
Quite generally, if there are two proofs for a theorem, you must keep going until you have derived each from the other, or until it becomes quite evident what variant conditions (and aids) have been used in the two proofs. Under a given set of conditions there can be but one simplest proof. Develop a theory of the method of proof in mathematics in general.

The 24th problem in my Paris lecture was to be: Criteria of simplicity, or proof of the greatest simplicity of certain proofs. Suppose I take the Wiles-Taylor proof of Fermat's Last Theorem, transpose the order of two adjacent but independent lemmas, then submit the result to Annals of Mathematics as "A New Proof of Fermat's Last TheoremWould I get away with it?Įquality of proofs is not merely a peer review issue, but also a concrete technical issue stressed by Hilbert in his 24thProblem (emphasis mine): Keywords: Combinatorial proof, Hilbert's 24th Problem, sequent calculus, Herbrand's Theorem The paper lifts a simple, strongly normalising cut elimination from combinatorial proofs to sequent calculus, factorising away the mechanical commutations of structural rules which litter traditional syntactic cut elimination. This sequel approaches Hilbert's 24th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for topological spaces. Proofs Without Syntax introduced polynomial-time checkable combinatorial proofs for classical proposi-tional logic. Towards Hilbert's 24th Problem: Combinatorial Proof Invariants Sequent calculus fails to be surjective onto combinatorial proofs: the paper extracts a semantically motivated closure of sequent calculus from which there is a surjection, pointing towards an abstract combinatorial refinement of Herbrand's theorem.Įlectronic Notes in Theoretical Computer Science 165 (2006) 37-63


Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for topological spaces. Proofs Without Syntax ] introduced polynomial-time checkable combinatorial proofs for classical propositional logic. Abstract of research paper on Computer and information sciences, author of scientific article - Dominic J.D.
