Proof and Knowledge in Mathematics

Proof and Knowledge in Mathematics

Proof and Knowledge in Mathematics

Proof and Knowledge in Mathematics

Synopsis

These questions arise from any attempt to discover an epistemology for mathematics. This collection of essays considers various questions concerning the nature of justification in mathematics and possible sources of that justification. Among these are the question of whether mathematical justification is a priori or a posteriori in character, whether logical and mathematical differ, and if formalization plays a significant role in mathematical justification,

Excerpt

The essays in this volume all deal with questions which arise in the course of trying to arrive at an epistemology for mathematics. Prominent among these are questions concerning the nature of justification in mathematics and its possible sources. Some of the essays in this volume treat these questions in a general way, while others focus on various more particular concerns relating to them. Among these more particular concerns, perhaps the most noteworthy are the following: (i) the question of the a priori versus a posteriori character of mathematical justification (including the question of the role of proof as a source of warrant in mathematics), (ii) the question of the character of mathematical reasoning or inference, especially the question of what role (if any) logic has to play in it, and (iii) the question of the epistemological importance of the formalizability of proof.

In "Proof as a source of truth," Michael Resnik addresses the general question of mathematical justification and its sources by taking up the question of how, if at all, a proof comes to provide a warrant for its conclusion. His argument makes use of the general framework developed in his earlier work on pattern recognition, in which mathematical objects are treated as patterns, and patterns as abstractions of material (and hence perceivable) things called "templates." in his view, proofs are to be seen as concerned with patterns, both establishing (as conclusions) and utilizing (as premises) laws that are structurally isomorphic to laws that have been verified for templates. in this way, templates become the gateway through which we gain epistemic access to the objects of mathematics dealt with in (realist) proofs.

This same question, albeit in its more familiar guise as that concerning the possible a priori character of mathematical

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