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

mathematics and physics: strategies of assimilation 427and every step in the calculation can be given a clear semantical interpretationin the world of set-theoretical and category-theoretic constructions.It would be a mistake, though, to see Rota’s interpretation as simply bestowinglegitimacy on a previously dubious mathematical technique. The interpretationbears fruit in other directions. The embedding of combinatorially definedsequences in the algebra of formal power series is heuristically very fertile, andenables a systematic treatment of special functions. Furthermore, the interpretationmakes contact with a large tract of more recent mathematics, such as categorytheory with its fruitful notion of adjoints. The marriage of 19th-centuryformal calculation with modern abstract theory produces a sturdy offspring.Let us look back over this example to see if we can extract some interestinggeneral features. In the case of the umbral calculus, as in many of the other caseswe discuss, computations are primary. However, the lack of a clear semanticsfor the objects manipulated leads mathematicians to treat the methods withdistrust, no matter how successful the computations.Rota’s reinterpretation of the umbral calculus has two features that we candetect elsewhere in the work of assimilation. First, the logic is reinterpreted.The umbral calculationis rewritten as(B + 1) 2 = B 2 + 2B 1 + 1 = B 2 + 2B 1 + 1L((U +1) 2 ) = L(U 2 + 2U + 1) = L(U 2 ) + 2L(U) + L(1) = B 2 + 2B 1 + 1.The introduction of the linear operator L allows the dubious move of raisingand lowering indices to be explained simply as a transition between twoisomorphic domains of calculation. Second, the reinterpretation involves theintroduction of objects of higher type, in this case linear functionals acting onthe space of polynomials in one variable.This reinterpretation of logic and the introduction of higher type entities toprovide a semantics is also visible in the theory of infinitesimals. Robinson’snonstandard universe is composed of equivalence classes of higher type objects,since the hyperreals are elements of an ultrapower of a standard model (Goldblatt,1988). Equations that involve treating infinitesimals as zero quantities(the kind of thing to which Berkeley raised objections), such as 2 + ɛ = 2, arereinterpreted in Robinson’s framework as 2 + ɛ ≈ 2, where x ≈ y means thatx and y are infinitely close (and hence have the same standard part). Interestingly,in the framework of smooth analysis, equations such as 2 + ɛ 2 = 2canbe literally true, where ɛ is a nilpotent infinitesimal; the price we have to payfor this literal interpretation of equations is the abandonment of classical logic.