New foundations for combinatorial hierarchy (partial) / Keith Bowden: Essential Frederick Parker-Rhodes construction

0:00 Thank you. I'm not sure that you've actually answered the question for me in quite the way I'd have done it But I'd say, well, here's A, something comes in, a reaction happens, and this is the result of the reaction and now I've got to ask is the result of the reaction the same as I started with and that's the same question all over again I mean it's not as you said that maybe nothing happened but that something happened and it gave the same guess I'll put it another way I said well you just duplicate A but then I have to reject it because you can't assume that you can duplicate anything So that means that you have to look at the output of the process and find out what happened. And there's got to be some discriminator out there that tells you whether they were the same or not. Otherwise, you're just back where you started. Yeah. Right. Right. And in the case of Benjamin, you have a discriminator. It's called the second law of femininity. Well, yeah. Well, without... And if that's not a discriminator, what the hell is it? Without either accepting or contradicting you, Peter, what you said there is that in the case of Megiddo, you've got a whole lot more out there which we haven't thought about yet. So you've got possibilities with that? Well, you won't let the possibility which we know from experience is something that we want to get there. I'm not saying we got there yet, but ultimately, ultimately we can get there. If we pursue that, if we pursue that, I'll give you a thought. One observation I would make is that Shannon said, questions always take time. So if one is worried about process or not process,

2:30 as soon as you say, well, there is something asking a question, is this the same or is it different? questions always face so let me now really get on to the beginning of the algorithm I've just given a couple of examples of binary operations which if I were presenting this to an audience of strangers I'd say getting familiar things now we'll find out what ours are and then at the end I'd say Will that be the same? About the basic algebra that we need. I want to use to the utmost this question of elements coming into play and so that one has some elements that are in play and some which aren't and put that beside the discrimination operation to see what we can get out of it. So let me begin with I have some element A which has turned up I don't want to say anything about what happened before this moment in the discussion but there it is. And so another one comes along and I'll call it B For the purpose of this bit of the calculation, I tell you, or we'll assume, that this B is different from A. If it's the same, then I can deal with that, but that just complicates the present calculation. So if I want to know about this binary operation, I can put in it on the application table. And I have a signal as well, a signal that the things are the same. Let me call the signal Z. Why should he call it Z? Why didn't he call it S or 1 or something? in the original Frederick construction

5:00 the signal was zero and I call my signal Z to remind me of that it will often turn out to be the identity in the structure and if I'm writing it multiplicatively it can be called one which conflicts a bit with Z but Z reminds me I mean it's sort of that Z it's a constant reminder of old Frederick telling me about this thing back in the late 50s or early 60s I venerate the Z there as we said AB is going to be something else so we'll call it C now let's think about BA and this is where this is where the new bit begins which I haven't laboured Frederick straightaway assumed because he wasn't doing this process he assumed that BA would be the same as AB and when Ted and I wrote our book I was so keen to make sure that what we put in would finish up with Frederick's structure that I took that on board straightaway in fact it may even be, I haven't looked it up lately it may even be that we said something like it's obvious that A B will be the same as B A or some such remark which unfortunately I think it is obviously false because the C is a statement of the process in which A has appeared already and something beautifully new comes in called B and you check whether it's the same or not whereas B, A is the same story only with B and A interchange so I cannot believe that the binary operation can be a commutated one and that's the basic difference between our construction and the original present one so this is the new bit yes this is the new bit so well it's half the new bit On the other hand, I would really be rather flying in the face of reason to say that BA and A be an utterly different thing with no connection at all.

7:30 so I think it's wise to say that BA is something different from C but associated with it and I'll call it C star and refer to C and C star as duels and the notion of duality is going to play a very important part so the elements are going to come up not as single elements but in dual pairs because of course then I construct the rest of the table in the usual sort of way well I've brought in C so I've got to put it here I've also brought in C dual so I must put that here and I must construct the rest of the table now I've got to do a little bit of argument about the dual operation and how it goes through the different multiplications because this idea of duality is the important one for me I should say before I go on though that, and this ties in with Peter's question about indistinguishables the notion of duality because, take it for granted for me that as well as this C of course things like CA and C star A and so on A star and B star are going to come in further down the line so they're all on the same footing and so I have to have a different distinction between elements from the one that I intuitively started with because obviously if I had two elements they might be quite different or they might be identical we're all familiar with those two but now there's an intermediate class in which each is a dual of the other so instead of a dichotomy between same and different there is a three, whatever you call it a triadic kind of thing which is a bit like the indistinguishable so we'll have that threeness coming in

10:00 of course quite a different threeness from the one that Peggy was talking about now when I look at the notes again because I don't want to get the sums wrong let's see how to complete a bit more of the table Suppose that we are looking, say, to put in something in that place. Well, we say, what can BC be? This is a way of, in another run of the thing, it might be that we had B there already and C has come in from outside and we want to check whether we're the same and so on. Well, I say, the only elements which are in play at the moment are A, B and C and of course they're dual So, B, C will either have to be A or it will have to be A, U but we don't know until we've looked at matters a bit further which one it's going to be whichever it is CB will of course be the other one but isn't BC also BAV well it's not associative we don't get any associative So you can't do any of those things Right, but you are saying that C is AB I'm saying C is AB, that's right, full stop Yes, but you don't know where to put the B in relation to that Time's getting on, so I think I'll take a shortcut I could tell people the whole details, but we probably don't want to hear them anyway what would turn out to be the case is that you cannot take BC as A star because if that were the case then you would by two more steps repeating the process be led to a contradiction so BC has to be A and CB

12:30 then would be A star and in fact what you'd find about this duality just by looking at what's possible in terms of what's in play and what isn't you find these beautiful results that if you, oh, let me do it in terms of general things instead of the A and B of these special elements if I have any two elements then x times y star is the same as if you've done xy and then took a little and so that's also equal to x star y by a similar argument and when you look at this you think surely I've seen something like that before so there must be a long time ago and then you think oh yes it was in the was it in the first year in my secondary school when I was taught how to multiply negative numbers x times minus y is minus xy, etc. So this duality has the same structure as the multiplication of the star is like a minus. What is the constraint you're working under? Do you want each product to be definitely non-associative? No, no, I'm not insisting on a non-associative. That just happened. the constraint I'm working on is any particular product I've only got this set of elements in play and so no product can bring another element in from outside unless it has the signal set of calls and you'll find And you want the whole core element system to be closed under Yes, right, yes, you could put that as closure Yeah, that's right. So, what I get here, of course, not surprisingly because I've constructed the lecture that way, what I get here just like your argument in terms of objects in planes and the duality concept is the Q-star, which I gave you as an example, the quaternions where you forcibly put ones down the diagonals. now there's happily there's still ten minutes in which I can tell you the real importance of this

15:00 I'll put it this way, I'll go back to my unreformed youth of a year or so or say two years ago, what I would have done then I'd have said, well, so now we've got firstly we've got the lowest level of hierarchy we're going to construct and it turns out this lowest level is Q star rather than S this would distress Ted a bit because he wants to have S there as this tie up with the spatial thing but it shouldn't be really because the A and A star or the B and B star and C and C star are just pairs of elements so we've got three pairs of elements which are dual not essentially different and so we've got the kind of symmetry of threeness which the which the quadratic group had it hasn't disappeared but we've got the structure which comes more directly just from realising that the thing is not commutated now what was I going to say oh yes, and then I would have gone on in my unrepentant days and said alright let's carry on, we've got A and B and C has come out as a result now suppose something else comes in say D and we can work a trick again we could look at A and D But you've cheated. Why don't you put the Z in your table now? I put Z, that's right, you know, along the top and down the side. There is no Z tab, the Z tab. Well, I could put... I mean, do you want to say it here? No, I want it down here. It's part of the room or the... There's the head there, yes? I want it down here on the left side. Down here? That's quite a fair thing. I know, I know what's happening. You've been corrupted by our letter.

17:30 Absolutely. Yes. She always wants to put every element in... and I just write an abbreviated version of the table and see if there happens to be an identity around it does remind me that I could spend a moment on a slightly different matter and that is since we now as well as being identical and being essentially different we have this duality there must be some sort of signal when two elements are dual and I call that signal Y because it's the next letter to Z in the alphabet and so when completing this table C, C star there would have to be I'd have to now have two signals and I'd have the Y there to say oh well if you bring that in and check it against that they're not essentially different So the system automatically grows a two-signal system. So it's very like a period of construction. I do think so. You've introduced what you might call proper mathematics. What? Something in mathematics has a well-defined sort of. Well, I agree with the first thing you said, Peter, but then you spoiled it. Well, no, no, I agree. It's like Twitter. Yes, no, I'm saying really it is very like Frederick. In fact, I believe it's more like Frederick's system than I've yet been able to convince Ted because I shall like to continue with convincing him. But now, as I say, suppose I want to go on in my unrepentant day and I'll bring in another element, D, and then A and D will be E and then the result of A and D and E being in play together will have a table which is exactly the same as this one so if I, I've got to have B star and A star there and then finally here will come D and then I get the bigger table ago. Now when you do that you find that the amount of argument which I put in up

20:00 till now will not be enough to finish that table off. You can find a lot of it but there are spaces where and in those spaces you can say well should I call that so and so there's no contradiction to doing it or shall I introduce a new and there's no reason why you shouldn't introduce a new element except that if you do the system never closes it shoots off to infinity so there is in terms of my present assumptions as distinct from the rather stronger ones that I used to make this construction only produces the first level it doesn't allow you just to go on by putting things further on it's inadequate for that and like the man who wrote about hyperbolic i say that's a good thing about my assumptions i cleared away some of the waste material which was there how do you get bigger structures well you only get the bigger structures by a level change that is you have collect the discriminately closed subsets at this stage of which there are three and then you look for the automorphisms of this structure which characterize these subsets in just the same way that Treadniss did, that is that when you apply the automorphisms to the subset it leaves its individual members I'm changed and it changes everything else in fact another of the peculiarities of the construction mathematicians seem curiously hard find curiously hard to grasp so now when you go up to the next level well then again it's just like Frederick going up to the next level he went up and he then got seven elements instead of three Well, at least the seven and three, they're important numbers, of course, physically, from the mathematician's point of view, they're a bit awkward, because the seven comes from the fact that the signal, which would make the eight, no, the three, sorry, the three here, three pairs,

22:30 it comes from the fact that you've left out you haven't counted the signals as one of the elements and you don't want to because it's the thing that sounds off the siren when you took those two things in but really of course the mathematician would say really you've got a group here and it's got four pairs of elements quaternions it's not a group a loop and is of order 8 these are 8 pairs so we've got 4 there and at the next level said Frederick did have 7 well the mathematician would turn that 7 into an 8 and put in the extra bit now as I'm getting right to the end of my time I'll leave a lot of the later part out but just say something about this next level which one then says a bit like this, only instead of having these two generators I now have got three generators, one for each of the VC subsets of this first level What will be the size of this next level? Well for Frederick it was of size eight, or seven if you want to signal off Here it turns out to be of size 64 because each element at the next level is I'll put it this way, it's just like it's an element just like one of Frederick's so the length of them but they can be multiplied in front by elements in the centre of the algebra and the centre of the algebra is in fact a group C2 cross C2 cross C2 which coincidentally is indeed the group that this next level constitutes for training so all the elements at this next level come in sets of eight and I would say in the analogy of the dual part these eight consist of an element

25:00 and it's seven new, or to be more suggestive physically, I'd say they come in octets and each octet corresponds to one element of the treasure. I think I'll stop at that point. I've not wanted to toggle onto level change which I ought to have done. I've said enough to fill me out. I'm going to give myself the privilege of asking the first question because this is the one I warned you about. It's a question and a suggestion and it might be called Pauly and the Ghost of Beauty. The power of matrices, they're really operators and you can represent in all sorts of ways, can be used to construct the quotennions, right? So that if however you construct the hierarchy, algebraically speaking, you found that at a certain level, maybe even the top level, you had one or more subsets, I won't call them groups because that would be offering to fortune, something that fortune ought not to have, subsets of entities which behave like Howley matrices, right? If you had that, this would be a sign that your construction was growing something like space because at least locally Howley goes to Quartonians and Quartonians summarise all the local properties of Euclidean geometry yes please comment I mean including the comment it's only taken you 24 years to realise that I think to make Sure, I like to hear something like that, but to make the most adverse comment in return that I can, I say we're not at this stage, we have no number field, we're conducting ourselves with groups, or loops, or something like that, purely discrete structures.

27:30 now if such and it happens that the structure here isn't quite the same as the Pauly Algebra because of my having forced the z's down the diagonal you know it's hard to take that difference too seriously although it is there but even if I have got if I had the group structure for Pauly Algebra I haven't got any ceasing choice, but well... You've got a minus one. You've got a ghost of a minus one. A ghost of a minus one. I'd also... Well, yeah. I'd also like to take you up about the powering matrices being tied on to space. I know they were I mean when the way they came in or that treatment of spin and so on the three things three of them that's not what I'm saying I'm saying the level up the Quartonians are tied to space well they do indeed because Hamilton made them do so, yes but you're saying I want numbers as well Yeah, it struck me that you end up in 64. In the light of our discussions yesterday of DNA and the structure of DNA, you have a table, a coding table of 64. Now that coding table was of course evolved. And that evolution process, if you think of the elements that making the DNA, are precisely things that come to catch. Now that closes physically at 64. If I'm correct, I may be wrong about this, but I seem to remember a statement of Mancad-Igons once, that you had to, for chemical stability, you had to have a three-level codon just to make the helix stable chemically.

30:00 And so you shut off, but you'll have then the elements in the table selected for whatever is the selected prebiotic mishmash, but it will depend on the kind of structures that are there. So that there might be already a physical analog of the process that you're talking about that leads up, and you might look at the regularities in the, after all this is completely determined now, the regularities of the 64 DNA code, and see, you know, symbols are there, you might see if your mathematical process has anything to do with it. with that. It's a different kind of clothing. And in fact, I have a sort of preliminary feeling that you ought to have something to do with it because of the generality of the way I set it up. Exactly. And your description is much closer to the evolutionary concept than anything I've heard of. More questions? Well, let me get back to Conway. Conway's thing is all about having models of theories in the language of sex, and he produces from this generalised non-standard analysis, or of the cereals, of the cereals and field, that you have these, for any theory, any theory within this analysis, has a field of automorphisms, and there's a unique birth altering. There's a unique birth altering of the automorphism. Yes, of the Autonautism, that I need to start talking about. Well, yes, but you will have noticed, perhaps as prompted the question, that in this presentation Conway has got left out.

32:30 And if I had gone on further he would still have been left out. Because I gradually over the years came round to realising that I brought Conway in and he helped, he I don't mean personally, but his construction helped me over a lot of gaps. but I now see that he along with some other stuff is part of the unnecessary mathematics which I'm cutting away well as I say to make it comprehensible to Ted but that's the sort of facetious way of putting it I'm cutting it away in order to only have the minimum which is necessary and it turns out that Conway is part of what's gone when you say instructions for numbers yeah that's right I mean it might come in like let me say again what I said at the beginning that I've pared it down making sure at every stage that I've got enough left for the calculation you see you're paring it down mathematically Now, what Ted said was that you should be physical and so you should be paring it down physically. Now, if you paring it down physically, you mean that every symbol you write down must have a measure. And every process you write down must have a measure. And because that makes things physical, really. Well, Peter, what you're doing there, I think, is, if I might say so without meaning something, you're being a bit bit Pope-like. Sure, your course of action will be a wonderful one, if one could possibly carry it out, but we're... That's just what we have done. Well, the other. It's precisely what we have done. It's precisely what happens in quantum order. And it works. It produces physics, real physics, in the form of magnetic resonance imaging machines.

35:00 okay what would be the last question well this is a comment this matter of it being related to physicality somehow related to physicality but open not uninterrupted but open to multiple interpretations it has to be otherwise it's not going to be in use right so you can't say it has a specific interpretation. It has a physical source. Otherwise, you're in trouble. I think that's the most useful comment. Yes. But if that's the case, we should be allowed to put X squared equals A in that. I'm not all thinking of that. No, I mean, I insist on, if I want, if my interpretation of what I'm doing is process, things coming in and checking them, then I can't have A to R equals A there sure you can okay, process is not necessarily an interpretation could I just say in conclusion that this is one way Ted and I together we've exhibited one way of trying to increase understanding of Of course there's another way which would appeal more to mathematicians and that would be to set it in a much more general framework and instead of cutting everything away as it were that this is essential to what he said. The world here is mathematics and there is the Friedrich construction in there and let's see. Well, but you're assuming mathematics then contains physics. And we make the opposite assumption. We make the assumption that physics contains mathematics. No, I'm not assuming that. That is, that mathematics is just a natural language like any other language. No, I'm not assuming that. And why shouldn't you assume that? Why should you assume there's some mythical space with mathematics in it that has nothing to do with the physical world, yet somehow we have access to it? Well, Vladimir Arnold says, physics... I don't care! But I want to get this quote over here.

37:30 Vladimir Arnold says, mathematics is physics with experiments that are really simple to perform. Let me just finish what I was going to say which is that actually Peter has falsified what I said but all I was going to say is one could look at the Frederick construction as a special case with much more generality and that point of view is the one that Keith is going to do this afternoon so you'll get both ways of looking at it Which begins at 3.30, everybody. 3.30, this afternoon. I'm going to talk about a paper that appears in this, which is perceived as a window, or last year's perceived as I should say, even though the work is actually new. The paper was somewhat draft. At the point in time we went to press, I think Clyde and I, I should say it's co-authored with Clyde, and especially in the early days, it consisted of me asking Clive questions and then Clive answering them, although I think we evolved a bit from there. I think in time we went to press, the paper seemed pretty solid. About two days ago, Clive and I pulled it to pieces, we then flipped back together again, and perhaps it's actually okay, there's some things it says that you need to fine-tune me a little. Before I start, just to remind you of some things, we talked a lot about groups and group points. This is a typical group most people have seen before, it's the Quaternium's Q, and it consists of these objects, which will assign funny symbols like minus k2, there's eight of them take any care and we assign another object which is within the set, so the set's closed under some symbol e equals k whatever we plug in there we'll get something out that's actually in the set of eight groups in particular properties as well as just closure. If you start by thinking about sets,

40:00 group hoids, semi-groups, legoids, that's for the dealing groups as well. I think these are the wrong way round, and of course the building at the top. So these are commutative groups, the building groups are commutative groups. What we're doing is we've gone up this, this hierarchy, is adding structure to sets. so the first thing we can add to a set we've got a set of eight things we can add a binary operator and we get something called a group odd this is largely what Clive was talking about bridge wise we can add associativity a unit that is something such that approximately one times a is a inverse and commutativity, A, B, as we know. Keith, these ones might be described as different levels. No, no, this hierarchy has got nothing to do with the combinatorial hierarchy at this stage. I've just thought of really about first-year pure maths at university or something like that. These things can be thought of as Christian Caspi's theory is to go from one to three's to the next one down and this transformation is called the functor and in this case it's the forgetful functor. We forget mutativity to get to here, we forget inverses to get to here, we forget units to get to here, we forget associativity to get to here and in the end we forget the operator altogether and we go back to sex. And for what purposes those sacks are finite? Oh yes, as Clive said everything that happens within these four walls is always finite. But yeah, but some of the things I'm going to say later in life are not. I need to be restricted. No, but you can have a group that that's happening within these walls that are not finite, of course. Oh yeah, I apologise. Of course, if I'm not standing up here

42:30 that is true. I can wear different hats at different times. One idea I want to put across is that the combinatorial hierarchy can be described at any of these levels, or at least there are different combinatorial hierarchies at different levels here. I'm perhaps talking too loosely for Clyde here. Oh, I mean, why that's the sex level, I know you can say very much. Oh, well, perhaps sex is a little bit weak. We can take... Well, you haven't got a binary operation. I'd have to think about that, thinking about the style of the hierarchy, I was in the paper and coming back to you. We can think of sex and we can think of collections of elements in the sex, and we can think of those collections as being the objects at the next level and then we can take collections of those subsets and think of those as being the objects at the next level and there's quite a picture we've got there quite pointing out that we might not have quite enough at the bottom level I mean, I think I remember I thought a letter accuses me of spending all my time talking about what I'm going to talk about I'll do the same thing to that. Thank you. Should I accuse you? Just to be consistent. There was also a lot of talk about generators, and I need to have a little bit more clarity about generators to say the things that we need to say, which is actually why I put this on here. it's always worth saying when talking about finite groups and people try to emphasise this enough in my opinion that the number of finite groups is actually incredibly finite there aren't actually very many finite groups with 8 elements in fact I think there's 3 could be 5 and as we go up Increasing the number of elements, the number of groups of a particular size does go up but not all that fast, at least at first.

45:00 One can buy a thing called the group analysis. Yeah? The government. It does these days. And some very peculiar things happen. But there's no way of predicting this. Well, it depends what you mean by that. Redenumerated from all the finite groups. Well, some very peculiar things happen. classified into about 12 different categories, except for the sporadic groups, on which one is called the monster and has 10 to the 17 elements or something ridiculous, and the other one is called the baby monster and has slightly less. They're very strange things even at this level. They're not as predictable as one might think. I'll put it that way. but I'll say something about generators as I say these things like minus I are actually symbols they don't really mean minus I quite so much as we might do if we were thinking about complex numbers or something like that but they do which is why we use these symbols but there are really very little other than symbols minus I is a symbol one might imagine like this, in some sense, from a subset of these things. How to do this? One might be surprised to remember what the generators of Q are. In fact, typically, they are, I mean, it might be a difference of I, J, and K, or what you mean. Or I, J, and K minus 1, or something like that. But in fact, I and J are sufficient. Because I times J is K. And I squared, I, J, I squared is minus 1. And minus 1 times E3 is E3 and the domain. So that's enough. And in fact, it's ambiguous. Because there There is no specific set of generators. J and K are just as good as the board. And this aspect is something that Clyde emphasizes every now and then, and then perhaps at other times we forget that it's a bit too much, but it really is crucial aspects of it that we have forgotten a little bit too much over the recent decades, and we'd like to emphasize those a little more

47:30 from now on perhaps. so generate this screw that paper it's interesting to think what happens with groups given any group it's quite often possible from the whole table to choose a subset of the elements such that this subset is closed. And then we have a group like so and choose a particular set of generators. And that set of generators generates the whole group. Think of a bigger group with more generators. Think what happens if we take a subset of the generators. That subset of the generators will generate a subgroup. It's not difficult to see that that's... Take any generators and start multiplying them together with a finite set, you're going to run out of new things that happen, and we're going to end up with something that flows. It's entirely within here, so it's a subgroup. It doesn't overlap, then. Generated by a subset. There's another one. Now, in the old terminology, in the world of the hierarchy, these things are actually DCSs, used subgroups of the group. What I'd like to emphasize is that there are DCSs which aren't generated by subsets of the generators. For instance, take a system which has generators A and B. Then A generates a subgroup which A, V, V, generate the whole thing, but think about A, V, A, V, A, V squared, A, V, A, V squared,

50:00 where A, B, and so on. It's not hard to see that this is closed. We've now gone rather with it's not generated by subgroup A and B. Why? It is generated by the subgroup A. Well, think, think. A generates that one. B generates that one. With B in there. A and B generate the whole thing. So that's the three possibilities we've got. Ah, I've seen it. I remember. This one is different. Okay. So, but this one, the thing generated by this one is a DCS. So, if we want to think about these ones, the ones generated by subsets of the generator as being special, perhaps we should give them a new name. So we're going to call them P, G, and I sometimes pronounce these peaks because they were at a time to primitively generated subgroups. So these two are primitively generated subgroups or subgroup or whatever, which are DCS's but this one is a DCS which isn't a Well, what's the history of this? of this is, we think this is what Frederick clearly meant originally. The idea has been followed along in the literature, and I think Clive managed to dig up three, four references to it. But three or four references out of the massive literature on the combinatorial hierarchy over the last 40 years is not a lot. And yet, for instance, without this idea of DGSS, we don't get the remarkable calculation of the found structure constant that Clive's And we don't follow the history back to Frederick's original construction problem. It is that the vast majority of mathematicians have no knowledge, that seems, maybe? Or no? I don't think PGS has come up that much in the literature. No, that's a good problem. I mean, that does seem to be the implication of what you're saying. Fine. I think, well, I think it's a bit weaker than that in saying the vast majority of mathematicians seem to have quite an accountable difficulty

52:30 in understanding the threat of construction, or that of all these misunderstandings. But would you say they deliberately ignore those kinds of things? I don't think it's correct. I don't want to discuss it now, but we should discuss it. Like, sorry about it. Could you repeat the definition of PGS? It's a subgroup generated by a subset of the generators. What do you mean by generator? Ah, good question. Okay, we'll come on to the test. That's the main point we can talk. I suddenly something new which I don't think it's relevant. We will talk, it's perhaps not. No, it's the real point. Well, if she's thinking about what we think she's thinking about. Yeah, yeah, yeah. You never know this. I think she is personally. Is A3 a PGS or not a PGS? The subgroup generated by the element A and B is a DCS, which is not a PGS, because it's not generated by a subset of the generator. Yes, the group generated by A and B are PGSs. Only what I said is not true. Well, they're all DCSs. Some of them are PGSs. And we want to get rid of the ones which are DCSs but not PGSs. and indeed the paper we published was a little bit draft there's phraseology in there that says that in the past we thought of the DCS's as being the fundamental entities of the hierarchy and that's not very accurate we want to now think that the PGS's as being the fundamental entities of the hierarchy but that as a statement in itself is too simplistic and we want to say what is essential? Frederick is the title of the paper. Well, Frederick is the Frederick we've been talking about all along, who is actually Bronwyn's Brenda. The construction is a denominator of hierarchy in its various forms. And what we want to find is the essence of this construction.

55:00 And this is the point of generalizing. We have some special cases and we want to understand what the essence of these special cases is. We can think what are more general cases that these are special cases and then we begin to see what the essence of something is. I want to talk about generalising from that. What is this essence? The one I hate is this one. Well, why did I stop doing this in the first place? One reason was that there's a very small number of people working on the hierarchy, and I thought perhaps it was some years ago when there were quite a large bunch of people doing this, and I thought that perhaps I'd try to do some things to massage the situation, shall we say. was to have a look at what had been done in terms of generalizing the hierarchy. I've had conversations with Arletta that she'll probably deny, in which we agreed that this seemed to be a rather slippery thing in the literature. If you tried to find out what it said in the literature about this, it seemed to jump away from you. And I wanted to try to tie it down a little more. My motivation was this, that actually I'm astonishingly impressed by what Clyde calls what Ted and I do now. I feel more impressed right at the moment than I did on the publication of the book. When the book came out, I thought, oh, it looks as though it's all very tied up. But having looked at the book quite a bit, I wasn't as comfortable as I am now. I very much like the construction as it is at the moment, so it seemed a good point in time to go back and look at some infrastructure, shall we say. The point is Clive's calculation of the fine structure constant. There are times now where I really believe that this is the fine structure constant, which is quite shocking.

57:30 It changes... It gets us all in the end, you know. Well, I felt like that at times before, but it didn't last very long, it seems to be lasting longer this time. So yeah, well that is a motivation, certainly. Yeah, another motivation was that really, I guess, Mike Heather, is it here? And Nick Rossiter's work on describing the hierarchy in terms of categories was bringing my attention back to that particular aspect of generalization. In fact, I've thought about this myself. somewhat unsuccessfully, Mike Wright had had quite a good go a long time ago, and again he said he couldn't really make it work, if you like, and dropped it, but certainly Mike and Nick starting again was one of my motivations. Physical understanding comes last. Why am I saying this? Well, because it's an aspect of the paper in its current form that strikes me worth thinking about. There have traditionally been two ways of thinking about the hierarchy. One is a rather ontological way. In that Client and Ted have often talked about entities moving from a set called the unknown. So a set called the known. And they've looked at this, at least in the past, as a picture of how this structure is a ontological kind of picture, except that the question has been raised as to who is the knower in this picture. On the other side, there's a sort of epistemological way of looking at things, which looks like this.

1:00:00 I look at the world through rose-tinted spectacles. As a consequence of this, the world looks pink. So here, I look at the world, through rose-tinted spectacles, the world looks pink. Well, I look at the world through whatever processes are involved in the process of observation. If the process of observation itself essentially contains structures that have certain mathematical aspects, it seems rather natural, actually, that the way you see the world will have those structures somewhere inside it. In some philosophers, the structures are all the structures that there are. Yeah, I think this argument as the world is entirely platonic... No, it isn't. The world is entirely... What's the word of one? Measurement. It's a bit silly. I think it's almost obvious to me that certain aspects of the world are constrained by platonic facts. At the same time, I mean, I think it's certainly true that there are some things around the world which aren't just mathematics. politics. However, what I'm going to lead on to towards the end of my talk is that something I've suspected for a long time, and that Mike Manthe in particular has hinted at, and Pierre's little talk about Mike Manthe's relationship to this whole picture was quite timely for me because although Clive and I and Pierre might not see exactly eye to eye on what Mike's done or hasn't done or whatever, I think there's something very very important there. One of my longer term plans is to go back and have a real good look at Mike's stuff and see if we can't massage that a little bit and make it a little tighter, and Clyde and I talked about this a little. So the aspect of the paper that makes me choose is that what we end up with in the paper is a picture of the hierarchy a sequence

1:02:30 of mapping between levels. And I'm not really thinking about stock rules or anything here, but I'll let a point out to me. I should really bring those back in. But I'm trying to sort of liberalise the whole thing and see how it might feel if we don't know about stock routes. So between each of these levels, there is a mapping which we call the level change operator, which maps from one level to a level. I think I note that a certain non-commutative picture of levels can be changed into a commutative picture of levels a process called abelionization at each level. And here we have now a commutative hierarchy. This is an obmuted hierarchy. In order for this to be true, it must be true that L times A is equal to A times L. So we could say the level change operator must commute with abelionization. Now, I think, again, rather facetiously, I've set this as an exercise in the paper, so if anybody reaches properly through a bit, I'd like to see the proof. But really, the picture I'm drawing here becomes very familiar. It looks like something from algebraic topology. And, as I say, Manthe's pointed in this direction before. Or if we think of these, there's a very familiar picture of mappings between dimensions of various braids here. And what it's pointing towards is a picture that looks like this. boundary operator on levels in algebraic topology we think about boundary operators such that that is true and there's a famous quote I think by Wheeler that says something like something like all physics is this I don't remember the quotings, actually. The boundary of the boundary is, you know.

1:05:00 Yeah, yeah, but he says, this is physics. Yeah. Yeah, I can't remember the quotings. Yeah. But the point I'm making is that there are a lot of hints now as to how to get from here to here. And Clive and I were talking about this a little earlier on. And it's something that has not been thought about more than an amuse in which we can think about, not necessarily the levels themselves, but the information between the levels and how this transforms between levels of various layers that may well look exactly like this. that it looks exactly like this then technically we have a chain complex and we have the sort of structures one would get in physics well the exterior derivatives do that yeah exterior could i just interrupt the moment if they were in the stuff i did this morning which i didn't get And indeed I have a level change, and the Arbelianization operator is my taking the skeleton, and what I proved is that indeed that diagram does come in. Well that extent to the very general case we have at home. I'm a bit more cautious now than I have been. Well, let me see. We now have this rather strong claim in the paper that DCSs are not the fundamental entities, rather PGSs are. I don't know what a strong implication that things which have been published in the past are a little bit flawed in the way they've presented things. Well, what I want to say now is both things are true in a sense. This statement is actually true, provided we take it in the right context. Brute side of our context, DCS is actually a lot of fundamental things because really these two things are identical. But they stop being identical in the context I'm going to...

1:07:30 The context is this. Let's ask, what is the combinatorial hierarchy? A combinatorial hierarchy indeed. It's a set of levels, a really set of levels. it is a level and this is the question that Ted rather famously asks Clive and Clive answers well a level is a group or a group void in fact I think he has to say that at the moment or at least something along those lines a specific of generators. It's not just Q, it's Q with I and J specifically stated as the generator. I'd like to say that a level is a specific set of generators plus an operator on those generators. Not the equivalent, but it has a difference. But other times I might even like to say a level is a specific set of generators plus a specific set of relations. which has a symmetry which I'm quite comfortable with. The generator plus operator is the one that has come up in thinking about the things we're thinking about. Now, if we specify the generators, the statement that I just rubbed off becomes true. If we specify the generators, it's certainly true that the fundamental entities of this particular common entirely are the PGSs. because the DCSs in general don't have a certain property we need in order to build a hierarchy. And that property is that for any subgroup, for any PGS or for any DCS, we need to be able to specify an auto-morphism that's defined by that, not do that in general for DCSs. We don't do it if the teacher looks uncomfortable because of something. Well, you sound as though you are explaining a mathematically impossible, but I know because we know of cases where it's impossible, I think. And once we know of cases where it's impossible, then...

1:10:00 Well, we've got the PGSs here, we've got the DCSs here, and we know we can always do it inside here, and we know that there are things here where we can't do it. Now, there may be things inside here where we can do it, but the fact is that we can't do it for all these ones, for us this one is guaranteed safe. The rider, which we'll talk about in a few minutes. I'll improvise. One thing which is stated over and over again in the paper in various different forms as we develop the story is the actual form of the level change operator. I'm not going to go into detail about that now. If anybody wants to, they should look at the paper and find all the mistakes. Are you going to at least reissue that paper to the hamper people through the... judgments at least that you've made in the paper that you don't feel happy with in their expectations. I could do that. I think we're going to say that the interpretation of the statement about the fundamental status of DGSs and DCSs needs a little massaging, as I've just said on the one hand. On the other hand there is a let me see, when the thing went to press I did have the feeling, and I'm not sure how strong was that it was pretty well wrapped up this story there are actually really two holes in it one is the one i've just talked about which is um actually relatively minor it's a matter of wording and um indeed if john anson's ever around i will i will emphasize the fact that the dcs's are in general the fundamental things whereas in emails within recently i've said well i'm not sure that's true anymore. Well, it is true, but what I said is also true in context. I don't think that's such a big deal, except so long as we don't forget it. Well, the other issue I'm going to talk about in more detail in a few minutes. I think there will be another version of the paper, but I don't think it will look anything like the version that exists at the moment. I think it wants a totally different angle to it now. Well, let me make one comment.

1:12:30 understand the paper, I do now, because I've listened to these two lectures. It's put flesh on the bones. I'm not terribly high with abstract algebra. I tend to forget the terminology and things like that. Well, fundamentally, without the change operators, it consists of two things. It consists of the observation that, and I've written them in a form that's very consistent with the traditional bit-stream view of the hierarchy here, so you can think of these as being matrices, but certainly in the form we're thinking of them now, they're not matrices at all. These are ultimorphisms on group-wise structures. Anyway, matrices are a special case of that, but we've generalized such that it is no longer true that it's valid to think of these in the old linear terms. Although, even though, as an H example, but this looks very simple. I should remember that these are matrices over 0-1 and an exclusive OR as the ring operators. So, I mean, this is just not overly everyday linear algebra, even at this level. But are you saying, in fact, there are no matrix representations of the whole thing? I think it is very unlikely that that's true. Now, that's less clear. In general, there are always matrix representations of root transformations. But that's associated with the... But that's associated with it. But what is X? X is... I think what they say is that's a good question. X is an arbitrary element at the lower level, or is it a separate element at the lower level?

1:15:00 It's a matter of what notation you want, but I would have read that with the usual pardonable gloss that was a universal quantifier in front of it. Because the quantifier is missing, because if it is for each X, what you have there... Well, the top one is for each X in the particular set. I see, because otherwise it is doing nothing. The bottom one is for any X. But if there is missing quantifier, then this A is doing nothing, because it's... Yeah, if you read our letter's point, please, you must say on that left-hand side, for all X belonging to something, in a subgroup. Yes, that's quite correct. We're thinking of DCS's. So for all X in a DCS, we want to be able to find an A such that this is true. Multiplying an element of the subgroup, or transforming the subgroup, using some transformation, think of it as a matrix or think of it as an orthomorphism, we want to get back when we start. Is X an element or a subject? That's what we were just talking about, X is an element. It's got to take any element back to itself. So what is this really doing? I thought the point was to look for automorphisms of discrimination. That's correct, but this is inside the DCS. We call this a fixer outside the DCS. We want to do the opposite thing. For any X, we want to make sure we haven't got X. Exactly, this is missing on the table. That's because I haven't thought that. What point of this is that for any subgroup we want to find an automorphism. It's certainly true that for any automorphism we have a subgroup because any automorphism, any mapping, will fix some of the elements of the group and not others. What we want to do is make this work the other way around. Now, the theorem that we're looking at says that this can only be done in general for

1:17:30 PGSs and not for DCSs. as we restrict the DCS's to be PGS's we can't construct a level change operator level change operator consists really of this bit and this bit which says that well, this is matrix algebra but we don't really know what A and B are A are just transformations that take this vector to this vector or this object to this object so as they're just transformations we really don't know how to combine them what we do know is that it's wrong to assume that this is an ordinary composition of homomorphisms. This is called the sum of homomorphisms. Well, the point is that we have to derive it. We can't put it in there. This operator here is derived such that the result here is equal to that. So, at the lower level, we've got this thing, which is adding an element Y to an element Z, at the lower level, but on the upper level, this reflects back to say how we have to bring all some autism together. Well, I understand that this is a definition of induced operation in all terms. On the right-hand side this class is operation on a lower level, on the left-hand side this is operation defined this way on the higher level. That's correct. So nothing changed. Well, nothing changed. The story is simply that the level change operator consists of this idea of relating the automorphisms to the subgroups, and this idea of the induced operator. This one doesn't change. This one's got the extra bit in it that says, we only note the definite we can do this for PGSs. And any fixed structure, any fixed hierarchy, any fixed levels, we can't do it through our and we know certain places where you definitely can't do it. On the other hand, for the original hierarchy and such five simple things, you can always find it. Which is one of the reasons it got forgotten about. Could I make up some remarks I would try that interruption I've forgotten to mention it to you this were to

1:20:00 be of a critical tone of mind and instead of instead of being like Arletta and saying oh yes that's just a usually used operation they were to say why should you choose that usually used operation, my answer would be that if the lower level is a discrimination I think, personally, that in this whole field of endeavor, it's interesting in its own right. I'm taking exactly the opposite approach to particularly Ted and to both of you guys in some extent, in that I'm looking at which might have physical implications, and it doesn't have to come from any particular understanding at the beginning there. walking around this territory and seeing what the territory feels like. I mean, the auto-protection of the theory, which, you know, produced an awful lot of quantum theory, are just like that. It's also related at the personal level to the continuance of identity. I interact with myself as I continue to view myself. So back to the constant thing. for a moment, there are perhaps some things I'd like to emphasize. The old deep-string hierarchy here got somewhat left behind in Clive's travel through the world of non-communativity, and very rapidly from that world, the world of non-associativity. So that in actual fact, at this point in time, there are three countries. I think you should call that math, rather than math. Well, Clive also calls this one the ACH, which bogged me for many years, because I thought This was associative. It's actually the combinatorial hierarchy with aspects. I wish I'd never used aspect as a word for. I mean, the existence of a man with that name is a totally different business. But then, well, this one's non-commutative. This one's non-associative. And what I wanted to emphasize a little is that this is one particular sense in which we are trying to generalize. So if this one's going to fit into the general structure, the general structure's got to be at least non-associative, which means, okay, we can think of this as a group void, but that categories aren't good enough.

1:22:30 because categories, well, now, my statement's what massaging is, which leads us on to the next talk, but I still want five minutes on this one first. There's another sense in which we've generalized that has been a fair amount done on the hierarchy as group structures in the past, but it's tend to concentrate on the actual specific group structures where they're here or here. And it's looked at each of the levels and it's found which group is here, which group is here, which group is here. Well, the paper we've got at the moment doesn't think about that initially at least. It just thinks there are transformations between groups. We don't specify what the groups are. So The more, the older work concentrates on actual concrete categories. The paper we have now is at the level of abstract categories. We don't talk about what the categories are until at least afterwards we say these are some examples of this general structure. So we've gone somewhat from the concrete to the abstract. We've also gone somewhat from, not to forget this, groups to group points, which is quite a big jump, actually. Although, now I want to go quite fast, there is a question about whether we can truly justify this group points. I think we can probably do it for groups, but this is the last thing I want to do. Finally, also worth saying that abstract groups are actually a specific category, a category of groups, and to be really general, we ought to be able to work in arbitrary categories, in abstract categories, not the particular group. Now, here's the crunch. What's wrong, sir? Actually, everything I've said, I think, is pretty comfortable in the paper, except that we've made this wrong assumption that for PGSs, it's true that x, y, z, and we

1:25:00 in place, finds a transformation that fixes the group and unfixes z, y, its complex. So if wxyz is the DCS or PGS we're thinking of, sorry, if wxyz is the group or level we're thinking of, and WX is the PCS or PGS, and we're claiming at least the PGSs, it's always true that we can construct an automorphism such that the subgroup is fixed, and the complement of the subgroup is unfixed. So we can take, so you can imagine with a whole stream of these, you just leave them as they are, and a whole stream of these, if you just rotate them around one place. We've done it. Unfortunately, there are problems with this. Right, here's one problem. What happens if there's only one element in the difference? This never happens, it's just crossed my mind. Is that impossible? I mean, you can't rotate one element around, you've still only got the one element. That is a problem. I haven't thought of this one before. It may be that this one never occurred. That's a nice way of getting rid of it. But, well, there's another problem. WX A, B, C. I'm running completely go back to this one think about the product of these A, B, C we'll take them around and the product of these B, C, A will be equal to that and we want to be able to let me see what I'm saying if the product is in here we don't want it fixed not that we don't want it fixed it's the top, we want it fixed so what I'm telling you is actually we haven't got a theorem this wants to work on it right, I've got two minutes

1:27:30 I want to read Clive's last note to me very quickly so in the last week I've been throwing this back at Clive and saying, look we thought we had this theorem we thought it was all wrapped up but well it doesn't seem to be and can you have a go at it and Clyde's reply is the theorem you get this and at the top it says the theorem and he just handed it to me and I walked off with it and I thought he meant it was done but actually the first line is I have been getting into a complex mess it's best if I explain why it comes from the being of generalizing Frederick. I have been guilty of jumping from one to the other without warning. To make it worse, they are inconsistent. In one way, the theorem is true, indeed trivial. In the other, I judge it to be very hard. Now, I think Clive and I agree that the problem is a theorem. Or at least we can massage the context such that those are true. For instance, I was fairly convinced we couldn't do it for grupoids at one point in time antiquity groups, I believe it was. So, well, we can always massage the context, hopefully, to find the context in which the theorem does exist. It is true, but it isn't clear how to do that at the moment. I won't read through the rest. There's more to it. But the point is that this whole thing is fairly well-rounded, except for this theorem. So, now, here's a puzzle for the mathematicians here. Thank you.