Seeing-as and Mathematical Creativity

This chapter provides Ludwig Wittgenstein's remarks on seeing-as with another aspect of his investigations in the philosophy of mathematics. It considers two examples of mathematical creativity from the history of mathematics, one from geometry and one from arithmetic. The chapter also considers one of the most important sources—perhaps the most important source—of philosophical methodology in Plato's Meno. It looks at the emergence of non-Euclidean geometry. The 'discovery' of irrational numbers was a key stage in the development of the mathematical concept of a number, and lying at the core of this development was a move that essentially required a shift of conceptual aspect. The influence of Greek geometry on philosophy is first revealed in Plato's Meno, the dialogue in which Socrates cross-examines a slave boy in an attempt to get him to 'recollect' the answer to a geometrical problem.