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

mathematics and physics: strategies of assimilation 425provide a consistent interpretation of all of the earlier calculations and proofs.Indeed, this follows from the logic of the situation. Abraham Robinson (1966,Chapter X) argued strongly that his modern infinitesimal analysis could beseen as providing a retrospective justification for the procedures of Leibniz andhis followers. Nevertheless, historians of mathematics such as Bos (1974) havelooked askance at this attempt at assimilating Leibnizian methods to moderntechniques. However, some of Bos’s criticisms of Robinson involve absurdand impossible demands—for example, his first criticism (Bos, 1974, p.83) isthat Robinson proves the existence of his infinitesimals, whereas Leibniz doesnot! The fact remains—no interpretation of dubious and inconsistent practicesinherited from the past can be perfect in every respect. The most we can askis that a rigorous interpretation reproduce at least some of the most importantcalculations and concepts of the older technique.16.4 The umbral calculusThe example of infinitesimal calculus is rather intricate. To illustrate someof the basic methods of assimilation, we shall look at a simpler case, theumbral or symbolic calculus. This calculus first made its appearance as acomputational device for manipulating sequences of constants (Blissard, 1861);for the history of the method, see Bell (1938). Later, it was rediscoveredby the number theorist and inventor of mathematical games Lucas (1876).The self-taught engineer and theorist of electromagnetism, Oliver Heavisidedeveloped a similar operational calculus in the course of solving the differentialequations arising in the theory of electromagnetism. The famous analystEdmund T. Whittaker rated Heaviside’s operational calculus as one of thethree most important mathematical discoveries of the late 19th century.However, Heaviside’s work was regarded with distrust until Bromwich gave arigorous interpretation of Heaviside’s operators as contour integrals.All of these developments were regarded as somewhat questionable whenthey first appeared. To see why, let us use the example of the computation ofthe Bernoulli numbers B n in the umbral calculus. These numbers, of whichthe first few are as follows:B 0 = 1, B 1 =− 1 2 , B 2 = 1 6 , B 3 = 0, B 4 =− 1 30 , B 5 = 0, B 6 = 1 42 , ... ,play an important role in number theory and combinatorics. I claim they aredefined by the umbral equation: (B + 1) n = B n ,forn ≥ 2, together with the