The Nature of Mathematical Explanation

Although in the past three decades interest in mathematical explanation revived, recent literature on the subject seems to neglect the strict connection between explanation and discovery. In this paper I sketch an alternative approach that takes such connection into account. My approach is a revised version of one originally considered by Descartes. The main difference is that my approach is in terms of the analytic method, which is a method of discovery prior to axiomatized mathematics, whereas Descartes’s approach is in terms of the analytic-synthetic method, which is a heuristic pattern in already axiomatized mathematics. 1. The Aristotle-Pólya tradition In two recent papers (Cellucci, 2005, 2006b) I challenged a claim of a long tradition, from Aristotle to Pólya, according to which there is a sharp distinction between two kinds of reasoning, demonstrative reasoning, that is, the deductive derivation of conclusions from premisses which are primitive and true in some sense of ‘true’, and non-demonstrative reasoning, that is, the non-deductive (inductive, analogical, etc.) derivation of conclusions from premisses which are not known to be true but are only ‘accepted opinions’, in the sense of Aristotle’s éndoxa. The former is essentially superior to the latter since it is cogent, whereas the latter is not cogent. This claim is untenable because, by Gödel’s incompleteness results, knowing that the premisses of demonstrative reasoning are true – either in Gödel’s strong sense that they express properties of objects independent of us or in Hilbert’s weak sense that they are consistent – is generally impossible. Thus premisses are only ‘accepted opinions’ in the sense explained above, and so have the same status as the premisses of non-demonstrative arguments (Cellucci, 2005, pp. 158-159). Moreover, deductive inferences can be justified only in the same ‘external’ non-absolute sense as non-deductive inferences (Cellucci, 2006b, pp. 231-232). Here I will consider another claim of the Aristotle-Pólya tradition, according to which, within demonstrative reasoning, there is a sharp distinction between two kinds of reasoning, the reasoning which shows why something is the case and the reasoning which only shows that something is the case. The former is essentially superior to the latter since it shows the cause, or reason, of the thing, thus providing an explanation of it, whereas the latter does not. 1 According to the Aristotle-Pólya tradition, ‘there are proofs and proofs, there are various ways of proving’ (Pólya, 1962-65, II, p. 126). Specifically, there is a sharp 1 The standard English translation for Aristotle’s ‘aitia’ is ‘cause’, which however has strong connotations. In what follows I will use ‘reason’ in place of ‘cause’ when possible.