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The justification of mathematical statements.

Peter Swinnerton-Dyer1

  • 1University of Cambridge Wilberforce Road, Cambridge CB3 OWB, UK. h.p.f.swinnerton-dyer@dpmms.cam.ac.uk

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|September 29, 2005
PubMed
Summary

Mathematical proofs are becoming too complex for humans to verify. This paper explores necessary compromises in mathematical proof standards due to complex proofs and computer involvement.

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Area of Science:

  • Pure Mathematics
  • Mathematical Logic
  • Foundations of Mathematics

Background:

  • Historically, mathematical proofs were expected to be fully verifiable by readers.
  • The early post-war era emphasized rigorous, checkable proofs in pure mathematics.
  • Modern mathematical research faces challenges to this traditional verification standard.

Purpose of the Study:

  • To discuss the impact of complex proofs and computer involvement on mathematical verification.
  • To explore the compromises required within the mathematical community.
  • To re-evaluate the standards of proof in contemporary mathematics.

Main Methods:

  • Conceptual analysis of proof verification.
  • Discussion of historical trends in mathematical rigor.
  • Examination of the role of computational tools in theorem proving.

Main Results:

  • Proof verification is increasingly challenging due to length and complexity.
  • Computer-assisted proofs introduce new paradigms for validation.
  • The mathematical community must adapt its standards for proof acceptance.

Conclusions:

  • The traditional model of proof verification is no longer universally feasible.
  • New strategies are needed to ensure the reliability of mathematical results.
  • Compromises in proof presentation and verification are inevitable and necessary.

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