Archives for category: Wolfram

The phenomenon of time dilation, though not noticeable in everyday circumstances, is not a mathematical trick but really exists. Corrections for time dilation are made regularly to keep the Global Positioning System from getting out of sync. The phenomenon becomes a good deal more comprehensible if we consider a network of ultimate events which does not change and spaces between then which can and do change. We are familiar with the notion that ‘time speeds up’ or ‘slows down’ when we are elated or anxious : the same, or very similar occurrences, play out differently according to our moods. Of course, it will be pointed out that the distances between events do not ‘actually change’ in such cases, only our perceptions. But essentially the same applies to ‘objective reality’. or would apply if our senses or instruments were accurate enough the selfsame events would slow down or speed up according to our viewpoint and relative state of motion.
RELATIVITY TIME DILATION DIAGRAMThis could easily be demonstrated by making a hinged ‘easel’ or double ladder which can be extended at will in one direction without altering the spacing in the other, lateral, direction. The ‘time dimension’ is down the page.The stars represent ultimate events, light flashes perhaps which are reflected back and forth in a mirror (though light flashes are made of trillions of ultimate events packed together). The slanting zig-zag line connects the ultimate events : they constitute an event-chain. By pulling the red right hand side of the double ladder outwards and extending it at the same time, we increase the difference between the ultimate events on this part of the ladder but do not increase the ‘lateral’ distance. These events would appear to an observer on the black upright plane as ‘stretched out’ and the angle we use represents the relative speed. As the angle approaches 90 degrees, i.e. the red section nearly becomes horizontal, the red part of the ladder has become enormously long. Setting different angles shows the extent of the time dilation for different relative speeds.
Note that in this diagram the corresponding space contraction is not shown since it is not this spatial dimension that is being contracted (though there will still be a contraction in the presumed direction of motion). We are to imagine someone flashing a torch across a spaceship and the light being reflected back. Any regular repeating event can be considered to be a ‘clock’.
What such a diagram does not show, however, is that, from the point of view of the red ladder, it is the other event-chain that will be stretched : the situation is reversible.

The idea for this diagram is not original : Stefan Wolfram has a similar more complicated set of diagrams on p. 524 of his great work A New Kind of Science. However, he makes the ‘event-lines’ continuous and does not use stars to mark ultimate events.  More elaborate models could actually be made and shown in science museums to demonstrate time dilation. There is, I think, nothing outrageous in the idea that the ‘distance between two events’ is variable : as stated we experience this ourselves all the time. What is shocking is the idea of the whole of Space/Time contracting and dilating. Ultimate events provide as it were the skeleton which shows up in an X-ray : distances between events are flesh that does not show up. There is ‘nothing’ between these events, nothing physical at any rate.        SH 


The Rise and Fall of Atomism

So-called ‘primitive’ societies by and large split the world into two, what one might call the Manifest (what we see, hear &c.) and the Unmanifest (what we don’t perceive directly but intuit or are subliminally aware of). For the ‘primitives’ everything originates in the Unmanifest, especially drastic and inexplicable changes like earthquakes, sudden storms, avalanches and so on,  but also more everyday but nonetheless mysterious occurrences like giving birth, changing a substance by heating it (i.e. cooking), growing up, aging, dying. The Unmanifest is understandably considered to be much more important than the Manifest — since the latter originates in the first but not vice-versa — and so the shaman, or his various successors, the ‘sage’, ‘prophet’, ‘initiate’ &c. claims to have special knowledge because he or she has ready access to the Unmanifest which normal people do not.  The shaman and more recently the priest is, or claims to be, an intermediary between the two realms, a sort of spiritual marriage broker. Ultimately, a single principle or ‘hidden force’ drives everything, what has been variously termed in different cultures mana, wakanda, ch’i ….  Mana is ‘what makes things go’, in particular what makes individuals more, or less, successful. If the cheetah can run faster than all other animals, it is because the cheetah has more mana ; if a warrior wins a power contest, it is because he has more mana, if a young woman has many more suitors than her rivals it is because she has more mana and so on. Charm and charisma are watered down modern versions of mana and, like mana, are felt to originate in the beyond, in the non here and now, in the Unmanifest. This ancient dualistic scheme is far from dead and is likely to re-appear in the most unexpected places despite the endless tut-tutting of rationalists and sceptics; as a belief system it is both plausible and comprehensible, even conceivably it contains a kernel of truth. As William James put it, “The darker, blinder strata of character are the only places in the world in which we catch real fact in the making”.
Our own Western civilization, however,  is founded on that of Ancient Greece (much more so than on ancient Palestine). The Greeks, the ones we take notice of at any rate, seem to have been the first people to have disregarded the Unmanifest entirely and to have considered that supernatural beings, whether they existed or not, were not needed if one wanted to understand the physical universe — basic natural processes properly understood sufficed (Note 1). Democritus of Abdera, whose works have unfortunately been lost,  kicked off a vast movement which has ultimately led to the building of the Hadron Particle Collider, with his amazing statement, reductionist if ever there was one, Nothing exists except atoms and void.

Atoms and void, however, proved to be not quite enough to describe the universe : Democritus’s whirling atoms and the solids they composed when they settled themselves down were seemingly subject to certain  ‘laws’ or ‘general principles’ such as the Law of the Lever or the Principle of Flotation, both clearly stated in quantitative form by Archimedes.  But a new symbolic language, that of higher mathematics, was required to talk about such things since the “Book of Nature is written in the language of mathematics” as Kepler, a Renaissance successor and great admirer of the Greeks,  put it. Geometry stipulated the basic shapes and forms to which the groups of atoms were confined when they combined together to form solids — and so successfully that, since the invention of the high definition microscope, ‘Platonic solids’ and other fantastical shapes studied by the Greek geometers can actually be seen today embodied in the arrangement of molecules in rock crystals and in the fossils of minute creatures known as radiolarians.

To all this Newton added the important notion of Force and gave it a precise meaning, namely the capacity to alter a body’s state of rest or constant straight line motion, either by way of contact (pushes and pulls) or, more mysteriously, by  ‘attraction’ which could operate at a distance through a vacuum. Nothing succeeds like success and by the middle of the nineteenth century Laplace had famously declared that he had “no need of that hypothesis”  — the existence of God — to explain the movements of heavenly bodies while Helmholtz declared that “all physical problems are reducible to mechanical problems” and thus, in principle, solvable by applying Newton’s Laws. Why stop there? The dreadful implication, spelled out by Hobbes and La Mettrie,  was that human thoughts and emotions, maybe life itself,  were also ultimately reducible to “matter and motion” and that it was only a question of time before everything would be completely explained scientifically.

The twentieth century has at once affirmed and destroyed the atomic hypothesis. Affirmed it because molecules and atoms, at one time considered by most physicists simply as useful fictions, can even be ‘seen’ with an electron tunnelling microscope and substances ‘one atom thick’ like graphene are actually being manufactured, or soon will be. However, atoms have turned out not to be indestructible or even indivisible as Newton and the  early scientists supposed.  Atomism and materialism have, by a curious circuitous route, led us back to a place not so very far from our original point of departure since the 20th century scientific buzzword, ‘energy’, has disquieting similarities to mana.  No one has ever seen or touched ‘Energy‘ any more that they have ever seen or touched mana. And, strictly speaking, energy in physics is ‘Potential Work’, i.e. Work which could be done but is not actually being done. (‘Work’ in physics has the precise meaning, Force × distance moved in the direction of the applied force.) Energy is thus not something actual at all, certainly not something perceptible by the senses or their extensions. Yet we are nonetheless often assured in popular (and not so popular) science books that “at bottom the universe is radiant energy”, whatever that means.

The present era thus exhibits the contradictory tendencies of being on the one hand militantly secular and ‘materialistic’ both in the acquisitive and the philosophic senses of the word, while the basis of all this development, good old solid ‘matter’ composed of  “hard, massy particles” (Newton)  and “extended bodies” (Descartes) has all but evaporated. When he wished to refute the idealist philosopher, Bishop Berkeley, Samuel Johnson famously kicked a stone, but it would seem that the Bishop  has had the last laugh.

A New Starting Point?

Since the wheel of thought concerning the physical universe has, in a sense, more or less turned full circle, a few brave souls have wondered whether, after all, ‘atoms‘ and ‘extended bodies’ were not the best starting point, and that one might do better starting with something else. But what? There was in the early 20th century a certain resurgence of ‘animism’ on the fringes of science and philosophy,  witness Bergson’s élan vital (‘Life-force’) , Dreisch’s ‘entelechy‘ and similar concepts. The problem with such theories is not that they are implausible — on the contrary they have strong intuitive appeal — but that they seem to be scientifically and technologically sterile, since it is not at all clear how such notions can be represented symbolically by mathematical (or other) symbols, let alone tested in laboratory conditions.
Einstein pinned his faith on ‘fields‘ and went so far as to state that “matter is merely a region where the field is particularly intense”. However, his attempt to unify physics was unsuccessful : unsuccessful for the layman because the ‘field‘ is an elusive concept at best, and unsuccessful for the physicist because Einstein never did succeed in unifying mathematically the four basic physical forces, gravity, electro-magnetism and the strong and weak nuclear forces.
More recently, there have been one or two valiant attempts to present and attempt to elucidate the universe in terms of ‘information’, even to view it as a vast computer. As far as I am concerned the weakness of such an approach is that it is so crudely anthropomorphic, projecting onto the universe the current human fascination with the latest technical invention: one hopes that the universe, or whatever is behind it, has better things to do than simply pile up and sift endless amounts of information like the Super Brains of Olaf Stapledon’s remarkable SF fantasy The Last and First Men.

The Event

During the Sixties and Seventies, at any rate within the booming counter-culture, there was a certain feeling that the West had somehow ‘got it wrong’ and was leading everyone towards disaster with its obsessive emphasis on material things and material explanations. The principal doctrine of the hippie movement was that experiences were more important than possessions — and the more outlandish the experiences the better.  Zen-style ‘Enlightenment’ suddenly seemed much more appealing than the Eighteenth century movement of the same name which spearheaded Europe into the secular, industrial era. A few physicists, such as Fritjof Capra, argued that, although classical physics was very materialistic, modern physics “wasn’t like that” and had strong similarities with the key ideas of eastern mysticism. However, though initially attracted, on further examination I found modern physics with wave/particle duality, quantum entanglement and uncertainty everywhere rather too weird, and what followed after, String Theory, completely unintelligible and for that matter devoid of the slightest experimental confirmation  so far.
Nonetheless, moving towards middle age, I realized with increasing alarm, given the highly technological era I had the misfortune to be born into,  that I had entirely forgotten all the (very) elementary mathematics I had reluctantly learned at school and set about remedying this.  I had no trouble with geometry and (whole) Number Theory, the Greek sciences, but found Calculus a major stumbling block, not because it was especially difficult as such but because its principles and procedures were completely unreasonable. D’Alembert is supposed to have said to a student who expressed some misgivings about manipulating infinitesimals, “Allez à l’avant; la foi vous viendra” (“Keep going, conviction will follow”), but in my case it never did. Typically, the acceleration (change of velocity) of a moving body is computed by supposing the velocity of the body to be constant during a certain short interval in time, we then reduce this interval ‘to the limit’ and, hey presto! we have the derivstive.  But if the particle is always moving, it is not at any fixed location, ever. In effect, ‘classical’ Calculus has its cake and eats it too —  something we all like doing if we can get away with it — since it sets (δx) to non-zero and zero simultaneously on opposite sides of the same equation. ‘Modern’, i.e. post mid nineteenth-century Calculus, solved the problem by the ingenious concept of a ‘limit’, the key idea in the whole of Analysis. Mathematically speaking, it is irrelevant whether or not a particular function actually attains  a given limit (assuming it exists) just so long as it approaches closer than any desired finite quantity. (For more specific details see a future post on my other website But what anyone with an enquiring mind wants to know is whether in reality the moving arrow actually attains its goal or whether the closing door ever actually slams shut (to use two examples mentioned by Zeno of Elea). As a matter of fact in neither case do they attain their objectives according to Calculus, modern or classical,  since, except in the most trivial case of a constant function, ‘taking the derivative’ involves throwing away non-zero terms on the Right Hand Side which, however puny, we have no right to get rid of. In any case, as Zeno of Elea pointed out over two thousand years ago, if the body is in motion it is not at a specific point, and if  situated exactly at a specific point, is not in motion. Calculus, traditional or modern, was, I decided, either unreasonable or unrealistic and no amount of teaching or reading has convinced me otherwise.
     This whole issue can, however, be easily resolved by the very natural supposition (natural to me at any rate) that intervals of time cannot be indefinitely diminished and that motion consists of a succession of stills in much the same way as a film we see in the cinema gives the illusion of movement. Calculus only works, inasmuch as it does work, if the increment in the independent variable is ‘very small’ compared to the level we are interested in, and the more careful textbooks warn the student against relying on Calculus in cases where the minimum size of the independent variable is actually known — for example  in molecular thermo-dynamics where it cannot be smaller than that of a single molecule.
All this will be examined in detail later but suffice it to say that I was immediately and entirely convinced that ‘time’ was not continuous, as it is always assumed to be in mathematics — indeed I had myself always felt it to be a succession of discrete moments —  and that there must be a minimal  ‘interval of time’ which, moreover, was absolute and did not depend on the position or motion of an imaginary observer. I was heartened when, in my random reading, I learned that nearly two thousand years ago, certain Indian Buddhist thinkers had advanced the same supposition and even apparently attempted to give an estimate of the size of such an ‘atom of time’. More recently, Whitrow, Stefan Wolfram and one or two others, have given estimates of the size of a chronon  based on the Planck limit. The essential was that I was suddenly relieved of an immense weight of senseless dogmatism concerning ‘continuous motion’, ‘infinitesimals’ and all the rest of it,  not to mention the insanities of Cantor’s imaginary ‘transfinite sets’. Not only that, I had the barebones of a physical schema that certain Indian sages had roughly mapped out thousands of years ago: ‘reality’ was at bottom composed of  events, not of objects, and these events were decomposable into ‘ultimate events’ (that they called dharmas) which had a fixed spatial and temporal extension. The world is the totality of events and not of things — Ultimate Event Theory was (re)born, though it has taken me decades to pluck up the courage to put the theory into the public domain, so enormous is the paradigm shift involved in these few innocuous sounding assumptions.    S.H. (Tuesday 28 June 2011)

Note 1 There exists, however, an extremely scholarly (but nonetheless readable) book, The Greeks and the Irrational by E.R. Dodds, which traces the history of a counter-current in Greek civilisation from Homer to Hellenistic times. The author, a professor of Greek and a one time President of the Psychical Research Society, asked himself the question, “Were the Greeks in fact quite so blind to the importance of non-rational factors in man’s experience and behaviour as is commonly assumed both by their apologists and by their critics?” The book in question is the result of his ponderings on the issue.