Can Eddington Be Rehabilitated?

0:00 All right. Thank you. I'm taking my time card because it's one of the most interesting resumes that have come across in years. What are we doing? What do you think? Well, it's about five minutes after half. Well, welcome to this week's seminar in philosophy and physics. To me it's really a very great pleasure to introduce Professor Clyde Kilmister to you. Professor Kilmister did his first degree in mathematics at Queen Mary College in London in 1944, went into the electronics industry, after the war did a PhD in mathematics, again at Queen Mary College, and then joined the mathematics department in King's College in 1950, and worked there until 1984, when he took early retirement. Professor Kilmister

2:30 sees himself basically as a general relativist, but when he needs to look at his publication record, particularly books, to see what an extraordinarily wide range of interest and confidence he has. I'm going to mention some of them. I'm absolutely sure I'll leave out some important ones. But books have been written by Professor Kilmister on rational mechanics, Hamiltonian dynamics, Lagrangian dynamics, language, logic, and mathematics, which is based on mathematical logic, the nature of the universe, rational mechanics, and a book on Bertrand Russell, not to mention special relativity and general relativity. Today we have a talk on Arthur Eddington, who made such important contributions, both experimental and theoretical, to general relativity. There's probably nobody in the world who's better, more prepared to speak to us on this. Can Arthur Eddington be rehabilitated, and should Thank you, Holly, for that embarrassing introduction. When I was having coffee just now over in the philosophy faculty, it reminded me that I was in Oxford many years ago and David Finkelstein gave a series of talks over there on discrete approaches to physics. and after the last of the talks I had a long session with him to explain what I was doing at the time what I thought was basically the same problem and he was very patient and listened to me to the end and then he advised me that though I might be right no one would listen unless I could get to the physics a bit quicker so you can decide this afternoon whether the intermediate 20 years have enabled me to do that or not I just want to say a word about the title I realised once I started preparing the talk that I was using the word rehabilitate in a different way from usual because I've been influenced by a play I saw some 10 or 12 years ago in which the villain was a property developer

5:00 who talked of rehabbing places which meant you take a downhill building, tear the inside out, put some smart flats in and leave the outside as it is. Well, I don't exactly mean that with Eddington but I do mean to ask whether one can take the outside shell that is the overall philosophical ideas and clean out the inside to give something which will be respectable. And I want to argue positively for the first half of my title and leave you to decide on the second half of whether he should be. I should say that I'm very happy for you to interrupt at any time during the talk if you want and if you don't do that then I shall talk for about an hour and leave leave time for the discussion. So the paper divides itself rather naturally into four parts. In the first part I'm going to remind you a bit of the history of how Eddington's reputation has suffered in recent years. Then I'll talk for a very short time on my own personal history as it relates to Eddington and so on. The third part, which is the main part of the paper gets down to the way that what I would call the Eddington problem could be tackled and then finally a very short conclusion so now as to the history to see how things have got into their present state if we leave aside Eddington's early astronomy actual looking through telescopes and things He achieved fame and, in some quarters, notoriety in the scientific community by the way he presented general relativity when it appeared. Then, the part he played in the Eclipse Expedition of 1919, which was widely seen at the time as confirming general relativity, I hope people have second thoughts about that now, earned him great and widespread respect with the general public as well he became a great man and I think he really rather liked that as perhaps I mean I would have done in his place now in 1923 in his book

7:30 The Mathematical Theory of Relativity he explained amongst the theory in general one thing on which he laid a lot of stress how all tensor equations are invariant. No talk here about group representations or any of that clever stuff which people do. He never seemed to have looked at things that way. But he saw that when you had tensor equations and then you did a transformation of whatever group it was you were concerned about you didn't finish with the same equations. He also conjectured, or maybe just made a common logical error, that conversely all invariant equations must be of tensor form, and that played quite an important part in his later thinking. By 1927, he'd noticed how things were beginning to change. He writes then, microscopic physics was the province of quantum theory but in 1923 as part of his earlier book you see this was little more than a collection of empirical rules which led to no coherent outlook the new quantum theory began with Heisenberg's researches in 1925 the conditions were becoming ripe for a unification with macroscopic relativity theory But Eddington's views about the completeness of the analysis of general relativity was suddenly upset in 1928 by Dirac's publication of his equation for the electron. At first, or so he claims, Eddington saw this as an opportunity. He says, To say that Dirac's equation was the first connecting link gives only a partial idea of its importance. to those who specialised in relativity theory I was soon convinced that this was the extension of relativity theory for which we had been waiting the way Eddington saw it he, Eddington had neglected certain algebraic structures that is Clifford algebras which Dirac had used to formulate his equation

10:00 Eddington concentrated on these algebraic structures I mean the rest of the Dirac paper he tended to ignore one shouldn't think too badly of him for this if you read the original Royal Society paper of Dirac where the Lorentz invariance is shown in a manner totally unlike the elegance that one's used to from the later Dirac the wood for the trees, and I think Eddington did that to a certain extent. Internal evidence suggests that Eddington started pretty soon after 1928 a lengthy series of trial and error investigations into algebraic structures, and this went on right into the 40s. It was protracted because Eddington, as he said to those who advised him to consult Modern Algebra, said, I like to work these things out for myself. Of course that takes longer. We lack any papers Eddington may have left on this, so I should just tempt Providence by attempting a bold reconstruction of the initial stages. Consider again his remark on the Iraq's challenge he emphasised C.G. Darwin's remark it is rather disconcerting to find that apparently something has slipped through the net he says Eddington says it was Darwin's insistence on this point in private conversation which led me to take up these investigations we don't know of course what the private conversations were but there's a letter that Darwin wrote as early as January the 11th, 1928, that is to say, about a week after the Royal Society paper appeared of Dirac's, wrote to Pauli, saying, I do not know if you've heard that Dirac has got a new system of wave equations which does the whole spinning electron correctly, Thomas Correction, Relativity and all. I have amused myself by putting them into vector form and the result is quite comical for it requires to express these four quantities

12:30 no less than two invariants, two four vectors and one six vector so to get Eddington's thinking one must go, I mean having heard something like that from Darwin one must go back from Dirac's paper this case of some sort of algebraic structure. I think he, on pieces of scrap paper which he then destroyed, he must have done something like this. He said to himself, suppose we have two elements, let's call them E and F, say, which combine in some way. So you have, You must remember, see, in all this, he wanted to work these things out for himself. Modern algebra is, as it were, a deliberately closed book to him. So, you have these two quantities, say, e and f, and they combine in some way, so you get a product. We could call it, say, f. Or, of course, the other way round, fe. eye on the fact that Dirac introduces anti-commuting quantities, he suggests that these quantities might be multiplied by a number. Well, in other words, really, he's doing the algebra over a number field, but he doesn't, he wouldn't have said that. So they might be multiplied by a number, ordinary number. And then, since these two products must have been related in some way, one could suppose that there was some number X which related those two then the next step in his argument I guess would have been to try and associate E and F with directions in space in order to come round to Darwin's remark about two invariants two four vectors and so on, so then you'd have these two third one here and then from a general feeling of the isotropy of space and so on these might have been associated the other way round and so as well as getting the fact that ef was x times fe by doing it the other way round he would have included the same equation

15:00 with the x on the other side and that would have suggested to him that x squared would have to be equal to one and so equals one wouldn't be very interesting, you take x equals minus one, and so Eddington, rather belatedly, rediscovered Hamilton's Quaternions by having two anti-commuting quantities and so on. Now, when he did that, if, I mean this is my guess, but when he did that he saw that he'd got three quantities, and because of these numbers which he wanted to import there would also be a unit element in the algebra so there'd be four altogether just as Dirac originally I'm sorry just as Hamilton originally did in the middle of the 19th century and this these four quantities were divided up into three of one kind that's E, F and a product, and one of another kind, that is the unit. So you have four altogether, and that four is in the form of three plus one. Now, that suggests, I think that suggests to a normal person, that there is probably, with the help of a little geometry, there's a peculiar appropriateness of this algebra for space and something else, shall we say not necessarily time and that's not a bit surprising because what all Eddington had done was rediscover Hamilton's quaternions and Hamilton devised them for that specific purpose describing three-dimensional geometry but I think he drew a much perhaps less reasonable but a much stronger conclusion from that pointed to a bigger prize in his mind because Eddington was an avowed Kantian and he couldn't fail to be excited at the prospect of a new approach which would complement Kant's discussion of three-dimensionality. The idea would be that Kant's transcendental exposition be replaced by one inside the theory the fact that space had three dimensions

17:30 would be a consequence of an algebraic structure about which of course one had got to say more than this but somehow one would have an argument to show the uniqueness of this structure and therefore the necessary three dimensionality of space and he actually put that in writing well not quite as explicitly as I put it but it is in writing in no less austere a journal than the Journal of the London Mathematical Society which is not the sort of place that one would go to for speculation normally the Dirac algebra itself of course is more than the Quaternions in fact it is just the direct product of two quaternion algebras and this fact actually does appear in later Eddington writings but not in the book I'm interested in the 1936 book with the relativity theory of protons and electrons but it does come up in fundamental theory so he eventually did come round to that simple way of putting it. But at any rate at this stage he saw that by taking two of these algebras you could then generate the one which Dirac had used and which has these 16 components which group themselves together in the way that Darwin was amused by. Now when you do this, suppose that you do take this direct product. Well, one can, I don't need to write all the symbols, it would be easier to do a little diagram, you see. If you represent, oh dear, my hand is shaking. You represent the, one algebra by a single, the unit element, and then the three, PF, and so on. And then the one, you're doing the product with another one, which is divided up in that way. And You have these 16 elements in here, of which there is one which is the product of the two unit elements, and then there are nine which are products of one of these in one algebra and one in another, so altogether ten. And then there are three which are the products of mixed kind of things,

20:00 with one unit element and one non-unit element in the two. So I would say that in the 16 elements are then divided into 10 which don't mix things and 6 which do. So 10 non-mixers and 6 mixers. That 10, of course, is in fact 3 squared plus 1 squared. And the other's needed to 6. Now, having done that, had Eddington, like the trained mathematician that he was, performed the usual mathematician's trick of, if something works, why not do it again? But this wasn't so arbitrary as it seems, because in this way he'd got the Clifford algebra with 16 elements, which splits up under automorphisms into, as Darwin said, two scalars, two vectors, and a six vector. And Eddington had the Riemann-Christophel text, because of his strong leanings to general relativity. And if he had to have an algebraic structure which would include that, then you've got to have the direct product of two of these Clifford algebras. So you turn the handle again, and then of course, the next time round, instead of getting three squared plus one squared of one kind, you get ten squared plus six squared of one kind, that's 136, and then 120 of the other kind. twice 6 times 10 and then you see Eddington began to get quite excited because he was aware of the long history of quantum mechanics obsession with spectroscopy where the fine structure constant that is 1 over, at that time, 1 over 136, comes up in experiment and in various parts of quantum mechanics.

22:30 Nowadays everyone agrees that this number and a few others are of basic importance, but their values, determined by experiment, are a bit of a mystery. The mystery was first noticed by Planck at the turn of the century, though he saw it as an opportunity. because he observed that his constant plus the speed of light of charging an electron and so on gave the possibility of setting up what he called an absolute system of units of physics. The only mystery about it is just why these numbers should have the particular values they do and null others. Dirac much later went on record as saying that quantum electrodynamics was an incomplete didn't calculate alpha, a fine structure constant, although its computational techniques depended on the smallness of alpha. Anyway, for good or ill, Eddington was sure he was on to something. Other people saw only the weakness of his argument, and the calculation, so-called, wasn't very well received. And matters were not helped at all when the experimental value for 1 over alpha moved up from 136 to 137 and Eddington argued for an extra degree of freedom which he'd got to add on to the value that he'd already got. Now, whatever the validity of this new argument and it's not so silly as it sounds at first as a PR move it was absolutely fatal and his position of eminence meant that criticisms weren't expressed forcibly in public Paulus for example wrote in a letter to Oscar Klein in 1929 note by the way I now regard the Eddington 136 work as complete nonsense more precisely as romantic poetry not physics but the public statements by people were much more restrained we shouldn't be too much on the side of the sceptics in this I think it's not so unusual in physics

25:00 to work from an experimental number back to some sort of explanation The old quantum theory worked for years on exactly such a basis. But what went wrong for Eddington was twofold. He had no underlying theory to say why he was getting the fine structure constant, only the very weak position that these Clifford algebras had come up in Dirac's paper. And as a result, he had no way of improving the value into a more correct estimate. he tried to tackle the first of these problems by using the algebraic structures he'd got mixing them in with orthodox quantum theory so he hoped to be able to say oh, here you see, this is where it comes in so he would be able to get to an equation in two ways and so identify the constant this never worked and that was not only I think because Eddington had no real expertise in quantum theory and his attack on the second problem was even less worthy he introduced the so-called bond factor this was suggested to him by someone called bond this factor was 137 over 136 and Eddington tried to argue for having it raised in different equations to various fractional powers well if you once are free to do that you can see that this is a fudging factor which will never fail you as long as you can make the powers whatever you like the limitation to public criticism suggests to me a reason why Eddington's response wasn't to seek to clarify his position but to go on to find other such numbers and so by 1936 to make a most extreme claim this is at the end of his relativity theory of protons and electrons he claims there is nothing in the whole system of laws of physics that cannot be deduced unambiguously from epistemological considerations well the response to this book was now indeed critical and people thought he'd gone off the rails

27:30 I don't think he had entirely yet but what he did as response to the criticism was to retreat to the observatory and he concentrated not as he should on making things clearer but on generating more and more numbers experimental numbers you see. I think it was only after 1936 that he finally left the rails and I'm not trying to argue for any possible rehabilitation of fundamental theory which appeared posthumously, only of the 1936 book. Now a very short bit on my personal history. let me start by clearing two things out of the way I think the Clifford Algebras notwithstanding Eddington's use of them are a red herring they just shaped Eddington's response to a certain situation nonetheless I do want to seek some algebraic approach and secondly I couldn't possibly try and argue for Eddington's extreme position that I quoted just now but one could take a less extreme position and suppose that rather than the whole of physics there may be some fundamental numbers which could be found in some way other than by experiment now this modification isn't without its own troubles it needs a criterion to determine which numbers are to be designated as fundamental and what I have to say from now on rests heavily on my long-term collaboration with Ted Bastin and initially on a peculiar idea put forward to us by Frederick Parker Rhodes back in the 50s of the last century Bastien and I were convinced that something could be salvaged from Eddington and that that something was a general program of finding some numbers perhaps those mentioned by Plunk We believe this program should be possible because it seemed to us a coherent position to see these numbers as logically prior to measurement and so casting their character as it were

30:00 on the subsequent measurements we hoped to do it in some papers in the mid-1950s by looking at certain Arbelian groups which we had isolated from some symmetry considerations which I wouldn't want to go into now we had the idea which partly derived from our reading of Eddington that there'd be a hierarchy of such groups but we couldn't puzzle out their precise relationship and this missing link was what was provided for us by Parker Rhodes in 1961 so that's the real beginning of the rather long and drawn out work that I've done since he devised a peculiar hierarchical structure was fitted. I'll say a little about this in a moment, but we don't want to spend too long on it. But for now, I just say the structure had successive levels, hierarchical, and the cumulative multiplicities as you went up through these levels were 3, 10, 137, and 10 to the 38 point something and the construction of the levels could not continue after that last one we saw this termination of his hierarchy as corresponding to the fact that gravitation is the weakest force and the inverse the inverse quantity corresponding to the fine structure instead of electromagnetism is of the order of 10 to the 38 and so the termination rather sold the construction to me in a way that the mere numbers by themselves wouldn't have done. Eddington had constructions which could be repeated over and over again and he was forced into the rather weak argument for stopping that physics doesn't seem to go any further complexity. But here the Parker Rhodes construction stopped and that seemed to be a good beginning for us. Of course we saw there were enormous problems. One problem was exactly how was

32:30 this peculiar structure related to physics and the second one as with Eddington was why was 137 not a very good approximation to the inverse fine structure constant because already by the 1960s one knew that it wasn't exactly 137 but a fairly small decimal part to it so the first of these problems is a problem in philosophy and physics the second one is essentially a mathematical problem and I'll say a few words about this second mathematical problem first because the fact that one can answer it successfully as I think is what motivates my long war on the first problem of making proper philosophical sense of the thing. Now to do this, I must first say just a little bit about the Parker Rhodes construction. I don't want to spend too long on this because the details aren't worth it. I do it differently now. Parker Rhodes based his construction in terms of vectors in the sense of linear algebra over the number field with two elements Z2, that is with the two elements 0 and 1 or as we now say, these vectors we call bit strings. He called addition over that field discrimination because two strings are the same as each other if and only zero. So, if you have two strings and you want to know whether they're the same, he would say, you add them together, and if it comes to naught, then you know you've got the same string. That was what he said. Of course, it's not a very consistent way of going about things, because if you're going to add them together, you've got to add the components in his formulation of it, and if you can do that, you can see whether they're the same or not, so you don't seem to need which we had to solve by looking at things differently. But that was the way he did the construction. So that makes zero a special element. It's a kind of sign that the two things are the same. Now in his construction, he begins with two elements,

35:00 and so it's sufficient, as there are only two of them, to take them as vectors of length two. say the vector 1-0 and the vector 0-1 and then you discriminate those two and you get the vector 1-1 and those three are an example of what Parker Rhodes calls a discriminately closed subset he means a set of these things which when you perform arbitrary discriminations on different members you don't get any more what such subsets are are really groups Arbelian groups with the identity element that is the zero left out because it's important for him to leave this out since the zero is this special element which tells you when things are the same it's not on the same footing as the others so from these two elements you've begun by generating three discriminately closed subsets two of them have one element only in that is the two individuals and the third one has those two plus the result of discriminating so that has three elements in so you've got three of these discriminately closed subsets and then Parker Rhodes' next move is what he calls going up a level by doing the following trick each discriminately closed subset is represented by a non-singular linear operator all of this over the field with two elements of course having just the elements of that discriminately closed subset as its set of eigenvectors with eigenvalues necessarily one since it's non-singular, they can't be zero and you've only got two elements so these linear operators are of course matrices two by two matrices at this stage so you can express them as bit strings of length four and then these three bit strings by discrimination generate seven discriminately closed subsets that's two cubed minus one and these now

37:30 that brings the total of discriminately closed subsets up to 10 at that point and you go up a level again by representing the 7 discriminately closed subsets by 7 4x4 non-singular matrices the various ones having exactly those discriminately closed subsets sets of eigenvectors and so you'll get 7 bit streams of length 16 now there's an important point at this stage in Parker Rose construction because what he says is you choose those 7 4x4 matrices to be linearly independent that means that when you get to the next stage will generate 2 to the 7th minus 1, that's 127 discriminately closed subsets. And so you bring the total up to 137. And then at the next stage, you've got about 10 to the 38 discriminately closed subsets, but you've only got 16 by 16 matrices to do it, so you couldn't possibly choose them linearly independent, and the construction stops. So that's the nature of the stock. I just would like to break off and tell a story because we published one or two things on this in the 60s and a joint paper with various John Amson and Pierre Noyes and Baxter and myself, not I think Parker Road, in which we stated that that was what what the construction was. Now, it is an interesting question to ask whether when you've got to choose this set of seven 4x4 matrices, whether you can choose them to be linearly independent so as to get the right thing at the next stage. And we were in the position where each author of the paper assumed that one of the other authors had a proof of this. But it was only when we got down to asking each other for this after it was in print that we found none of us had. However

40:00 we set to and we now have three proofs because we each managed to produce one. There is no problem about that. However it's not of any importance for this talk because the question of choosing them is the very thing that I want to cut out well now, keeping with this construction at first let me talk about the value of 1 over alpha or rather, let me talk about this quantity 137 now, well before I do that let me talk about 1 over alpha the experimental value of 1 over alpha nowadays it is 137.0360 there's about half a dozen experimental ways of determining it which give none of which of course give the same answers as always with experiments but they all agree to those four decimal places some of them are a bit lower 0, 3, 5, 9, 9, 7 or something like that now the first encouraging sign for us was in the way in which the Parker Rhodes construction went that 1 over 1, 3, 7 could be seen in this way as you looked at the hierarchy developing so long as you didn't go up to that top level where there were so very at the bottom 3 the 1 over 137 was the probability that as you chose another bit string it was one of those that you've got already because as long as you're staying up at those levels you've just got 137 of them in all there seemed to be a rational interpretation of some kind. The only trouble was that the basis of that was that you have to constrain the system in some way to operate at just the three lowest levels instead of going all the way up as it might be free to do. And you've no idea what constraint would mean in this system because we don't understand what the

42:30 system is at all so you haven't any notion of what constraint can get out of that by saying, well, OK, but suppose the system happens to stay at these three levels. What's the probability that it would do that? And will that then modify the 1 over 137? And it's fairly easy, and you'll need a little bit more in the way of assumptions, which I don't want to go into in detail, but it's fairly easy to show that Then, the 137 becomes 137.033. So, we have, um, and a bit more, you see. And that's an encouraging sign, because, after all, it might have been that when you try to improve matters in this way, to get rid of the constraint, the 137 might have gone up to 139 or 200 or something like that but it seemed to be behaving in an encouraging way and now the next encouragement that we got came from departing a bit from Parker Rhodes' original recipe this question of choosing the matrices at the second choice, the second level to make them linearly independent why should one do that? what should one be doing really? one should let the system take its course see what comes it's easy enough to see that on a reasonable kind of probabilistic basis the chances of the set of matrices linearly independent is roughly eight-ninths, the probability I mean, and one-ninth is the probability that they are, roughly speaking. So that's the sort of situation that you're in. Well, if instead of choosing them to be linearly independent, you just take what comes, then the correction to probability, which I've just talked about, has to be modified of it. And doing this is a long and tedious but essentially elementary calculation. Just

45:00 tell you the sort of calculation it is to show you that there's nothing arbitrary being introduced at this point. The sort of calculation one has to do is one has to look at all the various sets of matrices which would satisfy the algebraic conditions not the linearly independent there are about 75,000 of these and then you have to look for the linearly dependent ones they will have the character that when you add them together sometime you'll get zeros so you've got to look at all those possible zeros and it turns out that there is about I forget the exact numbers. I think there's about 47 cases that have to be looked at in detail. But it's not really as bad as that because you've got one or two simple theorems which do groups of those. So in the end, you're finishing up doing about 13 or 14 different rather lengthy calculations and saying, well, with these values, how could there be a zero? And how many zeros will there be? And that sort of thing. So it's long and tedious, but it's really nothing but arithmetic. Then the final result that comes out, well, not final result, but the result of doing that, is it turns out that instead of that, you get 0, 3, 5, 1. And actually, when I found that, I think I might be excused for beginning to get excited and thinking, now, this cannot be the nonsense people might make it out to be. The final stage, which is really the exciting one, was realising that Harker Rhodes in that construction, by doing everything additively over that field and so on, had made an unnecessary simplifying assumption involving commutativity. In fact, the operations involved don't have to be commutative and then you get a different set of numbers but they're closely related to the original and

47:30 you then have a further calculation which is essentially not so very different from the long and tedious one which I mentioned, I mean that serves as a useful basis but you find that certain small changes need to be made to it in order to take account of the fact that the discrimination need not be commutated and so finally as a result of doing that we get 0, 3, 6 0, 1, 1 as the basis values are And the value of that being so close to the current experimental values surely provides me with motivation for quite a lot of hard work on the main problem. So now I've come to the main problem, let's put aside the Parker Rhodes construction for the moment, make a fresh start, but of course bearing it at the back of our minds, because obviously if that provided, or a modification of it provided me with those numbers, then I've got to work my way back to it somehow. Let's consider instead the furious property of elementary particles that any two of them of the same kind, say two electrons are exactly alike and this property has a kind of logical status it's not the kind of similarity there is between cricket balls where, I mean they are cricket balls are exactly alike because the MCC decrees that they should be, but we know that if you take two actual cricket balls and weigh them sufficiently accurately, they won't be quite the same weight or same diameter, but the electrons are not like Essentially, in physics, this property of elementary particles is enshrined in the assumption that the particles are defined by a set of attributes, that is, quantum numbers. So that you've got the similarities captured there. Now, instead of assuming quantum numbers to start with, I want to take as a starting point an abstract construction process

50:00 which depends on this single criterion that two elements in the process are either the same or not. In other words, I want to take an intuitive idea of this discrimination, which Parker Rhodes talked about in that rather incoherent way, and say, that's my basic start. I can discriminate elements. it won't surprise you to find or to be told because I'm not going to go into the details that one finishes up then with a fairly quantum number like description but you can get more out of the structure than that and in fact amongst other things you can get these numbers and you can also get a structure which is not actually quite identical with the peculiar Parker Rhodes construction, but it's almost like it, sufficiently the same to give the same numerical results, and it's one in which his peculiar going up a level is no longer a strange and arbitrary trick that comes naturally from the construction. I don't know how much detail to go into for the construction I'll just show you how it begins anyway the construction process is like this the elements are constructed in a sequence and this sequence therefore defines an ordering the first one is produced and then a second and so on the construction has a finite and discrete character It's a little difficult to specify completely what I mean by that, but for the present I want to suggest two principles. Firstly, one of uniqueness, that for each position in the ordering there is just one element, and each element that's constructed in the process has a unique position and the second one which we started on fairly early we tend to call it a generalized ergodic principle that only what is allowed by the principles can turn up in the construction

52:30 and everything that is allowed by the principles must turn up eventually I was going to say if you wait long enough, I shouldn't say that though Construction is not supposed to be taking place in space and time, space and time having due course to be constructed out of the entities. So that provides certain linguistic difficulties about that. But if I use any time-like language, you'll understand it because of the nature of English which forces one into it. and I'd better emphasize again that the process is a self-generating one I mean there mustn't be any interference from mathematicians or by many white coats it just goes on now since every position in the ordering has just one element one can use those positions to refer to the element and every element is adjoined to ones which have been constructed already ready. Now, I think perhaps I could skip a bit here, although it might make it confusing, and go on to ask what sort of rules there would be when one discriminates between two elements. Suppose that you have an element, let's call it A, and then another element is constructed, provisionally we'll call that B and the essence of the process is to ask whether A and B are the same or not and as I say I'm taking that as a kind of intuitive given that I have a thing in the process which will do that now when I do, when I perform remove the I, I mean when the process performs this construction to determine whether A and B are the same or not, it has by so doing, it has generated another element and so from those two elements a and b if they are different another element is generated

55:00 I have to ask what position in the ordering does that element have well the process couldn't be actually at a or at b because the uniqueness required it should be one and if it were at one then it could equally be at the other the uniqueness would be destroyed but it can't be at a or b let me jump ahead these positions in the ordering I could very easily number off with the ordinal numbers 1, 2, 3, etc if I had already got elements 1 and 2 then the result of discriminating between them not, couldn't be at one or two, so the only unique position I could specify for it would be three. On the other hand, if I at some stage had one and three at some stage in this general flux of things, I had one and three and I asked, well, let's discriminate between them to see whether they're the same or not. Oh no, they're not the same. Now what is that construction? That construction has generated another element. Where would this element be? Well, the only unique position I could specify would be that it's the earliest one which isn't prohibited by not being 1 or 3 and that would be 2 so the 1 and 2 would give 3 and 1 and 3 would give 2 and so on and if what we are generating here is a structure which is closely analogous though not exactly the same as a construction given by John Conway in his book on numbers and gains where he generates fields of characteristic by a very similar process so the rule is that there are certain

57:30 positions in the ordering which are forbidden by various rules and then subject to not going into one of those forbidden positions you just put the newly constructed element one which is not forbidden. So you will generate these sort of numbers and I had argued about this over several pages but I think the time is getting on a bit so I would rather leave that for the discussion afterwards if you want it and say, well, what's happening here is, as I said, very like Conway's book where just such an operation when I say it's closely analogous really it's almost exactly the same the difference really is that Conway has nought in at the beginning and I'm keeping nought in reserve as a special sign that two things are equal so that's the only real difference between us I mean it's just the same as people talk in a slightly different field people talk about the natural numbers some people the natural numbers start with 1 and some it starts with 0 but for Conway it obviously starts with 0 and for me it starts with 1 so that's the basic difference now the nature of the operation between these numbers is all set out in Conway's book and it's what he calls Nim sum, referring to the use of this device in the game of Nim where you, a game I've never really had much fascination with, it's about taking or not taking the last element of something with different columns, but anyway the game part doesn't matter, the rule for the Nim sum comes like this, that if you Suppose you have two of these elements, M and N, say, and you want to discriminate between these, then the discrimination between them, which Conway would just write as M plus N, the rule for doing this operation is that you write M and N in the scale of 2,

1:00:00 and then you add them together mod 2. so that if I happen to have something which was in the third position the third thing which was generated and then I wanted to add it to something which was, say, in the sixth position then Conway's rule would be to say oh well, that means you're taking one plus two and then you're adding it to 2 plus 4 and you add these 2 plus 4, yes, that's all and then you add these mod 2 so the 2's disappear and you get 1 plus 4 which is 5 now another way of rewriting this and this is where we get back to the Parker Rhodes situation Instead of writing that as 1 plus 2, you could just use the scale of 2 business and say, well that's 1 and a 1 in a column vector. Here are the 1s, here are the 2s, here are the 2 squares and so on. And so what you're saying is, well, that's the same as writing this 1 plus 2 and this 2 plus 4, that would be 0, 1, 1. and then you add those over the field set too and you get one at the top and one at the bottom and not in the middle and so that's where your five comes from so what you're doing in this Nim San is really the same as Parker Rhodes was doing with his bit string it ties up that way except there is an important difference that is that he specified the length of his bit strings from the beginning Whereas these don't have a length specified. You could have gone down further and had noughts underneath if you had wanted. It's not exactly the same. That's what I meant earlier on by saying although we don't quite get Parker Road's scheme, we get something which is near enough that you get the numerical values coming out. Now, to have got the bit strings like this is only half the battle.

1:02:30 I'd better just say a few words about the question of level change because that comes up quite naturally in this process theory. One doesn't have to import it as something peculiar. the point is that one wants to talk about the discriminately closed subsets which Parker Rhodes introduced and so one's got now given such a subset to ask whether another one is the same subset or not these are the things you really want to talk about, the elements are rather shadowy creatures the system. The discriminately closed subsets are the things which are supposed to be of physical importance. Now, let me get things in the right order. One could start by saying, suppose one had such a set of elements and one wanted to ask whether a new element which has been generated is a member of that set or not now what would not be a possible way of going about this for the process would be to take this new one and say well let it test itself against each of these elements in the set S in succession this wouldn't work because after it had done one of those then it would have to pick another one and then before it did that it would have to ask well is this a different one from the one that I did before or not well I'll have to test and see whether that's the same one as I did and so we're off into an infinite regress in no time at all so it somehow got to test itself against the whole set at one one fell swoop. Now what is more, we're going to be concerned with discriminately closed subsets rather than individual elements and when you test a discriminately closed subset against one you've got already

1:05:00 the result ought to be a discriminately closed subset because those are the entities we're working with. That means result of doing the test of an element against S will involve you in a linear operator of some kind because it's an operator which takes discriminately closed subsets into discriminately closed subsets. So the operation of testing the new one against the whole set already involves you in a matrix as it were if you've gone into the bit string language the testing will be something involving such a matrix and of course this is really the same as the Parker Rhodes going up a level, these matrices are simply the ones which characterise the sets you've got already so the basis of my argument now so far We have in this, in my rather hurried exposition of this process way of doing things, we have a justification for the peculiar construction of Parker Rhodes from the idea that elementary particles are all exactly the same. the bit strings of course are of the nature of the general nature of quantum numbers which we mentioned to start with so the general process business throws up as well as quantum number like descriptions throws up the Parker Rhodes construction and therefore as a result of that the numbers So now let me try and conclude by saying what I think I've done and what I don't think I've done. What I think I've done is to establish that there is a basis which begins with ideas based on the practice of quantum mechanics for the peculiar Parker Rhodes construction.

1:07:30 and so for the calculation of the fine structure constant which now doesn't seem peculiar but seems a perfectly natural consequence of a detailed analysis of what it means to be an elementary particle. Now, if you are sceptical, you could be sceptical, I'm sure you will be, about whether the numbers which turn up in such a construction are indeed approaching the fine structure constant, rather than just some other number. and my answer to that is not Eddington's answer of trying to relate what I've done to orthodox physics but simply to point to the extraordinary exactness of the numerical value which has been found and to stress that this very exact value doesn't come out by means of any fudge factors in the Eddington thing but just by a rather tedious straightforward analysis of the thing now to what extent then to get back to the title to what extent does this rehabilitate well not very much of course certainly not completely what I've managed to do so far is to save something about and I had originally we used to call it Eddington's program that there are certain quantities, well at least there is one quantity which is logically prior to measurement and whose value can therefore be found before we do any measurements what I don't have any answer for at present is are there any other such quantities um the alpha was the one eddington picked originally he went on then and found three others well he found seven in his 1936 book but he needed uh four of those to set up a system of

1:10:00 unit uh three of them to set up a system of units so he only had four i mean he had to give the values of three of them in order order to fix his units and then he had four which he claimed to have one of them was the fine structure constant and that I'm arguing is a number which does come up as a logical consequence of this process another one which he found by several different and equally suspect methods was the mass ratio of the proton to the electron I have nothing to say about that from my point of view this afternoon I haven't been able to get anywhere with it in fact as far as I can say I wouldn't even be able to answer the question if you asked me if you were to say to me well what you seem to have shown is that the fine structure constant is not contingent necessarily this value what about the mass ratio and I'd have to say well I don't know if you want to think that the mass ratio is contingent in the sense that this universe has got got it as 1836 point whatever it is and another universe might have a different one I wouldn't be able to say whether I wanted to differ from you or not I know nothing about that but as far as the fine is I think we've rehabilitated one tiny fragment of Eddington's programme and possibly in the future we or someone else will manage to do a bit more. Now, I leave it to you to think whether he should be rehabilitated, whether the project is one which ought to be, maybe you think it should be forbidden on moral grounds. A clarifying question and then a word. So first a clarifying question, you set down this sort of quantum mechanical base in terms

1:12:30 of elementary particles characterized by their quantum numbers to find a sort of grounding for the kind of destruction you described, and then you discovered that it was almost identical to Conway's, I mean, historically speaking? Yes. Well, to sort of lay everything bare, the way this came about was, speaking of my personal history, as it were, or the history of the group, we puzzled over the Parker Rhodes construction for many years. when I look back on it it really does seem scandalous that so much of our time should have been spent on something we ought to have got clearer much earlier but it seemed puzzling at the time for example we did lots of calculations about operating on vectors with various matrices of noughts and ones and because you're in a finite situation you start with a vector and you keep operating with the same matrix and then you go around in a cycle and so you can ask all sorts of questions about what cycle lengths are possible and what aren't and you get kind of spectrums of possible cycle lengths and then if you were as speculative as we were at that time you could ask wild questions about whether these numbers of cycle lengths of fundamental particles, that's known at the time, they weren't so numerous then, and we did all sorts of things like that. Now, after some number of years of this, I came, well in fact it was when Conway's book was published, I had it to review, room, and I came upon his construction, and I thought, this is something like we, this is just something like we need in the Parker Rose construction. So I got hold of the Conway book, and then I suppose that, and I was sort of making a personal statement, I suppose the way it really went was that that provided me with the guide of how I should change the Parker Roads construction into this process business and in such a way that the Conway

1:15:00 construction came up naturally. Okay, well that's interesting. Okay, well my worry, and perhaps you can just tell me that my worry has to do with my weakness in terms of physics So a long, long time ago, I once did a problem set where it's just sort of basic kinematic sort of problems before you do dynamics. And the problem was that I was supposed to figure out the force on the edge of a door that's hanging on two hinges, right? So I'm given basically the length and height of the door and the mass of the door. And I knew that the target number that I wanted to get was 2.4 newtons. vector diagram in a way that I thought was well physically motivated, got 2.4 as my answer. Then in the problem set, there was another problem with a different door, different set of parameters. Reconstructed my diagram, and I got 2.4 newtons again. Then I started playing around with the construction and sort of convinced myself that I could plug any door into it, and I got 2.4 out every time without going to some really extreme set of parameters. So my worry is, is it possible, I mean, eventually I figured out that I had made a sort of a basic physics error. So my worry is that, so you have one target number that you hit with the construction. Is there something that happens to be an artifact of the construction that appears to be well physically motivated that gives you this almost exact agreement with the target number and perhaps it's not got the kind of connection with physics that, say, my construction for the door ultimately had basically no relationship with physics. Yeah, I think I see the question. I mean, this is a much more subtle question than the thought that I usually get about this. Which, because they're usually of the sort, well look here, 137 is h cross over e squared, if it takes c equals 1. But you haven't got any h cross or e squared in it, so how do you know there's any relation? That question is, I think it's an easy one to answer, by saying, oh no, no, no.

1:17:30 the expression of it as H cross over E squared comes up much later you've got to have a whole theory of electromagnetism and at least the old quantum theory or something to introduce H cross and you've got to bring all those in and then at that late stage you recognise that the thing we're talking about can be expressed as H cross over E squared well, this is a basic basic quantity which will come in any theory which starts from a quantum mechanical thing so, this is the usual mathematician's trick of not answering the question which I asked but one which he can answer now you're saying, how can I be sure that there wasn't some Well, I think the only thing I can say about that, I can point to the fact that there aren't any arbitrary choices in the construction. I mean, obviously there must be choices, because any construction would be like that. But these don't have any arbitrary character. they are just forced on one by realising that in the first step which got the point 033 just comes by thinking well, but you only get the 137 by having this constraint applied and yet you can't really apply constraints so what is the probability that without the constraint it will be like this it just happens to be like that and so that's the most arbitrary character of the thing the other if I look to see if there were any other choices I could have done differently I think they are much less arbitrary than that one so I'm just forced from the analysis of the whole process

1:20:00 business to finding this number and then because of that I feel that to point to the accuracy of it is a sufficient justification to show this must be the number that these physicists have been talking about all this time of course I think there's one other thing I could argue in my favour and that is that although this is called the fine structure constant and originally rose in looking at the fine structure of the hydrant spectrum yet indeed it comes up in all sorts of other calculations in QED and things which don't bear any relation apparently don't bear any relation to that original one it's always turning up which suggests even without my argument that it's some kind of fundamental quantity in the theory if I just break off for a moment when old Whittaker published Eddington's posthumous book Fundamental Theory either in the preface or perhaps it was in an article at the time I'm not sure he gave an analogy to the sort of things Eddington did which I as a young man I thought well, what nonsense. But over the years, I've come to think it's rather a good analogy. His analogy was with the value of pi in geometry. That is that there's an, you know, oddly enough, the first thing I was ever taught at school in physics was to find the value of pi by being given a rubber bung and had to roll it along a piece of paper and measure the circumference and then measure the I suppose the point of doing this was not because the value of pi was anything of any importance but it showed me how a very difficult experiment was because the values were so wrong there was this number which came up in various contexts later mathematics in lots of other contexts. And then along comes Archimedes and shows, well, shows it's between 3 and a 7th and 3 and 1071s, but he could have gone on and on with sufficient energy

1:22:30 and found an exact value. So that, in Whittaker's analogy, this determination of the, Archimedes' determination of the value of pi which stood in the same relation to geometry that Eddington's determinations and numbers stood to physics that was his analogy at the time and I think why I come to be more generous to Whittaker than I was as a rather brash young man is that I think as far as the fine structure constant is concerned it's probably rather a good analogy there is this number which people are measuring by half a dozen different methods and they seem to be see the interesting thing is the experimenters who are doing it in different methods have the view that perhaps they're not all measuring the same thing whereas to me that just seems nonsense I'm sure they're all measuring the same thing, they're measuring this basic constant which is somehow in the structure of the theory. We don't understand quite why it's in all these different places, but it's the same. It must be the same thing. And so, of course, that explains why they are getting nearly the same value. Any other questions? Yeah, sorry. All of that. Thank you, Mike. You made me an ounce with pi. While we talk about this, another number came up. which was Q to K-R metric ratio for electrons. The reason I was thinking about this is, in this case, you know, you get to a stage where at first you think it's going to be 137 exactly, and then you realize no for a more subtle analysis as these extra terms are. And that's sort of similar to what happened with Q. Because at first it was just an experimentally discovered number. They knew it was close to two. And then Dirac comes up with the Dirac equation, and it has t equals exactly two. And that was great, that was wonderful for Dirac. But, of course, it's not exactly two, and you go to the full one QED, and you get the calculational method, which gives you, you know, with greater difficulty more and more.

1:25:00 And perhaps I can sort of entertain some kind of fantasy of some mathematician in the absence of QED, coming up with a calculation method which zeroes in on G, and then that gets, you know, becomes embedded in on a field theory. Is that the sort of thing that you're imagining, that this, you know, eventually I would think would get fleshed out as the basis of a fuller quantum theory, is that the idea? Well, I could imagine that happening, but rather unwillingly, as it were. because that's a constant which seems to me to be very tied up with the sort of actual structure of QED in a way which I feel the fine structure constant isn't, although of course it comes up all over QED but it somehow well just my way of thinking it seems to be I don't like using the word but I'd say it's a more fundamental constant I mean Eddington uses the word fundamental about things everything that he finds is fundamental and if any if one asked him to find something else he would have said oh it's not fundamental so of course I can't do that I just had the feeling that that wouldn't be But, I mean, if someone did it, I wouldn't be unduly sceptical. I just want to have a little look at the calculation. So, I think this is related to the, well, the fine structure constant is h bar over e square. And given that you haven't started with those, how come you end up with it? thought is more that we seem to be able to imagine coherently defined theories which end up giving us very different numbers for the thing we measure when we measure the final

1:27:30 structure constant. We can imagine the world sort of working, but these constants being in, you know, if we just tune some of the fundamental parameters in our theory. Um, so, I just, I suppose I want to know how you can, I mean, if, if there is this sort of, um, almost a priori derivation of it, then there's got to be something wrong with that thought. So what, what's to be said against something like that? I think what I think I think my response to that would be sure, there are lots of constants around whose values for some extent contingent they could be altered a little bit and we'd then be in another universe where they were a bit different And what I'm claiming here, I mean, what I'm forced to claim here, is that alpha is not one of them. So its value isn't contingent in any sense. It's necessary. Could I just follow that up, Clive? I know there's another question, but just on this point. I mean, presumably, I imagine you've been asked many times about, well, what about the next decimal? I mean, it would be a wonderful thing if you could actually predict. Well, yes, I mean, I stick by the 011 on there, so I have predicted. Right. And how difficult will it be? I mean, does it involve really greatly more sophisticated... I think I'm pretty safe for a number of years, I think. There's a space-type experiment which is probably going to be done next year. Oh, well. Where they expect to get another decimal place. On the fine structure constant. Uh-huh. Which experiment is that? I just can't remember the name of it. It's one of these Lagos-Saudi ones. Anyway, Dennis, do you know that? Do you think that a similar construction would work for the other construction constants as well, for example, for the fine structure constant of the strong nuclear force, and how would you then manage that it's a running?

1:30:00 Well, no, the running character doesn't worry me I mean, sure, the fine structure constant has running values as the energy increases and so on But there is still the original constant at low energies in terms of which the running happens So that's just life But about strong nuclear and so on Well, I'm going to be rather shifty about this but back in the 1960s and 70s we thought that these other figures of 3 and 10 looked pretty hopeful that way only of course those various nuclear forces change rather rapidly with time as different experimenters and also get differently defined by different experimenters sometimes you find a two pi has been slipped in by somebody when someone else hasn't but we did have hopes at that time but at the moment I don't see any way of doing a calculation like this for those things the reason for that is So if these values were correct, all I would be able to do would be to point to the one-third or one-tenth or something like that. Well, corrected a little bit by the necessity of keeping the thing down at the bottom or the first two levels, which would also value a little bit, but not much. but I couldn't go on from there because the whole basis of the accuracy of this construction came from realising that when you get up to the next level you may not have linearly independent things and then you've got to look at all the details well that doesn't happen at those lower levels so I could make certain predictions but I wouldn't feel much confidence in them about there being certain constants one would be a quarter

1:32:30 and the other one would be something of the order not very different from a tenth but a little bit different and if someone suddenly said oh that's just what we wanted I'd be pleased but I'm not going to put myself forward expounding those things I just lack any confidence in it. If you have less confidence in such a procedure for the strong coupling constraint and are very sure about the electromagnetic coupling constraint, aren't you then forced to detect that electromagnetism is fundamentally different from itself? Well, yes, I would say so, yes. I mean, there are other constants in nature that also seem to turn up all over the place. Would it be fair to say it's the velocity of light that C is a constant that appears all over the place? Yeah, sure. I need, in order to derive anything in this, as Ollie said, I think a priori way, and I don't mind, I mean, term of abuse but probably still is in some quarters but I'm happy to accept it in connection with this but in order to do such an a priori derivation I need the dimensionless constant so the speed of light I can't do that in fact I've taken it as one because of my training from the very beginning but would you are there other dimensionalist constants that have a similar kind of universal feel to them well Eddington thought there were because he thought well the gravitational one corresponding to this is of course one I haven't expended much effort on the corresponding calculation there because I'm not entirely clear how to do it and it wouldn't really be worth that much of my time at present

1:35:00 because the gravitational constant isn't known all that accurately. So, I mean, you could interpret that as meaning Well, he doesn't quite know the answer if he wants to get there, and so he doesn't want to make a fool of himself. But that would be another one. You see, when you look at what Eddington did, he relates that there are a number of other dimensionless constants, mostly involving a radius of the universe, cosmological constants of various kinds. And actually I'm a bit in two minds about those because I feel that statements which are made about the universe and so on have a real smell of contingency about them which the fine structure constant doesn't have. So, I'm a bit suspicious about whether one could find such things. Any other...? Oh, I'm sorry, Barry. This notion of the sameness of fundamental particles, models. It seems to me to be a big step forward in being able to anchor your process in something which is obviously much more physical than has been started from scratch. It makes one feel that this might be part of a much more general process. Now I think the standard model needs about 19 inputs to it. It's an enormous number of inputs. I'm not quite sure how many it is. So my sort of first question is, is it that if we just have to put in a value that's found by experiment, we don't really understand the theory? And do we understand the theory better if we can calculate one of these inputs.

1:37:30 I think that's really what I want. Oh, there's one other thing. I have just been recently reminded of string theory rather strongly. And it did ring a bell when you spoke about a single particle and it being the same as the next one of it, and so on. The nice thing about the string theorist approach is that, again, it starts from a single entity, namely one, you always have the same single string when you start. Yes, I know there are terrific opponents to string theory, either for it or against it, I think. But, yes, I think my real question is if we can calculate one of the inputs, or some of the inputs. Yes. I think this is really a question in psychology, because understanding is a difficult concept, isn't it? I would certainly think that we understood it better if we could calculate. To get back to the standard model, I should say that the standard model is one of the things that we've been contemplating in the last six months or so, trying to see how we could get a bit nearer to it what we would hope would be of course, that if we get nearer to it, it would be without the arbitrary without the I know it's, that's a bit As you say, there are 19 or some such number of arbitrary bits. But if one could find any of them, for me, that would give me more confidence in the standard model than I have at the moment. And we're fairly actively engaged on trying to get to it. I had a question about your claim that there should be, there's got to be at least one number that's logically prior to measurement. Oh, all I, yeah, sorry, I didn't mean to claim there has to be one, I just claimed that

1:40:00 I found one. Right, oh I see. So there weren't, it's just that you've got this route independent of measurement to arrive at a particular number, it's not reflecting on the nature of measurement? No, no. No, no. Right. Right. Okay. Can I ask a different question that connects to the comparing, or asking the question about other coupling constants. I mean, you might think, looking at the parameters in the standard model, what this shows is that two parameters which look to be independent parameters in the theory are in fact not independent. But given... If that's really the case, then given the similarity of... I mean, shouldn't you expect a similar relationship to hold between H-bar and another coupling constant? I mean, you would expect them to be similarly not independent, and so there would be a similar... I'd expect, for example, that H-bar divided by big G MPME would be... of the order of 10 to the 38 so it's one of the other numbers which I'm concerned about but then I would need to, I need to elaborate what I've done here a little bit in order to be able to analyse the way it occurs sufficiently closely so that I'd be able to produce the the correct value to number of decimal places I mean just that the the 10 to the 38 odd whatever it is might turn out when I've analyzed it to be the correct value but it might not be in the same way that 137 wasn't the correct value so I'd actually need to do the analysis and I'm not actually quite clear what the nature of that analysis would be yet because I haven't tried it

1:42:30 but I mean it's not obvious to me it wouldn't be an exact reproduction of the analysis that's carried out in this case I'm not quite sure if I've understood correctly the philosophy behind the approach then you're not looking too closely for a mechanism to link these numbers to the places in the theory in which they appear. They are just fundamental. And you've got the link with a particle, and these particles that are identical. And apart from that, there doesn't seem to be anything much more connected to an electron. Given that, so if we did find, say, 19 numbers of similar type, why should I plug them into the standard model in the right way? 19 factorial or 19... And why should I connect this one to the fine structure constants in the first place and why shouldn't I connect it to something else? Right, to take the second part first This number turns up in a very basic way in this analysis And so one would expect to connect it with some other If it's to be connected with physics it would have to be to some number which has the same kind of basic character and the fine structure constant has got that character turning up in all sorts of different ways and always there it is as it were but I do rest a lot of the argument on the accuracy of the numerical value that the calculation has now to go back to your Is that all right about the fine structure constant? Would you be equally happy if the number you've got after your basic analysis had turned out to be pi times 137.06.001? Ah, yes, I see.

1:45:00 Or even pi squared pi to the power of 3. Ah, well, quite. Now, I see the point there. I mean I would have been less happy with pi squared and very unhappy if I had all sorts of things in I see now I think I see though a point here which has troubled me in the past not in my own work so much because we get convinced that you're right of course if I could go back to 1938, where Dirac published this curious paper in the Royal Society which said, oh, well, all the important physical constants are in one of three classes. Well, he addressed the material up a bit first. I think he said if one of these constants, like alpha for example any constant less than 1 you replace by its reciprocal so then you've got lots of constants none of which are less than 1 and then he said they occur in three groups there are those of order 1 and those of order 10 to 38 and then there's those of order 10 to 79 of order 1 1836 and quite big numbers like that but still there is a grouping of this kind and that certainly is what one observes now what always puzzled me about Dirac's paper and why I thought well if he hadn't been a fellow of the Royal Society already they never would have published it was that these numbers are just the ones that happen to come up in the way that one's done the theory and you feel, well isn't it odd one would have said to Dirac why do you group them like that suppose we had rewritten I don't know, rewritten electromagnetism in which we had redefined the constants

1:47:30 so that instead of these particular ones we took their logarithms or something like that we could alter by redefining the constants in various ways we could alter, we could make them come outside the group in some way so it's strange that the numbers which just happen to have come up should be grouped in this way now I find myself in exactly the same position that I felt Dirac was in then because I quite agree that there is something mysterious about the fact that I found the fine structure constant very accurately and I didn't find double the value and have to divide by two or anything like that. Now why is this? I don't think I can answer this it's just that when you asked it in that form it reminded me of my lost youth and what I thought about Dirac back in the 50s when I read his 1938 paper it seems that certain numbers do occur in a particular form and somehow it would I suppose all one could say is it would be very unnatural to have developed the theory so that these numbers turned up in a very different way and I'd qualify that certain numbers by saying well I'd apply it to one number anyway I'm not sure if I've asked the same question, or is the same question already If alpha is so strictly fixed, even over the space of possible universes, doesn't it then necessarily follow that the constants which are normally taken to build alpha are then correlated necessarily as well

1:50:00 so it would be that H-bar and C that H-bar couldn't have another value if C wouldn't have another value as well so it would be a much stronger connection between relativity and quantum mechanics than normal Yeah, I would say that. Certainly, yes. Well, there are three things involved here. There's not only C, but there's also E squared. I tend, because of my upbringing, to take C as one anyway. and so if I do that then that directs my attention to the other two and that would say there is a close a very close correlation between quantum mechanics and electromagnetic theory any other questions or queries? well I don't know about the rest of you At the beginning of this talk, I gave a very, very tiny probability to the claim that Eddington should be rehabilitated. And as a result of this talk, I feel that probability has increased. And I can give an exact figure for the factor. I've multiplied the probability by it, and it's 137.0360. But anyway, thank you very much, Guy. Thank you very much.