Hiroyuki Inaoka

In this paper, we would present an overview of the recent studies on the role of diagram in mathematics. Traditionally, mathematicians and philosophers had thought that diagram should not be used in mathematical proofs, because relying on diagram would cause to various types of fallacies. But recently, some logicians and philosophers try to show that diagram has a legitimate place in proving mathematical theorems. We would review such trends of studies and provide some perspective from viewpoint of philosophy of mathematics.

Gareth Evans proved that if two objects are indeterminately equal then they are different in reality. He insisted that this contradicts the assumption that there can be vague objects. However we show the consistency between Evans's proof and the existence of vague objects within classical logic. We formalize Evans's proof in a set theory without the axiom of extensionality, and we define a set to be vague if it violates extensionality with respect to some other set. There exist models of set theory where the axiom of extensionality does not hold, so this shows that there can be vague objects