Many children are incapable of becoming mathematicians who must none the less be taught mathematics; and mathematicians themselves are not all cast in the same mould. We have only to read their works to distinguish among them two kinds of minds - logicians like Weierstrass, for instance, and intuitionists like Riemann. There is the same difference among our students. Some prefer to treat their problems "by analysis," as they say, others "by geometry".I've often found this to be an interesting hypothesis. While I'm not a fan of viewing people as "types", I do have to wonder to what extent some individuals problems with math is an artifact of one type of approach to mathematics (say analysis) being emphasized over another.

It is quite useless to seek to change anything in this, and besides, it would not be desirable. It is well that there should be logicians and that there should be intuitionists. Who would venture to say whether he would prefer that Weierstrass had never written or that there had never been a Riemann? And so we must resign ourselves to the diversity of minds, or rather we must be glad of it.

There is a related hypothesis concerning the Pirahã I discussed here. While the Pirahã lack the ability (interest) to count, perhaps geometry might be more to their liking.

My own personal experience has found me distinctly preferring "analysis". While I enjoyed learning geometry, topology and the like I more or less stumbled my way through courses. If at all possible I would attempt to reduce the problem that I could solve "analytically" when a more "geometric" method would have been simpler.

In any event, Poincaré does make a good point regarding how both approaches are valuable. In many cases an analytic approach should be preferred while others require a geometric approach. In some cases problems can be solved by both approaches but one is often much simpler than the other.

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