Philosophy of Logic
One of the questions of mathematics is are we uncovering the ideal structure of reality or creating language. Yes. [i] I think of mathematics as a collection of tools or techniques. Socrates argued that all knowledge is innate. He took a young slave boy and interrogated him as to a proof of the Pythagorean theorem. Since the boy kept agreeing with Socrates, he must have known of this proof already. Socrates described a right triangle, then showed a square constructed from 4 of these identical triangles, then set the area of the square to the areas of the 4 triangles and the square contained in them. Then he solved for the Pythagorean theorem. Socrates was in the impossible position of arguing for ethics and logic in a polytheistic world, surrounded by the arbitrary gods. By proving that a slave had the same innate knowledge as the rest of us he was calling into question slavery. Euclid hated this proof of the Pythagorean theorem. The proof requires that you already know w...