Special Issue 64: The Limits of Mathematics

This edition deals with the various limitations of Mathematics from a variety of different scientific and philosophic angles, and features a fantastic guest paper by Abdul Malek, a Theoretical Physicist and Dialectician from Montreal, Canada.

It has taken me many decades to realise quite how limited Mathematics really is. I have the advantage of having been a gifted mathematician long before I switched to Physics. I made that significant change because Mathematics is a purely descriptive abstract discipline, of a very special type, and I wanted to really understand things rather than merely describe them in abstract form.

Unfortunately, as we shall see, Physics has become little more than an extension of Idealist Mathematics. Physics was converted into a Pluralist Science of Stabilities: and one driven idealistically by Purely Formal Laws.

No wonder it is in an untranscendable terminal impass as a Science! Indeed, we can legitimately go a great deal further, and insist that it no longer investigates Reality-as-is, but instead can only deliver a distorted formal reflection of that World: it is an investigation of Ideality - the infinite World of Pure Forms alone: the Abstract Realm of Mathematics.

In short, Physics can only be saved via a wholesale rethinking of Mathematics and how we use it.

The Limits of Mathematics

 

What is Form?

Sacred geometry?

Form: We think we know what it is!

And, we certainly know how to use it.

We have reached a remarkable sophistication in how we develop it.

Indeed, these positions have even led to it being generally promoted to delivering the essential Primary Causes in all of all of concrete Reality - and even, indeed, to the establishment of the Idealist philosophical standpoint, and the escalation of form's nature-and-laws, in Mathematics, to the position of Queen-of-the-Sciences, from its real position as its pragmatically useful Handmaiden.

But, it delivers, quite clearly, a wrong, or perhaps more accurately, a mistaken, path, overlaid-upon-and-hiding the more essential Route-to-Truth, and achieved by man's remarkable intelligence-and-ingenuity in purposely adjusting real situations to conform artificially, though never perfectly, to the consequent ideas and techniques of such a stance.

Early Man's intelligence coupled with his first intellectual tenet of "If it works, it is right!" - termed Pragmatism, had already equipped him to spread across the whole accessible World, as a successful hunter/gatherer. And, it has also underpinned all subsequent intellectual development, as it still does, to this day.

Initially, developments were slow in coming, as Man's only source of tools was sharp-edged, brittle-but-hard Flint, and his succession of discovered "cultures" were almost entirely to do with how he knapped pieces of Flint to produce variety and effectiveness in those tools, and, it took around 90% of Man's entire existence as a separate species (over 170,000 years) to arrive at the Neolithic Revolution. which began to change everything, culminating around 7,000 years ago with the first Human Civilisations.

By some 2,500 years ago, in Ancient Greece, the first entirely new addition to man's intellectual toolkit was, in fact, the recognition and study of Form.

It took the study of shapes into a wholly new level with what we now term Euclidian Geometry, and this in turn led to a whole series of other disciplines suggested by the success of that initial development of Mathematics.

It was in this following explosion of intellectual pursuits that Form, as dealt with in Mathematics, became essential. Indeed Formal Logic even includes that basis in its title.

But, in spite of its efficacy when coupled with a now well-developed Pragmatism and consequent effective technical achievements, it had a major flaw! It literally never worked perfectly in Reality-as-is!

All situations in Reality were clearly the results of many often-contending causes, and it took the union of Pragmatism-with-Form to begin to bring aspects of the real World into Man's better control. Situations were purposely simplified by removing as many contending-and-confusing factors as possible, and, thereafter, maintaining that control while the now clearly-revealed Form was extracted and studied.

Reality was firmly "pinned down" all the better to study it! BUT, was that studied-aspect exactly the same, when so artificially-isolated, as it was when embedded with others in Reality-as-is? The consensus decision was that it was indeed the same. So, implicitly, a crucial Enabling Principle had been inadvertently assumed.

It is, in fact, the Principle of Plurality: and it states that all factors, acting together, are independent of one another - they are indeed totally independent of all containing contexts. But, that is , most certainly, not true!

And, that fact is proved by successful use: for the extracted Form can ONLY be used successfully, if the very conditions of extraction are adequately replicated in use. And, Man, being a consummate pragmatist, could always achieve that requirement, and hence effectively use what had been revealed.

In spite of Plurality being incorrect, its inadequacies could be overcome by pragmatically optimised conditions. Technology was saved, but Science was sorely-damaged!

This was because extracted Forms were raised to the status of eternal Natural Laws, and that was quite-definitely untrue. And, consequently in Explanatory Theory such supposedly "eternal laws" were brought together in dealing with complex situations as totally FIXED components, and that was incorrect.

The edifice of "scientific explanations" was based upon an unsound premise and would, therefore, inevitably lead to both contradictions and impasse situations.

The philosopher Hegel had been right!

Though he was talking about thought, it definitely applied in Science Theory too, and his revelation of Dichotomous Pairs of totally contradictory concepts, revealed by unbridgeable impasses in Formal Reasoning, was true not only about Formal Logic, but also about all pluralistic Scientific Theory too.

A whole panoply of intellectual disciplines arose with exactly the same flaw, but only in the West. In India and ultimately in the East generally, the Buddha at about the same time as the ancient Greeks had implicitly assumed Plurality, had assumed Holism instead. The Principle of Holism took the opposite assumption - that "Everything affects everything else!" And, without any doubt, there is a greater content of Truth in this alternative.

Complex situations were seen as composed of multiple, always-varying factors, and though temporary dominances were indeed possible, and could last for significant periods of time, there always was an underlying dynamic which would, at some point, lead to dramatic changes, and crucially Developmental Changes too.

It was indeed, if an appropriate, investigative methodology could be developed, the ONLY basis for scientific Theory. The ultimate model of the pluralist view was Laplace's Clockwork Universe.

While, that of the developed holist view was closer to an Evolving Universe!

So, can we say what exactly the "Worship-of-Form" actually leads to, and why it has triumphed for so long?

As it turns out, we must answer the second question first. Form is true only when a single factor is acting in a simplified situation! It reveals a pristine-relation caused by a single physical cause. So, it is a legitimate extraction in such a situation.

BUT, even there, it is never primary, it is the result of the quite different and definitely primary Cause. And, this shows the Universality-of-Form: for the very same form can emerge in very differently-caused situations - and those actual causes do NOT have to be connected, just the resultant form is one of those which regularly occur all over the place! And hough, they can be used for description-and-prediction, they can never be used as the Sufficient Causes.

Now, why is this the case?

All versions of situations involving a particular cause are clearly related: but they are inevitably changed by other causal-factors acting-simultaneously. They therefore posess a family likeness: every one will be similar to the pristine-single-cause version - that is why they are all assumed to be the SAME! The pristine version is the Ideal Version of an adaptable Form.

As such it is fixed, but never happens as such in Reality. Indeed, the whole of Mathematics restricts itself solely to these Ideal Versions! And, all theorems, manipulations, and the rest are only true of those. We call that World Ideality: it is NOT Reality!

So, the process of finding-and-fitting measured data to a Mathematical Form is not only a simplification - due to the absolutely-necessary Farming-of-Context, but is also an idealisation of the actually-performing, modified-cause in the real situation. And, this puts major limitations upon systems of using these "fixed Laws" in production. Each law is limited to its own ideal context, so all productions necessarily involve sequences of ideal contexts, for each and every "law" used: it can be made to work, but only via this "factory-like system"! And, crucially, if those context-conditions are allowed to change, even marginally, or even if the process in a given context goes on for too long, it can cease to be appropriate, due to tiny and consequently uncontrolled factors, and the involved "Law" ceases to deliver, so the process inevitably fails.

Have you worked in a factory, if so you'll know exactly what I mean!

Now, in 1927 at the Solvay Conference, Niels Bohr and Werner Heisenberg unveiled their Copenhagen Interpretation of Quantum Theory, in which, to solve the problems of numerous impasses and anomalies in Sub Atomic Theory, ordained that Physical Explanations had not only failed in this area, but were also objectively impossible to take any further at the Sub Atomic Level, and proffered, instead, a wholly-mathematical-approach that could be made to successfully predict outcomes. 

Their main opponent was Albert Einstein, but though he argued for physical Explanations as paramount, he was also committed to mathematics as evidenced by his General Theory of Relativity, and the Copenhagenists won the day.

But, in the light of what I have revealed within this essay, it must be clear that this retreat was doomed from its outset. No wonder they had to invent Wave/Particle Duality, superposition and the illegitimate use of probabilities upon "simultaneous states of the dualist object", not to mention Quantum Entanglement and the Collapse of the wave function due to measurement.

With their gravely flawed philosophical stance, absolutely NO solution is possible. And, those errors are not just confined to Copenhagenists, but also to their Classical opponents too, and even those around de Broglie and Bohm's positions. The various amalgams of Idealism, Materialism and Pragmatism have to go, and the cornerstone of Plurality dumped for a Holist alternative.

Now that is, admittedly, a very Big Ask, but it has already commenced with the work of this theorist, Jim Schofield, to apply Hegel's solution to flawed premises in order to successfully solve every-single-one of the anomalies in the whole set of Double Slit Experiments. And, he did it by re-instituting the key omitted premise - the presence of a currently undetectable Universal Substrate throughout the Universe.

Remarkably, this assumption not only tackled those experiments, but also made possible the trascending of many other unexplained aareas such as quantised orbits within atoms and Electromagnetic propagation, as well as the subtending od concrete fields in what is usually assumed to be totally Empty space, and even Pair Productions and Pair Annihilations.

The "holistic" experimenter, Yves Couder, has even achieved quantised orbits in media at the macro level, and proved the role of such substrates as both general Sinks and Sources to provide energy in active situations, when its origin by the usual stances deliver nothing at all.

And, previous work by the holists Charles Darwin in his Origin of Species, and Stanley Miller in his Experiment to produce amino acids from an emulation of the Earth's Primeval Atmospheric processes, are both under review. First to extend Darwin's Natural Selection backwards into Non-living chemical reactions. And, secondly, with Miller's experiment to re-design it in order to monitor its internal developments as they occur.

Finally, a major and significant holist-dialectical critique of the whole History and development of philosophic Stances of Science is nearing completion.

The era of the dominance of Pure Form is finally coming to an end!

There Are No Laws of Nature!

There is a lot of evidence to suggest that this quote is misattributed to Euclid, though it is certainly an old phrase, and is the natural conclusion of the idealist view of natural law

All Ideal Laws of Nature are Bunk!

Exactly how then are natural behaviours caused?

"Is Quantum Mechanics involved in this area of study?" is a frequently asked question - but what does such a question actually imply? For example, in a recent news item in New Scientist magazine, a researcher wonders if an, as-yet-unknown, Law of Conservation is the reason for the stability of the proton.

Now, such a question from an actively working scientist is truly incredible! For, what is clearly assumed is that Reality behaves in the way that it does, in certain particular areas of study, entirely due to being directed to do so by purely abstract Laws.

For, that is exactly what is not only the basic assumption of the current interpretation of Quantum Physics, but within that consensus tradition, it is also asserted, in tandem, that a total rejection of Explanatory Theories is essential too. Instead of such Explanatory Theories, Formal Probabilistic Laws are, alone, supposed to actually cause observed behaviours, if that peculiar stance is accepted!

Now, is such a stance not clearly idealist?!

And, to make it even more ridiculous, a cursory consideration of the means Mankind uses to extract such "laws", clearly indicates that those means are such as to only-ever deliver an arranged-for result: hence the so-called Natural Laws are only extracted from very restricted, filtered and even "farmed" contexts - man-arranged Domains, which are expressly designed to suppress, remove or limit most present relations, to more clearly deliver only a single, targeted-for relation.

Hence, they are NOT the eternal Natural Laws that they are assumed to be.

They are currently existing relations, but ONLY in that specific, supplied context: elsewhere they will most certainly be different.

Now, long ago, this kind of idealist nonsense was allowed to be imported into Science via the devising of a special Enabling Principle. It isn't usually overtly stated, or even admitted, but the Principle of Plurality was long ago built, irremovably into the foundations of the standard approach, to make the above assumptions "true".

For, Plurality insisted that all individual relations within Reality are totally fixed: they don't change at all, so all the "farming" of contexts cannot change those relations, they can only effectively reveal them. But, I'm afraid, it just isn't true!

Innumerable examples can be given to show that the exact opposite principle - that of Holism, is much more generally applicable, and shows that Plurality, in fact, only holds in particular natural or man-made Stable situations: everywhere-else Holism is clearly evident, and simply must be assumed if qualitative changes, and the true creative development in nature, are ever to be adequately addressed.

So, these aspects imported an idealist strand into Man's investigations of concrete Reality. And, on revealing certain patterns within these stable, man-devised Domains, allowed them to be fitted up to pure formal patterns from Mathematics, evident in the measured data collected, to succinctly describe a particular found and isolated relation.

These Formulae were only ever Descriptions: they could, clearly, never be Explanations!

But, they had a very seductive property - they could be used to predict certain future outcomes reasonably accurately. And, this ability to be able to predict was extremely convincing to those who couldn't do such things, and the "predictor" was given high status, even though he often had absolutely NO idea why Reality behaved in such a way.

So, these so-called Natural Laws, which are assumed to drive Reality, are, actually, no such thing at all. They merely describe a pattern of simple, uneventful changes. Much more investigation will always be necessary to begin move towards answering the much more important question, "Why?"

Clearly, such formulae deliver NOT Natural Causative Laws, but, merely, widely-applicable and purely Formal Rules - very different things indeed! Remember the ONLY possible "explanation" delivered by the equation alone is, "Obeys this Rule!"

Not much of an explanation is it?

Now, it may well be asked, "Why do such clearly idealist "laws" ever fit Reality?". Well, two things must be made clear in answering that question.

FIRST: All such patterns must have arisen from material causes, but only if these particular causes are acting ALONE! That is why in scientific investigations, the primary task is to severely limit the Context, in an attempt to restrict the causation factors to a single one! Only if this is done, can such an ideal formal relation be revealed.

But, SECOND: The investigators invoke the Principle of Plurality, which, as explained earlier, assumes that sets of such "ideal laws" simply sum, to give the things we naturally observe in totally unfettered Reality!

And that assumption is WRONG!

Remember it isn't man-devised "Laws" that make things happen: it is physical causes, and these are always changed by context - in other words,

Reality is not Pluralist: it is Holist!

So, you cannot assume that an ideal law, revealed by extreme isolation, will remain exactly the same in complex situations - allowing simple addition of such laws to be assumed. They will, most certainly, have changed by physical causes in the combined context. The usually assumed Natural fixed Laws are transforming idealisations of the actual holist complexity!

Is Reason Merely Formal Logic? (Part 2)

Hill Arches by Henry Moore

Quantitative and Qualitative Changes

Though I didn’t realise during my formal education, Physics also vigorously excluded all areas involving any Qualitative Change. This was obscured by the obvious meticulous attention to quantitative change, which, in fact, seemed to be its primary concern, and most experiments were attempts to distil-out the most important quantitative relations between the significant factors involved. So, though such forms of Change were easily accommodated, both on the mathematical, and the physical, models of Reality, when things changed to become something else, the system avoided them like the plague. It simply was not able to deal with such phenomena.

Ultimately, of course, a more rigorous treatment of the Nature and Weaknesses of these types of Reason had to be addressed. And, such a treatment would have to do TWO COMPLETELY opposite things. It had to explain WHY the system worked so well (as in Euclid), and yet failed so profoundly as in the Copenhagen Interpretation of Quantum Physics (for example).

Somehow, such a study would have to deal with the “truth-within-falsity”! I can express the problem in no more appropriate way!

But, there were the to-be-expected confidences of youth beginning to grasp the World for the first time. The trajectory of the studies of my maturity turned out to be long and full of necessary detours.

Such a path was unavoidable, because I insisted on a multi-discipline approach. Instead of my writings being outpourings of my specialist studies, they had to be a search in themselves, and were, therefore, no synopsis of where I was, but, on the contrary, a running commentary on how I was travelling, and what I was discovering. My chosen path regularly took me into areas that I knew little about, but were obviously crucial to my chosen task. I, therefore, regularly dedicated whole periods to subjects, in which I had no experience, and these “detours” were perhaps THE most important part of my studies and profoundly changed my conceptions.

The best example of a similar, dramatic process elsewhere that I can give for this, was embodied in the excellent BBC TV series by Aubrey Manning entitled Earth Story, in which he, a life-long biologist, had his world “illuminated” by a serious study of Modern Geology. It transformed his understanding, and changed the “tempo” of his view.

In a similar way, I, a Physicist and Mathematician, have had my theoretical foundations regularly reformulated as a consequence of serious studies in Biology, History, Palaeontology, Philosophy, Pedagogy, Art, Marxian politics, Computing and many others.

Such a width of increasing knowledge and understanding cannot but undermine your narrow, purely discipline-based conceptions of the World. You are driven to an attempted synthesis, and this brings you constantly up against your often narrowly-based and unquestioned assumptions. You “realise” the short cuts and invalid extrapolations of your cherished methodologies, and you can only end up as a generalist rather than a specialist – what can only be called a philosopher.

The Crucial Inadequacies in Pluralist Reasoning

Let us relate this adult stage of my investigations. After a long knowledge-gathering phase, I started on my journey with the Paradoxes of Zeno.

Initial attempts were to “solve” his presented contradictions, and my skills in Mathematics convinced me that I could “explain away” his confusing conclusions. But what I discovered were the limitations of my own techniques and the profundity of his purposes. Throughout all the time since my first finding Zeno, I have regularly returned to these Paradoxes for a reappraisal, in the light of my ever-widening other studies. For surprisingly the thoughts of this Greek who lived 2,500 years ago, I found to resonate in all areas of human study. Zeno revealed the most important weaknesses of Reason, which until his revelations, no-one had addressed.

He showed that the two most obvious principles used in dealing with Space and Time were both inadequate alone. These were the “obvious” assumptions of Continuity and Descreteness, and both could be shown to lead to contradiction.

NOTE: Though I could give chapter and verse on these many occurrences, it would interrupt the flow of my overall argument here. So, I will, at this time mention only one area where the Paradoxes of Zeno transformed the attitudes of my students. I found it essential to introduce them into my teaching of the Calculus.

NOTE: I will NOT again be drawn into the innumerable “disproofs” of Zeno’s methods using modern mathematics and “even-more-modern” philosophical positions. He, after all, lived in the sixth century BC. Reading the achievements of the past from a NOW standpoint almost always throws out the baby along with the bathwater. The constant recurrence of Zeno’s Paradoxes is due to the fact that they contain profound truths and these MUST be addressed in every context, not circumvented.

Zeno of Elea

Not only was this work of Zeno profoundly important, but it also carried with it the opposite truth, that both these assumptions, and the Reason, based upon one or the other, could also, and did, produce profoundly true and useable extractions. Such methods were both true and yet clearly, and at the same time, also inherently flawed.

They were necessary and yet incorrect!

Now, needless to say, such contradictory conclusions wedded together in the same method (that of Zeno), did not entice queues of thinkers to embrace Zeno’s ideas. They usually simply ignored his Paradoxes as clever, spoiling tactics, and continued with their very recently discovered and obviously wonderful and dependable methods, taking care to use them ONLY in amenable areas of application.

“While he’s worrying about That, let us get on with the innumerable possibilities of This!”, was their attitude. A thousand descrete successes were more important than a single profound but almost unintelligible error.

Returning to Euclid’s Theorems (and Mathematics in general) what were his assumptions and premises?

Euclid assumed perfectly straight lines of zero thickness. They would be conceived of on perfectly flat planes of infinite extent. Any identified position on such a plane – a point – would also be of zero extent. Parallel Lines (evidently keeping the same distance apart) were pointing in the exact same direction: they were at 0o to one another and would never cross.

Now, do these assumptions truly reflect Reality? The answer is a clear “NO!”.

Yet, the whole structure turns out to be extremely useful, and the imperatives of the methodology so persuasive, that it is clear that this obvious “fiction” somehow delivered Truth of a kind!

Now, we can easily drop into a simplistic argument as to why this was the case with Euclid’s work.

We can say that Truth “on a certain scale” was embodied in the theorems. The thickness of the lines and dots could be disregarded, when distance, shape and geometric relations were the significant elements to be considered. Euclid’s simplifying assumptions were “empowering”, in that they threw away irrelevances in the search for higher spatial relations. And, of course, all this is true!

BUT, Mankind did NOT see it that way at the time!

The general consensus, among the users of this body of Knowledge, was that the essence of geometric reality had been extracted from the mire or blurring of inconsequential everyday Reality. The process was one of revealing the Essences out of which Reality was built!

But, that was certainly NOT what had been achieved!

Even at this early stage in Mankind’s development of methods of Thought, he had, in fact, invented the idea of a Model. In order to deal with Reality, an intelligent set of simplifications were always necessary. Any old simplifications would NOT do however. The removals had to be of detachable things which were not significant in the given area of study, and the revelations of its embedded truths.

Mankind learned that the division of Reality into areas, systems and indeed Parts, was useful.

He had unconsciously "invented" Plurality – the division of anything into its supposedly-independent component Parts, in order to make any sort of sense of it. Of course as soon as any Part was isolated, even conceptually, IT had to be addressed, and then the obvious next step was to consider its components too.

But, you can’t do that willy-nilly, and tumble into a possible infinite regress!

And, crucially, Reality is NOT composed of separable Parts!

In Essence, Reality is definitely holistic and not pluralistic. All elements are inter-related, inter-dependant and indeed mutually determining in the last analysis.

The isolation of Parts to explain things is merely a useful Model.

So, Mankind’s settling on Plurality presented him with many problems as well as useful models.

For it to work, he had to constrain Reality in dramatic ways. He had to “corral the beasts” in order to tame them. Plurality only worked if the context was either naturally constrained, or purposely controlled by Man himself in deliberate and appropriate ways.

Plurality was initially quite severely restricted to those areas of Reality, which were quite naturally constrained – naturally-stable sub divisions of Reality as a whole. And this certainly kept the progress to a relatively slow pace.

But it blossomed when Mankind’s knowledge reached sufficient proportions for quite small, unnatural areas to be physically constrained in MOST of its involved variables, purposely-chosen to be fixed values, and effectively “nailed to the floor” in order to “reveal” some hinted-at regularity embedded in Reality, which normally first exposed, and then hid, its beguiling regularities.

Plurality became profoundly useful with the invention of the “experimental set up”.

Now, I have dealt with such things at great length, elsewhere, so I will not repeat it all again here. What is required, in this essay, is to clarify the methodology of Mankind in attempting to deal with a holistic Reality. To do this he constrained Reality in a series of important ways, to identify “in isolation” some of its relations. He delineated his areas of study. He made clear (at least sometimes) his assumptions and premises. He physically constrained the area with often prodigious controls. He forced a series of changes in the magnitude of various parameters, noting their consequent effects on other values, and via these results extracted a relation, which he then generalised into an abstract formal equation.

This is the main Methodology of Science, and its power has been demonstrated a million times in the achievements of centuries of Technology.

But not everything is amenable to such forms of study. Once more, the methodology limited those areas to be studied to those that could be so constrained.

Science became not only a method, but also a defined area, wherein such a method could be applied. The Method could not but actually define its areas of application. And, where such physical controls were not possible, these methods could NOT be used. In addition, these same methods of scientific investigation ALSO defined those areas wherein its achievements could NOT be used.

Now, before I go any further, I cannot continue without making clear that the vast majority of human concerns were necessarily-omitted from this area of feasible investigation and use. Most things could NOT be treated in this way and so were not addressed by the scientists.

An equation may seem to relate two clear things in isolation, but that does not mean that, with it, we have in our hands one of the many “components” of Reality. To think that assumes that Reality is composed of separate Parts that exist independently of one another, and can be brought together to reproduce some area of the Whole. And that is certainly NOT true!

To use such equations (necessarily isolated, extracted and abstracted from a constrained piece of Reality), the EXACT SAME conditions must also be constructed for its effective and predictable USE! The equation would only be true with those precise defined conditions – defined on extraction.

So, every equation has its own Domain of Applicability, and if that Domain is not adequately delivered and maintained, the equation will FAIL. So, if we use the equation within that Domain it will work, but if those conditions are NOT in place, or if they change during use, for whatever reason, the equation will give increasingly more and more incorrect predictions until they are completely wrong!

Ask any school student of Science and he will be guaranteed to confirm this point. It’s why most of his experiments gave the wrong answers!

So, in applying our logical methods we have to clearly delineate our areas of application. We have to limit, in some way, our theatre of operations – either physically or conceptually, or most likely in both of these ways.

Even Reason is predicated upon a defined area with defined assumptions and premises, and Euclid’s Theorems are a perfect model of the method.

NOTE: The assumption that these extracted “truths” are eternal, and can be “summed” in some way to reproduce phenomena in Reality, first in Parts, and subsequently “as a Whole” is, of course, NOT PROVEN.

So, Man, via Control, found a way of bending certain constructed situations to producing predicted outcomes, and being both intelligent and flexible, as he undoubtedly is, he made them serve his needs (as well as fitting his conceived of "needs" to only he could deliver).

Surely, such a methodology is remarkable?

Well, yes it is, and is universally applauded as the solution to ALL problems in the modern world, where it is called “Science”, but is more accurately termed Technology!

And, could not this be considered as wholly sufficient?

The answer has to be a resounding, “NO!”.

Is Reason Merely Formal Logic? (Part 1)

Difference Engine

Let us begin to Define its Successor!

Introduction

As a scientist and mathematician, I have, to my own surprise, spent most of my adult life opposing the dramatic changes that have occurred in a part of my specialist area, which have taken it a million miles from the principles and premises that sustained it throughout the period of its greatest flowering, in the 18th and 19th centuries.

You may consider that, in saying this, I am harking back to the “Good Old Days”, in a similar way to the Romanticists, who rejected Science during the latter half of that period. You may even think that such a "backwards step" is an indication that I don’t understand the Great Revolution that has occurred, and long for the simplicities of that former stage, but that would be a mistaken conclusion! Indeed, compared to the group that I criticize, who are generally very narrow in knowledge and experience, outside of their own specialism, while I have studied well beyond my qualifications, and am amazed at the philosophical immaturity of almost all scientists.

Even the giants, who have contributed greatly to the achievements of Science, have proved to be infants when it comes to Philosophy. And, the reasons are easy to find. For, they all think that what they do in Science IS Philosophy, and are absolutely sure that they are the only ones in this "most basic" of all fields, and thus are those who can, indeed, instruct the philosophers themselves in this crucial area. 

Reactionary Science

But the categorisation of the modern changes in Physics as revolutionary, are as mistaken as those who similarly categorised Nazism in the same way. Both trends were wholly reactionary, and very dangerous! After all, it was this precise group of physicists that delivered exactly how to make the atomic bomb, without, in any way, investigating its immediate and lasting effects.

The changes marked a significant retreat on the part of the majority of my colleagues: delivering instead a significant and damaging decline that, only now, is beginning to be revealed as a major abandonment of all the key principles of scientific endeavour.

And, for what can only be described as an approach, which is both blatantly pragmatist, and, perhaps surprisingly, wholly idealist.

Science as explanation is now considered condemned (in Physics at least) as both impossible and ignorable self-kid, while the way-out speculations of the mathematical physicists’ apologists are given the title of “theories”, while we are expected to embrace unobservable Strings, Physical Singularities and Parallel Universes, yet sneered at for expecting sound, scientific explanations of everything in the World that can be studied.

No, the position which I demand is the correct one of Science-as-Explanation, and to combat the pragmatists and idealists, who have invaded our properly won territory, with their inexplicable formulae and mammoth speculations. We must deliver a theoretical drubbing, that will expose their lightweight approach for ever.

We must study Mathematics and Physics AND the versions of Logic and Reason which underpin them, and determine exactly how they should be correctly and productively related in delivering what Science is supposed to be about – Understanding!

What is Logic & Reason?

The earliest profound realisation of what was possible using disciplined thought was embodied in the system of Geometrical Theorems of Euclid, at the time of the ancient Greeks. His bringing together, in a meaningful and coherent sequence of all the discoveries of the Greeks, in this area, was the first significant achievement of Logic by means of a valid set of processes or manipulations. It was not only a coherent and consistent exposition of that extensive body of knowledge, but was also a demonstration of the power of Thinking itself.

Previously, Mankind had built up a collection of “truths”, about various aspects of the World, but they were generally unrelated fragments. The Greeks, and particularly Euclid, realised that they were indeed related, and the relations were to do with Form. From some, of se Forms, used-together, others could actually be derived via series of processes termed Proofs.

Euclid managed to organise them into a system, where from a small number of assumptions and premises, the Whole set could be systematically proven, in a clear series of logical steps.

This had never been achieved before, in any area of study, so, it was a profoundly important achievement and has, quite rightly, become a vital part of Education ever since.

It was for me, the most significant single piece of purely logical thinking I had ever been presented with, and so profoundly affected me, that I embraced it whole-heartedly, as the centre of my personal education, and quickly became adept at proving every one of the full set of theorems to which I was introduced.

I also thoroughly expected the rest of Mathematics to be similarly dealt with, and attempted to settle into very similar methods in all the subsequent areas of the subject that I was taught and used.

I actually chose to remember nothing, and instead took great delight in working out “everything” from first principles, whether in problem-solving situations, or, perhaps, surprisingly, even in Examinations. It was not long before I achieved my first 100% in a Mathematics Exam, and was subsequently “most disgusted” if I ever dropped below this “necessary” standard. After all, I wasn’t remembering things, neither was I, each and every time, showing my cleverness. I was simply deriving everything by well-oiled and well-learned methods which delivered "The Truth"! Mathematics seemed to be the Essence of disciplined Thinking, and surely ALL problems could be solved by such sound methods?

Of course, I was only 13 years old!

But, as you might expect, few other subjects were so easily susceptible to such methods.

Though, from that time onward, while at school, I never abandoned my “rational” approach, it was very soon evident that the World (and its study) was certainly not as pure and processable as the study of Mathematics might lead us to think.

(I only very much later realised that Mathematics existed in its own private world of Pure Form, which, in fact, set it well apart from the rest of my studies).

Yet, the strict Logic of Mathematics did, somehow, become the broader “Sweet Reason” of most other academic studies, the attitude embodied in my mathematical approach, even though it was not likely to be applicable in all problems of understanding in the World at large.

Indeed, there was, and could only be, a small selection of areas amenable to such a treatment.

My next significant revelation occurred, with my first encounter with Physics, which unlike Mathematics was NOT a monolithic World composed only of one thing, namely Pure Form, but, indeed, considered that its area of application was our surrounding World – everyday Reality itself, and which set as its task the explanation of all real-world phenomena.

But, though we were not limited to Mathematics or its absolute proofs, we were entirely subject to concrete Reality itself, and our techniques of revealing its properties, hidden relations and processes.

At least in my education, Physics was not Mathematics, but more like Natural Philosophy.

It explained things!

It found causal factors and investigated how they affected things and WHY! It named, studied and experimented to find sufficient to construct explanatory theories of exactly why things were the way that they were. This was clearly different to Mathematics. This was Reason in causal action!

Once more, we had a consequent approach which arose out of the physical division of the World into its supposed constituent Parts, and thereafter, via level within level, down to what were considered to be the most fundamental bases of everything.

To the young student, here was an approach which promised ALL! (As before, it was only later that I realised that Physics was no God-like passive, observational process, but, on the contrary, an effective interventionist means of controlling limited sections of the World in order to study them).

My World came (for me) to be built out of such logically amenable areas of study.

The Boundaries of Thought

Halo Rose - Bauhaus - Selbstportrait

The Self-imposed Limits of Human Thinking

When attempting to understand Reality, Mankind is always presented with an extremely puzzling set of apparently ubiquitous features, which seem to continually undermine such efforts. And, these problems usually stem from two sources.

The first is the complex, confusing and sometimes seemingly contradictory Nature of Reality itself. And, the second, as you have no doubt already guessed, is the Nature of Mankind itself.

You wouldn’t think that such would be the case, seeing as Man was created by, and is certainly a part of Reality, but, nevertheless, it is quite definitely so! Reality is not only incredibly complex, but it is also creative: it not only develops in complexity, but also, and crucially, it evolves, regularly, delivering wholly new and unique creations, and the only tool for revealing that nature is its own remarkable product of Evolution – Man himself!

Now, this paper will not be some dazzling literary effort, designed, expressly, to both entertain and beguile the reader, but, instead, is an important and informative attempt to inform that person, both about himself, and also the unavoidable diversions that Mankind inevitably takes in his efforts to make sense of his world.

And, most important in such a task, have to be the conditions that Mankind had to cope with, and which played a crucial formative role in his thinking. For they certainly made him, and developed him in ways appropriate to his means of life. And, that means that Mankind would be congenitally, ill-equipped by Natural Selection to undertake the quite different task of understanding things, and, consequently, of being able to explain them. Man was selected-for, throughout the vast majority of his existence, to be a hunter/gatherer, requiring very different mental processes to today. By the time he had turned to further paths, the forces of Natural Selection, and gene adaption, were over.

Man could NOT depend on his “appropriate genetic makeup” in dealing with the new problems he had begun to set himself. The usual forces of evolution no longer worked to equip Man in his new roles. Man had somehow, to do it for himself!

And, it has to be said that this has by no means been straightforward. Indeed, the forces of Natural Selection are never principled or planned, but entirely pragmatic – “If it works, it is right!” To this day it is the most important element in Man’s nature, and particularly in his thinking.

He is the epitome of pragmatism. And, the application of his undoubtedly superior intelligence is inevitably directed by this. He solves problems pragmatically, and often brilliantly, and clearly does it much better than all other known animals.

Now, that isn’t to say that Man, or at least some among his species, did not turn to tackling other questions. But, because of his nature, involving both his intelligence and his pragmatism, progress was neither direct nor easy. Indeed, though he found many physical pragmatic solutions to aid him, his explanations were, initially at least, very wide of the mark!

To begin to have any chance of doing that, he had to massage Reality in order to get any kind of a handle upon it. He just had to simplify it – ignoring all anomalies, and concentrate upon subsets that seemed to conform to extractable forms. The consequences, as you would imagine, were “temporarily useful”, but ONLY in chosen conditions. In the short term they could be pragmatically useful, but in the long term were bound to fail.

He had begun to make “conceptual bricks”, but they could only build the most flimsy of “explanatory erections”.

Indeed, Man was constantly prevented, by his own self-devised methods, or of getting a general grip upon Reality – for though encouraged by the successes he achieved, he was also bewildered by their failures, and the seeming contradictions that always came up. Of course, it isn’t easy “pulling yourself up by your own bootlaces”!

Let us be clear. Mankind initially made very slow progress for the vast majority of his existence as a separate species: yet modern man is exactly the same animal. That slow progress was not because he was unintelligent, but because he was too pragmatic: there just had to be a revolution in how he Thought! No recent new endowment has enabled his recent enormous developments, and his selected-for abilities, which were prodigious as a hunter/gatherer, but useless as a philosopher – as a thinker about himself and his World.

But, he did begin to re-invent himself, to a degree, by using his undoubted intelligence in new ways. What were essential for his future success, included thinking that could be adapted to other tasks, and perhaps the most helpful were ideas about Religion. Explanations could be put down to an all-powerful deity “on our side”, and such made the achievement of remarkable, and energising, common purposes in believers.

Of course, there were other crucial changes in his mode of life, which were vital in generating new thinking. The so-called Neolithic Revolution, wherein the cultivation of crops, and the rise of animal husbandry, allowed groups of human beings to not only stay where they were, but also to do it with other families. Then larger groups, and more diverse discussions, especially in a new or better way of life, were the trigger for a great deal of new thinking.

But, to see how he got to where he is now, and how he could go forward from here, he will certainly have to understand himself – what he presently does, how he thinks about, and what he must do to progress further.

For, the crises and impasses, in his understanding are by no means over yet. Without another, major revolution in his current thinking, he will grind to yet another crucial halt. 

We must bury forever the myth that amassing knowledge will be sufficient. What must also be developed is Understanding - knowing why.

And, that must be applied not only to the Reality he confronts, but also to himself!

Around 2,500 years ago, two new trends in thinking appeared, which have been crucial in the developments that Man has made in understanding his World. But, they seem to be mutually exclusive alternatives.

In Greece, the rationalist or reason-directed way of thinking – based upon the Principle of Plurality was established, and led to both Geometry and Formal Logic. Though, crucially, this naturally led to the assumption of eternal Natural Laws, which then summed to produce all consequent productions and behaviours. Such a stance meant that Reality was everywhere just a complication of unchanging Laws.

Meanwhile, in India, the philosophical methods of the Buddha took things in a very different direction, by conceiving that “everything affects everything else” – based upon the Principle of Holism.

Clearly, these two approaches were totally incompatible, as presented, and different cultures chose one or the other exclusively. These were, clearly, opposite to one another in many important ways, yet BOTH contained some aspects of the truth of situations – what we have come to call Objective Content. Neither was sufficient in itself, but, nevertheless, which led to significant progress in thinking – though each only in its own self-defined contexts. And, progress in either of these routes was compromised by Mankind’s built-in pragmatism, which crucially meant that Man could keep all that he found - including contradictory pairs of concepts, and could merely switch between alternatives, pragmatically, until he found the one that best fitted the given situation.

Now, though this pragmatic approach did, indeed, lead to the solving of many problems, it was anathema to the requirement of developing consistent, coherent and comprehensive understanding. It was very damaging indeed!

Now, worshippers of Logic, or even Science, will vigorously disagree, but I’m afraid that they will be wrong. Let us take the supposed “God of Understanding”, namely Logic, and see how that holds up.

Rodchenko

It was, of course, the achievement of the Greeks, and was, without doubt, both a brilliant and an essential development. But, it was only made possible by a strict and vigorous filtering of Reality in order to reveal its essences. In other words, Reality was NOT dealt with “as is”, but was processed by both simplification and idealisation to highlight certain then extractable features, which could then be directly investigated in their own terms alone, to attempt to formulate explanations of the phenomena involved.

This expression – “in their own terms alone” is crucial. For it took, what seemed to be, permanent features and assumed that they were always the same – whatever the context would be.

The expression, “The Greeks had a name for it” encapsulates this approach very well. For, by affixing a name, it made the named thing a “constant” component – as if naming it said what it was, and therefore how it would dependably behave. [You know what I mean – if someone can stick a label upon what you are, they assume they have a reliable handle on how you will behave: it is a very ancient “wisdom”]. And, of course, over a quite extended period, such a simplification would indeed suffice!

NOTE: To this day, there are those who will correct your attempts to explain something, with the interjection – “Oh, you mean colloquialism” (or some such contribution), assuming that the name, in Greek or Latin, for a phenomenon - inferring that its meaning is fully encapsulated in that name. It isn’t!

It showed itself constantly, for example, in the belief that all animal or plant species were immutable, and could not change into something else!

And, perhaps the epitome of this approach resides indubitably, in Mathematics – particularly in that lauded Greek achievement of Euclidian Geometry, where, at its very heart were Numbers which, by definition in “Counting”, cannot be but totally fixed. Indeed, Number is often the target, and essence, of simplification and the idea of eternal, unchanging things to be studied in the “fixed” terms alone!

But, in addition, the essential founding principle of such an approach can ONLY be that of Plurality! And, this is because, that Principle rejects evolutionary changes, and even developing Laws. And, it explains all differences in terms of various additive mixes of producing fixed components.

It led, inexorably, to the concept of Eternal Laws, on the one hand, considered incremental additional quantities as solely delivering emerging Quality, on the other. Formal Logic is entirely pluralistic!

Now, earlier in this paper, I compared Plurality with that of Holism – the stance of the Buddha, both of which originated at about 500 B.C. And, it was immediately clear that these stances could not be more contradictory!

The Constants, appearing in pluralist equations of phenomena, were the very same things as the ever-changing factors, of a holistic explanation. Clearly, one stance built the World ultimately out of fundamental eternals, while the other saw it as an evolution of components to ever new entities, properties, laws and Levels.

Now, posed like this, it is clear that the choice of Plurality – dominating for 2,300 years, and is still in charge today, and appears to be indisputable! Yet, Darwin’s Origin of Species was unavoidably holistic, as was Fred Hoyle’s Theory of the Evolution of Stars. And, we should not leave out Stanley Miller’s experimental investigation into the Origin of Life itself.

Even Yves Couder’s brilliant “Walker” Experiments, while not only completely holistic, for the first time breeched the Copenhagen dominance in Sub Atomic Physics.

Clearly, Holism had something crucial to offer.

And, believe it or not a holist stance has long been woven into Science - even though it seemingly arose entirely out of Plurality. Quite apart from the objective of finding formulateable pluralist laws, the equally important search for theoretical explanations of phenomena could never be delivered, materialistically, from mere formal descriptions. And, the only way to seek out such aspects of Reality had to be finding Causes in terms of entities and their evident properties – such clear materialistic incentives forced an alternative route to accompany the pluralist one. And, as it was also always extending the generality of things to ever wider areas, the particular limitations of pluralist relations, very quickly proved inadequate. Coherence, Consistency and comprehensiveness forced Theory into a holistic stance.

A dichotomous pair of contradictory stances grew up alongside one another, without, necessarily, blowing Science apart. A switching of ground (a very pragmatic method) managed to allow a surprising, yet fruitful marriage!

And, hence, perhaps less surprisingly, a third distinct stance arose within Science and grew in strength all the time. WE would call it Pragmatism, except that it was, to an extent, limited to the achievements of Science alone, where it was about effective use of the achievements of Science, and became known as Technology!

The practitioners involved, who came to be called Engineers, did not have to discover, but they had to successfully use the Laws of Science in productive ways!

Special Issue 38 - Thinking About Thought

Man & Reality V

Gloriana! 

So, mathematics is a great deal more than a mere toolbox. It is not just a rag-bag of pragmatic techniques. Then what is it?

My admission that I can spend major tracts of my life doing pure mathematics, doesn’t gel very well with it being a theory-less, quality-less and mightily abstract, yet man-made construction, does it? So, let us leave the pragmatic, head-down, carpet-fitting behind us and climb into the high uplands to breathe in the grandeur of true mathematics. To realise its qualities we have to reveal exactly what mathematics is really about.

Mathematics is the Science of Form!

Quite distinct from subject-centred studies such as Physics, Chemistry, Biology and the rest, it is not concerned with the causality of the world. It is concerned with disembodied Pattern! It isolates significant pattern from reality, or even from our own inventions, and finds its universal formal properties. It is the Logic of Pattern.

This isolating process refines the patterns to their minimal configurations, their ubiquitous essence, and allows their study without the confusing and inessential clutter of multiple overlays that abound in nature. Thus the idealisation of relations into mathematical forms (or formulae) is its essential feature. It is no wonder that Plato and other Greek philosophers waxed lyrical about ideal shapes and forms. Reality, as is, does not give up its secrets easily. They are hidden in a confusing matrix of contending forces, and a fog of what we would today call “noise”. But, nevertheless, constant glimpses (to tempt the curious) of significant relations were always being momentarily revealed and the extraction of these “truths” became, in time, an enticing mistress.

Exactly what these relations were caused by was not clear, though throughout mankind’s conscious history it has always been the question.

Nonetheless, millennia rolled by in which these magical relations were put down to gods and devils, and the never-ending detours of ritual appeasement of these forces were embarked upon. The clear social advantages of such co-operative and consensus beliefs entrenched the explanations, and mankind’s techniques and current knowledge always seemed incapable of an alternative approach. “You can’t pull yourself up by your own boot-laces”, as they say!

Antony Gormley - Another Time XII - 4

The Role of Belief

The processes involved were by no means scientific. The consensus belief system “seemed” to be confirmed by the society’s continued success and growth, but the coherence involved in a shared belief system was perhaps more significant than the contents of these beliefs in ensuring continued confidence and co-operation, and thus success in their endeavours. What always amazed me about the situation in the Second World War was the dedication and valour of the Japanese and Germans. In spite of seemingly insupportable reasons for going to war, and a twisted and reprehensible fascist ethos in both these countries, their soldiers sacrificed themselves in inordinate numbers for “Der Fuhrer”, or “The Emperor”, or even the “Fatherland”. How could this be? I think the answer is contained in my initial point about the strength and confidence imbued by a general and elitist belief system. Other examples litter history. The aggressors are invariably more radical in the way they carried out the business of war, and more confident, while the aggressed against fall away in disarray. All, I believe for the same reasons - a belief system. A primitive society exposed to the world for the first time, in a way that has its ideology, religion and self-belief shattered, dissolves into mediocrity and weakness. “Any coherent, confident belief system is better than none” - from a survival point of view.

However, by the time of the ancient Greeks things began to change. They were the first to see the possibility of another way. Without any of the modern techniques of scientific investigation, they nonetheless made profound gains. And where did they make them? In Mathematics! As a boy just started in Grammar school, I was introduced to the Geometry of Euclid, and was seduced by it. It was a surprising entity! Totally impossible abstractions were refined out of an “imperfect” reality, and studied in an ideal form. Circles were perfectly round, made of lines of zero thickness, while planes were perfectly flat and extended in all directions to infinity. Yet, a coherent structure could be erected on a handful of such assumptions. Mathematics had its first triumph! Form had been extracted and studied and shown to be good! And, all you needed to study it was paper and a pen (or in the Greek’s case – a sandy floor and a bit of twig). Yet, these same Greeks, also saw the pitfalls from the outset. It was Zeno who clearly revealed the dangers of idealised assumptions, and the consequences of their application to the limit. In spite of their epoch-making contribution, the Greeks could not develop their inventions without restriction. They were, so to speak, ahead of their time, and quickly found themselves in the same morass of spiritual causes as everyone else. Profound contributions were a long time coming after the Greeks.

NOTE: As each new abstraction was distilled from reality, and then investigated as an ideal system, the richness contained within what at first seemed to be quite modest assumptions, thrilled the practitioners and seduced them into seeing all as evidence of a co-ordinating mind. Certain areas were remarkable in that they did not seem available in nature at all. Figures, derived by mathematicians, such as the dodecahedron and the icosahedron were breathtaking constructions of order and beauty, yet they didn’t exist in nature. But surely, they couldn’t be pure invention. They had to express some profound, but hidden aspect of the world. Thus the mystification of mathematics was unavoidable, and the Pythagoreans spent lifetimes explaining the world in terms of their figures and forms.

In a surprising way, it was the emergence of proper measurement and experiment that allowed the breakthrough. These techniques revealed significant relations at every turn, and produced an avalanche of forms for mathematicians to extract and study. Though these developments led initially to the Sciences, they also “created” both Mathematics and mathematicians, and it was the latter who seemed to be dealing with the nitty-gritty of reality. Give them an inch and they regularly took a mile.

They had the advantage of not being shackled to demanding experiment and long-winded collection of data. Once delivered of a form, they could investigate the idealised version to their heart’s content. They even took to “creating” forms to investigate, and quickly showed that such studies were indeed possible. From quite early on, they embarked upon Number Theory – the most abstract aspect of mathematics, which studied Number itself, and led to the definition of Prime numbers, and the formulation of “rules” such as Fermat’s Last Theorem. Perhaps the most important feature of their “artificial” flights of fancy of mathematics, was their subsequent role in “fitting” these to new, “inexplicable” relations revealed by scientific experiment. Areas that were generally considered to be total inventions were found to “map” onto aspects of reality. This inversion of the normal relationship between maths and reality posed a new question. Could mathematics be the true essence of the Universe? 

Mathematicians soon became so confident in their ability to supply forms for every situation, that they embarked upon what can only be called maths-based speculation. They worked FROM maths forms to attempt “explanations” on a Cosmic Scale. The famed “Theory of Everything” is the current pinnacle of this line of work. From basic forms to do with oscillations of all kinds, arose the conception of “Strings”. These would be physically existing entities would could take on almost any oscillating mode. Like the old Fourier Analysis, it was postulated that these “entities” could then come together to produce the Universe, all by themselves! Now this aspect is not the only one in a breath-takingly broad phenomenon. In addition, maths formulae began to replace scientific explanation over an increasing range of situations. In particular, where analogous explanations seemed very difficult, or even impossible, they were immediately replaced by “reliable” formulae. “I don’t know why, but I certainly know how!” “I can tell you exactly what will happen in a given situation! Do I need an explanatory theory? Are they not always constructs anyway?”

Could not the universal forms of mathematics actually be the essence of reality? A world encapsulated in pure mathematical formulae might well be the epitome of Science.

This is the fifth in a new series exploring philosophy and mathematics: Man & Reality. The conclusion, Part VI, will be published here next Monday.