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Philosophy of Mathematics : An Introduction to the World of Proofs and Pictures

This example indicates that there is more than one way to look at the role of pictures in proof. The weakest claim is that a picture can serve as an aid to inspire a proof. The mathematician presenting the Bolzano proof could agree to this. A stronger position is that a picture may be taken to be an element of a formal proof. If one can specify syntactic rules for words that constitute a proof, then one might also be able to specify syntactic rules for pictures that constitute a proof. There are several recent works with this theme [1], [6]. The strongest position is that of the mathematician presenting the sketch of the function crossing the axis. This sketch is supposed to capture a typical representation of the true situation. It is a way to perceive the Platonic world of mathematical reality.