MATHEMATICS AND TDK WORLD 249
instance, on high pressure - and the older theories predict a behaviour
exemplified by the curve (A), while
the experimental new data show that
the actual curve is (B), such a result yr
would show unmistakably that the first / A
theory is not structurally correct. But,
in itself, this result does not affect the
correctness of a statement about the general structure of mathematics
which can account for both curves.
Until very lately, we had 3. very genuine problem in physics with the quantum phenomena which seem to proceed by discrete steps, while our mathematics is fundamentally based on assumptions of continuity. Here we had seemingly a serious structural discrepancy, which, however, has been satisfactorily overcome by the wave theory of the newer quantum mechanics, explained in Part X, where the discontinuities are accounted for, in spite of the use of differential equations and, therefore, of continuous mathematics.
But, if we start with fundamental assumptions of continuity, we always can account for discontinuities by introducing wave theories or some similar devices. Therefore, it is impossible, in our case, to argue from the wave theory (for instance) to the structure of mathematics, or vice versa, without a fundamental and independent general structural analysis, which alone can elucidate the problem at hand.
Mathematicians may object on the ground that the new revision of the foundation of mathematics, originated by Brouwer and Weyl, challenges the 'existence' of irrational numbers., and, therefore, destroys the very foundations of continuity and the legitimacy of existing mathematics.
In answer to such a criticism, we should notice, first, that the current 'continuity' is of two kinds. One is of a higher grade, and is usually called by this name; the other continuity is of a lower grade and is usually called 'compactness'. The new revision challenges the higher continuity, but does not affect compactness, which, as a result, will, perhaps, have to suffice in the future for all mathematics, since compactness is sufficient to meet all psycho-logical requirements, once the problems of 'infinity' are properly understood.
A structural independent analysis of mathematics, treated as a language and a form of human behaviour, establishes the similarity of this language to the undeniable structural characteristics of this world and of the human nervous system. These few and simple structural foundations are arrived at by inspection of known data and may be considered as well established.