> 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 believe that it is vital to have an understanding of sets to the extent of knowing what a Ven diagram is, as well as what intersections, unions, complements, etc., are. But do I really need
1+1=2, 2+1=3, 3+1=4,... or even 1,2,3,... seems more useful to me.
> 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.
Does any of that involve transfinite induction? -- Nil conscire sibi