A completely axiomatic theory purports to make no appeal to experience whatsoever though one doubts whether any such expositions are quite as ‘pure’ as their authors claim. Even Hilbert’s 20th century formalist version of Euclid, his Grundlagen der Geometrie, has been found wanting in this respect ─ “A 2003 effort by Meikle and Fleuriot to formalize the Grundlagen with a computer found that some of Hilbert’s proofs appear to rely on diagrams and geometric intuition” (Wikipedia).
What exactly is the axiomatic method anyway? It seems to have been invented by the Greeks and in essence it is simply a scheme that divides a subject into :
(1) that part which has to be taken for granted in order to get started at all ─ in Euclid the Axioms, Definitions and Postulates (Note 1); and
(2) that part which is derived by valid chains of reasoning from the first, namely the Theorems ─ Heath calls them ‘Propositions’.
A strictly deductive, axiomatic presentation of a scientific subject made perfect sense in the days when Western scientists believed that an all-powerful God had made the entire universe with the aid of a handful of mathematical formulae but one wonders whether it is really appropriate today when biology has become the leading science. Evolution proceeds via random mutation plus ruthless selection and human societies and/or individuals often seem to owe more to happenstance and experience than reasoning and logic. Few, if any, important discoveries in mathematics have been strictly deductive: I doubt if anyone ever sat down of an evening with the Axioms of von Neumann Set Theory in order to deduce something interesting and original, and certainly no one ever learned mathematics that way (except possibly a robot). For all that, the structural simplicity and elegance of the axiomatic method remains extremely appealing and is one of the reasons why Euclid’s Elements and Newton’s Principia are among the half dozen best-selling books of all time ─ though few people read them today.
Apart from the axioms which are an integral part of a science or branch of mathematics, there exist also certain methodological principles (or prejudices) which, properly speaking, don’t belong to the subject, but nonetheless determine the general approach and overshadow the whole work. These principles should, ideally, be stated at the outset though they rarely are.
There are two principles that I find I have used implicitly or explicitly throughout my attempts to kick-start Ultimate Event Theory. The first is Occam’s Razor, or the Principle of Parsimony, which in practice means preferring the simplest and most succinct explanation ‘other things being equal’. According to Russell, Occam, a mediaeval logician, never actually wrote that “Entities are not to be multiplied without necessity” (as he is usually quoted as stating), but he did write “It is pointless to do with more what can be done with less” which comes to much the same thing. The Principle of Parsimony is uncontroversial and very little needs to be said about it except that it is a principle that is, as it were, imposed on us by necessity rather than being in any way ‘self-evident’. We do not really have any right to assume that Nature always chooses the simplest solution: indeed it sometimes looks as if Nature enjoys complication just for the sake of it. Aristotle’s Physics is a good deal simpler than Newton’s and the latter’s much easier to visualize than Einstein’s: but the evidence so far seems to favour the more complicated theory.
The second most important principle that I employ may be called the Principle of Parmenides, since he first stated it in its most extreme form,
“If there were no limits, there would be nothing”.
In the context of Ultimate Event Theory this often becomes:
“If there were no limits, nothing would exist, except (possibly) the Locality itself”
and the slightly different “If there were no limits, nothing would persist”.
This may sound unexceptional but what I deduce from this principle is highly controversial, namely the necessity to expel the notion of actual infinity from science altogether, and likewise in mathematics (Note 2). The ‘infinite’ is by definition ‘limitless’ and so falls under the ban of this very sensible principle. Infinity has no basis in our sense experience since no one, with the exception of certain mystics, has ever claimed to have ‘known’ the infinite. And mystical experience, though perfectly valid and apparently extremely enjoyable, obviously requires careful assessment before it can be introduced into a theory, scientific or otherwise. In the majority of cases, it is clear that what mystics (think they) experience is not at all what mathematicians mean by the sign ∞ but is rather an alleged reality which is ‘non-finite’ in the sense that any form of measurement would be totally inappropriate and irrelevant. (It is closer to what Bohm calls the Implicate Order as opposed to the Explicate Order ─ unhappy names for a very useful dichotomy). In present-day science, ‘infinity’ simply functions as a sort of deus ex machina (Note 3) to get one out of a tight spot, and even then only temporarily. As far as I know, there is not a scrap of evidence to suggest that any known process or observable entity actually is either ‘infinitely large‘ or ‘infinitely small’. All energy exchanges are subject to quantum restrictions (i.e. come in finite packages) and all sorts of entities which were once regarded as ‘infinitely small’ such as atoms and molecules can now actually be ‘seen’, if only via an electron tunnelling microscope. Even the universe we live in, which for Newton and everyone else alive in his time, was ‘infinite’ in size, is sometimes thought today to have a finite current extent and is certainly thought to have a specific age (around 13.8 billion years). All that is left as a final bastion of the infinity delusion is space and time and even here one or two noted contemporary physicists (e.g. Lee Smolin and Fay Dowker) dare to suggest that the fabric of Space-Time may be ‘grainy’. But enough on this subject which, in my case, tends to become obsessive.
What can an axiomatic theory be expected to do? One thing it cannot be expected to do is to give specific quantitative results. Newton showed that the law of gravitation had to be an inverse square distance law but it was some time before a value could be attributed to the indispensable gravitational constant, G. And Eddington quite properly said that we could conclude simply by reasoning that in any physical universe there would have to be an upper bound for the speed of a particle or the transmission of information, but that we would not be able to deduce by reasoning alone the actual value of this upper bound (namely c ≈ 108 metres/second).
It is also legitimate, even in a broadly axiomatic presentation, to appeal to common experience from time to time, provided one does not abuse this facility. For example, de Sitter’s solution of Einstein’s field equations could not possibly apply to the universe we (think we) live in, since his solution required that such a ‘universe’ would be entirely empty of matter ─ which we believe not to be the case.
One would, however, require a broadly axiomatic theory to lead, by reasoning alone, to some results which, as it happens, we know to be correct, and also, if possible, to make certain other predictions that no rival theory had made. And a theory which embodies a very different ‘take’ on life and the world might still prove worthwhile stating even if it is destined to be promptly discarded: it might prepare the ground for other, more mature, theories by pointing in a certain unexpected direction. Predictive power is not the only goal and raison d’etre of a scientific theory : the old Ptolemaic astronomy was for a long time perfectly satisfactory as a predictive system and, according to Koestler, Copernicus’s original heliocentric system was no simpler. As a piece of kinematics, the Ptolemaic Earth-centred system was adequate and, with the addition of more epicycles could probably ‘give the right answer’ even today. However, Copernicus’s revolution paved the way for Galileo’s and Newton’s dynamical world-view in which the movements of planets were viewed in terms of applied forces and so proved far more fruitful.
It is also worth saying that a different world-view from the current established one may remain more satisfactory with respect to certain specific areas, while being utterly inadequate for other purposes. If one is completely honest, one would, I think, have to admit that the now completely discredited magical animistic world-view has a certain cogency and persuasiveness when applied to aberrant human behaviour: this is why we still talk meaningfully of charm, charisma, inspiration, luck, jinxes, fascination, fate ─ concepts that belong firmly to another era.
Finally, the world-views of other cultures and societies are not just historical curiosities : people in these societies had different priorities and may well have noticed, and subsequently sought to explain, things that modern man is unaware of. Ultimate Event Theory has its roots in the world-views of societies long dead and gone: in particular the world-view of certain Hinayana Buddhist monks in Northern India during the first few centuries of our era, and that of certain Native Amerindian tribes like the Hopi as reflected in the
structure of their languages (according to the Whorf-Sapir theory).
SH 26/09/19
Notes :
Note 1 The status of the fourth and last Euclidian subsection, the Definitions, is not entirely clear: they were supposed to be ‘informative’ only in the manner of an entry in a dictionary and “to have no existential import”. On the other hand, Russell concedes that “definitions are often nothing more than disguised axioms”.
Note 2 This is in line with Poincare’s categorical statement, “There is, and can be, no actual infinity”. Gauss, often considered the greatest mathematician of all time, said something similar.
Note 3 A deus ex machine was , in Greek tragedy, a supernatural being who was lowered onto the stage by a sort of crane and whose purpose was to ‘save’ the hero or heroine when no one else could.
Larry Constantine, in an insightful letter to the New Scientist (13 Aug 2011 p. 30), wrote : “Accounting for our universe by postulating infinite parallel universes or explaining the Big Bang as the collision of “branes” are not accounts at all, but merely ignorance swept under a cosmic rug — a rug which itself demands explanation but is in turn buried under still more rugs.”
Dharma 