Abstract
Drawing mainly from the Tractatus Logico-Philosophicus and his middle period writings, strategic issues and problems arising from Wittgenstein’s philosophy of mathematics are discussed. Topics have been so chosen as to assist mediation between the perspective of philosophers and that of mathematicians on their developing discipline. There is consideration of rules within arithmetic and geometry and Wittgenstein’s distinctive approach to number systems whether elementary or transfinite. Examples are presented to illuminate the relation between the meaning of an arithmetical generalisation or theorem and its proof. An attempt is made to meet directly some of Wittgenstein’s critical comments on the mathematical treatment of infinity and irrational numbers.
Highlights
The philosophy of mathematics was a major concern of Wittgenstein throughout his philosophical life and he expressed the opinion at one point that his greatest contribution to philosophy might lie there (Bangu, 2018)
Whatever its source, we find Wittgenstein by the early 1930s making the assertion that the meaning of an arithmetical generalization consists in its inductive proof
The doctrine that the meaning of an arithmetical generalization is given by its proof, as Wittgenstein recognized, gives rise to difficulties. Does this mean that none of us understands the unproved statement, “Every even number is the sum of two primes” prior to a proof being provided? in his writing, Wittgenstein attempts to make room for the notion of a meaningful but not yet proven mathematical sentence on a verificationist basis
Summary
The philosophy of mathematics was a major concern of Wittgenstein throughout his philosophical life and he expressed the opinion at one point that his greatest contribution to philosophy might lie there (Bangu, 2018). His writing and philosophical style which is aphoristic in the Tractatus evolves to become more argumentative in the middle period and more interlocutory in later writings While in his middle period Wittgenstein tends to use “the calculus conception” (which includes ways of determining a quantity or required expression), later on, it is “the language-game conception” (Gerrard, 1991, 1996). In his later writings, greater sense is given to the idea that mathematical propositions are dependent upon experience
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