 
Summary: Searching for proofs (and uncovering
capacities of the mathematical mind)1
Wilfried Sieg
Abstract. 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 GĻodel'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
