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The Philosophy of Mathematics (Part 1)

SERIES: The Philosophy of Mathematics
AUTHOR: Jim Schofield


This series of papers gives a brief introduction to the mathematical approach. They concentrate on a small number of currently important consequences of this implied philosophy. There had to be, of course, a historical element too, for this discipline is very old and has changed its viewpoint and its methods regularly over the centuries. From measurement and Arithmetic, via Logic and Geometry, the Greeks transformed the tricks of the pyramid builders and Sumerian "accountants" in the study of Form, and even into a still persisting maths-based philosophy of everything. The power beyond calculation tricks involved a process of isolation, extraction and then abstraction of relations glimpsed in Reality into algebraic equations. But these, for many hundreds of years, were confined to "ideal" areas of study (such as Geometry). It required the aquisition of the means to CONTROL individual areas within Reality to ensure accurate and consistent prediction, and this was not really acheived until the Renaissance period. By the time of Newton mankind was well equipped enough to tackle not only many straight forward areas, but even those which involved continuous qualitative Change. His researches (also seperately carried out by Leibnitz) resulted in the Calculus. Rates of change could be symbolised and used in relations, and this transformed Mathematics forever.

However, the techniques involved in ALL mathematics reflected back on how we conceived of Reality at large. The isolation and extraction phases involved control, selection and even dumping of many actually relevant factors in all studied situations. This not only confined use to artificially contrained Domains, but ensured total failure when these boundaries were transgressed.

Mathematics, though it was the legitimate study of Pure Form, also distorted our view of Reality, when we forced its equations to work there. The inevitable happened! Most mathematicians and even a sizable number of scientists began to see its revealed and abstracted equations as the essence of Reality itself: they became the components driving Reality, obscured only by complications and masking noise. The regular confusion around the ideas of Description, Prediction, Form and Cause did not help! Mathematics is ONLY the study of Pure Form, which it merely describes. Its proofs in theorems seemed to be explanations, but they were actually only the revelations of the full nature of a particular Form. To tackle real Explanation, mankind would have to look elsewhere.

His banker method was certainly that of Analogy. This technique mapped known sequences of occurence onto new, as yet unprocessed, but obviously very similar sequences elsewhere. To do this a new kind of Abstraction was involved, where quite unconnected entities and forces were mapped into one another to enable real analogies to be created. Their power was that they could traverse Domian boundaries. Their weakness was that they were at best only near Models, and could never be absolutely true. The critical property of these Models was that they contained true objective content, which, though not the full story, could certainly be used with confidence in most circumstances. In Sub-Atomic Physics, however, suitable analogies for what they found were not available in our everyday world, so these scientists dumped Explanation all together. A short but important diversion into the meaning of the "square root of minus one" was then necessary to indicate how Mathematics extended Number into non-numerical areas. The manipulative powers of Mathematics (with a few extensions) could be carried over into new areas. And, of course, this also explanded mankind's view of the legitimacy of its methods in even wider areas.

The consequences in Physics have been horrendous! The essential role of Analogisitc Explanation has been replalced by what can only be called maths-led speculation to wholly deleterious ideas and consequences.


1. In the Modern World Mathematics has taken centre stage in major areas of Science, which is just the latest of its many transformations throughout its history.

2. Mathematics was a surprisingly early development of Mankind once he embarked upon civilisation. It began the earliest writing to allow the recording of assets and was then essentially Arithmetic. But it was also a vital new direction, as it involved a clear and useful Abstraction.

3. And this meant that its techniques were available to all measured things, and were soon extended to both Area (land) and Volume (Liquids). Geometry was born in manipulating these abstract forms.

4. In combination with Formal Logic, the Greeks created the then most sophisticated forms of reasoning and proof. Geometry’s abstractions were not just simplification, but the “essence” of forms seen everywhere in Reality.

5. Its inventors clearly saw what they were doing as extracting the essence from Reality and discarding all inessential trivia. They soon set their sights on extracting the Whole Ideal World from its confused and messy location in Nature.

6. This Idealism was pragmatic, for their studies seemed to be useable in many ways in the Real world, and it soon became a World View, wherein the essence of everything was considered to be Mathematical.

7. And these ideas have persisted. With mathematics’ universal pragmatic uses we have continued to see it also as determining the Essence of all Reality too.

8. But different groups within society concentrated separately on these contrasting aspects. The pragmatic tricks were the tools of the builders and doersof antiquity, while the ideas of Essence were the realm of the philosophers.

9. Indeed, all the main strains in Modern Philosophy began with these early investigations into Mathematical Form, and its significance in the World at large.

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