around. They have another fixed element of the group, say P. Then A times P is a permutation of the elements, and so I take my permutation to P on the right. and so I call that a P-form and it will generate the whole thing you know well actually if you start doing a few examples

45:00 it's soon borne in on you that P-forms are actually associative so the whole thing has somehow collapsed in your face and it's not surprising that they're associative because if this permutation is multiplied by P on the right So that A dot B equals C means A P B equals C. Then it also means, since we operate in a group here, if I multiply on the right by P, it means that A P times B P equals C P. And all you've done here is label the elements of the group in some different order. got the same group again with different names for the players. So P-forms are trivial, you've got back to the original group. So if you go back to the question of the Q-forms, if Q happened to be in the centre of the group, then a Q-form would be the same as a P-form, so the Q-form would be associative as well so not all the Q-forms are going to be non-associative so they won't all be good candidates to help me out when I come upon a non-associative system but some of them are well that led me on to the next question if phi is a permutation which actually then introduces with the dot operation an associative system inclined to call phi trivial in the way that mathematicians do and well they use the words in various ways but sometimes oh that's trivial meaning they can't do it but this is one of those and a simple example for you to so now what I have shown but I won't bore you with it on the board that you can easily do it for yourself I think here's an exercise to show that

47:30 phi trivial if and only if G phi is isomorphic to G if it's trivial then you're going to get an associative system out because you're going to get a group and that group will be isomorphic to the original one so you just get G again as in the case of the P forms but slightly more generally second one now it gets a bit harder alright, that's an easy exercise what is the condition on phi that it should be trivial now this is a little bit harder because it obviously depends on the group G since if for example G is Arbelian then every phi is trivial so you have to take the group structure in order to find out this condition and I have done it I don't know whether to tell you about it or not I think perhaps I won't oh no, perhaps I'll just tell you a little so if you've got to take the group structure into account how are you going to do it? well the way I do it maybe this is very clumsy there's probably a cute way of doing it I instead of calling in the group elements A, B and C you need to be a bit more systematic so I'll call them A1, A2, A3 with suffixes And then if I want to put into my algebra the group, I've got to say how AI and AJ multiplied together in the original group. So I'd say, well, that gives me some other group element, which I could denote by a matrix, a permutation matrix for each AI, which tells me what sort of linear operation I'm doing on the AJ. so there'd be a matrix MI for this left multiplication which picks out a particular one of the A's

50:00 and similarly I could put in the permutation by means of the matrix obviously and then it will turn out quite easily that the condition for phi to be trivial is that the matrix version of phi commutes with each one of the ends which determine the multiplication of the group. So, that's actually quite a strong condition. It looks rather a strong condition on phi. You've got all these MIs for the left multiplication by the different elements, and phi, to be trivial, lot. So, in general, you'd think, well, not many permutations will be trivial. What is M sub i? Yeah, M sub i is this matrix, using the i as a label, and the j and k are the matrix suffices, and it's a permutation. It's that permutation matrix which corresponds to left multiplication by a i that's an easy one, now here is a harder example which I set you for homework and you could bring it in a year's time I don't know the answer to this one if you've got two permutations say phi and theta question will G5 be isomorphic to G3? Sometimes it will be apart from trivial cases because just think about it. Suppose that the group G is of order N. Then there are factorial N minus 1 permutations and they can't all be non-isomorphic because, for example, if n is 6 of order 6, there is one non-Arbelian group of order 6.

52:30 Factorial 6 minus 1 is 719 so there are 719 candidates for a g sub phi and the number of 6 by 6 Latin squares which aren't non-isomorphic can be found by looking it up in the book to be 22 so there's only 22 possibilities of non-isomorphic quasi-groups of order 6 and my 719 candidates There must be a lot of those many times over. So there must be some isomorphisms of that sort. Mind you, when you go up a little bit, it might not be so bad, because if you go up to order 8, which is as far as my list of Latin squares goes, their factorial 8 minus 1 is a bit over 120,000 possibilities of Gphys and the number of non-isomorphic Latin squares of order 8 is about just over 1.5 million so it's even conceivable that they might all be non-isomorphic but I don't suppose they are Anyway, that's the problem for next year. What's the condition for those to be isomorphic? Now, just to conclude, something which occurred to me only a month or so ago, and this really brings us round to where we started from, You could think of it as a bit of, the whole thing is a bit of a joke. Remember that Q form was written, I wrote, I wrote a Q form down like this, that A dot B equals C means Q A times B equals C in the original group. And that was one special case of the time mutation, which I called the Q form.

55:00 Well, of course, you could have rewritten this right-hand side by taking that Q into the form of Q minus 1 and put it over there. So you could do the permutation on the answer instead of one of the factors. And so one could generalize that to say, well, you see, you'd have a situation where A dot B equals C means is some permutation of the answer. So I use a different Greek letter just to, not to be confusing. And you might ask the question, well, if this original thing, instead of being a Q of Y, if this had been defined with some permutation of phi, what's the relationship between phi and the psi over here? And the answer is, that is not a good question to ask. because it's a hopelessly complicated thing to do because if you're permuting that one and you want to take it over to there, well, it will depend which bee you've got. So the whole thing is a problem. So this is essentially quite a different thing from permuting one of the factors. So we've got a new kind of tameness here. Oh, new kind. And indeed, of course, this is the one that I ought to have... I mean it's like those in medieval stories where there are guessing games you know someone comes in and says you ask me three questions and then I'll ask you three and it all goes fine for five steps except when the stranger asks him the last question the poor fellow thinks that was the one I ought to have asked him at the beginning now I shall be bewitched This is obviously the answer to the Maven's Prayer for my original property with the quaternions, because what I got was, oh, I got quaternions on the right-hand side, except they got the wrong things down the diagonal. So all I had to do was to have a permutation of the quaternions on the right-hand side in which the one and minus one got interchanged although the elements well the notation gets in the way a bit if you call the elements a minus a b minus b c minus c you leave all those as they are

57:30 for the one and minus one you swap them over and then you've got what you wanted so i really ought to have done it that way in the first place but i got misled by that lady over there and her the grouplets, into going through all this lovely stuff. Now, what I haven't done here, and I mentioned this earlier, was I started by thinking whether I should be able to give you a definition of what I mean by taming non-associativity. And I did correspond, both as I left around with John Amson, about this, and I had some rather clever-looking stuff first order um first order logic formulations and saying it will be tame if so and so but then i realized afterwards that really what i'm trying in doing that what i was trying to do was to give a precise mathematical definition of something which isn't a precise mathematical notion i mean the thing is tamed it's just like with wild animals it's tamed if it behaves in a taming is just something which turns the non-associative system into something which you don't feel uncomfortable with. And it's partly a subjective matter whether you feel uncomfortable or not. For example, if the permutation is rather complicated in order to work it out, well, then it won't be much help. So all I'm doing is just trying to help myself a bit with these systems. Thank you. Now what, what you are doing now, what, what bearing, believing, bearing, is it, on, on, on, on, on anything else? Yes, no, bearing is perhaps a strong word, strong, I wouldn't perhaps be able to answer the question in quite such a strong form. I was going to ask that question, then I thought it might be indiscreet. No, my answer is this, well, I mean, obviously some of this is just what mathematicians always do,

1:00:00 having started off on something you try and do other things because it's fun but to answer your question more precisely if I'm if as I'm trying to do I reconstruct the whole hierarchy without commutativity then when I get up to the next level where matters become more seriously non-associative I'm hoping that one of these taming devices will reduce it to something decent. What do you think for some device to understand non-associative of the higher levels? Yeah, that's right. I'm just trying to construct a tool for keeping it in check at the higher levels. Because could I just say to people, in case you get a wrong idea about these higher levels, I'm not so incompetent as you might think at the lower level you notice that the four elements of the quadratic group suddenly have become the eight of quaternions and you might jump to the conclusion and I know you might because I did that everything's going to be doubled but alas no, when you go up a level there are eight times as many elements as there were system and so it goes on getting worse so the higher levels are rather uncomfortable creatures to be working with and any little help like this will will be welcome do you have a table for second level yeah i have no you've taken us on a fascinating journey really um And I remember when I first saw this construction involving matrices, their eigenvalues and eigenvectors, how it was described, and it seemed to me extremely that the motivation was mysterious, not so much the results that came out, but the motivation. And it seems to me that what you've been doing is to make it look more friendly in more conventional mathematical categories, in other words, so that it may be possible to see physical motivations for things.

1:02:30 Physical motivations? which allow you to attach it to more conventional physics. For example, the structure of Howley matrices seems to have something to say about the three-dimensionality of space, the up, down, the sideways, and so on. And the proof of that is that it can act as a sort of basis called quaternions, which are more numerous and seem to have more degrees of freedom and so on. And one feels that the Pauli matrices in their non-commutativity and so on, and the fact that they collapse to scalars or unity, if you square them, are in some way connected to this thing. as well, in some way connected to the hierarchy. So that if you go on long rambles through different kinds of mathematics to try and find a representation of what the hierarchy does, it may be that you may also find a contact with the kind of mathematics that we have in physics. It may be. It may be, yes. The other thing I'd say about this, Tony, is what is friendly to one person isn't to another so that you see all of this sort of thing I mean Frederick's matrices and vectors and all that business was not at all friendly to me because I could do matrices alright but why was he doing these things and so my way of doing it is more friendly to me for Ted on the other hand Everything has to be essentially translatable into bit strings. No, it has to be translated into physics.

1:05:00 Well, yes, but via bit strings. Well, this is the impression I get from some of your letters. I mean, the way you put my question, which I worry about all the time, I have to see that knowing has to be associative or non-associative. Does it follow from the basic discrimination procedure? Yes, it follows from the basic discrimination procedure correctly understood, that it has to be non-associative. If it is to be a discrimination system in this sense, which I'm sure we want it to be. Yes, I think it would make it easier for the reader to stress that it's a point. Good point, yes. Please. Did you look, Clive, at the new multiplication tables that John Amson produced last year, which of course are non-associative? I wonder if you spent any time doing it. No, that's a good point. I could do that. I'm going this year is probably with John's worries about non-association, I'm also worried about it, I'm fairly happy, but John's uncomfortable, we've been really through the same process that you've been looking, and interestingly one of the things we were doing was permuting one of the elements of the product, in fact this was Louis Gidney's suggestion, precisely I hadn't associated it in my mind with grouplets until we started putting this on the... So it appears that they're taking exactly the same approach, which is amazing. That's good. We should... See, it surprises me to hear... Yeah, I mean, I'll take that, but that's very good. It surprised me to hear you say that you're not very worried about non-associativity because I regard it as a dreadful thing. I mean, and not just... but a thing which doesn't seem to come up much in the books about associativity

1:07:30 and which is what seems to make it so important is I mean, you say that the systems associative means this so that when you've got three elements you don't need brackets But then it follows from that that for any products you don't need brackets, whereas when you haven't got that to start with, then it gets worse every... However, what I perhaps should have said is that I'm not so worried about the sorts of associativity that we have got, because they appear to be tamable in the very same sorts of ways in the school. Yes, good. Well, I suppose that tameable would be, an example would be if there was some kind of closure on the associativity, that levels of products that were, say, beyond four, you only had the same kinds or groupings of non-associativity that you had at lower levels and things like that. Well, it's my mind-boggled as well. There is a way that this has been studied by various people. If you, for example, look at a four-fold product, then you'll find that all the different four-fold products that can be obtained from one of them by re-associating fit into a cycle of five. And so it looks like it ought to be equal as you come all the way back, perhaps that would be a simple kind of situation. Or it might not be equal. If you assume that it is equal, that's a higher level kind of associativity. It's not ordinary associativity. It's still non-associativity. And there's a way of kind of thinking about hierarchies of this, and making cell complexes for the different cycles that occur, and this has been done by a homotomy theorist, I think it was started by Jim Stashen for a long time ago, and it's still being used by all sorts of people. Oh, yes. Well, that's, I think, what I mean by closure.

1:10:00 It just came to me today whether it has any significance whether it has any philosophical significance for the whole system I mean, first you have philosophy that it's a process and it is a rising and so on and so on. And suddenly on such a basic level, the mathematical representations are not associated with whether it has any deeper meaning than just discussing mathematical divides. Whether it means that the mathematics of the universe is not associated, as I mean. All right. Which you believe is, sorry. It's non-associated, the more basic mathematics, which is sort of more accurate about the physical reality, it's non-associated. I said it as a joke last year, but it's suddenly, as far as we take what you do here seriously, over 50 years older, that perhaps it must be considered. This is the stuff of nightmares. Open Pandora's box by redefining category to non-associative category. Category is very simple, but it says the compositions of the arrows are associated. Ah, yes. So release that. And you get a simpler axiom system, and all horrors can have control. You see, because the only reason mathematics is mostly associated is because it's not easy to rule associated with mathematics. There is no other reason. I think that's absolutely right. Yes. The basic mathematics isn't associated. Exponentiation is associated. No, no. Differentiation, you could define, like what you have done. Differentiating, defining operation as differentiating. I mean a year or two years ago our letter pointed out to me that subtraction is not associative in elementary arithmetic. So there is a lot around.

1:12:30 And people, children has to sort of bend their mind to try to understand this stuff and the real reason is that it's not associated and you cannot tell them. that's the end of Clive Kilmister's talk Thank you.