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

mathematics and physics: strategies of assimilation 437with theorems deducible from the axioms of set theory, there still remainsthe non-formal question of justifying the axioms themselves, including newaxioms.Going beyond this, it would seem that if we define mathematics in thisnarrow way, that we would exclude from the sphere of mathematics mostmathematics from before 1900, surely an unwanted consequence.I am not arguing for relaxing the standards of rigour in mathematics.On the contrary, I am urging logicians and philosophers to look beyond theconventional boundaries of standard mathematics for mathematical work that isinteresting but nonrigorous, with a view to making it rigorous. The examples ofcalculus, distribution theory, complex function theory, and Brownian motionall show that some of the best mathematics can result from this process.Similarly, philosophers can surely find fruitful areas for their studies in areaslying beyond the usual set-theoretical pale.The Czech physicist Jan Klima has presented the activities of mathematiciansin a rather humiliating light:In the fight for new insights, the breaking brigades are marching in the frontrow. The vanguard that does not look to left nor to right, but simply forgesahead—those are the physicists. And behind them there are following the variouscanteen men, all kinds of stretcher bearers, who clear the dead bodies away, or,simply put, get things in order. Well, those are the mathematicians. (Zeidler,1995, p.373)By way of contrast, here is a quotation from the great Canadian mathematician,Robert Langlands:Field theories and especially conformally invariant field theories are becomingfamiliar to mathematicians, largely because of their influence on the study ofLie algebras and above all on topology. Nonetheless, in spite of the progress inconstructive quantum field theory during recent decades, many analytic problems,especially the existence of the scaling limit, are given short shrift. These problemsare difficult and fascinating and merit more attention. ... It is often overlookedthat the largely mathematical development of Newtonian mechanics in the 18thcentury was an essential prerequisite to the enormous physical advances of the19th and 20th centuries, that attempts to overcome mathematical obstacles maylead to concepts of physical significance, and that mathematicians, recalling thenames of d’Alembert, Lagrange, Hamilton and others, may aspire to nobler tasksthan those currently allotted to them. (Langlands, 1996)I hope that these inspiring words of Langlands may lead philosophers to lookbeyond the rather restricted range of topics in mathematics that currentlypreoccupy them.