e

The number e, which is equal to 2.71828…, might seem like something trivial—a play on numbers by mathematicians. Nothing could be further from the truth. It is defined as the limit of the expression (1 + 1/n)n as n becomes large without bound. What possible connection does this number have with other areas of mathematics? As it turns out, it forms the base of natural logarithms; it appears in equations describing growth and change; it surfaces in formulas for curves; it crops up frequently in probability theory; and it appears in formulas for calculating compound interest. It is another example of how the ideas in mathematics are not isolated ones, but highly interrelated. The purpose of this chapter is, in fact, to link e to other great ideas, showing how mathematical discovery forms a chain—a chain constructed with a handful of fundamental ideas that appear across time and space, finding form and explanation in the writings and musings of individual mathematicians.

Reasons Theory

After rejecting deflationism, the central further question is whether our rejections and acceptances of words, in the articulation process, are based on reasons. Reasons-theorists say “yes” and look for some mental state that gives us a reason for accepting/rejecting a formulation. One kind of reasons-theorist argues that our reasons come from some knowledge we have of our thought. Another kind of reasons-theorist argues that our reasons come from feelings that result from sub-personally matching our thought with our words. Contra the reasons-theorists, this chapter maintains that we cannot make sense of the bulk of our responses in the articulation process by assimilating them into the reasons framework. Resolving the puzzle calls for an alternative model of rational control—one that may be implicated in learning and numerous other epistemologically central activities, ranging from basic perceptual categorization to sophisticated mathematical discovery.

Lakatos, Imre (1922–74)

Imre Lakatos made important contributions to the philosophy of mathematics and of science. His ‘Proofs and Refutations’ (1963–4) develops a novel account of mathematical discovery. It shows that counterexamples (‘refutations’) play an important role in mathematics as well as in science and argues that both proofs and theorems are gradually improved by searching for counterexamples and by systematic ‘proof analysis’. His ‘methodology of scientific research programmes’ (which he presented as a ‘synthesis’ of the accounts of science given by Popper and by Kuhn) is based on the idea that science is best analysed, not in terms of single theories, but in terms of broader units called research programmes. Such programmes issue in particular theories, but in a way again governed by clear-cut heuristic principles. Lakatos claimed that his account supplies the sharp criteria of ‘progress’ and ‘degeneration’ missing from Kuhn’s account, and hence captures the ‘rationality’ of scientific development. Lakatos also articulated a ‘meta-methodology’ for appraising rival methodologies of science in terms of the ‘rational reconstructions’ of history they provide.