Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

understanding proofs 319theorem is true by exhibiting a proof; thus, it has been a primary task of thephilosophy of mathematics to clarify the notion of a mathematical proof, andto explain how such proofs are capable of providing appropriate mathematicalknowledge. Mathematical logic and the theory of formal axiomatic systemshave done a remarkably good job of addressing the first task, providing idealizedaccounts of the standards by which a proof is judged to be correct. This, in andof itself, does not address more difficult questions as to what ultimately justifiesa particular choice of axiomatic framework. But, as far as it goes, the moderntheory of deductive proof accords well with both intuition and mathematicalpractice, and has had practical applications in both mathematics and computerscience, to boot.But in mathematics one finds many types of judgment that extend beyondevaluations of correctness. For example, we seem to feel that there is adifference between knowing that a mathematical claim is true, and understandingwhy such a claim is true. Similarly, we may be able to convince ourselves thata proof is correct by checking each inference carefully, and yet still feel asthough we do not fully understand it. The words ‘definition’ and ‘concept’seem to have different connotations: someone may know the definition ofa group, without having fully understood the group concept. The fact thatthere is a gap between knowledge and understanding is made pointedly clearby the fact that one often finds dozens of published proofs of a theoremin the literature, all of which are deemed important contributions, evenafter the first one has been accepted as correct. Later proofs do not addto our knowledge that the resulting theorem is correct, but they somehowaugment our understanding. Our task is to make sense of this type ofcontribution.The observation that modern logic fails to account for important classes ofjudgments traces back to the early days of modern logic itself. Poincaré wrotein Science et Méthode (1908):Does understanding the demonstration of a theorem consist in examining eachof the syllogisms of which it is composed in succession, and being convincedthat it is correct and conforms to the rules of the game? In the same way, doesunderstanding a definition consist simply in recognizing that the meaning of allthe terms employed is already known, and being convinced that it involves nocontradiction?... Almost all are more exacting; they want to know not only whether all thesyllogisms of a demonstration are correct, but why they are linked together inone order rather than in another. As long as they appear to them engendered bycaprice, and not by an intelligence constantly conscious of the end to be attained,they do not think they have understood. (Book II, Chapter II, p. 118)