This post may surprise and even irritate some people here but I am sure that if any sort of discussion springs out of it, we will get to read some interesting opinions.
For the rest of the post, if I write "geometry", I mean "classical Euclidean geometry".
It is almost a sacrosanct obligation for any mathematician to know at least some of the basics of classical Euclidean geometry, and by that, I am referring to geometry without coordinates. However, it is also known -even though some people just can't accept it- that classical Euclidean geometry is one of the disciplines of mathematics that rely less on rigor and reason and more on perception and intuition. Now, not that intuition is not important, but, in my opinion, strict mathematics has to promote mathematical intuition and not vice versa.
Of course, I can't deny the benefits of being familiar with basic results such as the triangle inequality, the Pythagorean theorem or the parallelogram law.
However, since most of modern mathematics relies on rigor and when we can't prove something we want to be true we don't generally assume it, I have three arguments against classical geometry:
- It doesn't promote strict reasoning - contrary to popular belief. Calculus does so much more efficiently.
- It maybe provides fertile ground for false intuition on relations of size: people have a hard time dealing with infinity.
- The intuition it cultivates maybe provides ground for extremely counterproductive discussions revolving around somewhat dumb issues such as Zeno's paradox.
Question $\rightarrow$I need therefore some nice reason why people insist on teaching and learning geometry. I am obviously not talking about basic facts, but about more complex issues (eg math competitions stuff). The reason can be practical or aesthetic, but here is what will be considered a good answer:
- A practical reason. Random probably nonsensical example: "Many complex facts from classical geometry have exact analogues in partial differential equations."
- Various aesthetic reasons. I will need -in total, not in one question - at least five of those and their clarity and beauty -in a loose sense- have to be on par with facts such as the Bolzano-Weierstrass theorem, the Banach fixed point theorem, the Schroeder-Bernstein theorem or Cantor's diagonal argument. Non-example: "if four circles the diameters of which are in arithmetic progression intersect in..." etc. Further non-example: Simson line.
- An answer will not be considered good if it is in the following form: "Geometry provides motivation for such and such a branch of mathematics". I can find enough motivation in mathematics without geometry.
The only thing I can think of is that geometry cultivates imagination and creativity in constructions. The rest is yours. Thanks in advance.
Notes:Through this question I am more interested in finding out why I should learn geometry rather than why people should teach geometry in general. However, an answer to both would be nice. Also, the point I am trying to make with that long text is that even though learning geometry sums its merits, maybe the way geometry is taught does not really do much good, because I don't know of any involved facts of geometry that are of interest in any other field of mathematics