SUSPICION OF THOUGHT By: RICHARD J.KOSCIEJEW

SUSPICION OF THOUGHT

Presented by: RICHARD J.KOSCIEJEW

A ‘theory’ usually emerges as a body of (supposed) truths that are nearly organized, making the theory difficult to survey or study as a whole. The axiomatic method is an idea for organizing a theory (Hilbert, 1970): One tries to select from which all the others can be seen to be deductively inferablle. This makes the theory rather more tractable since, in a sense, all the truths are contained in those few. In a theory so organized, the few truths from which all others are deductively inferred are called ‘axioms’. David Hilbert (1862-1943) had argued that, just as algebraic and differential equations, which were used to study mathematical and physical processes, could themselves be made mathematical objects, so axiomatic theories, like algebraic and differential equations, which are means of representing physical processes and mathematical structures, could be made objects of mathematical investigation.

In the tradition (as in Leibniz, 1704), many philosophers had the conviction that all truths, or all truths about a particular domain, followed from a few principles. These principles were taken to be either metaphysically priori or epistemologically priori or both. They were taken to be entities of such a nature that what exists is ‘caused’ by them. When the principles were epistemologically priori, that is, as axioms, either they were taken to be epistemologically privileged (e.g., self-evident, not needing to be demonstrated) or (again, inclusive ‘or’) to be such that all truths do indeed follow from them (by deductive inferences) of truth.

Gödel (1984) showed - in the spirit of Hilberret6, treating axiomatic theories as themselves mathematical objects - that mathematics, and even a small part of mathematics, elementary number theory, could not be axiomatized, that, more precisely, any class of axioms which is such that we could effectively decide, of and proposition, whether or no t it was in that class¸, would be too small to capture all of the truths.