Proof, Logic, and Formalization

Proof, Logic, and Formalization

Proof, Logic, and Formalization

Proof, Logic, and Formalization

Synopsis

The mathematical proof is the most important form of justification in mathematics. It is not, however, the only kind of justification for mathematical propositions. The existence of other forms, some of very significant strength, places a question mark over the prominence given to proof within mathematics. This collection of essays, by leading figures working within the philosophy of mathematics, is a response to the challenge of understanding the nature and role of the proof.

Excerpt

Among the features of mathematical thought that most invite the philosopher's attention is its regimentation of justificatory procedures; its canonization of proof as the preferred form of justification. This regimentation is made the more striking by the fact that it is clearly possible to have other kinds of justification (even some of very great strength) for mathematical propositions; a fact which at least suggests that concern for truth and/or certainty alone cannot account for the prominence given to proof within mathematics. This being so, arriving at some understanding of the nature and role of proof becomes one of the primary challenges facing the philosophy of mathematics. It is this challenge which forms the motivating concern of this volume, and to which it is intended to constitute a partial response (on a variety of different fronts).

One possibility, of course, is that it signifies nothing distinctive in the epistemic aims of mathematics at all, but is rather the outgrowth of certain purely social or historical forces. However, since regimentation of justificatory practice is the kind of thing that is at all likely to arise only out of consideration for epistemic ideals, to attribute it to nonepistemic, socio-historical factors would require special argumentation, and there is currently little if any reason to suppose such argumentation might be given. Thus, accounts of the ascendancy of proof in mathematics that make appeal to some epistemic ideal(s) would appear to be more promising.

One consideration which seems to have weighed heavily in favor of proof as the preferred form of justification is the traditional concern of mathematics for rigor. This drive, in turn, seems to have derived from one of two (in some ways related)

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