On 31 Aug 2012, at 09:57, John Doty <email@example.com> wrote:
> On Thursday, August 30, 2012 2:09:19 AM UTC-6, Alexei Boulbitch wrote: > >> Let me just point out that the origin of this interesting and passionate discussion was the >> question of what should be the content and tools of the mathematical education for students in >> non-mathematical specialities at present, observing that since long computers have become the >> reality of our world. > > The nature of mathematics is very relevant here. If mathematics is the product of a magical sense that detects objects in a supernatural world, then it is impossible to scientifically approach the problem of teaching mathematics. But if mathematics is a product of the natural processes of cognition that humans use to find their way in the world, teaching mathematics is similar to other kinds of teaching. Then, the methods and insights of cognitive science are likely to be of use in answering your question. >
The question of how human beings learn mathematics and whether the natural (not "super-natural") world is governed by laws which can be described by means of mathematics are two quite different (although related) questions. Mathematical "pPlatonism" is only concerned with the latter question, it asserts that mathematical theorems (in some sense) hold in the real world and not just in human minds. How human beings learn and discover them is a rather different matter.
To idea that human beings acquire mathematics as part of a "natural processes of cognition that humans use to find their way in the world" and at the same time mathematics "has nothing to do" with this world, sounds to me like a very feeble one, but since the issue is metaphysical, probably harmless. However the idea that by using "insights of cognitive science" you can turn people who have trouble adding fractions into Riemann's and Poincare's sounds to me about as "useful" as Marx's idea that once you "abolish" private property all human beings will become geniuses and every kitchen maid will be able to run the State. And, if you are not claiming that everyone can be a Newton, Gauss, Riemann, or Poincare etc. (with proper, "cognitive science" guided education) then why not? Is it possible that mathematical ability may be largely inherited? Why, what evolutionary purpose would that serve?