Archives for category: Calculus

CALCULUS

“He who examines things in their growth and first origins, obtains the clearest view of them” Aristotle.

Calculus was developed mainly in order to deal with two seemingly intractable problems: (1) how to estimate accurately the areas and volumes of irregularly shaped figures and (2) how to predict physical behaviour once you know the initial conditions and the ‘rates of change’.
We humans have a strong penchant towards visualizing distances and areas in terms of straight lines, squares and rectangles ― I have sometimes wondered whether there might be an amoeba-type civilization which would do the reverse, visualizing straight lines as consisting of curves, and rectangles as extreme versions of ellipses. ‘Geo-metria’ (lit. ‘land measurement’) was, according to Herodotus, first developed by the Egyptians for taxation purposes. Now, once you have chosen a standard unit of distance for a straight line and a standard square as a unit of area, it becomes a relatively simple matter to evaluate the length of any straight line and any rectangle (provided they are not too large or too distant, of course). Taking things a giant step forward, various Greek mathematicians, notably Archimedes, wondered whether one could in like manner estimate accurately the ‘length’ of arbitrary curves and the areas of arbitrarily shaped expanses.

At first sight, this seems impossible. A curve such as the circumference of a circle is not a straight line and never will become one. However, by making your unit of length progressively smaller and smaller, you can ‘measure’ a given curve by seeing how many equal little straight lines are needed to ‘cover’ it as nearly as possible. Lacking power tools, I remember once deciding to reduce a piece of wood of square section to a cylinder using a hand plane and repeatedly running across the edges. This took me a very long time indeed but I did see the piece of wood becoming progressively more and more cylindrical before my eyes. One could view a circle as the ‘limiting case’ of a regular polygon with an absolutely enormous number of sides which is basically how Archimedes went about things with his ‘method of exhaustion’ (Note 1).

It is important to stop at this point and ask under what conditions this stratagem is likely to work. The most important requirement is the ability to make your original base unit progressively smaller at each successive trial measurement while keeping them proportionate to each other. Though there is no need to drag in the infinite which the Greeks avoided like the plague, we do need to suppose that we can reduce in a regular manner our original unit of length indefinitely, say by halving it at each trial. In practice, this is never possible and craftsmen and engineers have to call a halt at some stage, though, hopefully, only when an acceptable level of precision has been attained. This is the point historically where mathematics and technology part company since mathematics typically deals with the ‘ideal’ case, not with what is realizable or directly observable. With the Greeks, the gulf between observable physical reality and the mathematical model has started to widen.

What about (2), predicting physical behaviour when you know the initial conditions and the ‘rates of change’? This was the great achievement of the age of Leibnitz and Newton. Newton seems to have invented his version of the Calculus in order to show, amongst other things, that planetary orbits had to be ellipses, as Kepler had found was in fact the case for Mars. Knowing the orbit, one could predict where a given planet or comet would be at a given time. Now, a ‘rate of change’ is not an independently ‘real’ entity: it is a ratio of two more fundamental items. Velocity, our best known ‘rate of change’, does not have its own unit in the SI system ― but the metre (the unit of distance) and the second (the unit of time) are internationally agreed basic units. So we define speed in terms of metres per second.

Now, the distance covered in a given time by a body is easy enough to estimate if the body’s motion is in a straight line and does not increase or decrease; but what about the case where velocity is changing from one moment to the next? As long as we have a reliable correlation between distance and time, preferably in the form of an algebraic formula y = f(t), Newton and others showed that we can cope with this case in somewhat the same way as the Greeks coped with irregular shapes. The trick is to assume that the supposedly ever-changing velocity is constant (and thus representable by a straight line) over a very brief interval of time. Then we add up the distances covered in all the relevant time intervals. In effect, what the age of Newton did was to transfer the exhaustion procedure of Archimedes from the domain of statics to dynamics. Calculus does the impossible twice over: the Integral Calculus ‘squares the circle’, i.e. gives its area in terms of so many unit squares, while the Differential Calculus allows us to predict the exact whereabouts of something that is perpetually on the move (and thus never has a fixed position).

For this procedure to work, it must be possible, at least in principle, to reduce all spatial and temporal intervals indefinitely. Is physical reality actually like this? The post-Renaissance physicists and mathematicians seem to have assumed that it was, though such assumptions were rarely made explicit. Leibnitz got round the problem mathematically by positing ‘infinitesimals’ and ultimate ratios between them : his ‘Infinitesimal Calculus’ gloriously “has its cake and eats it too”. For, in practice, when dealing with an ‘infinitesimal’, we are (or were once) at liberty to regard it as entirely negligible in extent when this suits our purposes, while never permitting it to be strictly zero since division by zero is meaningless. Already in Newton’s own lifetime, Bishop Berkeley pointed out the illogicality of the procedure, as indeed of the very concept of ‘instantaneous velocity’.

The justification of the procedure was essentially that it seemed to work magnificently in most cases. Why did it work? Calculus typically deals with cases where there are two levels, a ‘micro’ scale’ and a ‘macro scale’ which is all that is directly observable to humans ― the world of seconds, metres, kilos and so on. If a macro-scale property or entity is believed to increase by micro-scale chunks, we can (sometimes) safely discard all terms involving δt (or δx) which appear on the Right Hand Side but still have a ‘micro/micro’ ratio on the Left Hand Side of the equation (Note 2). This ‘original sin’ of Calculus was only cleaned up in the late 19th century by the key concept of the mathematical limit. But there was a price to pay: the mathematical model had become even further away removed from observable physical reality.

The artful concept of a limit does away with the need for infinitesimals as such. An indefinitely extendable sequence or series is said to ‘converge to a limit’ if the gap between the suggested limit and any and every term after a certain point is less than any proposed non-negative quantity. For example, it would seem that the sequence ½; 1/3; ¼……1/n gets closer and closer to zero as n increases, since for any proposed gap, we can do better by making n twice as large and 1/n twice as small. This definition gets round problem of actual division by zero.

But what the mathematician does not address is whether in actual fact a given process ever actually attains the mathematical limit (Note 3), or how near it gets to it. In a working machine, for example, the input energy cannot be indefinitely reduced and still give an output, because there comes a point when the input is not capable of overcoming internal friction and the machine stalls. All energy exchange is now known to be ‘quantized’ ― but, oddly, ‘space’ and ‘time’ are to this day still treated as being ‘continuous’ (which I do not believe they are). In practice, there is almost always a gulf between how things ought to behave according to the mathematical treatment and the way things actually do or can behave. Today, because of computers, the trend is towards slogging it out numerically to a given level of precision rather than using fancy analytic techniques. Calculus is still used even in cases where the minimal value of the independent variable is actually known. In population studies and thermo-dynamics, for example, the increase δx or δn cannot be less than a single person, or a single molecule. But if we are dealing with hundreds of millions of people or molecules, Calculus treatment still gives satisfactory results. Over some three hundred years or so Calculus has evolved from being an ingenious but logically flawed branch of applied mathematics to being a logically impeccable branch of pure mathematics that is rarely if ever directly embodied in real world conditions.                                         SH

 

 

 

Note 1 It is still a subject of controversy whether Archimedes can really be said to have invented what we now call the Integral Calculus, but certainly he was very close.

Note 2 Suppose we have two variables, one of which depends on the other. The dependent variable is usually noted as y while the independent variable is, in the context of dynamics, usually t (for time). We believe, or suppose, that any change in t, no matter how tiny, will result in a corresponding increase (or decrease) in y the dependent variable. We then narrow down the temporal interval δt to get closer and closer to what happens at a particular ‘moment’, and take the ‘final’ ratio which we call dy/dt. The trouble is that we need to completely get rid of δt on the Right Hand Side but keep it non-zero on the Left Hand Side because dy/0 is meaningless ― it would correspond to the ‘velocity’ of a body when it is completely at rest.

Note 3   Contrary to what is generally believed, practically all the sequences we are interested in do not actually attain the limit to which they are said to converge. Mathematically, this does no9t matter — but logically and physically it often does.

 

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Space-Time

Minkowski, Einstein’s old teacher of mathematics, inaugurated  the hybrid ‘Space-Time’ which is now on everyone’s lips. In an address delivered not long before his death in 1908 he said the now famous lines,

“Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”

         But why should Minkowski, and whole generations of scientists, have ever thought that ‘space’ and ‘time’ could be completely separate in the first place? Certain consequences of a belief in ‘Space-Time’ in General Relativity do turn out to be  scarcely credible, but there is nothing weird or paradoxical per se about the idea of ‘time’ being a so-called fourth dimension. To specify an event accurately it is convenient to give three spatial coordinates which tell you how far the occurrence of this event is, or will be, along three different directions relative to an agreed  fixed point.  If I want to meet someone in a city laid out like a grid as New York is (more or less), I need to specify the street, say Fifth Avenue, the number of the building and the floor (how high above the ground it is). But this by itself will not be enough for a successful meet-up : I also need to give the time of the proposed rendez-vous, say, three o-clock in the afternoon. The wonder is, not that science has been obliged to bring time into the picture, but that it was possible for so long to avoid mentioning it (Note 1)

Succession

 Now, if you start off with ‘events’, which are by definition ‘punctual’ and impermanent, rather than things or ‘matter’ you cannot avoid bringing time into the picture from the start: indeed one  might be inclined to say that ‘time’ is a good deal more important than space. Events happen ‘before’ or ‘after’ each other; what happened yesterday preceded what happened this morning, and you read the previous sentence before you started on the current one. The very idea of ‘simultaneous’ events, events that have occurrence ‘at the same time’, is a tricky concept even without bringing Special Relativity into the picture. But  the idea of succession is both clearcut and basic and one could, as a first bash, even define ‘simultaneous’ events negatively as bona fide  occurrences that are not temporally ordered.

So, when I started trying to elaborate an ‘event-orientated’  world-view, I felt I absolutely had to have succession as a primary ingredient : if anything it came higher up the list than ‘space’. Originally I tried to kick off with a small number of basic assumptions (axioms or postulates) which seemed absolutely unavoidable. One such assumption was that most events are ‘ordered temporally’, that they have occurrence successively, ‘one after the other’ ─ with the small exception of so-called ‘simultaneous events’. Causality also seemed to be something I could not possibly do without and causality is very much tied up with succession since it is usually the prior event that is seen as ‘causing’ the other event in a causal pair. Again, one might tentatively  defined ‘simultaneous events’ as events which cannot have a direct causal bond, i.e. function as cause and effect (Note 2). And,  in an era innocent of Special Relativity and light cones, one might well define space as  the totality of all distinct events that are not temporally ordered.

From an ‘event-based’ viewpoint,  chopping up reality into ‘space’ and ‘time’ is not fundamental : all we require is a ‘place’ where events can and do have occurrence, an  Event Locality. Such a Locality starts off empty of events but has the capacity to receive them, indeed I have come to regard ultimate events as in some sense concretisations or condensations of an underlying substratum.

 Difference between Space and Time

 There is, however, a problem with having a single indivisible entity whether we call it ‘Space-Time’ or simply ‘the Locality’. The two parts or aspects of this creature are not at all equivalent. Although I believe, as some physicists have suggested, that, at a certain level, ‘space’ is ‘grainy’, it certainly appears to be continuous : we do not notice any dividing line, let alone a gap, between the different spatial ‘dimensions’ or between different spatial regions. We don’t have to ‘add’ the dimension height to pre-existing dimensions of length and width for example : experience always provides us with a three-dimensional physical block of reality (Note 3). And the fact that the choice of directions, up/down, left/right and so on, is more often than not completely arbitrary suggests that physical reality does not have inbuilt directions, is ‘all-of-a-piece’.

Another point worth mentioning is that we seem to have a strong sense of being ‘at rest’ spatially : not only are we ‘where we are’, and not where we are not, but we actually feel this to be the case. Indeed we tend to consider ourselves to be at rest even when we know we are moving : when in a train we consider that it is the other things, the countryside, that are in motion, not us. It is indeed this that gives Galileo’s seminal concept of inertia its force and plausibility; in practice all we notice is a flagrant disturbance of the ‘rest’ sensation, i.e. an ‘acceleration’.

What about time? Now it is true that time is often said to ‘flow’ and we do not notice any clearcut temporal demarcation lines any more than we notice spatial ones. Nonetheless, I would argue that it is much less natural and plausible to consider ‘time’ as a continuum because we have such a strong sense of sequence. We continually break up time into ‘moments’ which occur ‘one before the other’ even though the extent of the moment varies or is left vague. Sense of sequence is part of our world and since our impressions are themselves bona fide events even if only subjective ones, it would appear that sequence is a real feature of the physical world. There is in practice always an arrow of time, an arrow which points from the non-actual to the actual. Moreover, the process of ‘actualization’ is not reversible : an event that has occurrence cannot be ‘de-occurred’ as it were (Note 4).

And it is noteworthy that one very seldom feels oneself to be ‘at rest’ temporally, i.e. completely unaware of succession and variation. The sensation is so rare that it is often classed as  ‘mystical’, the feeling of being ‘out of time’ of which T.S. Eliot writes so eloquently in The Four Quartets. Heroin and certain other drugs,  by restricting one’s attention to the present moment and the recent past, likewise ‘abolish time’, hence their appeal. In the normal way,  even when deprived of all external physical stimuli, one still retains the sensation of there being a momentum and direction to one’s own thoughts and inner processes : one idea or internal sensation follows another and one never has any trouble assigning order (in the sense of sequence) to one’s inner feelings and thoughts. It is now thought that the brain uses parallel processing on a big scale but, if so, we are largely unaware of this incessant multi-tasking. Descartes in his thought experiment of being entirely cut off from the outside world and considering what he simply could not doubt, might well have concluded that sequence, rather than the (intemporal) thinking ego, was the one item that could not be dispensed with. For one can temporarily disbelieve in one’s existence as a particular person but not in the endless succession of thoughts and subjective sensations that stream through one’s mind/brain.

All this will be dismissed as belonging to psychology rather than physics. But our sense impressions and thoughts are rooted in our physiology and should not be waved aside for that very reason : in a sense they are the most important and inescapable ‘things’ we have for without them we would be zombies. Physical theories that deny sequence, that consider the laws of physics to be ‘perfectly reversible’, are both implausible and seemingly  unliveable, so great is our sense of ‘before and after’. Einstein towards the end of his life decided that it followed from General Relativity that everything happened in an ‘eternal present’. He took this idea seriously enough to mention it in a letter to the son of his college friend, Besso, on receiving news of the latter’s death, writing “For those of us who believe in physics, this separation between past, present and future is only an illusion, however tenacious”.

Breaks in Time

 If, then, we accept succession as an unavoidable feature of lived reality, are we to suppose that one moment shifts seamlessly into the next without any noticeable demarcation lines, let alone gaps? Practically all physicists, even those who toy with the idea that Space-time is in some sense ‘grainy’, seem to be stuck with the concept of a continuum. “There is time, but there is not really any notion of a moment in time. There are only processes that follow one another by causal necessity” as Lee Smolin puts it in Three Roads to Quantum Gravity..

But I cannot see how this can possibly be the case, and this is precisely why the ‘time dimension’ of the Event Locality is so different from the spatial one. If I shift my attention from two items in a landscape, from a rock and its immediate neighbourhood to a tree, there is no sense that the tree displaces the rock : the two items can peaceably co-exist and do not interfere with each other. But if one moment follows another, it displaces it, pushes it out of the way, as it were, since past and present moments, prior and subsequent events, cannot by definition co-exist ─ except perhaps in the inert way they might be seen to co-exist in an Einsteinian perpetual now. And all the attributes and particular features of a given moment must themselves disappear to make way for what follows. We do not usually see this happening, of course, because most of the time the very same objects are recreated and our senses do not register the transition. We only notice change when a completely different physical feature replaces another one, but the same principle must apply even if the same feature is recreated identically. Since a single moment is, in its physical manifestation, three-dimensional, all these three dimensions must go if a new moment comes into being.

Whether there is an appreciable gap between moments apart from there being a definite change is an open question. In the first sketch of Ultimate Event Theory I attribute a fixed extent to the minimal temporal interval, the ksana, and I allow for the possibility of flexible gaps between ksanas. The phenomenon of  time dilation is interpreted as the widening of the gap between ksanas rather than as an extension of the ‘length’ of a ksana itself. This feature, however, is not absolutely essential to the general theory.

What we actually perceive and consider to constitute  a ‘moment’ is, of course, a block containing millions of ksanas since the length of a ksana must be extremely small (around the Planck scale). However, it would seem that ksanas do form blocks and that there are transitions between blocks and that sometimes, if only subliminally, we are aware of these gaps. Instead of being a flowing river, ‘time’ is more like beads on a string though the best image would be a three-dimensional shape pricked out in coloured lights that is switched on and off incessantly.

Mosaic Time

Temporal succession is either a real feature of the world or it is not, I cannot see that there is a possible third position. In Einstein’s universe “everything that can have occurrence already has occurrence” to put things in event terms. “In the ‘block universe’ conception of general relativity….the present moment has no meaning ─ all that exists is the whole history of the universe at once, timelessly. When laws of physics are represented mathematically, causal processes which are the activity of time are represented by timeless logical implications…. Mathematical objects, being timeless, don’t mhave present moments, futures or pasts”  (Lee Smolin, It’s Time to Rewrite time in New Scientist 20 April 2014)  

This means that there is no free will since what has occurrence cannot be changed, cannot be ‘de-occurred’. It also makes causality redundant as Lee Smolin states. One could indeed focus on certain pairs of events and baptise them ‘cause and effect’ but, since they both have occurrence, neither of them has brought the other about, nor has a third ‘previous’ event brought both of them about simultaneously. Causality becomes of no account since it is not needed.

Even a little acquaintance with Special Relativity leads one to conclude that it is impossible to establish a universally valid ‘now’. Instead we have the two light cones, one leading back to the past and one to the future (the observer’s future), and a large region classed as ‘elsewhere’. It is notorious that the order of events in ‘elsewhere’, viewed from inside a particular light cone, is not fixed for all observers : for one observer it can be said that event A precedes event B and for another that event B precedes A. This indeterminacy if of little or no practical consequence since there is (within SR) no possibility of interaction between the two regions. However, it does mean that it is on the face of it impossible to speak of a universally valid ‘now’ ─ although physicists do use expressions like the “present state of the universe”.

I personally cannot conceive of a ‘universe’ or a life or indeed anything at all without succession being built into it : the timeless world of mathematics is not reality but a ‘take’ on reality. The only way to conceptually save succession while accepting some of the more secure aspects of Relativity would seem to be to have some sort of ‘mosaic time’, physical reality split up into zones. How exactly these zones, which are themselves subjective in that they depend on a real or imagined ‘observer’, fit together is not a question I can answer though certain areas of research into general relativity can presumably be taken over into UET.  One could perhaps define the next best thing to a universal ‘now’ by taking a weighted average of all possible time zones : Eddington suggested something along these lines though he neglected to give any details. Note that if physical reality is a mosaic rather than a continuum, it would in principle be possible to shift the arrangement of particular tesserae in a small way, exchange one with another and so on.                     SH 23/01/15

 

 Note 1 Time was left out of the picture for so long, or at any rate neglected, because the first ‘science’ to be developed to a high degree of precision in the West was geometry. And the truths of (Euclidian) geometry, if they are truths at all, are ‘timeless’ which is why Plato prized geometry above all other branches of knowledge except philosophy. Inscribe a triangle in a circle with the diameter as base line and you will always find that it is right-angled. And if you don’t, this is to be attributed to careless drawing and measurement : in an ‘ideal’ Platonic world such an angle has to be a right angle. How do we know? Because the theorem has been proved.

This concentration on space rather than time meant that although the Greeks set out the basic principles  of statics, the West had to wait another 1,600 years or so before Galileo more or less invented the science of dynamics from scratch. And the prestige of Euclid and the associated static view of phenomena remained so great that Newton, perversely to our eyes, cast his Principia into a cumbrous geometrical mould using copious geometrical diagrams even though he had already invented a ‘mathematics of motion’, the Calculus.

 Note 2   Kant did in point of fact defend the idea of ‘simultaneous causation’ where each of two ‘simultaneous’ events affects the other ‘at the same time’. He gave the example of a ball resting on a cushion arguing that the ball presses down on the cushion for the same amount of time as the cushion is deformed by the presence of the ball. And if we take Newton’s Third Law as operating exactly at the same time on or between two different objects, we have to accept the possibility of simultaneous causation.

Within Ultimate Event Theory, what would normally be  called ‘causality’ is (sometimes) referred to as ‘Dominance’. I chose this term precisely because it signifies an unequal relation between two events, one event, referred to as the ‘cause’, as it were ‘dominating’ the other, the ‘effect’. In most, though perhaps not all, cases of causal relations I believe there really is priority and succession despite Newton’s Third Law. I would conceive of the ball pressing on the cushion as occurring at least a brief moment before its effect ─ though this is admittedly debatable. One could introduce the category of ‘Equal Dominance’ to cover cases of  Kant’s ‘simultaneous causality’ between two events.

Note 3  I have always found the idea of Flatland, which is routinely trotted out in popular books on Relativity, completely unconvincing. I can more readily conceive of there being more than three spatial dimensions as there being a world with less than three : a line, any line, always has some width and height.

 Note 4. If it is possible for an event in the future to have an effect ‘now’, this can only be because the ‘future’ event has already somehow already occurred, whereas intermediate events between ‘now’ and ‘then’ have not. I cannot conceive of a ‘non-event’ having any kind of causal repercussion — except, of course, in the trivial sense that current wishes or hopes about the future might affect our behaviour. Such wishes and desires belong to the present or recent past, not to the future.

 

 

The Rise and Fall of Atomism

So-called ‘primitive’ soceties by and large split the world into two, what I call the Manifest (what we see, hear &c.) and the Unmanifest (what we don’t but in some way seem to be aware of). For the ‘primitives’ everything originates in the Unmanifest, especially the really drastic and inexplicable changes like earthquakes, storms, floods, but also more everyday but nonetheless mysterious occurrences like giving birth, changing food by heating it, growing up, dying. The Unmanifest is much more important than the Manifest and the shaman, or his various successors, the ‘sage’, ‘prophet’, ‘initiate’, and so forth,  claims to have special knowledge because he or she has ready access to the Unmanifest which normal people do not. Ultimately, a single principle or ‘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 them more, or less, successful. If the cheetah can run faster than all other animals, it is because the cheetah has more mana and the same goes for the racing car; it is because he has more mana that a warrior wins a contest, because a young woman has more mana than her rivals that a young girl  has more suitors, and so on. Charm and charisma are watered down modern versions of mana.
Our civilization 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 required if one wanted to understand the universe and its physical processes. 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 settles themselves were seemingly subject to certain  ‘laws’ or ‘natural principles’ such as the Law of the Lever or the Principle of Flotation, both clearly stated by the brilliant Archimedes.  A new symbolic language, that of higher mathematics, was needed to talk about such things since the “Book of Nature is written in the language of mathematics” as Kepler, a Renaissance successor and admirer of the Greeks,  put it. Geometry stipulated the basic shapes and forms to which the groups of atoms were confined when they settled down — and so successfully that, since the invention of the high definition microscope, ‘Platonic solids’ and other fantastical shapes studied by the Greeks can actually be seen 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 unaccelerated 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 and Helmholtz had declared that “all physical problems were reducible to mechanical problems” and thus, in principle, solvable by applying Newton’s Laws. As to everything else, not only spirituality but even human thoughts and emotions, the implication was (and Hobbes and La Mettrie even spelled this out categorically) that such matters were also ultimately reducible to “matter and motion” and that it was only a matter of time before everything would be completely explained.
The twentieth century has at once affirmed and destroyed the atomic hypothesis. Affirmed it because molecules and atoms have been shown to exist and can even be ‘seen’ on an electron microscope. They are, moreover, undoubtedly involved in most, possibly all, physical processes including mental processes. However, atoms have turned out not to be indestructible or even indivisible as the early scientists supposed.  Atomism and materialism have, by a curious circular route, led us back to a place not so very far from our original point of departure since the new 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 signifies in physics ‘Potential Work’, i.e. Work which could be done but is not actually being done, where Work has the precise meaning, Force × distance moved in the direction of the applied force. (We are nonetheless constantly assured in popular and not so popular 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 Bishop Berkeley 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 one might do better starting with something else. What? On the fringes of science and philosophy, there was for a while a certain resurgence of ‘animism’ in the form of 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 clear how such notions can be represented symbolically by mathematical (or other) symbols and 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 force.
More recently, there have been several attempts to present and attempt to elucidate the universe in terms of ‘information’, even to view it as a vast computer (though one wonders quite how literally we are supposed to take this). 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 computing : one hopes that the universe or whatever is behind it has better things to do than simply pile up and sift information like the Super Brains of Olaf Stapledon’s remarkable fantasy Last and First Men.

The Event

During the Sixties and Seventies, at any rate within the booming counter-culture, there was, for a while, the feeling that the West had somehow ‘got it wrong’ and was leading everyone towards disaster with its obsessive emphasis on material things. 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’ seemed more to the point than the Eighteenth century movement of the same name which had 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 confirmation  so far.
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 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 difficult as such but because its principles and procedures were so 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 in the neighbourhood of a particular moment in time, then reducing this interval as much as possible. And the velocity of the same body is computed by supposing its position to be fixed (stationary) during a particular small interval of time and then reducing this interval likewise. In effect, ‘classical’ Calculus had its cake and eating it too —  something we all like doing if we can get away with it — by setting (δx) at non-zero and zero simultaneously in 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, it is irrelevant whether or not a particular function actually attains  a given limit (assuming it exists) just so long as it approaches it more and more closely. (For more specific details see a future post on my other website www.originsofmathematics.com). But what anyone with an enquiring mind wants to know is whether in reality the moving arrow actually attains its goal or the closing door actually slams shut (to use two examples mentioned by Zeno). 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 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. I don’t see how one can quarrel with the logic of that, Calculus or no Calculus.
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 suddenly convinced that ‘time’ was not continuous, as it is always assumed to be in mathematics — indeed I had always felt it to be a succession of moments —  and that there was indeed a minimal possible ‘interval of time’ and which, moreover, was absolute and not dependent on the position or motion of an observer. I was heartened when, subsequently, I read that nearly two thousand years ago, certain Indian thinkers had advanced the same supposition and even, in one or two cases, 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 as will be examined in due course. The essential was that I suddenly had the barebones of a physical schema : ‘reality’ was composed of  events, not of objects, and these events were decomposable into ‘ultimate events’ which had a fixed spatial and temporal extension. Ultimate Event Theory was born, though it has taken me some twenty-five years 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)