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

mathematics and physics: strategies of assimilation 431the origin. This definition has the advantage of being closer to the originaldefinition of the physicists.Schwartz’s theory of distributions also fits the general pattern that we havedetected in our two previous examples. The common feature of the examplesof the Dirac delta function, infinitesimals, and the umbral calculus is that theexplications given for the anomalous objects and reasoning patterns involvingthem is what may be described as pushing down higher order objects. In otherwords, we take higher order objects, existing higher up in the type hierarchy,and promote them to new objects on the bottom level. This general patterndescribes an enormous number of constructions.The process we have just described is closely related to the method ofadjoining ideal elements familiar from the history of mathematics. This idea isvery clearly described by Martin Davis:This is a time-honored and significant mathematical idea. One simplifies thetheory of certain mathematical objects by assuming the existence of additional‘ideal’ objects as well. Examples are the embedding of algebraic integers in ideals,the construction of the complex number system, and the introduction of pointsat infinity in projective geometry. (Davis, 1977, p.1)Ken Manders (1989) has given a general account of this method as a strategy forunifying and simplifying concepts in mathematics. Here we are emphasizingthe role of the method as a means of giving a rigorous interpretation to dubiouscalculations.16.6 Strategies of interpretationEach of the three examples that I discussed above demonstrate the secondstrategy of assimilation that I described in my introductory chapter. The theoryof distributions, nonstandard analysis, and the modern umbral calculus all arosefrom the idea of constructing new conceptual schemes to validate anomalouscalculations, and all of them have proved useful and fruitful techniques thathave led to new mathematics.Sir Michael Atiyah (1995) has given a more complex and nuanced analysisof mathematical strategies. He lists four (rather than two), strategies thatmathematicians can adopt towards ideas emerging from the physics community.The first is a version of the first strategy described above, namely thatmathematicians should ‘take the heuristic results ‘‘discovered’’ by physicists andtry to give rigorous proofs by other methods. Here the emphasis is on ignoringthe physics background and only paying attention to mathematical results that