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

426 alasdair urquhartinitial condition B 0 = 1. By this is meant: expand the left-hand side by thebinomial theorem, then replace all of the terms B k by B k . Then the result isthe usual recursion equation defining the Bernoulli numbers. Blissard’s originalexplanation is not fully satisfactory. He writes:According to this method, quantities are considered as divided into two sorts,actual and representative. A representative quantity, indicated by the use of acapital letter without a subindex, as A, B, ... P, Q, ...U, V , ...is such that U n isconventionally held to be equivalent to, and may be replaced by U n . (Blissard,1861, p.280)Blissard states clearly that he is operating in a two-sorted logic, with two kindsof variables. Furthermore, the two types of variables are linked by the equationU n = U n ,whereU is the representative variable corresponding to the actualvariable U. This transition between terms involving the two types of variableis known in the literature of the umbral calculus as ‘raising and loweringof indices’. The remainder of his proposal, however, is somewhat unclear,because the rules of operation for the representative variables are left tacit,though some of them may be gleaned from Blissard’s practice. Furthermore,the nature of the representative quantities was unclear, even when it wasplain that the method produced the right answers, used with appropriatecaution.All of this was cleared up in a fully satisfactory way, mostly through the effortsof Gian-Carlo Rota. Rota placed the calculus in the framework of modernabstract linear algebra, and gave a rigorous foundation for the calculations(Rota, 1975). The framework is simple and elegant, and can be explained asfollows (Roman, 1984). Let P be the algebra of polynomials in one variableU over the complex numbers, and consider the vector space of all linearfunctionals over P. Any such linear functional L is determined by its values onthe basis polynomials U 0 , U 1 , U 2 , ..., and consequently can be correlated withthe formal power series:f (U) = L(U 0 ) + L(U 1 )U + L(U2 )2U 2 + L(U3 )6U 3 +···+ L(Uk )U k + ... .k!If we use the representative term U n as an abbreviation for L(U n ),thenitisclear that the computation involving the Bernoulli numbers can be justified,and the mysterious equation from which we started can be interpreted as sayingL((U + 1) n ) = L(U n ). Thus, we have found a way of rewriting the originalcomputations so that every step makes sense. The representative variables arerevealed as shorthand expressions for the coefficients of formal power series,