The Development of the Foundations of Mathematical Analysis from Euler to Riemann (0262070340)

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The key to the presentation is motivation ... The mathematical teaching literature in often criticized for prejudice of presentation in favor of current conventions, or simply for being boring; and the reason for these deficiencies seems to be that, however skilfully the subject matter is presented and the examples are chosen, there is always a fatal lack of problem-background for it all. One may learn of this theorem or that property, but the problems which they are supposed to be solving are not mentioned. ... Competing approaches and their alleged inferiority are not mentioned or, worse, dismissed as unreliable or inadequate without the reasons for this action being explained." (pp. ix-x)

Unfortunately the book largely fails to live up to this goal. It is a standard historical narrative driven by the whim of its author more than anything else. Insights of the type indicated in the above quotation can only be extracted by doing all the work that the author should have done had he really meant what he wrote in the introduction.

When I say that this is a standard historical narrative I mean that the author is too much of an academic historian to ever state any interesting theses or claims. I shall do what I can to remedy this defect of the profession by arguing for the following thesis: the formal theory of differentiation and integration did not correct any pre-formal intuitions; rather it corrected only post-formal errors. The nonsense one often hears about how "everybody thought" that any continuous function was differentiable almost everywhere and so on is nothing but propaganda invented to serve the ideologues currently in power.

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