Hatto von Aquitanien wrote: > Virgil wrote: > > >>If the technicalities of set theory were as easy to learn as Russell >>seems to think it ought to be then everyone would learn it in grade >>school. In fact, it, like many specialities, usually takes years of >>study for one to become really good at it. > > > A good number of people can master concepts of mathematics sufficiently to > solve difficult problems in, say, fluid mechanics without much grasp of set > theory. That makes me wonder if set theory really is fundamental to > mathematics. I ask what it is that underlies the pragmatic application of > sophisticated mathematics. What are the intuitive assumptions these people > have made, and how are they manipulating ideas? I believe what I'm asking > is, what are the anthropological foundations of mathematics?
I have (tried) working in fluid dynamics for a number of years, and certainly some sort of naive set theory is indispensible. I already indicated this in another post where I talked about problems in showing which differential equations have unique solutions and which don't. Basically anything beyond numerical simulations or solving simple laminar flows requires sophisticated techniques that involve objects much more abstract than the real numbers (namely things like measure spaces and Besov spaces and things like that). A good example is the result of Caffarelli, Kohn and Nirenberg which looks at the maximal possible Hausdorff dimension of the set of singularities in the solution to the Navier-Stokes equation.