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What Is a Proof?

Alan Bundy, Mateja Jamnik and Andrew Fugard
Philosophical Transactions: Mathematical, Physical and Engineering Sciences
Vol. 363, No. 1835, The Nature of Mathematical Proof (Oct. 15, 2005), pp. 2377-2391
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/30039733
Page Count: 15
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What Is a Proof?
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Abstract

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.

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