A Philosophy of Mathematics

A Philosophy of Mathematics

A Philosophy of Mathematics

A Philosophy of Mathematics

Excerpt

In glancing back over the development in our understanding of mathematics, we are impressed with the tremendous distance we have come. Contemporary discussions of the nature of mathematics tend to take one of two forms: (a) epistemological, and (b) elemental. Elemental investigation into mathematics had for the most part itself become another branch of mathematics. We can say that the impetus to an investigation of mathematics along elementalistic lines really got under way with Hilbert's metamathematics and Russell's attempt to deduce mathematics from (logic or) logistics. The subsequent unification of these two approaches (never really different) has produced a literature in the field which is growing larger and more difficult of comprehension. The increase in the number of symbolisms, the different techniques employed have caused confusion. Today, mathematics is not certain whether or not it should embrace elementalistics as a branch of itself, or to leave it to logistics -- and logistics cannot answer this question either. As a consequence, the Journal of Symbolic Logic has appeared to offer a medium for publication of results in this field.

The logistic analysis of mathematics has developed into this "elemental mathematics" -- a new, and it appears to be, most fundamental branch of mathematics. Elemental mathematics purports to be an analysis of the root concepts such as number, function, or operator involved in mathematics. It appears to out-metamathematicize metamathematics. There is the increasing danger that elemental mathematics is taking itself outside the study of the philosophy of mathematics. But be that as it may, it is casting a great deal of light upon questions in the philosophy of mathematics. Perhaps these questions are meaningless, as logical positivism would have us believe. But even if they are, their very meaninglessness is of interest, for we may still ask the nature, form, and criteria of meaningless question.

But meanwhile, what of the student who is beginning the study of mathematical philosophy? There are at present no elementary introductions to this field. The student must dive into the heart of things and after struggling some years, comes to one of two conclusions: (a) it's all nonsense, (b) it's beyond him. In either case, he fails to see the depths of the subject and the light it throws on many problems . . .

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