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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:

  1. It doesn't promote strict reasoning - contrary to popular belief. Calculus does so much more efficiently.
  2. It maybe provides fertile ground for false intuition on relations of size: people have a hard time dealing with infinity.
  3. 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

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    $\begingroup$ I think by going so far to prohibit several answers, many of which a lot of people would already consider good motivation for teaching Euclidean geometry, this question is too close to a rant to be a good question on this site. $\endgroup$
    – rschwieb
    Jan 16 '14 at 16:06
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    $\begingroup$ I don't get the point you make about rigour. Firstly, any branch of math can be taught rigorously or non-rigorously, or anything in between. The level of rigour is up to the teacher. Secondly, I don't see how it relates the rest of the question; if you're looking for motivation as to why geometry should be taught, why both going off on that tangent about rigour? $\endgroup$ Jan 16 '14 at 16:06
  • $\begingroup$ @rschwieb, I think there is a good question buried beneath the rant, but it needs a major rewrite. $\endgroup$ Jan 16 '14 at 16:07
  • $\begingroup$ @rschwieb I can't help it, I need a reason which will motivate me to study geometry, and motivation for other branches of mathematics or (imo) uninteresting facts about unnatural -in the sense that they don't arise naturally- figures is not going to help. $\endgroup$
    – Steve Pap
    Jan 16 '14 at 16:12
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    $\begingroup$ @StevePap I think you could say the same for 15 year olds and calculus, which you pushed as a better alternative. Nobody is claiming that geometry perfectly teaches rigor at that age, but I think a lot of people will agree that it is a good first-approximation, and that it is relatively accessible. For one thing, it strongly appeals to our experience with shapes and distances, which children already have some experience with. $\endgroup$
    – rschwieb
    Jan 16 '14 at 16:17
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Rigor

Despite the ancient foundations of geometry not being rigorous by modern day standards, there are rigorous systems for Euclidean geometry now. One can either teach using these new systems, and one can also gain a lot of educational value by comparing the old system to the new one, as Hartshorne does in his wonderful book. This study clearly does promote reasoning from axioms and previous propositions, so I have to strongly disagree with your assertion to the contrary.

False intuition

It maybe provides fertile ground for false intuition on relations of size. It also provides fertile ground to dispel false intuitions and reliance on pictures, and reinforce reasoning from axioms and principles. This is a double-edged complaint.

Counterproductivity(?)

The intuition it cultivates maybe provides ground for extremely counterproductive discussions revolving around somewhat dumb issues such as Zeno's paradox.

Or it introduces the student to a famous philosophical problem, and how one might engage in intelligent discussion about such things.

Summary

Of course it is important to not teach wrong things and/or false intuitions. But that does not mean they should be avoided entirely. Rather they should be intelligently incorporated into an overall discussion. It is really rough to find the right educators and the requisite amount of time to do this in a very good way. That's the struggle between ideal and practical at work. The best we can hope for is to produce better teachers and give the students more time.

Aside from geometries direct applications, proofs in geometry facilitate the development of what constitutes a proof and basic logic.

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    $\begingroup$ I just borrowed this book from the university library since they didn't have Hartshorne's. The first few chapters discuss Euclid's postulates, Hilbert's axiomatisation (and in particular the parallel postulate) and the difference between them. $\endgroup$
    – Arthur
    Jan 16 '14 at 16:18
  • $\begingroup$ I've skimmed that one: it's also pretty good. $\endgroup$
    – rschwieb
    Jan 16 '14 at 16:19
  • $\begingroup$ However, you can't start from Hartshorne, or can you? High school students are not taught rigorous geometry, at least not where I live! $\endgroup$
    – Steve Pap
    Jan 16 '14 at 16:26
  • $\begingroup$ One more point: the question was not to convince me that we should not completely abandon geometry -I am convinced-, the question was to give me some reasons why I should learn extensive geometry according to two patterns. Thanks for the information, though, I will definitely check Hartshorne's book. $\endgroup$
    – Steve Pap
    Jan 16 '14 at 16:28
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    $\begingroup$ @StevePap Yeah maybe so. Have you ever studied points of concurrency? I always thought the four basic theorems about concurrence of certain lines in triangles were pretty surprising. Another interesting theorem in geometry is Desargues's theorem. Desargues's theorem in particular has a deep implications beyond the diagram. $\endgroup$
    – rschwieb
    Jan 16 '14 at 19:15

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