Contentious Geometry: Ancient Greek Arguments on the Validity of Proofs

For those who enjoy the beauty, clarity, and simplicity of Euclid's Elements, it might be strange to think that Greek geometry was more like a battlefield than a schoolroom; but such it was. Greek geometry arose in a climate of strong loyalties, firm convictions, and vigorous public debate. Its authors laid the foundations of a mathematical tradition that we still recognize and participate in, but they often disagreed about how to do it. These disagreements, which touch on profound questions about the nature of mathematics, are best exemplified in criticisms of others' proofs.

In this talk, we will look at works by Plato, Aristotle, Archimedes, Eratosthenes, and others for evidence of three areas where a proof could be criticized (apart from basic mistakes). First, foundational questions: a proof may not be valid if it does not adhere to the first principles, definitions, and axioms of geometry. Second, methodological questions: a proof may not be valid if it uses disputed methods of solution, such as mechanical curves. Third, formal questions: a proof may not be valid if is not expressed in conventional terms. These questions hint at broader ongoing debates in the background of Greek mathematics, but the struggles of ancient authors to articulate the rules and conditions for doing "good" geometry remain both historically informative and relevant to our philosophies of mathematics today.