Preaxiomatic Mathematical Reasoning: An Algebraic Approach

摘要: In their correspondence on the nature of axioms, Frege and Hilbert clashed over question how best to understand axiomatic mathematical theories and, in particular, nonlogical terminology occurring axioms. According Frege, axioms are viewed as attempts assert fundamental truths about a previously given subject matter. disagreed emphatically, holding that contextually define matter; thus, so long an axiom system implies no contradiction, its be thought true. This paper considers whether it is possible extend Hilbert’s “algebraic” view preaxiomatic reasoning, where our concepts not yet pinned down by definitions. I argue that, even at stage, informal characterizations determinate enough viewing setting well-defined “problems” with “solutions” remains illuminating.