Logic, Mathematics, and Philosophy
Abstract
Logic occupies a special position at the intersection of mathematics and philosophy because mathematics has repeatedly provided both the paradigm and the occasion for transformations in logic. The historical movement considered here runs from Greek diagrammatic practice and Aristotelian term logic, through Descartes’ symbolic algebra and Kant’s quantificational logic, to nineteenth-century deduction from defined concepts and Frege’s Begriffsschrift. The guiding problem is how mathematical knowledge can be at once strictly deductive and genuinely ampliative. Frege’s distinction between Sinn and Bedeutung, and between objectivity and relation to objects, makes possible a renewed answer: mathematical reasoning need not concern mathematical objects, but can reveal necessary relations among mathematical concepts. A mathematical language can display the contents of such concepts in signs that function both as a lingua and as a calculus ratiocinator. Some proofs extend knowledge because definitions do not merely supply premises; in fruitful cases they license materially valid, necessary inferences not licensed by logic alone. Defined concepts, theorems, and chains of reasoning can thereby be understood as intelligible unities: real wholes of real parts. Frege’s logic thus offers not only a new account of mathematical knowledge, but also a possible transformation of mathematical practice, teaching, and discovery.
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