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

the boundary between mathematics and physics 411physicists into a quasi-empirical mode of operation, and also pushed themin the direction of purely formal calculations. At the same time, the mathematicianswere attracted by increasing rigour and abstraction. Emblematicof the period is the inception in the mid-1930s of the series of booksby the Franco–American mathematical collective who published under thename of Nicolas Bourbaki. In the foundational volume of his encyclopedia,Bourbaki (2004) writes in the preface: ‘This series of volumes, ...takes up mathematics at the beginning, and gives complete proofs’. Partlyas a result of Bourbaki’s influence, the language employed by mathematiciansand physicists continued to diverge. The physicists, brought up in theolder framework of traditional calculus, with its emphasis on formal manipulationsof integrals and infinite series, to a considerable degree no longershared a common language with the mathematicians. Freeman Dyson (1972)remarked sadly:As a working physicist, I am acutely aware of the fact that the marriage betweenmathematics and physics, which was so enormously fruitful in past centuries, hasrecently ended in divorce. Discussing this divorce, the physicist Res Jost remarkedthe other day, ‘As usual in such affairs, one of the two parties has clearly gotthe worst of it.’ During the last twenty years we have seen mathematics rushingahead in a golden age of luxuriant growth, while theoretical physics left on itsown has become a little shabby and peevish.However, in an unexpected and exciting development, the apparentlydiverse streams of mathematics and physics are beginning to reconverge. It isnot quite clear what led to these new developments, but there seem to havebeen forces acting on both sides of the divide. On the side of physics, thesuccess in the 1970s of the Standard Model of elementary particles, formulatedin the language of quantum field theory, led to an astonishingly successfulframework in which to fit the unruly zoo of elementary particles that had beendiscovered experimentally in the decades following the Second World War.The scope and success of this triumph of theoretical physics is remarkable.The SU(3) × SU(2) × U(1) model has been confirmed repeatedly, and isconsistent with virtually all physics down to the scales probed by currentparticle accelerators, roughly 10 −16 cm. The very success of this model has ledto a change of emphasis. A great deal of current effort is being devoted to stringtheory. However, the scale of string theory is roughly 20 orders of magnitudesmaller, of the order of the Planck length, 1.6 × 10 −33 cm. Consequently,there appears to be no hope of direct experimental tests of this theory. Instead,the physicists are guided to an increasing extent by aesthetic criteria moreand more resembling those used by pure mathematicians. On the other sideof the divide, mathematicians seem to have grown tired of the abstraction