Algebra or Geometry

Axioms

The debate I am entering here is not a new one. For me it is a surprising fact, that geometry has been based on axioms since the days of Euclid, more than two thousand years ago. These axioms may not stand the standards of today, but nevertheless they were axioms. It was Hilbert who sorted them out in his famous book about geometry, and made them rigorous.

On the contrary, the axioms for algebra and analysis are quite recent. Until about 1850, discussion about arguments in analysis was on vague grounds. Riemann himself used infinitely small numbers, and it was much later that the epsilon-procedures in use today were established. A solid foundation on set theory came even later than that. The axiomatic approach to infinitely small numbers (non-standard analysis) is a very recent invention.

Nowadays, axiomatic geometry is no longer taught, and proofs in geometry in the school are based on „known or obvious facts“. We can call these proofs local proofs, since no profound axiomatic system is used. In the schools, the same, however, is true for analysis.

And it has to be that way! If anyone ever teaches axiomatic geometry to students, he will make the experience that this is quite a difficult world for them. This kind of rigor seems to be completely unsuited for beginner students, let alone for school kids.

However, the first year student in mathematics is indeed faced with a rigorous axiomatic system, at least in most countries I know. Most of the time, analysis is taught starting from some form of axioms for the real numbers.

So what is the difference between rigor in geometry, and rigor in analysis? I actually cannot give a complete answer to this, especially if I consider the historical background mentioned above. Abstract geometrical thinking is thought of as weird and useless, despite the fact that there are useful non-euclidean geometries. Maybe the algebraic computations with coordinates are so engraved into the learner that he cannot even imagine to let this go. Maybe there is no desire to study geometric proofs at all. Or maybe, geometry is too much about proofs, and too little about numerical applications. I do not know!

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