Proof as experiment in Wittgenstein
pp. 191-204
in: Gila Hanna, Hans N. Jahnke, Helmut Pulte (eds), Explanation and proof in mathematics, Berlin, Springer, 2010Abstract
Ludwig Wittgenstein allows us to contrast two ways of regarding proof. On the one hand, a proof can and ought to be regarded as a picture that meets the requirement of being surveyable, as exemplified by a calculation on a sheet of paper. Here, what was proved serves as an identity-criterion for the proof; indeed, only the proof as a surveyable whole can tell us what was proved. On the other hand, a proof can be regarded as an experiment, necessarily so if one wants to understand the productive and creative aspects of proof. In analogy to scientific experiments, proof as experiment refers to the experience of undergoing the proof, as exemplified by reductio ad absurdum or negative proof. Here, the conclusion of the proof does not add a conclusion to the premises but leads to the rejection of a premise and changes the domain of the imaginable. The proof shows us what was proved in that it implicates us in a certain experience at the end of which we see things differently: that is, we evaluate certain commitments, mathematical procedures or hypotheses differently and therefore, in a sense, live in a different world.
