History of mathematics illuminates philosophy of mathematics: Riemann, Weierstrass and mathematical understanding (Q6623924)

When mathematicians begin to explore a new area, they may appeal to concepts -- understanding, explanation, natural definition, rigour, and even harmony, beauty, etc. -- whose meaning in the new context is still developing. By focusing on a nineteenth-century case, the Riemann and Weierstrass approaches to elliptic functions and their generalisations, and appealing to a concept of fruitfulness, taken as a (generally tacit) prediction that an approach will lead to further discoveries, the paper explores how the area, and the language used to discuss it, developed. When something appears to work, is it a method (i.e. of more general significance) or a trick (something that happens just in this particular case)? A method is fruitful, a trick (perhaps better, an oddity) is not. The illustrative examples of this important difference are particularly clear. The outcome is that many things debated philosophically in the early days of a subject area may become clear only as the historical processes of debate, argument, proof, and development have had time to play out.\N\NFor the entire collection see [Zbl 1537.01004].