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What is a proof?

Alan Bundy1, Mateja Jamnik, Andrew Fugard

  • 1University of Edinburgh, School of Informatics Appleton Tower, Crichton Street, Edinburgh EH8 9LE, UK. a.bundy@ed.ac.uk

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|September 29, 2005
PubMed
Summary
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This study explores how errors in mathematical proofs, like Euler's Theorem, persist despite known counter-examples. It introduces schematic proofs as a cognitive model to explain these persistent errors in mathematical reasoning.

Area of Science:

  • Philosophy of Mathematics
  • History of Mathematics
  • Cognitive Science

Background:

  • Traditional logic-based definitions of mathematical proof are a recent development.
  • Earlier mathematical proofs often differed in nature and rigor.
  • Euler's Theorem has a history of faulty proofs and rational reconstructions.

Purpose of the Study:

  • To investigate why errors in mathematical proofs remain undetected for extended periods.
  • To explore the possibility of proofs involving partially defined concepts.
  • To develop a logic-based explanation for these phenomena in mathematical discovery.

Main Methods:

  • Analysis of historical mathematical proofs, focusing on Euler's Theorem.
  • Examination of Imre Lakatos' rational reconstruction of the history of this proof.

Related Experiment Videos

  • Introduction and exploration of the concept of schematic proofs.

    • Main Results:

    • Schematic proofs are proposed as a cognitive model for human proof construction.
    • This model can account for the persistence of errors in mathematical proofs.
    • It offers insights into how proofs with undefined concepts can arise and persist.

    Conclusions:

    • Mathematical proof has evolved, with earlier forms differing from modern logic-based standards.
    • Schematic proofs provide a framework for understanding the cognitive processes behind proof generation and error propagation.
    • Further research into schematic proofs can illuminate the history and psychology of mathematical reasoning.