Searching for proofs (and uncovering capacities of the mathematical mind)

What is it that shapes mathematical arguments into proofs that are intelligible to us, and what is it that allows us to find proofs efficiently? — This is the informal question I intend to address by investigating, on the one hand, the abstract ways of the axiomatic method in modern mathematics and, on the other hand, the concrete ways of proof construction suggested by modern proof theory. These theoretical investigations are complemented by experimentation with the proof search algorithm AProS. It searches for natural deduction proofs in pure logic; it can be extended directly to cover elementary parts of set theory and to find abstract proofs of Gödel’s incompleteness theorems. The subtle interaction between understanding and reasoning, i.e., between introducing concepts and proving theorems, is crucial. It suggests principles for structuring proofs conceptually and brings out the dynamic role of leading ideas. Hilbert’s work provides a perspective that allows us to weave these strands into a fascinating intellectual fabric and to connect, in novel and surprising ways, classical themes with deep contemporary problems. The connections reach from proof theory through computer science and cognitive psychology to the philosophy of mathematics and all the way back. 1 Historical perspective It is definitely counter to the standard view of Hilbert’s formalist perspective on mathematics that I associate his work with uncovering aspects of the mathematical mind; I hope you will see that he played indeed a pivotal role. He was deeply influenced by Dedekind and Kronecker; he connected these extraordinary mathematicians of the 19th century to two equally remarkable logicians of the 20th century, Gödel and Turing. The character of that connection is determined by Hilbert’s focus on the axiomatic method and the associated consistency problem. What a remarkable path it is: emerging from the radical transformation of mathematics in the ∗This essay is dedicated to Grigori Mints on the occasion of his 70th birthday. Over the course of many years we have been discussing the fruitfulness of searching directly for natural deduction proofs. He and his Russian colleagues took already in 1965 a systematic and important step for propositional logic; see the co-authored paper (Shanin, et al. 1965), but also (Mints 1969) and the description of further work in (Maslov, Mints, and Orevkov 1983).