Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

422 alasdair urquhartA time-hallowed form of philosophical writing, that continues to be popularto the present day, is the commentary on a classical text. Typically, thewriter proposes an ‘interpretation’ of this text, and defends it, often withgreat polemical vigour, against competing interpretations. The commentatorfrequently seems convinced that the interpretation offered is the only correctone, and all the earlier attempts were completely incorrect and illusory. Itis an odd and quite surprising fact that the same philosophers who teachtheir undergraduate students Quine’s sceptical arguments about the notionof meaning, and describe them as great contributions to modern analyticphilosophy, also write such polemical articles of historical commentary.The idea that there is a ‘correct’ view as to what a given philosophermight have meant in a puzzling passage seems taken for granted in suchdebates.The rules of the game in such debates between would-be exegetes are farfrom obvious—it is not clear, for example, just what evidence could be countedas settling the dispute. However, the history of mathematics provides us withsome interesting examples where puzzling, even contradictory, computationswere later given consistent interpretations. In this case, at least, the rules of thegame have greater clarity than in the case of philosophical writings. The criteriaof success are fairly simple—to provide a consistent conceptual framework inwhich the puzzling computations make sense.16.3 Varieties of infinitesimalsPerhaps the most famous example of such a development lies in the area ofcalculus. The invention of the calculus is linked indissolubly to that of earlymodern physics; it is notorious that the early work was lacking in rigour, asnew results and ideas appeared in a flood. However, in spite of early attemptsat making Newtonian calculus rigorous, such as the work of Colin MacLaurin(1742), a fully satisfactory foundation for the differential and integral calculusdid not appear until the mid-19th century. Even so, the older tradition ofinfinitesimals, even though supposedly banished from rigorous discourse bythe new limit concepts, exerted a seemingly irresistible attraction on practicalmathematicians, and the folk tradition of more or less naive reasoning withinfinitesimals is far from dead even today, as can be seen (for example) froma set of introductory lectures on celestial mechanics by Nathaniel Grossman(1996). In the opening remarks of his first chapter, he says: ‘While these objectsare indispensable to applied mathematicians and appliers of mathematics, their