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

mathematics and physics: strategies of assimilation 429experiment or computer simulation, then they are much less inclined thanthe pure mathematicians to examine the pedigree of a successful piece ofmachinery for calculation. This appears most flagrantly in the case of quantumfield theory. In this case, we have a calculational device (Feynman diagrams)that produces answers that agree to nine decimal places with the measuredvalue of certain physical quantities. As a consequence, the physicists havecomplete trust in the technique, even though it appears that it still seems toelude a full mathematical formulation (Johnson and Lapidus, 2000).The situation, though, is mathematically unstable, because the physicistscan often compute certain quantities, and ‘prove’ them to their own satisfaction,while employing the powerful but unrigorous methods peculiarto their own fields. In the introductory chapter, I mentioned two strategiesthat mathematicians can adopt in dealing with anomalous practicesof reasoning and computation. The first is to prove all of the establishedresults by conventional means. This is of course a perfectly reasonable idea,though it still leaves unanswered the question: just why do these outrageoustechniques work?The entities that appear in these calculations, such as infinitesimals orrepresentative variables, are logically anomalous objects. Anomalous objectsin my sense are a little like Imre Lakatos’ monsters, but anomalous in aneven stronger sense, namely not just falling outside a standard definition, butviolating fundamental mathematical principles.An interesting example of an anomalous object is the Dirac delta function.When it first made its appearance in the physics literature, it was explained asa function δ defined on the real line so that it was zero everywhere except atthe origin, but the integral of the function was 1:∫δ(x)dx = 1.All this is very plausible from the physical point of view. If we think of δ asrepresenting the mass density function of a point particle of mass 1 situated atthe origin, then the delta function has exactly the right properties. The troubleis, there is no such function. It’s very easy to show using standard calculus thatthere simply is no such function defined on the classical real line.We are caught in a dilemma. The physicists have postulated an object that,classically speaking, simply doesn’t exist, a logical contradiction. There arevarious attitudes that one can take to this.The first is simply to dismiss the unruly objects as nonsense, a commonattitude among mathematicians. It is not hard to cite mathematicians who tookjust such an attitude to objects like the Dirac delta function. For example,