Euler developed a program which aimed to transform analysis into an autonomous discipline and reorganize the whole of mathematics around it. The implementation of this program presented many difficulties, and the result was not entirely satisfactory. Many of these difficulties concerned the integral calculus. In this paper, we deal with some topics relevant to understand Euler’s conception of analysis and how he developed and implemented his program. In particular, we examine Euler’s contribution to the construction of differential equations and his notion of indefinite integrals and general integrals. We also deal with two remarkable difficulties of Euler’s program. The first concerns singular integrals, which were considered as paradoxical by Euler since they seemed to violate the generality of certain results. The second regards the explicitly use of the geometric representation and meaning of definite integrals, which was gone against his program. We clarify the nature of these difficulties and show that Euler never thought that they undermined his conception of mathematics and that a different foundation was necessary for analysis.
Communicated by Jesper Lützen.
There are many ways to consider the philosophy of history. In this article, I claim that one of the most viable approaches to the philosophy of history today is that of critical theory of history, inspired by Reinhart Koselleck. Critical theory of history is based on what I call known history, history as it has been established and expounded by historians. What it contributes—its added value, so to speak—is a reflection on the categories employed to think about historical experience at its different levels, not only as a narrative but also as a series of events: their origins, contexts, terminology, functions (theoretical or practical), and, finally, eventual relevance.
Jouni-Matti Kuukkanen has written an important book. It directly confronts a key theoretical dilemma that has shadowed debate in historiography for several decades: histories cannot be written without using some narrative structure or other, but epistemological evaluation cannot be applied to narratives qua narrative. Thus, if empirical inquiry takes the form of a history, then it cannot be rationally evaluable, and if rationally evaluable, empirical inquiry cannot be in the form of a history. Kuukkanen's book both directly confronts and proposes a strategy for surmounting this tired and tiresome theoretical barrier. Kuukkanen deserves great credit for attempting to reshape a long-stalled debate in a way that enables the theoretical options to be imagined anew. Yet his structuring of the oppositional tendencies engenders some ongoing problems regarding how to understand the philosophical stakes and options. This review argues that achieving Kuukkanen's postnarrativist future requires going back to past epistemic concerns discarded because they were tied to conceptions of logic and explanation that could not be reconciled with narrative form. Kuukkanen practices postnarrativism but still preaches a prenarrativist conception of logic. To reach his promised future, to actually overcome the dilemma that he rightly seeks to transcend, one must actually have the courage of Kuukkanen's pragmatist convictions.