What is a proof?

To those brought up in a logic-based tradition there seems to be a simple and clear definition of proof. But this is largely a twentieth century invention; many earlier proofs had a different nature. We will look particularly at the faulty proof of Euler's Theorem and Lakatos' rational rec...

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Bibliographic Details
Published in:Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences physical, and engineering sciences, 2005-10, Vol.363 (1835), p.2377-2391
Main Authors: Bundy, Alan, Jamnik, Mateja, Fugard, Andrew
Format: Article
Language:English
Subjects:
Algorithms
Automated Theorem Proving
Axioms
Constructive Omega Rule
Cubes
Culture
Logical theorems
Mathematical Computing
Mathematical functions
Mathematical induction
Mathematical Proof
Mathematical theorems
Mathematics
Models, Theoretical
Numerical Analysis, Computer-Assisted
Polyhedrons
Reasoning
Schematic Proof
Software
Software Validation
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Summary:To those brought up in a logic-based tradition there seems to be a simple and clear definition of proof. But this is largely a twentieth century invention; many earlier proofs had a different nature. We will look particularly at the faulty proof of Euler's Theorem and Lakatos' rational reconstruction of the history of this proof. We will ask: how is it possible for the errors in a faulty proof to remain undetected for several years-even when counter-examples to it are known? How is it possible to have a proof about concepts that are only partially defined? And can we give a logic-based account of such phenomena? We introduce the concept of schematic proofs and argue that they offer a possible cognitive model for the human construction of proofs in mathematics. In particular, we show how they can account for persistent errors in proofs.
ISSN:1364-503X
1471-2962
DOI:10.1098/rsta.2005.1651