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

mathematics and physics: strategies of assimilation 423language has been long banished from the usual calculus books because theywere considered to be ‘‘unrigorous’’ ’ (Grossman, 1996, p.1).Philosophers who are interested in the foundations of mathematics know thatin the 1960s, Abraham Robinson created a consistent theory of infinitesimalsthat has found significant applications in many areas of mathematics. Whatis not so well known is that there are several theories of infinitesimals, sothat competing interpretations of the classical passages involving infinitesimalscan be given divergent interpretations. Of these theories, the most interestingfrom the philosophical point of view is the theory of smooth infinitesimalanalysis (Moerdijk and Reyes, 1991; Bell, 1998). In this theory, all functionsare continuous—it follows from this that the logic is of necessity non-classical,since with the help of the law of excluded middle we can define the ‘blipfunction’ that is 1 at the real number 0, and 0 everywhere else.Abandoning classical logic may seem a high price to pay, but in fact theresulting theory is very rich. In the ‘smooth world’ not only are all functionscontinuous, but in addition, the principle of microlinearity, or microaffineness(Bell, 1998,p.23) holds. This principle says that if we examine the infinitesimalneighbourhood of a point on a curve in the smooth world, then the curve,restricted to that neighbourhood, is a straight line! In other words, in such auniverse, the idea of early writers on the calculus, such as Isaac Barrow, thatcurves are made out of ‘linelets’ is here literally true. Smooth infinitesimalanalysis provides some retrospective justification for the idea of a curve asa polygon with infinitely many sides that played a central part in the earlyLeibnizian calculus (Bos, 1974, p.15).What is more, the nature of infinitesimals in this theory enables the basictechniques of differential calculus to be reduced to simple algebra. In the earlywritings on the differential and integral calculus, it was common to neglecthigher order infinitesimals in calculations. That is to say, if ɛ is an infinitesimalquantity, then at a certain stage in the calculation, early writers such as Newtontreat ɛ 2 as if it were equal to zero (even though at other stages in the calculation,ɛ must be treated as nonzero). This apparent inconsistency was the target ofBerkeley’s stinging criticisms in The Analyst (Berkeley, 1734). Strange to say,these inconsistencies disappear in the smooth worlds invented by the categorytheorists. In this theory, a nilpotent infinitesimal ɛ is one that satisfies ɛ 2 = 0. Itfollows from the microlinearity postulate that in a smooth world, the axiom¬∀x(x 2 = 0 ⇒ x = 0) holds (Bell, 1998, pp. 23–24, 103). Here, the fact thatthe logic is non-classical (intuitionistic) saves us from contradiction, because wecannot infer from this fact that ∃x(x 2 = 0 ∧ x ̸= 0); this last formula results inan outright contradiction, whereas the theory of smooth analysis is known to beconsistent by means of the models constructed in Moerdijk and Reyes (1991).