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

278 jamie tappendenmathematics have had a profound effect on the emergence of the styles ofmathematical reasoning that evolved in the subsequent century.My goal here is to describe the Riemann–Dedekind approach to ‘‘essentialcharacteristic properties’’ and indicate some of the mathematics that gives itsubstance. Along the way, I’ll spell out why I regard this as a promising andphilosophically profound strategy for arriving at an account of naturalness inmathematical classification that can coalesce with an account of properties anddefinitions in general. To set the stage, I’ll first discuss the current situation in thegeneral metaphysics of properties as it pertains to the naturalness of mathematicalproperties. The point of the context-setting will be to explain why the waythings currently stand—especially the role of metaphysical intuitions, and thestock of examples used as reference points (plus the potential examples thatare not used)—make it difficult to address mathematical properties in anilluminating way.10.2 Analytic metaphysics: the ‘rules of the game’and the method of intuitionsEven if we acknowledge the Port Royal principle of Chapter 9,andensurethatour account of mathematical ‘classification and definition’ pays due heed to ‘thesubject matter being discussed’, we should work toward a synoptic treatment ofmathematical and non-mathematical cases. It’s unlikely that mathematical andnon-mathematical reasoning are so disjoint as to exclude interesting points ofoverlap. In recent decades there has been a revival of old-fashioned metaphysicaldebates about the reality of universals, the artificial/natural distinction, andcognate topics. It might seem initially promising to draw on these debatesto illuminate the questions appearing in the survey essay. In this section I’llillustrate why the methods accepted as defining the ‘rules of the game’ in therelevant areas of contemporary analytic metaphysics are unlikely to help us asthings currently stand. This will force a different perspective on the problem;I’ll explore one possibility that centers on inductive practices of conjecture andverification in the subsequent section.Consider again the example from the survey essay: In algebraic numbertheory, the definition ‘a ̸= 1 is prime if, whenever a | bc then a | b or a | c’ is,in an important way, the ‘correct’, or ‘proper’ definition of ‘prime number’,and the school definition ‘n ̸= 1 is prime if it is evenly divided by only 1 andn’. is comparatively accidental. As in the suggested comparison—the changein the definition of kinetic energy made necessary by relativity theory—we