Preaxiomatic mathematical reasoning
an algebraic approach
pp. 47-57
in: Gila Hanna, Hans N. Jahnke, Helmut Pulte (eds), Explanation and proof in mathematics, Berlin, Springer, 2010Abstract
In their correspondence on the nature of axioms, Frege and Hilbert clashed over the question of how best to understand axiomatic mathematical theories and, in particular, the nonlogical terminology occurring in axioms. According to Frege, axioms are best viewed as attempts to assert fundamental truths about a previously given subject matter. Hilbert disagreed emphatically, holding that axioms contextually define their subject matter; thus, so long as an axiom system implies no contradiction, its axioms are to be thought of as true. This paper considers whether it is possible to extend Hilbert's "algebraic" view of axioms to preaxiomatic mathematical reasoning, where our mathematical concepts are not yet pinned down by axiomatic definitions. I argue that, even at the preaxiomatic stage, our informal characterizations of mathematical concepts are determinate enough that viewing our mathematical theories as setting well-defined "problems' with mathematical concepts as 'solutions' remains illuminating.
