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

what structuralism achieves 365get the intuition. There is no serious question of whether the methods areintuitive. They are to people who grasp them. Keep in mind how intuitionfunctions in mathematics. Italian algebraic geometry into the 1920s wasfamously intuitive. Focusing on complicated four-dimensional configurationsin higher-dimensional spaces, it relied on long years of expert experience toreplace explicit proof or calculation. Outsiders were unable to follow it. It roseto a level which ‘even the Maestri were unable to sustain’ (Reid, 1990, p.114).One person’s intuition can be another’s nightmare, and expert intuition bydefinition does not come easily.The more interesting question is: are the new methods number theory andtopology or a foreign import? Coordinate geometry is a paradigm of ‘impure’mathematics where questions about points and lines are answered using realnumbers and differential calculus. It is an evident change of topic while noone denies its value. Homology and other categorical methods do not changethe topic.Compare computer spreadsheets. Some people find them ungainly monsters.But spreadsheets are merely a format for what has to be done anyway: Data mustbe categorized, and some relations between categories must be routinized. Thegeneral theory of homological methods today is thousands of pages long andgrowing—and deep new facts are revealed by it. Yet experts who wrote thosepages and use cohomology throughout their work say it should be used easily asan organizing device. Functorial homology is pure topology, or group theory,or number theory, or other mathematics, depending on which functors it uses.The main stake in discussions of purity is that pure proofs are meant tobe desirable for their own sake, as being more intrinsic to their subjects.One century of great efforts proved the prime number theorem in arithmetic byusing deep complex analysis. Many people still wanted a purely arithmeticproof and Chapter 7 of this volume shows how another fifty years of greatefforts found one pp. 189–191.¹⁵ All purely arithmetic proofs of it to date arelonger and harder than the analytic proof, but they are valued for their purityand genuinely new insights. The situation in homology is just the reverse.Routine methods can eliminate homology from specific number theoretic ortopological theorems. Mathematicians eliminate it in practice only when theyhave an easier alternative.For example, a ‘very utilitarian description of the Galois cohomology neededin Wiles’s proof’ defines certain cohomology groups by elementary constructionsknown well before group cohomology—indeed since Noether in the¹⁵ In Detlefsen’s terms I claim homology introduces no remote ideas but merely organizes ideas alreadydirectly involved in number theory and topology.