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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
Summary
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.
- 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.