Seminar on History and Philosophy of Science

The proofs of Euclid's Elements are often said to contain implicit assumptions about basic geometric facts. Specifically, in contrast to modern theories of elementary geometry, the Elements does not contain axioms for order relations (e.g. lying between, lying inside). Recent analyses of Euclid's arguments have aimed to show that nevertheless a systematic proof method underlies them. The basic idea is that Euclid uses geometric diagrams in a controlled and principled way to record order information. After presenting how on this account Euclid's proof method differs from modern axiomatic approaches to elementary geometry, I examine the account's relevance to broader questions on informal mathematical proofs and axiomatic analyses of them.