Abstract

Is pure mathematics - arithmetic as well as geometry - reducible to formal logic? Russell answered in the affirmative, considering this so significant as to constitute a fatal blow to Kant's synthetic-apriori philosophy of mathematics. But either pure arithmetic and pure geometry include the full, extra-logical content of their unique axioms and hence their unique theorems, or they do not. If they do, then this reductionism is trivially unsound. It they do not - if they include only the logic of demonstration and exclude everything else - then it is trivially true, but insignificant. In fact this would accomplish no reduction at all, but rather a harmless formalization.