[This is a transcription of a book I took babysitting when we belonged to a babysitting circle; the entries are dated from 1981 to 1990. I filled it with night thoughts.] -------------------------------------------------------------------------- 3/1/81 It is a truism that many of the great stories concern a quest: one thinks of the Holy Grail, Monkey, the Lord of the Rings, and the case is sufficiently proved. And what could better round off a work of fiction than bringing a quest to a successful connclusion, unless it be another chapter or two describing the fruits of that success? Some real-life quests of modern times include Peter Mattheisen's journey to the Crystal Mountain in Nepal, and Robyn Davidson's trek with camels from Alice Springs to the Indian Ocean. (Include John Hillaby's several journeys too, if you will.) The most obvious difference between these and the classic tales is that the adventurers' accounts end on a depressed note, despite the achievement in most cases of the objective (though Peter Mattheisen didn't actually see a snow leopard, and John Hillaby didn't walk the Appalachian Trail). Why? On looking at it more deeply, the question reveals many levels. The quests I have cited resemble their fictional counterparts, at a deeper level, more closely than they resemble attempts to travel down the Ganges by hovercraft or break the world water speed record: the real goal is spiritual, not physical at all. In the stories, this is confused by the fact that physical objects (the Grail, the Ring, the Scriptures) have spiritual power, a circumstance which apparently cannot happen in the real world as it is now. Bit the real function of these objects is as symbols, as Mattheisen's snow leopard is a symbol. In his case, he seeks satori, or enlightenment; for the others, the goal is not so easily categorised, though no less powerful for that. Davidson says, as she begins her adventure, that she really wants to bring off something she has set her heart on, but she discovers on the way that there is more to it than this. Then one gets the uneasy feeling that the analytical knife has been applied in the wrong place. What distinguishes these quests from the others is that they use unaided human muscle rather than the internal combustion engine, surely? In each case, the hero has an introspective nature, and is driven slightly potty by the solitude and hardship of the journey, surely? Or the individual cases are too dissimilar for any generalisation -- Davidson didn't attain her quest because she didn't know what she was seeking; Mattheisen knew well enough (in words) and still failed. Or, Mattheisen got on with his partner, George Schaller (a comrade, but quite remote), but hs own ego beguiled him wiith pictures of him sitting at the Lama's feet or trimming the butter lamps; while Davidson's solitude was destroyed by her involvement with her partner, Rick Smolan, and in a different way that with her dog and camels, so that the only undemanding travelling companion she found, Mister Eddy, called forth at first inappropriate responses. And yet ... Could it have been other than this? A successful book must be built of creative tension, and a person must be a considerable egotist to write a book about himself. If I understand the nature of the goal that these two in their separate journeys were seeking -- and as has been made clear I believe there is a common goal -- then it is only truly attained by extinction of the self. Sorrow springs from desire, and desire is desire for attachment; and any attachment, be it to camels, unexpected lovers, dead wives, or priests of alien religion in dark monasteries in the high Himalayas, is at bottom attachment to self, the self that, such as it could, loved them. And while it may be true that the physical hardships of travel, like fasting, may weaken the ego to the point where it can be annihilated, that doesn't seem to have happened in these cases. At least, I can't see that the moments regarded by the authors as most satisfactory followed close on the heels of thse of greatest privation or danger. There is another, more central, objection. The very idea of a quest is surely a self striving for an external goal; what could be more different, nay, opposed to ego extinction? On a quest, surely each minute, each footstep, has value only in bringing you nearer to the goal, while time spent otherwise than in travel is time wasted; and surely there must be an enormous anticlimax if the goal is attained, and frustration and regret if it is not. This throws the whole question into new relief. Why are so many of the heroic stories based on a quest, if this is so; and why does it seem so natural to us that a journey should be the setting for a search for enlightenment? Eliot, paraphrasing part of the Bhagavad-Gita, reminds us that while on a journey we should not think of past and future, we are not the person who departed nor yet the one who will arrive; we are told, not farewell, but fare forward. In the same passage Eliot makes it clear that the ultimate destination is death. Perhaps it is no coincidence that death is our most powerful symbol for the extinction of self. Again, Somerset Maugham, in The Razor's Edge, describes his hero setting off on a pilgrimage, certain that he is fitter and stronger than his companions and bound to reach the destination. He does not. It is not because his faith is less strong; he is so concerned with his destination, while they are concerned only with putting one foot in front of the other. Having posed the question, I am at a loss to give it an answer. Clues abound. The Buddha's enlightenment can be seen as the successful conclusion of his quest for an answer to the questions of sickness, injury and death; but it can also be seen as the commencement of his ministry, whic continued for more years than all his life up to that moment. Again, if we are to use death as our adviser, as so many teachers tell us, then it becomes not the ultimate destination but a thing of the moment, or even closer, the shadow at the climax of The Hollow Men. Perhaps the death urged on us by these teachers is to be equated inn some sense with the non-ego stressed by others. This would mean that, for the quest to be successful, the goal is attained not once, but at every moment, at every step along the way -- and this begins to look very much like what some thologians have to say about the death of Christ (there's that word again, closing the circle) -- ; well, perhaps not every step, not for us mere mortals. In the words of Mr Goad's lesson to Thetis Blacker, eternity is always now, but now isn't always eternity. Comments: The piece has reached its end now. I had no idea when I started it that it would be about death. One could draw more closely a parallel between the lives of Jesus and the Buddha. Each of them, while in his early thirties, brought to a successful conclusion his quest; in each case the goal of the quest was to overcome death. But the story of Jesus is like one of the classic fictional tales: a couple of chapters to show the fruits of success, and then the story closes. For the Buddha, however, the remarks made earlier apply. One could also make the passage from The Dry Salvages more central: the greatest Christian poet of this century interpreting the words of a classical Hindu poem. East meets West, indeed. If the piece were to be reworked into a more public form, a little more background about the journeys of the two modern adventurers should be given, but much of the discussion of detailed reasons why they failed could be left out. If you succeed in your quest, you die; but if you die, you don't necessarily succeed in your quest. -------------------------------------------------------------------------- 10/1/81 In Baltimore Zoo there were three polar bears Twenty-five years of age Contented they lived their declining years Shambling round their cage. The Department inspector came snooping one day Said Them bears gotta go or else you must pay For a new filtration machine A hundred and twenty-five grand There's too many bugs in the water But it comes from the Baltimore water supply The water that you and I drink (Chorus) Red tape red tape it's killing us all (x 2) A workman was buried while digging a hole The fire brigade were called They dug for three hours and they got him out whole He was alive and that's all The Government man sued the fire brigade You've broken the law and now you must pay Your men weren't correctly equipped No metal to metal screw jack shorings While working below five feet You should have just left him to die in the hole It's cheaper than messing with us (Chorus) A ski ranch was built in the Rockies so high And winter is when you can go You get there by chairlift there's no other way It's buried in twelve feet of snow The Government man came a-calling to say You can't have a licence to run it today You haven't got wheelchair toilets I haven't got wheelchair toilets??? You haven't got wheelchair toilets. A man in a wheelchair who wanted to ski Jusst wouldn't be able to go (Chorus) -------------------------------------------------------------------------- 17/1/81 A trailer for a radio programme on sleep this evening mentioned a scientist who based his theories on study of that small minority of people who need an hour or less of sleep a day. Very properly, he was criticised by his colleagues for arguing from such special cases to the general. And yet, wouldn't it be nice to be one of that minority? As Sheila said, it would be like living two lives at once. All one would miss out on would be dreams; maybe not even that. It is hard for us to admit that someone else can do something that is in principle impossible for us. Far more reassuring if the difference is only one of degree, and if it is possible to imagine some kind of training programme that would bring us to a comparable level. So is it true that the average human could be trained to exist without sleep? If so (and it seems very likely to me), this is surely an instance of a very much more general phenomenon, the enormous unused and unsuspected potential that people have. There are so many instances of this: people who perform superhuman feats in emergencies or in war; the amazing bodily control, levitation, etc., displayed by the cheaper Indian holy men (and the fact that one culture can produce such a relatively high proportion of these people suggests that a corresponding proportion in any culture would be capable of such feats); the steadily declining athletics records; and so on. Just this week I've been very closely involved with a less dramatic example of the same thing, when Jeanne (who could easily pass herself off as an approaching-middle-aged suburban housewife) suddenly wrote four or five very good songs. (And, for the record, whatever may or may not have been her inspiration, Sheila, on first hearing the first of them, thought it was one of mine.) This seems to lead this tangled account to its next topic, creativity. I have said, too often perhaps, that creativity at its best is a property of the unconscious; that it ticks over, or cooks, or whatever is the correct metaphor, until at the appropriate time it is ready. I still believe that's basically right, but there is more to it. For a start, even when the alarm goes off, or the dish is ready to serve, it doesn't often appear of its own accord. What happens, rather, is that one sits down with a blank sheet of paper, and whatever it is, loosened up by some preliminary doodling, pours out onto the page; under similar circumstances earlier, it had not come out. This isn't a big obstacle, but the next is more serious. It appears more clearly in songwriting than in my favourite example, theorem-proving; for songwriting is more influenced by external events. Like John Lennon, bits of my songs come from newspaper articles (though the process is more wholesale with me). The article is read, and the song comes, usually immediately and quickly. Now it isn't possible that the song had been cooking for months. My explanation of this is not easy to explain, perhaps not possible. What cooks is not the song or theorem itself but some kind of vaguely Chomskian "deep structure" corresponding to it, but something which can still be nudged into different forms by external circumstances. Now we are getting towards some very exciting ideas. I have believed for some time, and tried to put in words to Jaap, that these ideas can reconcile the conflicting views that mathematics is discovered or invented, and also explain why some mathematics is more "significant" than other. This is a Chomskian interpretation of the cliche that "mathematics is a language". The language in question is rather a multiplicity of languages, analogous to natural languages, derived from a common "deep structure" which is a property of humankind. We do mathematics the way we do, we discover facts, we find that old theorems remain true although ideas of proofs change, we observe the fantastic correspondences between parts of the subject that have grown up quite independently, because of this deep structure; its manifestations must be in some rather vague sense "isomorphic". But I run the risk here of boxing off mathematics separate from other forms of human creativity, or more generally of human endeavour. While this may be valid for some kinds of scientific enquiry, it gets in the way of action in the real world. For instance, it must be granted that I am reasonably efficient at encouraging mathematical creativity. Surely, the same general processes can work for other purposes too. Thus, I should be able to write good songs, by making use of these processes. While I was at school, I once debated with Brian Blinco the proposition that there are specific talents, which he denied. I asked whether a certain forward in the First XV could have played in the First XI if he had set his mind to it; a completely impossible circumstance I thought, but Brian claimed that he certainly could. This was a koan which I have not yet fully resolved. -------------------------------------------------------------------------- 31/1/81 How should I best adapt to the fact that I have very little free time, and use that time to improve my writing (assuming that this is what I want to do)? Two trains of thought pull different ways. On the one hand, I write best when in unusual situations, out of the everyday rut, and by this I don't mean only that I am inspired by the exaltation of the soul this produces, but also that the mechanical process of writing proceeds most smoothly under these circumstances as well. Witness my completing my novel in Sydney -- not that it was satisfactory, but I couldn't do it at all at home -- and the account of my Norfolk trip, for two examples of this. Indeed, the mere fact of writing that paragraph has made me more convinced of its truth; it was included originally merely for contact with what follows, bt now I wonder if the truth may be the other way round. The other factor is that, of course, practice makes perfect. Theories of creativity that I have often rehearsed, parts of which come from various sources, say that a great artist must be so much master of his instrument or medium that his thoughts are transmitted instantly and faithfully to his fingers without being filtered through the barrier of inadequate technique -- nay, more, there should be no differentiation at all between thought and execution, no mind-body duality. There's a problem to come back to here. But first, how best to practise? It's clear that one can't live on the heights; they would quickly become ruts. In any lifestyle, however free it first appears, most of what is done is habit, be it habit for what wasn't done yesterday, or "letting the dice decide", or whatever. So the first argument doesn't suggest a way to live, unless it be grasping at any opportunity that arises. Then, more than ever, it would seem necessary to be perpetually prepared, as well as alert; to be in practice all the time. But what is the best way to practise? Is it to write every day about the day's events, however mundane, as Sheila does, perhaps investing the mundane with a touch of magic? Or is it to write on set topics, as undergraduates in arts subjects do? Or to write anything that comes to mind, as I seem to be doing here? The middle one of the three would seem best, though it differs frrom the first only in the increased variety, and from the third only by separating the choice of topic from the writing. (In fact, I had given some thought to the first entry in this book before I wrote it, and the second was a pure exercise, writing a song on a topic selected from the newspaper, as I did in 1975 with "Snowy Owls".) For me, the choice of subject or plot is at least as difficult as the actual writing, and perhaps I need practice at that. On the surface, it would seem that this is just what Sheila's method wouldn't give; a little deeper, it would give practice in selecting subjects from the milieu of day-to-day. But of course they would be based in fact, not invented; and invention is the hardest thing for me to do. I am so tied to "reality". Incidentally, earlier this week I read the things I wrote in Australia in 1971-2 and in Oxford shortly thereafter. A good example of the best I can hope for with this method: all fantasies, but all based in reality (such as Lake Annand and the Old Toll Bar, Beerwah, the rock pools at Alexandra Headland, and so on.) Perhaps that is the right sort of practice for me. I should to back and say something about that thread I left dangling. To put it in the context of guitar-playing: how do you distinguish between the master who plays as I describe, and the person who knows a few set patterns and has the ability to throw them in at the right place? I find it invidious that there should be a clear distinction. Yet Idries Shah, in his story of the prayerful man, clearly says that there is a difference, one which only someone who has reached the master's level can appreciate. I was worried by something similar that Alan Watts says, in The Way of Zen, on distinguishing true enlightenment from affected buffoonery. Again he seems in no doubt that there is a clear difference. This seems to me fraught with danger. If the beginner cannot tell the difference, there is the risk that he will go to the charlatan for instruction. Up to the point where he receives his own enlightenment, he won't be able to see the charlatan for what he is, and so will proceed just as well on his teaching as on that of the master. Under the circumstances, there seems no doubt that he is just as likely to attain enlightenment under the charlatan; so perhaps the man isn't really such a charlatan after all. In fact, I would say that, just as there is far more in a good novel than its author deliberately put there, so no teacher can consciously foresee the outcome of all his teaching; he should certainly take no more credit for his pupils than the author for his books. In any case, perhaps this month's several events have given me enough of a jolt to take writing seriously. One long-standing exercise was to be my Sgt Pepper story. That will be a little different; it will require some forward planning before I sit down to write it. But perhaps it will get done this year. -------------------------------------------------------------------------- 31/1/81 Won't you come for a ride in my aeroplane If you have some time to spare We'll get a good view if it doesn't rain There's plenty to see from up there Come see my world, it's all I've got The oceans and seas so blue The mountains so high the deserts so hot I offer it all to you I offer it all to you my love That's all that I can do Whatever I have that's mine to give I offer it all to you Won't you take a peep in my crystal ball If you've nothing else to see Come look through the hole in the garden wall And see what might come to be If what you see is dark or deep Or if it's here and now If it makes you smile or makes you weep I offer it all to you I offer it all to you my love That's all that I can do Whatever I have that's mine to give I offer it all to you Won't you come upstairs to my tiny room For coffee or maybe some wine Forgive the clutter, I haven't a broom There isn't much, but it's mine My heart and soul hang on the wall And all I hold as true And if you think you could like me at all I offer it all to you I offer it all to you my love That's all that I can do Whatever I have that's mine to give I offer it all to you ---------- [I had thought of making the last verse something along the lines of "Tiptoe upstairs and see your baby".] -------------------------------------------------------------------------- 8/2/81 Sometimes songs don't come. How like mathematics the creation of songs is! For many years there was no great incentive to write songs. Nobody would perform them but me, and I know that nobody would come to hear the sound of my voice. Then starting up with Jeanne and David brought the first new song in quite a long time, "I wish that I were young again" (or whatever it's called). In two ways it was Sheila who called it forth. First, she saw first that we should be doing our own numbers; second, she specifically asked me to write a song with a simple melody (and I don't remember for certain, but she may even have proposed the subject). I wrote the first verse in the shower after an evening's singing; the melody, however, refused to be bound by her request. The rest of the song came later. It was this, much more than "Green River Wyoming", that inspired or challenged Jeanne into her frenzy of creativity. Of course, one could have foreseen the outcome. Her songs, the product of one steeped in the Radio 2 morning programme, were far more commercial than mine, and suited her voice better as well, so that of our first tape (to which we each contributed five) it was clear that hers were to be the ones we should use to sell ourselves. Then the other business happened and brought the enterprise to an end. The relevant point is that it threw me from a posture of striving after more commercial songs right back into the position of not needing to write songs at all, with the added disincentive that listening to the tape could always stir up the old feeling that I had been cheated out of something. All I have is this: Figure with guitar Haunts me now Figure with guitar -------------------------------------------------------------------------- 11/2/81 An image from a walk round Christ Church Meadow last week. The sun was low and a heavy bank of clouds above the horizon left a slit which was illuminated to a golden glow along the western rim. On the river, rews trained for torpids, chopping the water into tiny ripples. I stood for some time on the bank watching the ripples race across towards me. As each reflected its perfect circle of the world, it was ringed about with a golden line like a neon light; these lines swelled and shrank to extinction in the play of ripples. Allusions and interpretations jostl in the mind. The Hindu view of the world as the play of the void; the many references to reflections and mirrors, to wind ruffling the surface of a lake, in esoteric literature. Rather more important to me was the simple fact that I could react to such a sight at all. This time last year, when I walked round the meadow on a day of brightness and high cloud, I felt quite empty. -------------------------------------------------------------------------- 13/2/81 (the first of three Friday the Thirteenths this year) How difficult to be a vicar, who must preach a new sermon every Sunday. Here am I, completely lost for inspiration of that kind after just a few weeks. As I run out of things of a universal kind to say, my thoughts invariably turn back towards the centre; I could write endlessly about myself. Then again, the same comments as I've made about songwriting and mathematics could surely apply to preaching as well, or even writing in a book like this. Nobody is stuck with a fixed amount of creativity which must be eked out to last him a lifetime; or more precisely, our potential is so much greater than we ever realise, that any sort of practice aimed at increasing our fluency is bound to increase our output as well. -------------------------------------------------------------------------- 20/2/81 Change: how is it to be brought about? The only real change in my mathematical life in the last seven years has been my foray into infinite permutation groups. Lately I've found myself wondering whether I shouldn't restrict myself to another such mathematical redirection rather than something more drastic like becoming a pop star or having an affair. So, how did it happen last time? The first work on highly homogeneous groups was done at about the same time that I wrote "Parallelisms". Thus, the logical course now would seem to be to write a monograph on infinite permutation groups. Of course, there is the time factor: I would haev to do it without the luxury of two hours a day spent in undistracted serenity on a train. I guess the answer must be to find the odd half-hours before lunch, between tutorials, etc.; just the times when I've often promised myself I'd get work done (and did indeed write part of my novel, the year before last). The other factor would be John McDermott, if I could talk him into collaborating; he might be willing to do some of the stuff I know least well, like finitary and cofinitary grous, and perhaps wreath products. Writing a book has another valuable spin-off. It thrusts under your nose questions which you have never considered before, an answer to which is needed for tidiness in exposition. In this case, the first such questions have already surfaced, concerning normal subgroups of multiply transitive groups. Here is just one. A k-transitive group can have a sharply (k-1)-transitive normal subgroup for k=2 or k=3, but not for k at least 5; what about k=4? At present my proposed layout goes roughly like this: 1. Introduction (definitions, coset spaces) 2. Primitive groups (and related concepts: strongly primitive, strong, generously transitive) 3. Multiply transitive groups (just examples) 4. Normal subgroups of multiply transitive groups 5. Multiply primitive groups (and related concepts, k-set primitive and k-Steiner primitive groups) 6. Multiply homogeneous groups (my theorem and examples) 7. Groups with finitely many orbits on k-sets (basic results, Macpherson's theorem, examples and questions) 8. Imprimitive groups and wreath products 9. Finitary permutation groups (Neumann's results, and Wielandt's version of Jordan) 10. Cofinitary permutation groups (examples, Hall and Yoshizawa, Frobenius and Zassenhaus groups) 11. Ordered permutation groups (following Holland and Glass; speculation about similar theories elsewhere) -------------------------------------------------------------------------- 23/2/81 Two clocks on the wall, like the chronometers in Slessor's poem. A busy cuckoo clock, more than five minutes fast, achieving almost three ticks to every two of the other, its weight visibly descending when it announces the hour; and an elegant clock in a rectangular case, its pendulum large and deliberate and protected from the vagaries of draughts by a glass door, less than a minute slow, announcing the hour in booming tones. I jumped when I heard it first, imagining myself not alone. Since I've given them my attention, the ticks have become obsessive. I've tried them three against two, five against four; there is no easy fit. Each count gives a shifting rhythm that appears as a series of straitjackets rather than a continuous flow. So our time on earth is measured out. It is easy to see the approximate rhythms. In my own life, there is a seven-year cycle: pre-school; primary school; secondary school and university; research; teaching. Of course the fit isn't exact. One can draw the boundaries differently, say by including London in the fourth period, or use different criteria, say defining the third period by running, or the start of the fifth by Hester's birth. All these tempt one to say, at such and such a moment, my life of carefree irresponsibility came to an end, or, everything was different when I came to England. It is less easy to see these moments as really insignificant, the ticks of one or other clock in a slow evolution of a pattern of life, nothing more. Perhaps, again, the real significancce attaches to other things again, things whose importance is lost on us at the time they happen. Most likely none of these views is quite right. For a complex analytic function, knowledge of the values of it and its derivatives at a single point determines it throughout its region of definition. One might say that time past and time future are conntained in each present moment. Of course we actually model time as one-dimensional, and real functions have no such nice properties unless we make the somewhat articifial assumption of analyticity, which in this case amounts to assuming exactly what we are trying to infer. However, Roger Penrose and others would have us believe that our real world is simply the real part of a complex analytic manifold, and its physical laws are forced by the complex analytic structure of this manifold; this obviously overcomes the above objection. This theory is clearly no more than an analogy, but it has some pretty features; especially the motivating one that all change is reflected in the present, and the spatial equivalent that I can know the universe without moving from this spot. Perhaps its worst disadvantage, one it shares with almost all mathematical models of the world, is that it makes things too obvious, thereby hiding more subtle features of the truth. -------------------------------------------------------------------------- 1/3/81 The book will require some thought, especially regarding scale. How far back to go? This is especially relevant to the question of examples, which will take up a great part of it anyway. Thus, for example, Tits' examples require a little knowledge of free groups, McDonough's a little more, Schleiermacher's yet more -- does one take this knowledge for granted, or put in an introduction to free groups? If the former, the book will make heavy demands on readers; but if the latter, the the same choice must be faced again when you come, for example, to Adian's periodic products, and now if you put in an introduction you double the length of the book while if you leave it out you are guilty of gross inconsistency. Perhaps it is necessary to decide which things can be explained in a reasonable amount of time, and explain them, and relegate the others to starred sections. Great problems too in deciding how much to put in about imprimitive groups. In the finite case, any transitive group is contained in a finite wreath product of primitive groups. The same holds for infinite groups if both maximal and minimal blocks exist, e.g. finitary groups. In some special cases, the blocks through a point form a chain -- e.g. ordered groups, if "blocks" means "convex blocks". But in general, one needs the wreath product over an arbitrary partially ordered set to embed the group. Is it worth the trouble? Nevertheless, I have to discipline myself to work through all this stuff to decide whether to include, condense or omit each piece. The beneficial effects continue. The question noted on 20/2/81 has a negative answer, with quite an attractive proof. I suspect that the question "Does an imprimitive normal subgroup of a 2-transitive group necessarily contain a regular normal subgroup?" has a negative answer, and that a counterexample could be found by constructing a free resolvable Steiner system with automorphisms, as I already did in my construction of free Steiner systems with automorphisms to get my first provably k-transitive but not (k+1)-transitive groups. However, I'm not too convinced about this -- it would appear to be a "free structure" which was forced to satisfy a "structural restriction". (The normal subgroup would possess minimal blocks ad so would have all 2-point stabilisers trivial.) One of the most exciting features of infinite multiply-transitive (and other) groups compared to the finite case is that the borderline between true and false is so much more subtle that one never knows which way to jump. I once rashly conjectured that a k-homogeneous group not acting on a linear or circular order must be k-transitive. This is false; but it must be (k/2)-transitive, and I still conjecture that it must be (k-1)-transitive. Then again, Dugald Macpherson has shown that if a primitive group is not highly homogeneous the the number of orbits on k-sets grows faster than polynomially; but must it grow exponentially (as is the case in all the examples I know)? Come to that, is there any upper bound on growth rate? (Obviously not directly, but prhaps in terms of the recursive function theorists' hierarchy.) There are lots of unexplored questions in this area: how do operations like wreathing with a finite group (above or below) affect growth rates? Then there is the ring theory question, is e prime in A^G, or is A^G an integral domain, given that G has no finite orbits? If so, is A^G a polynomial ring over the rationals? How many elements of each degree do you need to put into a polynomial basis? It seems to be that if there are m_i elements of degree i, the generating function for the numbers of orbits on k-sets should be the product of (1-x^i)^{m_i}. It would seem that the exponential growth rate for the n's would imply that the m's must also be growing (e.g. if all m_i=1 then n_k is the partition function.) Question: Can one deduce the m's from the n's? By a nice formula? (For local orders one gets 1,0,1,0,2,1,4,4,... How clear it is that I find writing mathematics in this book so much easier than writing the secular sermons with which I started out. Do growth rates offer an apparent escape from change and decay? Of course the answer must be looked for in a much wider context. Why do mathematics at all? But nothing I've read, and nothing I've found while meditating, has suggested that I shouldn't do it; and stuff such as I've just described has such an obvious appeal! Footnote. I should add to the last mathematical question: What does the condition that the m's are non-negative mean for the n's? Does classical commutative algebra give answers to these questions? -------------------------------------------------------------------------- 4/3/81 A lesson in the way such things work. After writing down the formula in the last entry concerning the relationship between the number of generators of each degree of a polynomial algebra and the dimensions of its homogeneous components, I was looking in the enumeration chapter of Beineke and Wilson to see whether there was a formula of any sort for the number of graphs with a given number of edges. Not finding one, I looked through the rest of the chapter. There was exactly the same formula, relating the number of connected graphs to the nummber of all graphs, and again, relating the number of trees to the number of forets. The first of these is directly relevant, of course, but in any case there seemed to be a general principle here. So it's a simple matter of taking, in the graph case, the characteristic functions of connected graphs as generators. Then an arbitrary graph is built out of connected graphs just as a monomial is built out of its factors, so there are the same number of monomials of degree k and graphs on k vertices; thus it is enough to prove the monomials independent. A lovely trick does this. Partially order the graphs by innvolvement, and extend this to a total order; it's easy to see that the resulting matrix is lower triangular, with non-zero diagonal entries. The same works in more general situations. One needs two things: a notion of connectedness, so that any structure is uniquely the disjoint union of connected components; and a notion of involvement, a partial order with the property that, if a structure is partitioned in any way, it involves the direct sum of the induced substructures on the parts. This handles the homogeneous k-uniform hypergraph corresponding to the collection of all finite k-uniform hypergraphs, and also all of Lachlan and Woodrow's homogeneous graphs except the disjoint union of finitely many complete graphs, which are easily dealt with otherwise (up to complementation). For a symmetric group on k-sets, the structures are k-uniform hypergraphs with a fixed number of edges. Here connectedness works in the same way, but involvement requires modification: since any two graphs hae the same number of edges, we put one less than another if the partition of the number of edges induced by the components of the second refines that of the first; within each partition, the order is immaterial. THe one I'd really like to do now is the automorphism group of the countable dene local order. Here one has two features -- a comparatively slow growth rate, and a fairly simple formula for n_k. It might be possible to get a formula for m_k, assuming the algebra to be polynomial. I haven't sorted that out yet; I don't immediately see the right concept of connectedness (is it defined in terms of the equivalence relation using the colouring? Hardly, since in this case each equivalence class is totally ordered, while it is the set of classes that carries a non-trivial structure.) Perhaps an entirely new method is needed. Then again there are projective and affine spaces. One final remark. If A^G is polynomial, not only is e prime in A^G, but it is so in A^H for any supergroup H of G (even though A^H needn't be polynomial). -------------------------------------------------------------------------- 8/3/81 It's really magic how a picture works on you. We bought one from Jean Jones yesterday. We spent the best part of an hor at her house, up and down the stairs, backwards, forwards, and round, narrowing the choice down to two Oxford and two similar Dartmoor, finally taking the stormier Dartmoor. I wasn't sure it was better than the other; it seemed to need viewing from further away, when of course its impact was less. In any case, the constraints of hanging it force you to look at it across the width of a room. Now it's here, and this afternoon Sheila and I sat for quite a while looking at it. At first, large parts of it were just an impressionistic blur. Then, all of a sudden, the blur began taking form in a most incredible way. Rocks and bushes along the stream bed, small shadowed depressions on the hillside, fine detail of distant green, even the flow of water. Each time my eye went round the picture, another detail demanded my attention. Now it is for me a very clear and fine-wrought scene, and how I could ever have thought of it as a blur simply defines comprehension. One gets so much more out of any work of art by making it part of one's life, if only for a while. Music is the most portable; it fins its own way into your head, and you are powerless to turn it off. Paintings can be put up on the wall and absorbed painlessly when your defences are down. Words present the greatest problem. If you read a powerful novel, you may think about it, even dream about it, for a little while, but soon the details fade from mind. It is not good to read the novel many times; in fact, in almost all cases I've found it actually harmful. I don't kno what answer to take. This is a question of some importance, since there are novels, like "The Solid Mandala" and "The Glass Bead Game", which seem to me to deserve such treatment. Clearly haiku is the best verse-form for this treatment! Which leads on to another recent thought. It would be great fun, and I think rewarding, to play at writing linked verse with something of the spirit in which Go is played, and perhaps a dash of that of Finchley Central as well. Players alternate turns. There are no rules, but it is expected that each player's contribution will be two to four short lines, or thereabouts, and that it will be linked to the other player's last verse in some unspecified way. It is also expected that the verse will be composed quickly and written down without alterations or erasures. No points are awarded, but the writing of the verses should give pleasure like the handling of the Go pieces. -------------------------------------------------------------------------- 11/3/81 A shadow in the street -- is it coming or going? ---- Lamplight on foliage always takes me back to my student days. -------------------------------------------------------------------------- 18/3/81 A combination of end-of-term and the first sunshine for ages has the effect of making me cycle very much faster and more effortlessly. There's a little more than that; the blossom is out, but has only become noticeable since the sun has shone on it. It's ungentle March weather; the sunlight precarious above the banking clouds, wind coasting or driving, rain or worse always threatening. -------------------------------------------------------------------------- undated Back to babysitting -- and, one way and another, many of the entries in this book have been written while babysitting, at home or for someone in the circle. Thus, here I am again with the "figure with guitar" on the wall, reminding me of one particular song that went no further than the chorus, indeed actually went backward (it had an unsuccessful first verse when it was written). Yesterday evening, while writing the last piece during Sheila's absence, I transferred my songs onto a cassette, including (and especially) "Oblations", which I still think is super. But the history of the recent singing bout has largely been rooted in the past. Of the five of my songs on the JPD tape, three were written before I met Jeanne and David, one long enough ago to be on my own tape; a fourth involved just putting words to a guitar riff which also is quite old. In the same time, Jeanne has written siz songs and fragments of others. But that's not a fair comparison; I have two fragments, one song that isn't our style, and one that requires a male singer, written in the period. The real differences, though, are first that Jeanne is writing for herself, and will happily put past lovers, present fantasies, etc., in her songs; while I, if I write for us at all, write for someone who actively disapproves of what I consider the most important part of my style. Hence the emphasis throughout this book on the "five-finger exercise" aspect. There is more than an element of truth in saying that I can only succeed in the game by writing Jeanne's songs better than she does. But that isn't at all what I started out to say, if there was really any such thing. The general mood, I guess, had something to do with the pot plants, prints on the wall, goldfish, executive toys on the television, and all the rest of it. -------------------------------------------------------------------------- 27/4/81 In an attempt to settle the issue alluded to earlier in this book, I kept a diary of my American trip. Since the trip was so structured that my only free time was at bedtime (almost without exception), this meant starting while I was tired, and continuing as I drifted downhill into sleep; and it was quite noticeable, as I started writing each evening, how much neater my writing came out (then) than the last few lines of the previous evening. In the event, there was little to tell about the trip except the mathematics. The exception to this was my stopover in (and trip through) the Cascades. The place was chosen by chance; a combination of uease at news of eighteen inches of snow on the pass, irritation at missing spectacular scenery in the dark, and creeping tiredness after 22 hours awake, much of it spent travelling, led me to cross the bridge in Skykoming Junction and check into the Skylark Motel for the night. On waking early next morning I walked for two hours, out along the Old Cascade Highway, the turning off onto Miller River Road, then turning off up a track up a logged slope past a waterfall, writing my initials in the snow on a stump, then turning back. The details (views of snow-capped mountains hemming in the narrow alpine valley and, at the near point, lilies, yellow violets, flowers and birds of many kinds) are all in the diary, as is an account of the magnificent breakfast of juice, steak and eggs and hashbrowns, and coffee before reboarding the bus. Apart from this, there were just two brief bits of sightseeing. A trip along Snake River canyon was a bit disappointing, because of the pouring rain and the "taming" of the canyon for a dam, though the view from the top of Lewiston Grade was impressive. And across the country in New Jersey, George Washington Rock was a surprise, mainly because of the flatness of the land over which it looks. One thing the trip did was to confirm my current theory about coping with jet lag. The two rules are, first, don't sleep on the plane, and second, switch to destination time as soon as possible and keep to it. Thus, don't go to bed in mid-afternoon, stay up until bedtime and (if rule 1 has been followed) you will be tired enough to stand a chance of sleeping through until morning. Nothing in the way of musical (or even independent mathematical) thoughts emerged on the trip. Not too surprising. For the first week I was content to be lazy, to sit and accept input and not worry too much about processing it; for the second, I had to make the most of the opportunities and the people around me. Apart from working out the book in some detail, Bill and I cracked our heads on some very specific geometric problems; and I got Ron interested in colour schemes. -------------------------------------------------------------------------- 15/5/81 Using the ideas aout connectedness given before, one has the following staggering result. let G = H wr S, where S is an infinite symmetric group. Then A^G is a polynomial ring whose homogeneous generators are in 1-1 correspondence with the homogeneous basis elements of H. For the proof, one lets "connected" (for a G-orbt on k-sets) mean "contained in a single block", the "direct sum" of connected components be obtained by choosing them in distict blocks, and the "involvement" partial order is refinement of partitions, with orbits giving the same partition incomparable. It still staggers me that A^G is a polynomial ring no matter what bad behaviour A^H has! Now it is possible to write down the generating function for A^G (say N^+(x) = sum n^+_kx^k) in terms of that of H (N(x)= sum n_kx^k): one has N^+(x) = product (1-x^k)^{n_k}. Define N^(0)(x) = 1+x, N^(s+1)(x) = N^(s)+(x). Then N^(s)(x) is the generating function for the wreath product of s symmetric groups. We have n_k^(1) = 1 for all k, n_k^(2) = p(k) (the partition function), so we might call n_k^(s) the s-th iterated partition function. Now one can assert that if Omega has s infinite composition factors (as G-space), then n_k(G) >= n_k^(s). Question 1: How big is n_k^(s)? Question 2: The composition factors of a G-spacce are not, of course, uniquely determined; even the number of infinite ones is not determined. (In the real affine plane take flags -> points -> 1 and flags -> lines -> parallel classes -> 1.) But, for groups with finitely many orbits on k-sets for all k, is the number of infinite composition factors determined? (Of course, an answer to Question 1 would give a uniform upper bound for this number.) [Then follows some incorrect speculation about Question 1.] The answer to Question 2 is "no". Take an infinite-dimensional projective group over GF(2). We have point pairs (= flags) -> points -> 1 and point-pairs -> lines -> 1. -------------------------------------------------------------------------- 15/5/81 Autumn had squandered its riches until the gutters were choked with its husks The tears of an afternoon shower were hanging on threatening twigs in the dusk Refracting the dayglow to crystalline sparks Or fireflies piercing the gathering dark A ringing of footfalls marks A lone virgin Preoccupied In the desolate street. Autun had left me unmoved, had squandered its harvest of riches in vain My heart was hardened, I'd never be victim of romance and poetry again And waiting beside the moss-covered wall I heard the ringing of her footfalls And saw, beneath her shawl A lone virgin Preoccupied In the desolate street. A gaslight threw a pallid hue As she reached the hanging branch The vision struc, she reached to pluck The jewels from out of the tree And she was held in Autumn's spell And nobody saw it but me. Autumnal sorrow convulsed her then as the rain and her tears wet her face A desperate sob shook the universe, and me in my hiding-place And then, as suddenly as it came, The mood had passed, she was nce again A girl so proper and plain A lone virgin Preoccupied In the desolate street. -------------------------------------------------------------------------- 22/5/81 Of course, the biggest recent development for me, matematically, was finding my own path to Dugald Macpherson's theorem. My proof uses my own result on 4-hmogeneous but not 3-transitive groups in an essential way, and also some other more general facts: if G is k-transitive and a k-point stabiliser has only finitely many orbits, then the union of its finite orbits is a block of a Steinre system; and a subgroup of finite index in a 2-homogeneous group is primitive. In adddition, I've been going through problems for next week's seminar. Virtually nothing in the way of fresh developments, except the realisation that my question about whether the permutation a -> f(a) on the roots of the polynomial f^(n)(x)-x lies in the Galois group is nonsense. I still think there is something to be had in that question, but am much less sure what it is. If n=2 and deeg f = 2, then the discriminants of the two quadratics f(x)-x and (f^(2)(x)-x)/(f(x)-x) differ by 4. Why? In selecting the problems I tried to pick just those which could be explained briefly and were hopefully not impossibly difficult. I wondered about giving the car park problem at the start, and revealing the answer at the end; but I think not, some may have heard it, and others might really be distracted by it. I've always used problem sets to stop and evaluate my own directions. This one doesn't really serve that end. (I have felt the temptatio, more than once recently, to devote more energy to cleaning up comparatively small problems, but so far I've managed by and large to resist it. It will be observed that some of my collaborations deal with things of this kkind, while others are the most ground-breaking work I've done.) But perhaps it's not out of place to think about those wider questions again. A long-standing one has been to tackle harmonic analysis with a view to proving some kind of orbit theorem for the real projective plane. I will get down to this one day. Also, I think that establishing an exponential growth rate in Macpherson's theorem is well worth doing (and perhaps within reach now). My interest in the geometry of the exceptional groups of Lie type had a boost from Alan Sprague's theorem characterising geometries with two L's radiating from a point: wouldn't it be nice to characterise those with an arbitrary finite number of L's radiating from a point? (But this won't be true without something extra, since there are thin counterexamples; one needs some Tits-type simple connectedness. But so what? I wouldn't mind assuming that!) Ultrahomogeneous tournaments sound special, but I think that is a good problem. Use of inverse limits of Hjelmslev planes to get at structure (and maybe orbit theorems) for some infinite structures, e.g. planes over local rings. Slightly more generally, could it be that Wednesday's telephone call from Dana Scott is the intervention of fate that I had been looking for? Some time in a department of logic and computer science would be almost guaranteed to put me into a neew groove, and one which I would find very congenial. -------------------------------------------------------------------------- 6/6/81 (ch) We are the space invaders We are the space invaders We hope you'll welcome us with open arms We've been around the Galaxy And sampled all its wonders We are the space invaders We are the space invaders We hope you'll welcome us with open arms We'll tell you of the double moon Of Aldebaran Three That rises in a purple sky Above a crimson sea We'll tell you of the ring of fire Round Sirius' dark companion Where asteroids like icy hail Go hurtling to oblivion (ch) And you will show us all the joy And wonder of your planet The Himalayas, coral reefs, And Colorado canyon The polar ice, Sahara's dunes The forests and the cities Your planet in its glory and Its wonder and its pity (ch) (first five lines then) We are the space invaders We are the space invaders We hoped you'd welcome us with open arms But all you do is shoot us down To maximise your numbers -------------------------------------------------------------------------- 17/6/81 The recurring lesson of the end of Trinity Term is that survival is only possible through self-indulgence. This year it seems to mean spending four evenings in eighth week in front of the television watching "A Town like Alice" and not trying to get anything else done in the intervening time (having been let down again by the prospect of playing in front of an audience, having been deserted by Sheila almost every weeknight this fortnight, having been alarmed by my own reaction to a taxi driver who almost ran me down this morning, having got so much business waiting to be negotiated). A small self-indulgence to balance a great deal of external pressure. The equilibrium is only jut being kept. -------------------------------------------------------------------------- 26/6/81 Just to show how things change. I went for a run last Sunday. Felt great. On the grassy towpath I passed a man with a dog. For no reason at all, perhaps because he thought I should have waited on his pleasure, he began shouting and swearing at me. His dog was so big that I dared not go back that way. But I regarded him with equanimity. Compare the last entry. The weather was warm, the elders in bloom, so that you were never in doubt when you were approaching one. I had enough nettle stings that after a bath I tingled in spots all over. A weird feelinng but one containing a surprising amount of relaxation. -------------------------------------------------------------------------- 26/6/81 Rain stains my face Grass binds my feet Evening breezes chill me to the bone Surrounded by space No-one to meet Evening sees me running all alone Ever since my heart has turned to stone Today, today, today you've gone away You've gone, you've gone, you've left me here alone [And at this point it has become too complex. The first part of this was "Today, ..., away" with second line unspecified, perhaps a repeat of the first, and the third a very small image like "And the worm sleeps in the bitter apple", with variants of this in repeats. The second section was something like "You took away my memories/And dreams of yesterday" and then something referring to tomorrow ad leading back into today. Keep it simple!] -------------------------------------------------------------------------- 26/6/81 (ch) I'm sitting here doing my jigsaw puzzle all the while I'm thinking of you I'm sitting here getting it all together so the picture will look like new There's a piece of my heart gone forever missing since the day I lost it to you I'm sitting here doing my jigsaw puzzle all the while I'm thinking of you. Once I could see the whole big picture not a cloud was in my sky Then you came along and broke it to pieces with just oe flash of your eyes You had me down and you had me reeling picking them up off the floor A thousand pieces of my life you gave me and you had me begging for more (ch) You mixed me up right good and proper and now you're getting me straight Night after night we're putting it together and I can hardly wait For the day I ask you to complete the picture I'll put a ring on your hand And on that day I tell you all my puzzles will be solved by a wedding band (ch) -------------------------------------------------------------------------- 26/6/81 Hall and Knight or z + b + x = y + b + z by E. V. Rieu When he was young his cousins used to say of Mr Knight: "This boy will write an Algebra or looks as if he might." And sure enough, when Mr Knight had grown to be a man, He purchased pen and paper and an inkpot, and began. But he very soon discovered that he couldn't write at all, And his heart was filled with yearnings for a certain Mr Hall; Till, after many years of doubt, he sent his friend a card: "Have tried to write an Algebra but find it very hard." Now Mr Hall himself had tried to write a book for schools, But suffered from a handicap: he didn't know the rules. So when he heard from Mr Knight and understood his gist, He answered him by telegram: "Delighted to assist." So Mr Hall and Mr Knight they took a house together, And they worked away at algebra in any kind of weather, Determined not to give it up until they had evolved A problem so constructed that it never could be solved. "How hard it is," said Mr Knight, "To hide the fact from youth That x and y are equal: it is such an obvious truth!" "It is," said Mr Hall, "but if we gave a b to each, We'd put the problem well beyond our little victims' reach. Or are you anxious, Mr Knight, lest any boy should see The utter superfluity of this repeated b?" "I scarcely fear it", he replied and scratched his grizzled head, "But perhaps it would be safer if to b we added z." "A brilliant stroke!" said Hall and added z to either side; Then looked at his accomplice with a flush of happy pride. And Knight, he winked at Hall (a very pardonable lapse) And they printed off the Algebra and sold it to the chaps. (from "A Puffin Quartet of Poets") -------------------------------------------------------------------------- 6/7/81 I've now cracked the Rubik cube. It came about virtually by accident. We bought one for the children on Thursday, and of course the first thing that happened was that they played with it for a few minutes and then brought it alonng saying "Daddy, put it right." I had to stall. I've had one in my office for nearly a year, and had never got around to solving it. Perhaps I knew too much. I knew the monstrous order of the group, and the estimates for the numbers of moves required by good algorithms, and tended to think that only a superman could manage it. But the children of course would not be gainsaid. So over the weekend I spent some time playing with permutations on 48 ymbols until I could do it, in between the usual weekend activities of playing with the children, gardening, cooking Sunday lunch, swimming, singing, etc., etc. Now I can do it in 20 minutes or so and cook the children's tea at the same time. The secret of course is to aim for simplicity. My algorithm uses five ingredients: permutations of corner cubes with cycle structure 2^2 (two of these for good measure, but in fact I only need to use one); 3-cycles on the edge cubes; flips of two adjacent edge cubes; and complementary twists of two corner cubes at distance 2. These five are simple expressions involving commutators of generators -- only two types of commutators are required. For a while I was worried by the fact that I culd only twist corner cubes at distance 2. This meant, of course, that I might conceivably be left with two adjacent corner cubes twisted. Fortunately, on my first trial this is just what happened. I spent a little while looking for a new move to handle this, until it finally struck me that a simple conjugation would do it. As it is, it is now staggering just how easy the cube is! The only thing about which I'm not yet certain is whether, at the stage where I have to deal with the eight edge cubes below the top layer, it is best to compute within A_8 on paper or to use triial-and-error algorithms on the actual cube. I started with the first, but on the last try I used the second, and I think it may prove faster as I develop better intuitive feel for what will work. The same applies to the top face, where the somewhat tortuous algorithms used further down can probably be much improved by intuitive trial-and-error. -------------------------------------------------------------------------- undated Three invocations of the Lotus Sutra, with commentaries 1. Nam-myoho-renge-kyo Nam-myoho-renge-kyo is the expression of the ultimate truth of life. It also pronounces the essential reality of life. Nam derives from Sanskrit and means dedication, or the perfect relation of ne's own life with eternal truth. Similarly the word "religion", which was derived from Latin, originally meant to "bind strongly to something", and so it is also encompassed by the word nam. Yet the sigificance of nam is twofold. One is, as mentioned above, to dedicate one's own life to, or to become one with, the eternal truth of life. The other is to draw infinite energy from this source and take positive action towards relieving the suffering of others. What is the eternal truth that one can identify with one's own life? It is Myoho-renge-kyo, the title of the Lotus Sutra as it was translated into Chinese. Myoho literally means the mystic law. Myo (mystic) signifies "incomprehensible", and ho means "law". Myoho is the law which lies behind the incomprehensible realm of life. However, this is but one interpretation of Myoho. In another, myo indicates substance of eternal truth, and ho means all of the phenomena brought about by myo. In terms of the principle of ichinen sanzen (three thousand conditions of life within each moment of one's existence), the three thousand changing aspects correspond to ho, and the underlying reality (ichinen) of these changing aspects is myo. All existence at one time or another assumes a physical condition with shape, size and vital energy, and at other times assumes an incorporeal state (called ku in Buddhism). No matter how the fundamental reality may change, it is itself eternal. Phenomena (ho) are changeable, but deep within all phenomena there lies a constant reality. This reality is called myo. Renge means the lotus flower. Buddhism takes the lotus to explain the profound law of causality because the lotus produces its flowers and seeds at the same time. The lotus is therefore the symbol of simultaneous cause and effect. Simultaneous cause and effect means that essentially our future can be determined by present causes. Thus the law of cause and effect is also the principle of personal responsibility for one's own destiny. However, because the innermost depths of our life are independent of the karma accumulated by our past deeds, we can create true happiness, irrespective of karma. This is also represented by another quality of the lotus. Its pure blossos spring forth from a muddy swamp, yet they are undefiled by the mud. In other words, the innermost nature of our life remains untainted despite the evil causes we may have made. Renge thus means to reveal the most fundamental nature of the reality of life. Finally, kyo indicates sutras, or the teachings of a Buddha. In the broader sense, it includes the activities of all living beings and of all phenomena in the universe. The Chinese character for kyo also means the warp of cloth, symbolising the continuity of life throughout the past, present and future. Yasuji Kirimina, Fundamentals of Buddhism [sic], Nichiren Shosu International Center. ---------- 2. Om Mani Padme Hum Pronounced in Tibet Aum-Mani-Pay-May-Hung, this mantra may be translated: Om! The Jewel in the Heart of the Lotus! Hum! The deep, resonant Om is all sound and silence throughout time, the roar of eternity and also the great stillness of pure being; when intoned with the prescribed vibrations, it invokes the All that is otherwise inexpressible. The mani is the "adamantine diamond" of the Void -- the primordial, pure and indestructible essence of existence beyond all matter or even anti-matter, all phenomena, all change, and all becoming. Padme -- the Lotus -- is the world of phenomena, samsara, unfolding with spiritual progress to reveal beneath the leaves of delusion the mani-jewel of nirvana, that lies not apart from daily life but at its heart. Hum has no literal meaning, and it is variously interpreted (as is all this great mantra, about which whole volumes have been written). Perhaps it is simply a rhythmic exhortation, completing the mantra and inspiring the chanter, a declaration of being, of Is-ness, symbolised by the Buddha's gesture of touching the earth at the moment of enlightenment. It is! It exists! All that is or was or will ever be is right here in this moment! Now! Peter Matthiessen, The Snow Leopard, Chatto and Windus. ---------- 3. Om Mani Padme Hum OM, symbolising the origin, the Supreme Source, the Dharmakaya, the Absolute, is a powerfully creative word often held to be the sum of all the sounds in the universe -- the harmony of the spheres, perhaps. MANI PADME (jewel in the lotus) signifies such pairs of concepts as: the essential wisdom lying at the heart of Buddhist doctrine; the esoteric wisdom of the Vajrayana contained within the exoteric Mahayana philosophy; Mind contained within our minds; the eternal in the temporal; the Buddha in our hearts; the goal (supreme wisdom) and the means (compassion); and, if I may be permitted to draw an inference, the Christ Within who dwells in the mind of the Christian mystics. HUM is the conditioned in the unconditioned (being to OM as Te is to Tao in Taoist philosophy); it represents limitless reality embodied within the limits of the individual being, thus it unites every separate being and object with universal OM; it is the deathless in the ephemeral, besides being a word of great power that destroys all ego-born hindrances to understanding. John Blofeld, Mantras. It must be said that he follows it by saying: Such interpretations are naturally of interest, but it is necessary to stress that reflection upon the symbolism forms no part of the contemplative practice. The mantric syllables cannot produce their full effect upon the deepest levels of the adept's consciousness if his mind is cluttered with verbal concepts. Reflective thought must be transcended, abandoned. ---- But yet -- talking about the practice must be done on a more speculative philosophical level than the practice itself. If oone is to help others, in any other way than by silet example or silent service (the latter risky, the former arguably no help), one must do it at the verbal level. ------------------------------------------------------------------------ 18/11/81 Babysitting again, after such a long gap, and out of the habit of writing, in the middle of a tiring term, I don't quite know what to day. All evening I dithered between starting writing in this book and doing some mathematics, and as a result did neither, browsing in the earth's last mysteries, and John DOnne, and so forth. The pressing of time has made it difficult for me to meditate this term. And yet I really did feel in need of the restitutio of meditation yesterday, when things were especially rushed -- perhaps I should just take comfort that I am open enough to notice this, though of course that isn't really very satisfactory. -------------------------------------------------------------------------- 5/12/81 On the night I wrote the last entry, tired as I was, I proved that if a group i 3-hmogeneous but not 2-primitive then it preserves or reverses a linear order. It was one of those proofs, exciting at the time, that didn't survive the night. And the harder I tried to repair it, the worse it became. Writing x|yz to mean that y and z are in the same G_x-block, the hard case is when just one of x|yz, y|zx, z|xy holds. (Three or none are both clearly impossible, while two easily leads to a betweenness relation.) Well, there are three possible types for a 4-set: A: a|bc, a|bd, a|cd, b|cd; B: a|cd, b|cd, c|ab, d|ab; C: a|bc, b|cd, c|da, d|ab. Whether it was a stroke of luck or not I don't know, but at first I missed case C. If C doesn't occur, then the relation on G_x blocks defined by B < B' if z|xz (for y in B and z in B') is well-defined. It is trivially a total order. About the structure of a block itself this says nothing except that its stabiliser isn't 2-primitive. So what is more obvious than to plug the group into itself? This gies a beautiful example which is universal for relations x|yz with the t=1 condition and all 4-sets of type A or B. I haven't yet managed to enumerate its substructures -- however I can show that a 3-homogeneous but not 2-primitive group with slower than exponential growth preserves a betweenness, which is what I really wanted anyway. The real puzzle is the mysterious type C 4-set. Can it occur? If so, there must be another, quite different, kind of example; if not, one must go beyond 5-sets to rule it out. I'm writing this at the Molloys'. I've wrritten before of their two clocks -- in fact they have three, but when I arrived only two of them were ticking, and just after Traude and Pat left the cuckoo stopped, so I have only the leisurely deep-voiced chiming clock for company tonight. As the number of filled pages increases, so too does the temptation to browse over old entries rather than writing new ones. The first entry in this book was written eleven months ago, so one would expect the wheel to have turned nearly full circle. Indeed I am now re-reading "Tracks", having just finished "The Snow Leopard" (day by day, as it happened), and halfway through "101 Zen stories", read aloud at bedtime to Sheila (except when not, as when I'm late through babysitting or rehearsing with Jeanne and David or, as last night, out (at the Merton Christmas Ball)). I had started them with good intentions, doing yoga as well, and hoping that when the sequence was completed, by around my birthday, I would be further advanced in meditation than I've managed before -- but alas, term gets in the way of such hopes. Things are very much easier when it's not the dead of winter. The greatest strain lately has been the necessity of getting up while it's still dark, to dress and breakfast the children (while Sheila lies in bed), change James, and put the washing on, in time to get Hester to school. Is the answer to delay my bedtime? This would open a gulf between Sheila and me, but perhaps for my own sanity I need some sort of isolation of this nature. But then, is the resulting strain in the mornings harmful? My natural reaction is to ridicule this, but of course, continuous pressure might have quite a different effect from the more dramatic peaks. To return to the Zen stories: what I am not doing right there is failing to carry each story with me all day. I'm not sure how best to do this. I managed it to some extent the first time I read them. Part of the trouble is that there is no space in my mind for them which isn't intruded upon by work. Perhaps it is necessary to internalise this isolating barrier. Another thing: at the Alternative Bazaar last weekend, my eye was caught by a farm cooperative offering residential weekends, of which most were for work or discussion but some were quiet weekends. What a luxury, and for me at present one which is virtually unattainable! But these longings have begun rising independently, for example a craving for a long walk on my own. If I were to do something like this, it would require preparation for best effect: practice in meditation and/or mantras so that my mind doesn't simply flit about, and perhaps some theme to keep at the back of consciousness. It is now 9:50. At 10:00, in a strange house, with a loudly ticking clock, I will put this book aside and attempt to meditate. Having written that, it will be no surprise if little more gets written. Lack of sleep may well turn meditation into that kind of suspended judgment I've experienced so often, as in the time returning from Oberwolfach on the overnight car ferry when the buzz of conversation about me turned into fantastic mathematics. -------------------------------------------------------------------------- 5/12/81 Paul Colinvaux, "Why big fierce animals are rare", Pelican, 1980. Brian Rice and Tony Evans, "The English Sunrise", Matthews Miller Dunbar, London, 1972. -------------------------------------------------------------------------- 5/12/81 Decay clouds The smell of marigolds With damp earth Until cleared by frost Winter's low sun Raises waves On the still meadow Branches bare of leaves Allow lamplight To fall unhindered On stone -------------------------------------------------------------------------- 5/12/81 Jesus was 33 when he died, Sakyamuni 35 when he attained enlightenment, Goethe 37 when he first visited Italy. I understand that a ten-year-old girl gained the top mark in this year's Oxford entrance exam in mathematics. Keep perspective. Don't use birthdays as markers. Death is the adviser, not the advent of the second half of your threescore and ten. Don't panic. Be aware constantly. Make peace part of yourself. Tread lightly. A Elbereth Gilthoniel Om mani padme hum Hare Krishna, Hare Rama Lord Jesus Christ, Son of God, have mercy on me, a sinner The medium is the massage A path with a heart The middle way -------------------------------------------------------------------------- 23/12/1981 Cosmology: over the frost-fingered pines the pallid sun A coal train hauls carboniferous sun through modern snow -------------------------------------------------------------------------- 23/12/1981 It is easier to print the new snow than to write on a clean page Under foggy skies, where has the yellow snow, the blue snow gone? -------------------------------------------------------------------------- 16/1/1982 Meditation comes a little easier now, at least the bare bones of being able to turn away from distraction. Does that mean that the time is coming to meditate on a subject rather than just empty the mind? At present I see two things. One, that continued practice at emptying the mind is essential (and enjoyable too). The second is that there may be a way out of the dilemma I encountered last time I reached this point, that if you take a subject for meditation, condense it to a short mantra, and recite it over and over, then it ceases to have meaning, whereas if you try to think about it your mind is functioning in the wrong mode. As with research in mathematics, you impress every detail of the problem on your mind by a period of intensive thought, and then let it cook -- and I expect that, with proper direction, it cooks best during the mind-stilling exercise of meditation. Needless to say, a technique that would work best if you could devote virtually full time to study and meditation, but one that might achieve something even in my circumstances. Another parallel between mathematics and Zen (or rather, the same parallel viewed rather differently) is the koan of the relation between groups and trees that has come to light recently, stemming from 18/11/1981 (and observe that following an entry of such downbeat character I could strike so rich a vein of mathematics). I have done my best to treat it as such; late at night I have lain wrestling with the question, I've done a weaving depicting the boron trees on up to five end vertices, I've bought Sloane's book of sequences and started my own ... No enlightenment has come yet, but the progress to that point was perhaps breakthrough enough. There are many loose ends that could be followed up here -- characterisation of 5-homogeneous but not 4-transitive groups, axiomatisation of the other tree relations, etc. -------------------------------------------------------------------------- 16/1/1982 Some are mathematicians, some are carpenters' wives Bob Dylan -------------------------------------------------------------------------- 16/1/1982 Following on from the last entry, I had it in miind to write an account of the history of that bit of research. Now that term has started, it is almost certain that I won't get it done, and the enterprise smacks of vanity anyway, but there have been some remarkable features; for example, Dominic giving me the reference "trees" for the numbers I'd computed for the 3-homogeneous group, from a preprint version of Sloane, this being correct to I think eight terms but failing for the ninth, and yet not being a red herring as things turned out. There may be more of this to come as well, over and above the developments I just referred to. Also one could mention that the number of orbits on ordered k-tuples was worked out because such things came up naturally in considering wreath products, ad it was recognition of this sequence in Joyal as "commutative bracketings" that led to the identification and (matching it up with trees) the construction of the transitive extension. Also the part played by two mistakes -- the original false proof, and the non-recognition of type C quadruples. -------------------------------------------------------------------------- 13/2/1982 Twice now, reading a Russell Hoban novel has made me want to write a novel myself. ("Turtle Diary" on the bus back from Cambridge, and "The Lion of Boaz-Jachin and Jachin-Boaz" mostly last night.) There must be something in me that vibrates at his frequency. After "Turtle Diary" I even planned a book that Sheila and I would write jointly, each writing from the point of view of one of the protagonists, discussing the episode but not actually reading the other's part until the first version is finished. I think I would like to write another novel. But I still think there are good bits in the one I did write, and it seems a pity to abandon it. (I've no idea where it is now -- Sheila borrowed it quite a long time ago, a year and a half maybe, and I don't know what she did with it.) I don't want to rewrite it completely, but since there is such a break in style at the point where I left it and came back to it, I wouldn't mind having another go at the interplanetary bit. I always felt that it needed considerable expansion, perhaps with more of a plot, still ending in the same way but going through a lot more convolutions before getting there. And, for that matter, if I were to write another, what sort of a story should it have? Dinosaurs, perhaps? (Use what knowledge comes your way!) Talking of projects, the allotment is actually getting dug at the moment. At least, the digging is the easy part; getting the couch roots out is a fearsome task, and will take at least three times as long. Of course, if this job is not done, then any other work put in is wasted; the couch is so overpowering. Today I felt, for the first time in abut two years, that it might be possible to keep it under control. The strategy is not to plant anything at all until this enormous amount of work is done, and to go in for labour-saving crops like potatoes and jerusalem artichokes which have the added virtue of getting the soil dug painlessly in the autumn (and hopefully, as some say, keeping down the couch too). Getting back to the novel, it can't be a relatively simple Starship Enterprise type story. The aliens must be so alien that normal communication is impossible, because this is how they have been set up. Of course, if somehow some sympathy for them could be generated, it would add something to their final destructio. Also, an adventure where the participants were early on in some kind of paradise would give the possibility of their falling further into the uttermost hell in which they end. -------------------------------------------------------------------------- 13/2/1982 Out on the meadow, showers come, showers go; but if you stay indoors, rain starts, rain stops. Evening: girls call their horses home, no-one calls me. -------------------------------------------------------------------------- 13/2/1982 A house in which I haven't babysat, or even ventured, before. Two tall thin statues of men stand in the fire surround, one of birds on a small high shelf. Nothing else fits together. The grandfather clock has stopped at 11:39 and has its door open; a plate pictures a Chinese mountain; tradescantia tumbles like a waterfall onto a white candle; a monochromatic canal scene on the chimney breast faces a fern in an imitation Wedgwood pot in a macrame hanger dangling from a wall light with rustic wood support and red shades, not matching the red paper shade on the central light at the other end of the room. A railway line covers most of the floor. The books cover everything thinly, only Dick Francis and H. S. Merriman being multiply represented. One speaker carries a school photograph; the other, a model Concorde. I can't keep still here. I listen to music -- Beatles, Wings, John Williams, Davy Graham, some of which kept the kids awake -- read bits of books, try to write, or think, or sometiing else. The Chinese plate suggests the Sixth Patriarch of Zen fleeing from the monastery after his succession has been disputed, or, more generally, somebody fleeing from somewhere possessing secret knowledge which he hasn't yet fully integrated into his personality; the crises he meets on the way draw out of him the things he has learnt, and finally he realises why it was that he was fleeing. Or perhaps this isn't the right ending. Maybe he comes to doubt more and more that there ever was a monastery. A kind of Razor's Edge in reverse. It could also be that he helps others unwittingly; things he says or does resolve problems for others, but he sees no more than anyone else would see of another's mind. At the same time it could be that when he tries to help others he hurts them, and the harder he tries the more he hurts them. But how can be be "saved" if he shrugs off the monastic discipline? Some dreadful prvation and tragedy had better befall him, leaving him holed up in a cave where he stays for a long time, gradually regaining discipline and simplifying his life. Perhaps too, like the Buddha, he could be unaware of death until this personal tragedy brings it home to himm. But it is so much easier to make his experiences teach him; the point of this tale is that they remind him of what he knew already but had forgotten. Perhaps he could bear a jewel with magical properties. His mission, unknown to him, is to carry this jewel from the monastery, across the mountains, to a land where it has been prophesied that a holy man will come from the East bearing a sacred jewel. It is from using the magic of the jewel to try to help others that much mischief arises. The disaster could be that a girl he meets along the way, who chooses to come with him, heavily pregnant, falls from a ravine during a blizzard, and after a long taxing climb down he finds her body being eaten by scavenging animals. When he reaches the country in the Wes, after some adventures he is recognised as a holy man. (Perhaps he could be robbed and only later establish his identity as the bringer of the jewel.) Then, seeing the jewel established in a magic-worshipping cult, a far cry from what he has discovered in the mountains, he faces his last trial, the attempt to teach the true way; in this too he is mocked, as a distorted account of his sayings becomes a gospel of the cult he abhors. Finally he is drawn even further west to the sea and to the realisation that he must cast off the attachment to preaching a other attachments. The girl could be his half-sister. He grows up in a village in the foothills of the mountains, and is sent to the monastery in the plain after his father dies. On travelling west he has an inkling that he is near his home without knowing for certain. He meets the girl, they decide to travel together, and afterwards an old man in the next village tells him her story -- daughter of a wwoman who was the mistress of a man who had a wife and child in another village; the man died and the woman brought up her daughter in an isolated hut until she too died. Oh yes, the man's child was sent away to a monastery in the east and never seen again. -------------------------------------------------------------------------- undated If that's the effect Russell Hoban has on me, what about Gregory Benford's "Timescape"? I've just finished it, having been very immersed. He does much better than Fred Hoyle at writing a story full of real physics done by real physicists; somehow it comes out not at all crackpot-like. Gripping. But how could you put mathematics into a novel in the same sort of way? The physicists have the best of things there, since the "connection between the microscopic and the macroscopic" for them works at the level of basic theory: take the existence of a hypothetical new particle the tachyon, and make a 400-page novel out of it, because its implicatios are that energy and information can be sent into the past, that the future can be changed, etc. (Of course, this isn't to deny that there's a lot of good novel-writing in there too.) But if a molecular biologist were to do a similar sort of thing then the macroscopic aspect would be just gadgetry, unless you were very careful. I was thinking today, though, that Hofstadter's anti-Lucas argument is an atttempt to get round the following sticky point. Those who claim on "physical" grounds that we are machines are under pressure to identify the reading heads and tape squares of the Turing machine with the hardware of the brain, neurons and ganglia and such; but the operation of this hardware is not "deterministic" and apparently not even pre-programmed. Put against this my view from way back that free will is an illusionn caused by the fact that most of the brain's workings are unconscious: if we attempt to chase the "I" that made any single decision, the "I" very soon disappears into the unconscious. (These are meant to throw the paradox into sharp relief, as Newtonian mechanics and Maxwell's electromagnetism once did.) I don't really remember exactly what Hofstadter said; but as I recall, he was claiming that a machine could do anything a person could, not that people are "just" machines: said otherwise, consciousness arises naturally in a system having sufficiently much complexity. It was this I found unconvincing. It was, in the end, completely undemonstrable, because I cannot demonstrate consciousness in any other being. The argument for assigning consciousness to other people rests on the observation that they are like me; there is no logical reason to do so. Again, there is no logical reason to extend consciousness to a highly complex computer; but this time the argument from similarity does not apply. The figure with guitar has gone, replaced by a blue and yellow nocturne over the sofa. Words to describe the awfulness of walking home pushing a bicycle with a punctured back tyre and laden with groceries through the persistent icy rain won't come. I'm in bad shape at the moment, steamrollered by term. I don't think I've been as bad-tempered as I sometimes am under these circumstances but there hasn't really been a lot of communication going on with Sheila, or with the kids either for that matter. No, the figure with guitar has just crossed to the other side of the room. I'm desperately tired. -------------------------------------------------------------------------- 2/4/1982 Funny how history repeats. A pretty awful trip back from town -- not raining, but too much stuff from Sainsbury's, so one orange juice carton burst, the bag split open and had to be piled on top of the basket, the washing-up liquid discharged all over me, the handle of another bag broke, etc. I can still feel the tension in my left wrist: must relax it later. What a record collection! With enormous difficulty, I restricted myself to four: Joni Mitchell, Joan Armatrading, Mike Oldfield, and Bob Dylan. One can't take in more than that. And it interferes with my writing too -- but I'm so tired I can't write straight anyway. On the way to Oberwolfach I wrote the first paragraph of the novel, but I couldn't get any further. I couldn't really see the place I was writing about. I have the broad outline in some detail now, but not the details. Should I try to write it shorter, twenty pages or so? -------------------------------------------------------------------------- 2/4/1982 After the frost-killed shrubs have been cut down the garden looks empty Daffodils didn't expect to awaken to a hard sky and bitter rain One tree, yet without buds is etched in black on the silver-grey sky -------------------------------------------------------------------------- 2/4/1982 Do my guesses sometimes hit near the mark? It is quite extraordinary to glance back at things that I have written, to see some thought displayed there that seems to have a hint of truth in it. Apparently it must have passed through my mind once, yet I haven't assimilated it into my own life. While in Oberwolfach last week, I woke early one morning, rose, dressed, and set off up the hill. It was the morning of the day I was to give my talk, and the walk was partly to settle my ideas, which were still in a state of flux: I'd twice changed the emphasis of my talk to bring it into line with the interests of others at the conference. As I walked, at the back of my mind, and becoming more and more insistent, came the strains of "... living through a million years of crying until you realise the art of dying." It came like a message from outside. Such a message was in perfect accord with the first paragraph of "Leaf by Niggle", that I had read on the train (in lieu of writing my novel), about the little man called Niggle who had to make a long journey, and put off making preparation for it. I am going to die. It could be any time. Of that I felt quite sure as I put one foot after another up the unrelenting hillside, and there was a strong conviction that it would be soon. Was I prepared for my journey? How does one prepare? Niggle only had time to pick up a bag that was lying in the hall, and its contents were of no use to him. But he didn't know where he was going, and had little idea what he would need anyway. The art of dying. As I endeavoured to still my mind, I came without realising it to a fork in the path, where one track led upward much more steeply than the other. Outside my conscious control, my feet took the right path, the steeper one. A good omen? The morning was very clear, very still, and very cold. The mountainside was bathed in the song of the river over its rocks in the valley, almost drowning out the occasional vehicle on the road. As I descended, trying to hold the stillness in my mind, I came to a point where the path ran between two dense stands of pine trees. The right-hand stand particularly caught my attention. The darkness there appeared as a physical presence. I stopped, and stepped into the shadow. The calm, cool darkness helped to create a mood of reverence. Trees stood apparently at random, but it was clear to me that each was in its appointed place. They were the participants in some ceremony of worship which (it was obvious to me) was conducted standing rather than kneeling. I stood, surrounded by friends, and worshipped with them for a while. I didn't need to knnow any details about them; I knew that each was an individual, and that was enough. After a while I knew that it was time for me to continue on my way. For the rest of the walk, I was back in the everyday world. The peaceful feeling persisted; I noticed, as I might have done any time, the curious phenomenon of casting several clear shadows before sunrise, as dawn light poured in through windows in the forest. But no more circuits needed completing. My talk was no better or worse than many others I've given; I didn't die, and the feeling of the imminence of my death left me. Was it a lesson that I was expected to learn and act upon? If so, how? By conjuring up the feeling at other times (if this is possible), by carrying the intellectual content always close to awareness, or in some completely different way? Was it a reminder, given out of grace? Perhaps the most radical idea is that I really did die there in the forest, that death is not one big final barrier but a condition always present in life, breaking through into awareness on a few precious occasions. May it even be that the final death is a recapitulation and summing up of all these little deaths? -------------------------------------------------------------------------- 24/4/1982 Just to sit back, to think, or not; to remember what it is that I must keep in mind; this is the luxury of these evenings of babysitting. I have a real need for this. Coming home today, I was feeling desperately tired. Though burdened with a load of compost, and cycling into the wind, I knew that my bicycle shouldn't be going as slowly as it was. (My bicycle is actually a very good barometer of my mental states, in this way -- though, of course, it contributes to them too.) But I didn't want to get home and crawl into bed. When I thought about that, the answer came immediately: I go to bed every day; if that were the cure, I'd be cured. At the time, I couldn't say just what it was I did want. (Term began today. Dominic said at lunchtime that I didn't look well. I'd dedicated the morning to getting a load of things off my desk before the tumult of term; in the event, I'd got just three done, two of them having come by the morning's mail anyway. That's the background; it hints at the phobia of getting down to all those tasks that blocks up my mind at the moment.) Now, I welcome the idea of death less than I fear the onset of instability, even insanity. I read about Jesus yesterday -- another terrible thing, I forgot about the College mmeeting -- in Humphrey Carpenter's book in the OUP Past Masters series. The prevailing theme, at least in terms of what Jesus probably said and did, was that he was no liberal, setting personal freedom and fulfilment above the law; rather, his message was that every letter of the law must be obeyed, but that the law is not enough; God demands total commitment, everything, from us. While there may be a sense in which murder is wrong, it is certainly not in the sense that its wrongness gives us liberty to condemn the murderer; the idea that there could be anything for us to condemn must be totally foreign, no part of our make-up. I use the plural here; but in reality, of course, if anyone, however close to me, fails to meet this standard, the idea that they are imperfect of fall short in any way must be totally foreign to me. It seems that only so can Jesus uphold the law against adultery and yet get the woman out of a very tricky situation as he did. A subsidiary theme was Jesus' frequent insistence that the coming of the Kingdom of Heaven is only a few months or weeks away, and the way this is played down by commentators from the Gospel writers to the present day, since it obviously didn't happen. Did Jesus really believe it in the "objective" sense? Surely not. So the way out is that our own experience of the Kingdom is at hand, and more, perhaps is happening all the time. Put "death" for "coming of the Kingdom", and this fits very well with the last entry. Or, if you don't like that, it needn't be "death". Suzuki claims somewhere that Jesus' words to the thief on the Cross have been mistranslated, and should read "Today (i.e. now) you are in Paradise". One who can be aware of that while undergoing such barbaric torture has surely realised the art of dying. But then again, thoug that has a tremendous heroic quality about it, and makes my own problems pale into insignificance, might I not at least suggest that keeping this awareness of the Kingdom through the drudgery of everyday life presents its own difficulties? This is what is essential, and it successful practice surely makes the other much easier. I've wondered before just what successful practice might mean -- how do you get to the state of constant awareness, if indeed it is possible? Is it just grasping every positive opportunity that comes, or is it necessary to batter at the door during the drudgery too? (Is this what meditation is?) -------------------------------------------------------------------------- 24/4/1982 Taming the Green Knight -- planting seeds in the dusty earth. ---- When the dead enter the earth, is it theirs for all time? and is your body mine? ---- To see the world, for an instant, as they saw it -- they, whose empty tomb stands set about with trees on the high downs -- impossible? or true? ---- The pine trees were my friends. It was they who taught me that one can pray standing. ---- The geometry of flamingoes in flight, The arithmetic of pines at dawn, Helps you count clouds at night, And see rainbows on a dewy lawn. -------------------------------------------------------------------------- 30/6/1982 A long absence, as the book was lost. It has now turned up again. I didn't miss it much. Today I just finished reading more hagiography, Monica Furlong's biiography of Thomas Merton. Important for me in many ways (e.g. "Christianity was his language") but just one comment of a different sort -- should I make notes on a book like that? (The purpose being, as with all study, to help me remember. As it is, I tear through it, and things that seem important as I pass don't remain. But on the other hand, there was little in it of the sort that requires note-taking -- I don't need to be an expert on the details of Thomas Merton's life -- whereas the important effects don't occur at that level at all. The question is, though, how to smooth the path of these important effects.) -------------------------------------------------------------------------- 22/7/1982 Like teasel flowers, transients on their seed-head, the summer passes over us. ---- Does the new moon burn itself biting the sun? ---- We sing love songs, knowing no others, while, above us clouds cascade. -------------------------------------------------------------------------- 9/8/1982 Ring road, 8/8/1982 Amid harvest-ripe fields Set in deep green hedgerows, The angry yelp of air brakes. Beside the busy ring road Two picnickers load the debris Into separate cars. The white undersides of leaves Wind-tossed against firmer green Blaze in the late sunshine. A red traffic sign Strikes home with its warning Of changing priorities. How important are the echoes Of factory chimney shapes In the children's playground toys? -------------------------------------------------------------------------- 9/8/1982 Fragments Figure with guitar haunts me now ---- Rain stains my face Grass binds my feet Evening breezes chill me to the bone ---- Today, today, Today you've gone away And the worm sleeps in the bitter apple. ---- Footnote: The 2nd, 3rd and 5th from 24/4/1982 are also fragments rather than haiku. -------------------------------------------------------------------------- 22/8/1982 Television images of Zanskar -- Valleys like flowers, Mountains like flames. -------------------------------------------------------------------------- 24/8/1982 The words of Lao Tzu Only provide puzzles, But plants relax me without speech. Brief commentary: The Tao Te Ching, both in the text and the editorial comment, is unsettling. Claimed as the most important sourcebook on pre-Buddhist mysticism, it contains some of this, but also much that is almost Macchiavellian political science. On a visit to the Botanic Gardens today I suddenly realised that I must set aside time to go there and sit. It was almost an understanding that here is the source of peace. The same feeling in a different form came when I saw the plants in the living room tonight. Footnote: Te = Arete? -------------------------------------------------------------------------- 29/8/1982 This mysterious world -- Seven butterflies on the dahlias In a tiny front garden -------------------------------------------------------------------------- 29/8/1982 I don't properly understand, and can't put into words, the impression the seasons make on me. I've said elsewhere that there's no such thing as spring; late winter suddenly becomes early summer, and it is impossible to put a buffer zone between them. This year, the different parts of summer had their own character, even if not always clearly delineated. The garden changed from neat rows into a jungle, the corner of Wolvercote Green looked and smelt like a plethora of green; that was early summer. Then the clearest line that one could ever find was about a fortnight ago, when the sun suddenly lost its strength, the berries were red and purple and black, the air carried a chill. -------------------------------------------------------------------------- 30/8/1982 Today the children and I made a fire in the living room. On the way home Hester was lagging and I knew we were in for trouble. I said to a wet bedraggled Neill that we'd put on a fire (i.e. an electric fan heater) when we got back. I knew he would assume a real fire even as I said it; there had been talk of this before. They wrote letters to Father Christmas and sent them up the chimney. An interesting set-up; just how real is it to them? Hester knows that Father Christmas isn't real, but plays along and gets very involved; Neill was fascinated when I said that, if the letter burned, the ink would go to the North Pole; James believes that Father Christmas is coming tonight. Hester asked for a colouring-in poster; Neill, for Star Wars toys; and James, for a car. -------------------------------------------------------------------------- 2/9/1982 Why do these dead petals fallen haphazardly on a sheet of waste paper spell my name? Brief commentary: In the front garden this morning was a piece of rubbish, apparently a transfer backing sheet thrown over from next door, waterproof and white. Three marigold petals made my initials, exactly like a signature, in the bottom right corner, and insistently asked the question of the poem. They drew attention to the rest of the sheet; it had a few smudges and dewdrops scattered about, and could have been a delicate work of art; but there was nothing in them to suggest that they specifically gave the answer. Perhaps it was the transitoriness of this particular art form. I said to Sheila not lonng ago that, just as weaving was her thing, perhaps string figures were mine. (No matter how perfectly you make them, you destroy them in a few seconds; and, to some extent, the art is in the making.) I think that the uneasiness produced by this find is not so much that I feel I shuld be able to find a specific answer as that I feel I should be directed to something, and these dated revelations don't fully satisfy. -------------------------------------------------------------------------- 2/9/1982 Dahlia petals scattered on the steps of the bus -- a harbinger of autumn ---- On the day the swallows gathered on the wires the weather finally turned hot -------------------------------------------------------------------------- 8/9/1982 Haiku from the other book Flimsy silver flakes, sunset-lit, flicker in the soundless solar wind, like frozen snowflakes. ---- A car wrecker's yard spilling over into a graveyard. ---- Flowers falling over stone reflected like jewels in the muddy water ---- Riding beside the canal the sun rides in the water beside me ---- On the quiet towpath I heard the rain sing on the surface of the water ---- Flowers in shadow are blue until you touch them and turn them to daisies ---- A tree is more solid than a house; when you move, you know it ---- A steeple once seen across moorland haunts me now as I cut my hands on rough grass ---- To travel to japan in search of Zen, is like ... Fill in your favourite simile ---- A train across an open field, a factory across the canal, a pothole in the cycle path ---- The afternoon stillness is painted on the canvas of the incessant traffic noise ---- An open window, a curtain flapping, high above the street ---- Sitting on Primrose Hill faced with city and sky -- yet I think of poetry. ---- Commentary: The first one describes honesty pods, and is thus not self-contained. The next three were written on a train journey, and the last nine while invigilating an examination and meditating on the jewel in the heart of the lotus, or as I put it then: Does even this lotus have a jewel in its heart? -------------------------------------------------------------------------- 28/9/1982 After the equinox the morning sun shone straight down the churchyard path -------------------------------------------------------------------------- 6/11/1982 Tolkien and Dyson put to Lewis the argument that the story of Jesus is a myth, which moves us in the way that myths do, but happens to be true. Idries Shah asserts that the wise know that folk tales can work on their hearers at a on-verbal level, even when they have become textually corrupt, and even if their hearers believe that the stories are historic. These two passages (which I might look up and transcribe) provide an interesting contrast. If I understand them correctly, Tolkien and Shah would have many points of disagreement about the nature and function of fairy stories. And yet, when I try to write down the points on which they would disagree, they come out as saying the same thing. Tolkien believed that there is a sense in which his stories are true -- "Leaf by Niggle" makes this clear. Shah does too, despite some of his comments about the story of Nasrudin's ass, for example. And however much one tries to dismiss Tolkien's beliefs as pie in the sky, they are not that, but refer to the here and now. His expression -- I don't remember it exactly -- for the "turn" in a story is describing an immediate experience. His biography makes clear just how personally he was involved in all his activities. Writing a preface for an edition of a mediaeval text could become an apologia for the style of "The Lord of the Rings". I suppose that part of what I'm struggling with here is the weight of "established church" attitudes that Tolkien carries -- not only the morality of "The Silmarillion" and "The Lord of the Rings", but things like a conversation he had with Tom Braun about missionaries. I shy away from dogma (or is it only some dogma?). It becomes necessary to note that I've used the Jesus prayer as a meditation mantra several times recently. The very first time was the most successful; bit when I was in a kind of threshold state, Sheila burst into the room to telephone her mother, and I certainly don't yet have the concentration to ignore that! Until then, there had been a feeling of "coming home" about it. Back to something like the main topic: in a Cambridge bookshop yesterday I found a book by the man who has "proved" that Castaneda's works are a hoax. I read the cover blurb, no more. Although he didn't seem to be dismissing it out of hand for that reason, he did appear to miss a rather important point. Castaneda's work has been hailed as significant by a lot of people. As with any work, but perhaps even more than most, it could only have this effect if it could reach these people at a level and in a way that they are ready to accept. If Castaneda could do that without the guidance of a seer well versed in ancient lore, that only makes his achievement the greater. I find it compelling, and I find myself quite ready to accept the incredible things he narrates. (That is not the same as believing it to be true, a totally irrelevant consideration here.) But, consciously, I regard the detailed instructions about seeing, dreaming, stalking, etc. very much more valuable than either the personal conflicts or the detailed accounts of the visions (though of course it may well be that either or both of these work on me subconsciously). I have the feeling that a similar pair of contrasting passages could be extracted from this and Joan Halifax's book -- which, incidentally, loses some of its value for me because the word "shamanize" grates in a way that "see" (in italics) doesn't. I must start making an anthology. I have in mind keeping pieces half a page or a page long, perhaps in complementary pairs or even sequences. -------------------------------------------------------------------------- 6/11/1982 Up that hill, past those rows of houses, to get to the airport -- and there was nobody waiting, anyway -- -------------------------------------------------------------------------- 6/11/1982 At the last College meeting, we discussed a proposal for an Emeritus Fellowship. It was stated that our custom is: less than ten years, no chance; more than twenty years, a virtual certainty; between ten and twenty, well, we have to think about it. It didn't occur to me then, but later, that come the New Year I will have been a fellow for ten years, three as a JRF and seven as a tutorial fellow. The thought struck home while babysitting, sitting on the same sofa on which I have previously written about seven-year intervals. -------------------------------------------------------------------------- undated "The Listing Attic" and "The Unstrung Harp", Edward Gorey, Abelard, London, 1974. There was a young man who appeared To his friends with a full growth of beard; They at once said "Although We can't say why it's so, The effect is uncommonly weird." There was a young woman named Ells Who was subject to curious spells. When got up very oddly She'd cry out things ungodly By the palms in expensive hotels. A headstrong young woman in Ealing Threw her two weeks' old child at the ceiling; When quizzed why she did, She said, "To be rid Of a strange, overpowering feeling." ---- Hottentottenpotentatentantenattentat: an attempt on the life of the aunt of a Hottentot potentate. Fritz Spiegl -------------------------------------------------------------------------- 21/11/1982 I've just gone through the calculation to verify the median law for a cubic curve. A couple of slips, one major, led to me sitting for a while staring at a blank sheet of paper, not wanting to plunge into the calculation of 324 terms -- I even started and wrote down the first three, but got them wrong -- but when I noticed the mistake it was easy. Take the quadrilateral to have vertices (1,0,0), (0,1,0), (1,1,1) and (0,0,1); thus the cubic is ax^2y + bxy^2 + cy^2z + dyz^2 + ez^2x + fzx^2 + gxyz = 0, where g=-(a+b+c+d+e+f). Easy calculation gives the "midpoints" of the sides as (b,-a,0), (b+c,e+f,b+c), (d+e,d+e,a+b) and (-e,0,f). Now the easiest way to finish is as follows. Observe that the second and fourth satisfy fx - cy = by - ez, while the first and third satisfy by - ez = dz - ax; so at the point of intersection of the medians, these three quantities are equal. Hence (x-y)z(fx-cy) + (y-z)x(by-ez) + (z-x)y(dz-ax) = 0, that is, the intersection lies on the curve. The reason for this is Peter Neumann's observation that cubic curves contain arcs of high intricacy. If a cubic curve has an inflexion, then it has an abelian group structure with the inflexion as identity. (This follows from the median law, which was why I was checking it.) If there is a tangent from the inflexion to another point on the curve, then the group has even order, and so has a subgroup of index 2. The coset of this subgroup contains no collinear triples (these have sum 0) and so is an arc of size about q/2. But being contained in a cubic curve, it meets any conic in at most six points, giving a lower bound of about q/12 for the intricacy, compared with an upper bound of about q/5. The Fermat curve x^3+y^3+z^3=0 always has inflexions; for q congruent to 2 mod 3 it has q+1 points, and so (if q is odd) gives an arc of size (q+1)/2. Note that any arc produced in this way has the property that its intersections with conics form a 5-design with lambda=1 and blocks of sizes 5 or 6; moreover, the "generic" block size might be expected to be 6.it would be fascinating if, for example, for certain congruences on q one always had block size 6 and produced a Steiner system S(5,6,(q+1)/2). I'm almost tempted to work the case q=23 by hand to see if the S(5,6,12) is obtained. An infinite family of Steiner systems with t=5 would be quite a find, if they're there. Further calculation for the case q=11 showed that the six points outside the subgroup of index 2 do lie on a conic. -------------------------------------------------------------------------- undated What a long time without an entry in the book! No babysitting since last November -- no wonder I don't get so much work done these days. Well, that's not quite true, but only a very few babysits, and almost all of them short ones. I've really done physical work for my keep tonight. I spent about 3 1/2 hours typing the bulk of my "graphIX" program onto the computer. Almost all works amazingly well. I gave up when having a little bit of trouble getting the cursor subroutine working: for some reason, the computer doesn't like working out INT(SQR((x(2)-x(1))^2+(y(2)-y(1))^2)), and I was too tired to find out what was hanging it up. But everything else -- the store and show routines, global and local parameter changes, user-defined graphics, even the spiral -- work perfectly. I can't believe this isn't a remarkably good package. For commercial use it would only require better error-trapping and perhaps some abbreviation of the keywords. Bt this isn't a place for work, because of the lodger (who watches television and sat hovering all the while I was working -- it must have been somewhat boring for him) , even were it not for having been to a schools dinner last night, so that I'm more than ready for my bed. -------------------------------------------------------------------------- 12/11/1983 That was undated, but still a long time ago -- for one thing, the program has changed its name since then. Tonight, a late sit, giving me the chance to finish reading Dave Lischka's draft thesis -- a weight off my mind, but more remains -- but not leaving the energy for much more. Something new is growing, though. A feeling of casualness, absence of striving; not because of liberation, but rather just the harsh lesson that striving takes too heavy a toll. The Dales' houes has old carved figures and a new print, a brass toy and a jug of honesty and dried flowers. -------------------------------------------------------------------------- 20/2/1984 I've just dscovered an important Universal Truth. Consider this list of translations of "anti-dandruff" into several European languages. ANTI DANDRUFF COMBAT LES PELLICULES GEGEN SCHUPPEN ANTIFORFOR ANTI ROOS ELIMINA LA CASPA MOT MJ\"ALL ANTI\Pi ITY PI\Delta IKO Notice that four of the eight use the word "anti", and the others are equally likely to have a word recognisable to an English speaker. But on the other hand, the word for "dandruff" changes beyond recognition even between closely related languages. -------------------------------------------------------------------------- A footnote to the undated piece between 3/2/1982 and 2/4/1982 re Hofstadter's argument that consciousness is an emergent property of a sufficiently complex system. (See how I'm mastering the terminology of Sheila's subject?) I was reminded, by something I read recently, that when I was a child I used to draw pictures of enormously complicated machines with myriad wheels, belts, gears, axles, etc. The assumption was (and I think it is an assumption that I queried even then, and not just with hindsight) that motion, or the generation of energy, was an emergent property of a sufficiently complex machine. The reason for this was simple. At the time, I had no idea of where the motiveforce for something like a car engine was generated, and the function of the complexity was really to hide my ignorance. What better metaphor with which to attack Hofstadter? The reading was Arnold Pacey's book on the history of technology, in particular the passage where he explains how similar explorations were advanced for the operation of complex systems of pulleys; the assumption being, as I understand it, that natural laws could be circumvented by a sufficiently high degree of artificiality in the system, enabling it to produce force out of nothing. The scientific method, personified by Galileo, demonstrated that in fact there were simple natural laws at work in such systems, and that the gain in force was paid for by a loss in speed. So psychology is still in a pre-Galilean state! But there is something else interesting here, the attempt to communicate with myself as a child. Just what did I know, and in what sense did I know it, in those days? For example, as a child I knew how to sum geometric progressions with common ratio 2 or 3. (I can remember that I remember the occasion when my father posed to me the problem about the invention of chess, and when I had told him that it was necessary to sum the series, he had begun to explain that there was a simpler way, when I surprised him by describing it to him. I can also remember seeing the pattern for the case of common ratio 3 as I rode home following the cows, in much the same way as when I run south along the canal towpath between the A34 and Duke's Lock I remember some of my thoughts about orbits on k-sets from two years ago, my last serious bid for fitness. But I digress. Through the gradations of mathematical knowledge (I could give a rigorous proof, I could see the pattern which could be proved by induction, I had a feel for what was going on, I had guessed what the answer was, I had obsered the rule in some cases), where was I then? Where am I now, on other matters? -------------------------------------------------------------------------- 23/2/1984 Everything changes, even mathematical truth -- but how sure was I that I had it right before? I'm referring to the cycle structure of automorphisms and almost-automorphisms of the random graph. Some time ago I observed that the larger of these didn't refute Peter's conjecture about embeddability in the homeomorphism group of the rational world, because his necessary and sufficient condition on cycle types for the latter was necessary for the former. (At least, it doesn't refute it this way!) This much is easy, and I think I still believe it. Then I decided, on further thought, that this condition was also sufficient in both Aut(R) and Aut*(R). However, that was half a dozen changes of mind ago. I think I now have a credible assertion about exactly which cycle types occur; but it is much more complicated, and is not the same for the two groups. (In particular, Aut*(R) contains elements with one fixed point and one cyycle -- bbut these are not conjugate to their inverses -- while obviously no such element can occur in Aut(R).) Note that an element of Aut*(R) with n fixed points and the rest infinite cycles has at least 2^{n-1} infinite cycles, this bound being achieved according to my current calculations, so the stabilisers of n points are mutually non-isomorphic (as permutation groups). But it is also true that each is isomorphic to a highly transitive subgroup of the original group. The process had an element of experimental science about it. It's not too difficult to decide, of a given cycle structure, whether or not it can occur. So one can ask questions of nature and try, from the answers, to deduce the general rule. Having formulated a rule, one can try to prove it, but beware: the commitment to your guess will certainly be strong enough to lead you into beliveing weak or fallacious arguments, and even a proven rule may be refuted by further experimentation! -------------------------------------------------------------------------- 10/3/1984 Once upon a time there was someone, who claims to have some special relationship with me, who ran 5000 metres in 15 minutes 3.6 seconds at the McGillivray ground in Perth, Australia on 26 May 1966, and subsequently ran 10000 metres in 31 minutes 20.8 seconds at Sydney University on 23 May the following year. How did I do it? At present I am almost crippled by pains in my left hip and right calf, as a result of trying to get fit enough not to make a complete fool of myself in a half marathon in Reading in a fortnight's time. Prior to these pains, which have slowly grown in intensity to the point where I can hardly even walk, I was running six miles on the road, fairly flat, in just under 39 minutes at my best, and really having to work for it. And this is not just a slight difference. I'm now 25% below my best ever, having once reckoned that at my best I was between 10% and 15% behind the best in the world. Does this mean that my earlier form is twice as inaccessible to me now as, say, an Olympic gold medal was then? But reading my old training diary, nothing comes across more clearly than what a quitter I was then. My best time ever was the cross-country season in 1966, so let's lk at that. On 4 June I mapped out a winter build-up as follows: Sunday: Mt Coot-tha [9 1/2 miles including a very substantial hill] Monday: 12 x 520 yards with walk 130 yards rest between each Tuesday: Fartlek+ [about 5 miles] Wednesday: Mt Coot-tha circuit [about 13 miles] Thursday: (?) ["my busy day"] Friday: Fartlek+, unless competing on Saturday Saturday: Race, or (?) What happened? On 5 Jue I did more than indicated (3 miles road in just less than 16 minute, then Mt Coot-tha in 60:20, then a 660 in 1:34). On Monday I had a sore tendon and quit halfway through. This kept me out of action for the rest of the week. Competed on Saturday. Sunday to Tuesday again nothing, this time the excuse being a weekend at home and then "tired and upset stomach". Wednesday I did the Mt Coot-tha cirtuit, but nothing on Thursday or Friday, and about a third of a training session on Saturday because my jock strap broke! Other excuses: read a novel, stone-bruise, and more days off without comment. So June passed. In July I had five idle days out of the first 15m complaining of the stone-bruise, and managed to miss one race by running to the course and arriving too late. I actually managed second in the State titles by taking an opponent just before halfway and holdng another off for almost the entire second half, but it was only 36:06 for 10km. Things seemed to be looking up in the second half of July, apart from the persistent stone bruise, but again August saw me laid off for seven of the first fifteen days. However, somehow my times were tumbling, and my 34:12 for 10km against Alan Blackburn showed me how he could be beaten. I did nothing on August 18th, 19th, 22nd, 23rd or 25th, and only derisory training on 20th, 24th and 26th. And then the miracle of Inter-Varsity on the 27th where things just clicked and I led the field home in 32:57, having made my move as prearranged in the third quarter to pull Blackburn's sting and accidentally eaten the whole field. A novel or a film would have ended there, but the notebook continues for another year, mostly downhill with a slight reprieve for the Inter-Varsity track 10000 in May 1967. Almost the last thing in the book is the 1967 Inter-Varsity cross-country, where I finished 23rd in 38:08. (Though it is fair to add that the weather was bitterly cold and the course rough and slippery, in complete contrast to the previous year.) The very lasat thing is a resolutionn to "try to get fit and try for a good season next year". Yet even reading that book makes my pen and my pulse run faster and eases the pain in my calf. I am made to run, and even the most painful outings now bring great joy. Not only that, but even the disasters have a cathartic effect. When I pulled my calf muscle outside Bladon church, at the furthest point of my 12 mile circuit, I railed and ranted against my leg, my job, cars that wouldn't stop, the bitter wind, Sheila, even the children. I've said elsewhere that I'm much more aware now that the totality of me is involved with running, and I think the mistake that day was setting out after having been hassled by the children all day, much too tense, despite warning signs that I should have been able to read. The other really bad day, last Sunday, once round the ring road (the first half, the running sweet, but new aches and pains cropping up on all sides; then all of a sudden, virtual muscle seizure, complete loss of pace), had also been on a day full of hassle and tension. And even this has an unexpected side. I really believe that I'm not so much a quitter now as I was. In the last 16 days, since I've been keeping my training diary, I have gone out on nine of them; not brilliant, but when you weigh in the demands of my job, which leave late evenings as the only possible time for running on most weekdays, the temperature at that time of the evening, the stretched weekends with Sheila often away and the children hyperactive, and lately the state of my legs, against the freedom and warmth of my undergraduate life, you see that those nine outings, on which I covered nearly seventy miles, have been quite an achievement. And on not a single one of them have I run less than I intended, when I set out, and on scarcely any have I even let the pace drop below the best I could manage for the distance, despite muscle pains worse than anything I have experienced before. Sometimes, before setting out, I could scarcely walk, and I felt a fool running in a kind of limp over the railway bridge, waiting for the muscles to thaw out enough for me to stretch my stride to a decent pace. And I still cann't do better than a pace I would have laughed at as a schoolboy; but I am not a schoolboy, and running to my present limits brings me a large part of what joy I find in my life at present. The other thing is yoga. Almost every day I haven't run, and some of those I have, I've put in 20 minutes, either in the evening (though difficult now because of having two carpets up and all other rooms as junkpiles because of the builder) or snatched from the jaws of a day's work. If only the loosening effect on my muscles could really dissipate the tension. But the one time I tried it before I set out (the terrible ring road session), I was much too impatient, and the relaxing effect was nil. But work on it; perhaps, if not salvation, at least some improvement lies that way. So, enough for self-indulgence, now I begin ... -------------------------------------------------------------------------- 7/1/1985 A new year (and I haven't made any resolutions yet!); halfway through the book (or nearly so). What happens at this point? I've just been writing up my runnning book, spending a mere two pages on the interval between stopping writing in it on 27/5/1984 and resuming on 6/1/1985. So many details left out: runing across town in Montreal to the Olympic stadium in a thunderstorm; running round Wascana Lake in Regina, coming unexpectedly on a green oval with cricketers in whites; running in Eugene, the runners' Mecca (where a runner had been assassinated the week before I was there) -- two runs: one up Spencer Butte in cloud, sunlight through the pines, a mountain that has haunted my dreams for 11 1/2 years; the other in rain and gale with two mathematicians, myself by far the fastest, falling asleep as I ran --; running in Calgary with the little kit I'd brought, amazing the locals by going out in shorts and T-shirt with the temperature -9 (though I had my revenge the next day, when the Chinook blew and the temperature rose twenty degrees, and they were too hot in their elaborate all-over suits), and amazing them too with mentions of my best times, to such an extent that they wouldn't even discount them for having been done done sixteen years ago; and, most of all, running in Pasadena, the incredible five miles up Lake, the road getting steeper and steeper, and then the lovely feel of running in the mountains, from desert up (if I was lucky) to pine forest at Henninger Flats or above Echo Mountain. The last time I ran up there was the day we flew home; I cheated and had Sheila drive me up to the start of the trail, and I made further up than ever before, and could have gone on much further still but for the time being short. But perhaps the most significant thing, hinted at above, was Ivan Rival in Calgary. Somehow he forced me into the role of pundit. He (and everyone else there, and in Eugene too) started and ended each session with 15 minutes' stretching (to counteract the compression involved in pounding the roads). This might be a good idea; it's not something I've ever tried (except for a little yoga last year whe I was injured, which didn't help), but instead of expounding further, he said that he assumed that people who had been running for twenty years didn't need such things, they were only useful for the latecomers. He questioned me persistently about any changes I'd ever made in my running style, and encouraged me to talk aboout whether I do mathematics when I run (he doesn't), and more generally about the relationship between mathematics and running. These are theories that I abandoned some years ago. (In fact, part of the conscious reason for writing my novel was to write these things out of my system.) So I was somewhat reluctant to get involved, and I don't think I put the idea across as well as I might have done. (Back in the "Road Runner" days, I would pontificate about such matters at the drop of a hat!) The argument was that what running and mathematics share is that both are a form of meditation, and (in apparent contradiction) both have elements of "practice" and "performance" about them. To take the second point first. For running, at least on the surface, this is clear; probing a little deeper, one sees other things. You don't jut run 13 miles even-pace every other day to train for a half-marathon. A race has moments of drama, moments when the (easily-attained) steady rhythm doesn't serve you well, when you need, say, a sudden burst of power for tactical reasons, or you need to be able to hang on against the pain for an extended period. You can easily call on the resources, because you have rehearsed them all in training (fartlek, fast fives, etc.). As with any such practice, no verbalisation or intention is needed; the need calls forth the performance automatically, as the musician plays the note even before he hears it consciously. In mathematics, this aspect is less clear. But I've written elsewhere of the technique of immersing yourself in a problem, familiarising yourself with all aspects of it until it becomes your constant companion, until you cry it aloud in traffic jams and wake at night from bad dreams about it; and then letting it stew subconsciously. The final act in this play can also be helped. Sit down with a pencil and plenty of paper, start writing, and let it come out. Now I think the analogy is fairly clear. Again, anything you might want in the performance has bee thoroughly rehearsed, and you don't have to distract yourself thinking about what it is you may want. I have, to a very great extent, turned away from competition and back solely to training, since the Blues match debacle. (Now I feel again the stirrings of the desire to compete.) What is training without the expectation of competition? I think it is the first theme I introduced before: meditation. If meditation is training, then competition can only be enlightenment, except that, unlike the Zen monk, we are expected to produce it on demand! This can't be done, which is why competition has its unsatisfactory side (as "I could have got more out of myself" or "Today just wasn't a good day"). But, if you run without thought of competition, further, if you regard competition as no more than another training run, and each training run as demanding full concentration and exertion, then, if the analogy holds good, you will, sooner or laterm reach the moment corresponding to satori, the performance that takes you entirely out of yourself; and, because you can't fail to be a bit keyed up by competition, it is more likely to happen in a real race than on an everyday training run. I think that is roughly what I tried to summarise in a few words for Ivan on our way back from the dressing room in Calgary. I don't think all of that came across. Maybe what I said had some effect on him; I've had a letter from him in which he mentions trying to push his stride out a little, with some sucess. But it is more likely that the encounter will leave its mark on me. Already it has driven me back ito those old long-buried speculations (and almost certainly they appear here in slightly different guise). And, if nothing else, I've been out running three times since Christmas eve, despite a fearsome cold (shich now seems to be over, and I hope that the fates don't work in such a way that that comment tempts them beyond endurance!). The thing I've most noticed on these runs is the ability to keep going, at quite a good pace, despite the pain. All three runs were faster than I was doing even at my best last Spring, which is an encouraging sign if it reflects better motivation and not just the hangover of Californian fitness. And, from an entirely negative point of view: will the thought of Ivan going out running in a Calgary winter sustain me when conditions are bleak here and help keep me going out? Certainly I found it easier there than here, but it is difficult to disentangle the effect of the clear sky and sunshine from that of being a visitor with something to prove. -------------------------------------------------------------------------- 19/1/1985 No time for yoga tonight; they will be back at 11, or so they say. Neither do I seem greatly moved to fill this book with writing. I had intended to do some work but divided my time between Thomas Merton's biography and Andrew Penell's microdrive routines. An unsatisfactory wee has filled me with anger and despair once again, this terrible destructive process that gets me deeper into the mire each time. Sheila away in Shetland, walking on frozen peat-bogs under blue skies on an island where it rains on three-quarters of the days; the Kochs arriving on Thursday with a van not big enough to clear their possessions off the shelf; the children's refusal to go swimming that morning (defused eventually by the irrelevance that nobody went); frozen drainpipes; spending Saturday morning downtown shoe-shopping or watching "Ghostbusters"; my inability to find the necessary stuff for the accountant despite having carefully sorted it all out before going away. (This last probably the heaviest burden, worse even than the weight of editing sitting on my desk.) But if I can't even cope while on leave, however will I manage when I'm back in harness? I didn't write anything about my surroundings while in America. Tonight, I brought with me a little-used and early notebook on the trip. It contains five problems on infinite permutation groups, brief lecture notes on the amalgamation property, constructions, and normal subgroups of multiply-transitive groups; and in the back, notes on regular subgroups of the automorphism group of Woodrow's graph, a shopping list, some calculations of probabilities of all odds and all evens in a random sum-free set when the natural numbers are ordered 2,1,4,3,6,5,..., children's pocket-money accounts, and doodles, mainly of Union Jacks. For such a notebook of mine, remarkably sparse. I tell everyone that the time spent in Pasadena was productive for me. Was it? What exactly did I get underway? First and foremost, ubiquitous relational structures -- while I had considered this from a measure-theoretic viewpoint before (and got to the same point, more or less), both absolute and categorical ubiquity are new, as are their interconnections with logical concepts and (most importantly) the realisation of the significance of category here. There are some pointers, e.g. a comment by Alan Mekler on the relationship with finite forcing. But I'd like to be able to prove my conjecture on absolutely ubiquitous graphs (in a way, just a loose end, but one whose resolution would give important perspective). I got chunks of the book written. This is probably the most important, and I must now burden myself with the task of word-processing a chunk of it. Of the people there, time was too short to get anything of significance begun with anyone; Peter Frankl's problems are probably much too hard (if he can't solve them, who can?) and neither Michael's nor Phil's stuff had me hooked. The plane of order 10 was an unfortunate waste of time; good for some people perhaps, but not for me. Sum-free sets -- a technical problem, but at least it provided evidence that I could apply myself to a technical problem and make some progress on it. THis week, can I not begin to discipline myself to work? Divide my time between clearing my desk and typing the book. But I'm haunted by unfinished trivia -- evaluation of w(43) and of f(33), or further complete sum-free sets of residues. All these would be pleasure to investigate, hence distraction. I am very very sleepy. I don't think these slightly pessimistic night thoughts will help me out, so I'll read instead. -------------------------------------------------------------------------- 16/10/1985 After quite a long gap, tonight I have the inkling of a little, nibbling feeling of return to the things that drove me when I used to write in this book. Not self-analysis, thank God, but perhaps "a little bit of madness that is poetry". This feeling is, most likely, brought on by reading Peter Ackroyd's biography of T. S. Eliot. It's actually a very bad book in some ways, despite the good reviews. For one thing, it is so jumpy; by adopting a strict chronological approach, in which each page is headed by the year it discusses, large concerns are impossible, and each must be discussed briefly, discarded, and taken up again and again, giving a desperately fragmented account. Then again, he has taken a very narrow view of his subject; he rightly disparages attempts to read real events and people into the poems (though he does it with isolated images -- he suggests that "Dust in the air suspended Marks the place where a story ended" describes the aftermath of a bombing raid in the Blitz), but he makes little attempt to get inside the poetry at all. (And, I add, even when he does, I find his interpretations desperately unsatisfying.) Finally, of course, there is the imbalance between the discussion of Eliot's criticism, where the positions are stated in detail, and that of his poetry, restricted maily to externals such as its critical or popular reception. I had intended to re-read the poems at the chronologically appropriate points in the narative; but I couldn't find "Selected poems" until tonight, and I've now read up to the end of the war. So, in the event, I read "Four Quartets" aloud to myself. It was a different person reading it, and I heard it in a different way, and caught different echoes (for example, "yet being someone other", which someone recently -- Laurens van der Post? -- used as the title of an autobiography). One aspect that came across was the koan-like nature of many of the lines. As well as the deliberate paradoxes, I cite here the abrupt changes in feeling (such as "Go, go, go, said the bird: human kind Cannot bear very much reality") or, better, since it moves from the abstract to the concrete, "So the darkness shall be the light, and the stillness the dancing. Whispers of runing streams, and winter lighting" which recall something like "What is the Buddha?" "Three pounds of flax." Another echo, of something derived from Ackroyd. Earlier in the book he describes Eliot's conviction, apparently derived in part from Bradley, that outside a narrow circle all is formlessness and terror, and the integrity of the circle is only maintained by discipline and classical values. This void erupts very clearly in East Coker II. But this rather goes agains what has always, for me, been the crucial theme of the poem, namely, that "... to apprehend The point of intersection of the timeless With time, is an occupation for the saint" -- how does the saint do this, if his "... lifetime's death in love, Ardour and selflessness in self-surrender" is a device for maintaining his boundaries against the threatening void? But I prefer to take Eliot's words directly than to swallow Ackroyd's interpretation, and to believe that if Eliot held that belief (and there certainly is plenty of "formlessness" in the earlier poems), he had taken his advice that "Old men ought to be explorers" and faced the void more honestly by the time he wrote "Four Quartets". Certainly the Eastern elements of the poem (not only the Zen cited, but the characteristically Buddhist form taken by some of the discourses -- or is that from the Athenasia creed? -- the commentary on the Bhagavad-Gita, and the whole mood of the poems) reinforce this view. Look closely at the irruption of the void in East Coker. It follows the bit aboout the limitations of knowledge derived from experience, and the newness of every moment, and precedes another comment on the weakness of old men. Another thing struck me, a reminder of an earlier discovery of mine. When I was doing the quotes for the chapter-headings in "Parallelisms", ten years ago, for the chapter on edge-colourings I was forced to use "And hollyhocks that aim too high Red into grey and tumble down" because it is (I think) the only reference to colour in the whole sequence. The imagery is largely sound (especially music) and movement (dancing); roses have no colour and no smell, but are there only to transform into flame at the end (or indeed, to be burnt-out a little earlier); while the whole poem is full of things unseen, be they children in the shrubbery, waterfalls, or whatever. Back (reluctantly) to Ackroyd's book. The consistent picture which emerges is misery: ill health, a disastrous marriage, shyness, even hangovers, are all carefully charted. We only get hints of what prodded him into producing poetry, with a very strong implication that all the negativity in his life held him back. Is that really true? Walking up here tonight, street lights cut swathes through the fog lying on meadow and city; at the top of the bridge arch I passed a stranger. While I was an undergraduate, nearly twenty years ago, I was torn apart by reading "The New Poetic: Yeats to Eliot" by C. K. Stead. I don't really remember why. (I don't think I ever knew.) I don't believe it was just because he made negative remarks about a guiding beacon I'd just discovered. He described Eliot's "theory" of poetic composition, I think, and implied that this for of composition is seen in its purest in "The Waste Land", and is strait-jacketed by the conscious purpose in "Four Quartets". Perhaps I should look at this again. The first thing I feel inclined to say is that Ackroyd stresses that Eliot was very reluctant to be dogmatic about the meaning of his poetry (or, presumably, its purpose), and that in any case I find the critics' description of the conscious purpose to be at worst misguided, at best subsidiary and not compelling comared to what the poems have always communicated to me. Furthermore, look at the process of poetic composition. As I understand what I remember that Stead said that Eliot said, poetry bubbles up from the uconscious after it has been pressure-cooked in there for some time, but it was consciously put in there at first; also, unstructured bubblings may well require conscious supervision (and the extent of Eliot's revisions clearly suggests that he knew this). I believe that this is quite accurate and that, moreover, it equally describes mathematical creation. Indeed, I feel this so strongly that I deliberately attack research problems in just this way, consciously assimilating the problem in detail and then letting it lie fallow. But anyway, if Eliot worked this way, then the Quartets are not noticeably less "pure" than "The Waste Land", except perhaps for Little Gidding, which is perhaps my least favourite anyway. But why did Stead upset me so much then, while now Ackroyd has little effect? Some contributing factors: I'm more thick-skinned now, and oscillate less between the heights and the depths; Ackroyd steers well clear of the hazardous area of where Eliot's poetry came from, merely comparing it with defecation. (Would that have upset me then? I don't think so, and hope not.) Also, I'm now further from poetry myself. On the theory above, poetry would be expected to emerge some time after I had imprinted on a problem whose solution must be in poetic form. I haven't been doing that! On evenings like this, I find it quite irresistible to look back over past entries in the book. One impression is overwhelming, and he said it better than I can: "You say I am repeating Something I have said before. I shall say it again. Shall I say it again?" -------------------------------------------------------------------------- 26/10/1985 10:18pm, the night the clocks go back, and the toughest babysit I've ever had to do: children who, however appealing, lack the concept that people mean what they say and are not their slaves. So we had a solid hour of all-inn rioting, then three-quarters of an hour of reading bedtime stories, then a quarter of an hour of lower-key rioting, then a wet bed to change, and only now are they prepared to leave me alone (though still chattering away, not entirely amicably, up there). I was feeling good at odd moments today (though not when I was having lunch in Wimpy with the kids), but this has destroyed my mood. An ominous thing happened today. I ran for the third time a program designed to see whether there could be a finite set of integers with more sums than differences. The first run, in Pasca, found none in {0,...,n} with n <= 18; the second, in machine code, found none for n <= 23. But today I found one (in fact four) with n = 14, and several for larger values of n. Judging from my notes, I omitted a single crucial line of Pascal and two lines of assembler; rather than bringing everything to an undignified halt, they had the effect of biasing the outcome radically against sums. Of course, there is no doubt which is right; a single presumed example can easily be checked, and I have done so for this one. In addition, the last program was the only one I modified to list sets of zero discrepancy, and it did handle these correctly for small n, providing a comforting check. Now this program, since validated thus, can have its (presumed reliable) data used in testing further programs, for example the one I'm writing tonight, an entirely rewritten program in assembler to find the maximum discrepancy. But the general unreliability is worrying. This problem came indirectly fro complete sum-free sets. After Dominic found asymmetric examples, we wondered whether the proportion of odd (asymmetric) elements in such a set is boundsd by a positive fraction (less than one) of the size of the set. If S is complete sum-free, then of course every element outside S is a sum of two elements in S; but, if x in S is odd, the -x is not in S and is ot a difference; so it would suffice to prove that the excess of sums over differeces in a set mod n is at most a fraction (less than one) of the size of the set. Of course, if this is true, then it is true in the integers without modulus (or take counterexamples with sufficiently high modulus). I observed that this implies the much stronger statement that any set of integers has at least as many differences as sums. (Taking a counterexample and copying it into the residue classes congruent to 0 or 1 mod 3 doubles the size of the set, but triples the sizes of the sum and difference sets and so the discrepancy; repeating ad nauseum establishes the result.) The one minor triumph I've had lately has been writing a disc function in Pascal that allows compiled programs to access the disc; a version of Hisoft's TIN and TOUT. (Of course, {$F ...} would be more useful to me by far, but that is beyond me at present.) Having written a BASIC interface, trimmed for the very limited space below the Pascal, I ventured forth into the more elaborate project. I made a mistake. I'd lent the manual to John McSorley, and so I wasn't able to check anything -- it turned out that I was terminating INLIE code with an uncalled-for RET -- and it wasn't until just getting into bed in the second-best guest room at St John's College, Cambridge, on Wednesday night that I realised my mistake. But it was lucky, as it allowed me to correct two mistakes that were not my fault. First, the IY register must be loaded with ERR_SP before calling the BASIC simulation. Second, it turns up its nose at SAVE "..." CODE x,y, if x and y are numeral-valued strings, so I was forced to be ingenious and write a variables buffer to go alongside the BASIC buffer. Seems to work, so far! Anyway, and obviously, I'm so tired now that I can hardly keep my eyes open, and this house has no soft surfaces to recline, or even to sit, on -- even the chairs are hard and uncomfortable. But I had to knock off and prowl around the house for a while; for, as is obvious, I was not in control of what I was writing. This is a sensation I've had at various times. One of these was on my trip to Canada in summer 1984; bits of it are reflected in the trip diary, but other bits got scrubbed out. I would be describing the events of the day and, as my eyes closed, my pen would trace words which seemed to convey meaning of some sort, but were right off the point and quite unintelligible. One of these occasions blends, in memory, with the unreality of the scene I was describing at that moment, a lake where, looking straight down into the water, you saw that it was rust-red, whereas looking at an angle it was blue and reflecting. -------------------------------------------------------------------------- undated Still falling apart. I have actually managed, over the last week, to get a lot done -- JCMP business, JRF committee business (hampered by others who wouldn't answer their phones or reply to letters), admissions business, etc. -- and yet have failed to get the sense of achievement I deserve for this, because of things not done. Several papers are sitting on the word processor awaiting minor amendments before being submitted, and I can't find the time to get down and do them. I have written long letters to Michel Deza, Neil Calkin, Simon Thomas, but not quite finished them, so they sit around waiting to be sent. I've been slow marking first-year work, slow refereeing papers, slow producing problems for the second year; and it is all unutterably depressing. This book wasn't meant as a diary. Perhaps, by giving a detailed picture of how things stand at irregular intervals, it actually serves the function of a diary more effectively. But it was meant to consider topics. I think it fails to do that because, on an evening of babysitting, I lack the strength for any major project. Tonight I'm back at the (much better behaved!) Gresswells'. At 9:45, Julie came silently downstairs and into the room behind me, giving me qute a shock; but apart from that, it's absolutely soundless. They have, set up in the living room, an Atari 600XL computer (I don't know if it's new -- the asking price now must be almost nothing) connected to an ancient, blurred, colour television built into a wooden cabinet that looks more like a cocktail cabinet. (I almost said "reproduction" -- made in the image of a Georgian television set?) I played with it for a bit. No manuals or software were in evidence, so I couldn't persuade it to do graphics or sound. I tried a program, but it didn't like being told to RETURN, so I gave up in disgust. Neil Calkin had asked two questions about su-free sets. First, given a sum-free set of integers, is it necessarily contained in a complete sum-free set mod n for some n, and how large must n be? Second, what is the average size of a complete sum-free set mod n? This reminded me that I couldn't even say what is the expected density of a random sum-free subset of {1,...,n}, and I had waned to redo some of my old calculations on this question. -------------------------------------------------------------------------- 15/7/1986 Under the roof of the barn It was hot and airless. Dust swirled in the half-light The specks igniting As they crossed a beam of sunlight From a nail-hole in the galvanised iron roof. Far below, on the floor, The beam spilt an egg of light And I knew that, if a branch outside intervened Or if the moon took a bite out of the sun, The egg of light would show it faithfully. Eclipses were the best; Spots of light all over the floor All became identical crescents. Those nail-holes were peep-holes Into the great mystery of the universe. ---- For a while I sat in the gloom Listening to Beethoven's late quartets And Bartok's Hungarian sketches While writing machine code To find the period of sum-free sets. Then I turned on the light And wrote poetry. -------------------------------------------------------------------------- 19/7/1986 Tonight it's the Brewsters', where so far (after 3/4 hour) I haven't yet succumbed to the temptation to play the guitar or the piano, listen to music, read books, or look at pictures. I'm sure I will later. They are out for another 2 1/2 hours. Bad times. I haven't run for three days; I haven't done any yoga for months; I'm trying again with sum-free sets, writing long-runnning programs to try to find the periods. I'm beginning to have doubts about the truth of my periodicity conjecture. Using a more efficient method of coding, I've checked some of the sequences up to several hundred thousand without establishing periodicity. (Instead of storing 0s and 1s, I store the differences between successive 1s; this multiplies both space and time bounds by a factor equal to the density of the sequence.) But on the theoretical side I'm still totally stuck there. I'm also stuck on the enumeration question; I've decided that it should be possible to establish the square root of 2 as the exponential constant. (Looked at another way this says that the set of sum-free sets has Hausdorff dimension 1/2; from this point of view, we're getting near to things like fractals, chaos, etc., which seem to be the latest buzzwords. Also having got a repaired and upgraded disc interface from Technology Research, I've found that it leaves me no room for a program underneath the Pascal compiler to handle saving and loading of files, so I have to rewrite it in machine code and put it somewhere else; and then there is some problem about getting the new interface to work when called from machine code. At least TR have provided me with an alternative version which is supposed to work, though I haven't tried it yet. This sort of hacking is bad for the health! -------------------------------------------------------------------------- undated The story of my getting here tonight almost defies belief. Babysitting for the Gidneys, but with no contact with them; it was all arranged at second hand by Addy Gresswell. So, having checked their address on the babysitting list (21 Thorncliffe), I set off on my bike. First intimation of trouble -- the house in darkness. Nobody answered my ring. Could I have got the number wrong? Down to the Banbury Road end of Thorncliffe to find a phone box, either to check in the phone book, or to call Sheila to confirm the address. There was a phone box, without a queue. Went in. The handset had been ripped off. No book. Feeling of deja vu. (This morning I forgot my season ticket; the 7th phone box I tried actually worked. The others were either phonecard only, or 999 calls only.) So I hurried up to Summertown. Two phone boxes. Long queues outside both. No phone books. Turned away in disgust and who should walk past but Malcolm. He took me into NAG where he had a phone book in his office. The place was in darkness -- we used my bike lights to find the right keys. There they were in the book; the address, 52 Victoria. So I set off once again and belted up there. For the second time, the business of hunting house numbers. I actually knocked on the wrong door, number 50, to the puzzlement of the occupants. That was because 52 was also in pitch darkness, with For Sale signs up. I nearly despaired and went home. But there was a phone box at the end of Victoria Road, not vandalised. First problem -- it wouldn't take 5p's, and I had no 10's. Second problem: Sheila was on the phone. By the time I got a reversed charge call through to her, she had been speaking to John Gidney, and had discovered their new address, 102 Linkside, in other words, closer to our house than to my position at the time. But I jumped on and set off yet again. I was less than half an hour late. The Gidneys should have been abject, but they preferred to blame it on Addy. I think we cross them off the list of people I'm prepared to babysit for. I don't know how they do it, but they manage to have a house totally without any place to sit down. You have to stand to answer the phone, and if you sit on the floor leaning against the bed that passes for a sofa, you do your back in. -------------------------------------------------------------------------- Christmas 1990 A new pen -- a new start? This beautiful, but heavy, Waterman pen has at least one effect which is immediately apparent: it encourages me to write well! Perhaps because of its weight, I tend to leave it on the page, so that a cursive capital I in one stroke cmes more naturally thann a serifed I which requires three. The pen really needs longer acquaintance, but I think we'll get on well. It can use either cartridges or ink out of a bottle, with an ingenious screw-pump filling mechanism. I'm not really givinng this writing my full attention, since the beautiful "Powaqqatsi" is on television. More later, maybe. (later) That is all the television I want to watch over Christmas, though I very much doubt it will be all I do watch. That was excellent television: no dialogue, no plot, but the images cutting into one another created stories and moods; the photography was marvellous, and Philip Glass' music absolutely magic. Everything else on the box seems vapid by comparison. --------------------------------------------------------------------------