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

introduction 9consists in trying to dismantle the foundationalist filter. The second, the parsconstruens, provides philosophical analyses of a few case studies from mainstreammathematics of the last seventy years. His major case studies come from theinterplay between mathematics and computer science and from n-dimensionalalgebras and algebraic topology.The pars destruens shares with Lakatos and some of his followers a stronganti-logical and anti-foundational polemic. This has unfortunately damagedthe reception of Corfield’s book and has drawn attention away from the goodthings contained in it. It is not my intention here to address the significanceof what Corfield calls the ‘foundationalist filter’ or to rebut the argumentsgiven by Corfield to dismantle it (on this see Bays (2004) andPaseau(2005)).Let me just mention that a very heated debate on this topic took place inOctober 2003 on the FOM (Foundations of Mathematics) email list. The parsdestruens in Corfield’s book is limited to some arguments in the introduction.Most of the book is devoted to showing by example, as it were, what aphilosophy of mathematics could do and how it could expand the range oftopics to be investigated. This new philosophy of mathematics, a philosophyof ‘real mathematics’ aims at the following goals:Continuing Lakatos’ approach, researchers here believe that a philosophy ofmathematics should concern itself with what leading mathematicians of theirday have achieved, how their styles of reasoning evolve, how they justify thecourse along which they steer their programmes, what constitute obstacles totheir programmes, how they come to view a domain as worthy of study andhow their ideas shape and are shaped by the concerns of physicists and otherscientists. (p. 10)This opens up a large program which pursues, among other things, thedialectical nature of mathematical developments, the logic of discovery inmathematics, the applicability of mathematics to the natural sciences, thenature of mathematical modeling, and what accounts for the fruitfulness ofcertain concepts in mathematics.More precisely, here is a list of topics that motivate large chunks of Corfield’sbook:1) Why are some mathematical entities important, natural, and fruitful whileothers are not?2) What accounts for the connectivity of mathematics? How is it thatconcepts developed in one part of mathematics suddenly turn out to beconnected to apparently unrelated concepts in other areas?3) Why are computer proofs unable to provide the sort of understanding atwhich mathematicians aim?