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

410 alasdair urquhartworld. However, this point of view brings with it considerable difficulties, thatare the fundamental subject of this chapter.The basic difficulty is this: if we construe these models simply as mathematicalobjects, then we have to face the fact that physicists do not employ normalmathematical methods in investigating them. Rather, the methods that theyuse are frequently so far from normal mathematical practice that it is sometimesnot clear that the objects themselves are even mathematically well defined.15.2 A renewal of vowsThroughout most of their history, mathematics and physics have been closelyintertwined, as can be seen by looking at the history of ancient astronomy(Neugebauer, 1975). Furthermore, if we look at work by mathematicians andphysicists from the early 19th century, the standards of mathematical rigourused by both groups of researchers appear indistinguishable. For example, CarlFriedrich Gauss has a high reputation among mathematicians for the rigourof his mathematical reasoning, but a glance at his famous treatise of 1827 ondifferential geometry (Gauss, 2005) shows that it makes free and uninhibiteduse of the infinitesimal quantities that later mathematicians were to regard withfear and loathing.The middle years of the 20th century saw an unprecedented divergencebetween the two communities of researchers. As Faddeev (2006)remarks,intheearlier part of this century, mathematical physics was not distinguished from theoreticalphysics—both Henri Poincaré and Albert Einstein were called mathematicalphysicists. However, with the expansion of theoretical physics in the1920sand1930s, a separation between the two areas occurred. Whereas ‘mathematicalphysics’ came to be understood as a somewhat restricted area, confinedto the study of mathematical techniques such as the solution of partial differentialequations and the calculus of variations, the theoretical physicists themselves,caught up in trying to understand the physics of atoms and elementary particles,began to move further and further away from the confines of mathematicalrigour. Although there are notable examples of interactions between mathematiciansand physicists in the middle part of the 20th century, such as theelucidation of quantum mechanics in terms of linear operators on Hilbert spaceand the application of fibre bundles to non-Abelian gauge theory, in general theconcepts and standards of the two fields showed an ever increasing divergence.The pressure of experimental results, notably the explosion of new elementaryparticles discovered during the 1950s and1960s, forced theoretical