Pure and Applied?

Philosophically, Mathematics has shown itself to be a very unusual discipline, in that it seems to be investigating Forms found in Reality, but is, in fact, more to do with how we humans conceive of, extract and handle such Forms in our thinking.

Historically, the recognition of such Forms in Reality-as-is was never a straightforward discovery - directly extractable as such from that source. For, to even recognise what we were seeing in that complex and varying context, Man just had to both simplify and idealise what he glimpsed, into something both fixed and intelligible. For such Forms as we arrived at, were not what we saw, for they were invariably imperfect and often somewhat transitory, so observers felt they were seeking an underlying perfection below the really existing complexity, and they, with experience and over time, became very adept at the necessary processes involved to reveal perfected and unchanging patterns.

As already mentioned, though seen as unearthing these Forms, they were always both simplifying and idealising them, into “ideal versions”, and, thereafter, studying them instead.

The believed excuse for such processing, was that these Pure Forms did seem to exist “out there”, and the processes of extraction were considered to be mere “tidying up”, and, thereby, releasing the crucial Forms from a natural context of confusing and inessential “noise”, caused by a complexity of other diverse causes, which could be removed to reveal the “real determining heart” of a situation. And, having extracted these causal essences, they could be investigated and their important intrinsic properties revealed.

It was, historically, the initial beginning of Science, even if the actual causality had been inverted, and effects labelled as the actual causes!

Yet surprisingly, many of these initial conceptions have been, at least partially, retained ever since!

And, again surprisingly, this discipline was the very first that Mankind was able to construct into what seemed to be a self-consistent set of relations and rules underlying Reality. And this was so universally taken on by those involved in such things, that, from the outset, these Forms were given Causal Attributes - real things were seen as behaving as they did, because they were obeying the eternal Laws of these Forms.

Hence, these first steps were clearly idealist, and not materialist, in the march of the human intellect.

NOTE: It ought to be mentioned at this stage, that the almighty retreat, in 20th century Sub Atomic Physics, embodied in the infamous Copenhagen Interpretation of Quantum Theory, was merely a retrenchment back to this ancient idealist stance.

Henceforth, in that area, Form was deemed to be the cause of ALL phenomena, and Equations replaced physical explanations almost completely.

So, in Ancient Greece, truly remarkable strides were made, particularly in Geometry, which was immediately available for investigation via drawing, and this amazing development ended up with what we now call Euclidian Geometry – which is still taught as a cornerstone of Mathematics worldwide.

But, it did have major drawbacks! And, these are not only in its idealist stance, but also in its major simplification by imposing eternality upon all its contributing Forms. They were fixed! And, the implications of this, along with their endowment of being also the cause of phenomena, had deleterious implications for the real study of Reality.

Clearly, the extrapolation of this assumption onto many other non-mathematical ideas was inevitable as well as being profoundly mistaken. For though Forms could be “found” in real situations, they were never the determining causes for those situations, and as the real determinators changed, so did the evident Forms.

Forms, as such, were permanent (that is as Formal abstractions), but their real existence was always temporary and never causal!

The consequences were extremely damaging: most things were dealt with as unchanging, and the basic tenet of Formal Logic, which was an intellectual product of this development – that is A = A – the Identity Relation, cast in stone the un-changeability of the ideas and elements involved in this major extension of what had been learned in Mathematics.

NOTE: That this is still around and propagated, was revealed in a book entitled A Certain Ambiguity by Guarav Suri and Hartosh Singh Bal, published only a few years ago (2010 - Ed).

But after, maybe, 2,000 years of Greek Science, based upon such a position, a breech was made in the then towering edifice of that “science-based-upon-Logic”, into one based upon careful observation of concrete Reality. Finally, a new approach was developed, which was, in fact, materialist Science.

Perhaps surprisingly, the new approach prospered in tandem with a rejuvenated Mathematics, because Science was now based upon quantitative measurements, and hence was regularly delivering dependable data sets, clearly also revealing Forms to be extracted and formulated into useable Laws. And, of course, the “ideal experts” for doing this, were the mathematicians, who by this juncture had amassed a truly remarkable number of Forms “to fit all possible patterns”.

An extremely fruitful cooperation developed between idealist mathematicians and materialist scientists, and sometime, and somewhere, something was bound to give! A unifying concept arose that was that of Natural Law. When experimental data was turned into an Equation, by the mathematicians, it was agreed by both parties to be a causing and eternal Law – the scientists had in fact succumbed to the idealist position of the mathematicians.

But, this purely pragmatic compromise was never a solution. Indeed, it was yet another example of a Dichotomous Pair of contradictory conceptions, which couldn’t both be true!

Yet, without a transcendence of the ever-evident theoretical impasse, the two groups pragmatically “agreed to differ” (at least partially), and both stances were kept – using one rather than the other, when it was clearly productive to do so.

And, though the idealism of Mathematics affected Science, the materialism of Science also affected Mathematics, and the result of this was the wholly different discipline of Technology – the application of scientific discoveries in advantageous inventions and devices.

These technologists were not interested in Theory, and they were also not enamoured of Pure Mathematics either.

They required a kind of Mathematics that enabled their purposes – they claimed Applied Mathematics as their own vital Toolkit, and indeed, regularly invented new tricks to facilitate their objectives, whether or not they conformed to either theoretical stance. They were completely pragmatic and nothing more.

Interestingly, the three, closely-related disciplines, not only went their own ways, but on quite different philosophical bases.

Science was primarily materialist.

Mathematics was entirely idealist (termed Pure Mathematics).

Technology stuck to Applied Mathematics, but was entirely pragmatic – “if it works, it is right!”