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

introduction 17influenced by set-theoretic foundations and then proceeds to argue that onlydetailed attention to the structuralism embodied in the practice (unlike otherphilosophical structuralisms) can account for certain aspects of contemporarymathematics, such as the ‘unifying spirit’ that pervades it. In his research paperhe looks at schemes as a tool for pursuing Weil’s conjectures in number theory asa case study for seeing how ‘structuralism’ works in practice. In the process hedraws an impressive fresco of how structuralist and categorial ideas developedfrom Noether through Eilenberg and Mac Lane to Grothendieck.Finally, the last chapter is on the philosophical problems posed by somerecent developments in Mathematical Physics and how they impact puremathematics. In the third quarter of the 20th century what seemed like aninevitable divorce between physics and pure mathematics turned into anexciting renewal of vows. Developments in pure mathematics turned out to beincredibly fruitful in mathematical physics and, vice versa, highly speculativedevelopments in mathematical physics turned out to bear extremely fruitfulresults in mathematics (for instance in low-dimensional topology). However,the standards of acceptability between the two disciplines are very different.Alasdair Urquhart (University of Toronto) describes in his introduction someof the main features of this renewed interaction and the philosophical problemsposed by a variety of physical arguments which, despite their fruitfulness, turnout to be less than rigorous. This is pursued in his research paper where severalexamples of ‘non-rigorous’ proofs in mathematics and physics are discussedwith the suggestion that logicians and mathematicians should not dismissthese developments but rather try to make sense of these unruly parts of themathematical universe and to bring the physicists’ insights into the realm ofrigorous argument.5 A comparison with previous developmentsThe time has come to articulate how this collection differs from previoustraditions of work in philosophy of mathematical practice. Let us begin withthe Lakatos tradition.There are certainly a remarkable number of differences. First of all, Lakatosand many of the Lakatosians (for instance, Lakatos (1976), Kitcher (1984))were quite concerned with metaphilosophical issues such as: How do historyand philosophy of mathematics fit together? How does mathematics grow? Isthe process of growth rational? The aim of the authors in the collection ismuch more restricted. While not dismissing these questions, we think a good