1
$\begingroup$

The five postulates (axioms) are:

  1. "To draw a straight line from any point to any point."

  2. "To produce [extend] a finite straight line continuously in a straight line."

  3. "To describe a circle with any centre and distance [radius]."

  4. "That all right angles are equal to one another."

  5. "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

What's wrong with them? Which axiomatic system is being used nowadays? Hilbert's or SMSG (School Mathematics Study Group)? I believe in the case of SMSG the list of axioms contains some redundancy. Why do people say Euclid's axioms are 'far from being sound', even if they are all still (I guess) believed to be true? If there's something wrong with them, then maybe our better (Hilbert's or whatever) axiomatic system contains some false statements?

$\endgroup$
  • 1
    $\begingroup$ For example the third axiom is the only one about circles. So strictly speaking, it is hard to prove anything about circles $\endgroup$ – Hagen von Eitzen Apr 21 '15 at 19:35
  • $\begingroup$ Can you cite a source that finds issue with them? Most criticism I've heard has centered around extra necessary assumptions; things that let us conclude "Yes, these two circles definitely have a non-empty intersection" in proposition 1, for example. $\endgroup$ – pjs36 Apr 21 '15 at 19:36
5
$\begingroup$

There's nothing wrong with Euclid's postulates per se; the main problem is that they're not sufficient to prove all of the theorems that he claims to prove. (A lesser problem is that they aren't stated quite precisely enough for modern tastes, but that's easily remedied.) In every modern axiom system (e.g., Hilbert's, Birkhoff's, and SMSG), each of Euclid's postulates (suitably translated into modern language) is provable as a theorem, which shows that Euclid's postulates are consistent. You can find an extensive discussion of these ideas in my book Axiomatic Geometry.

$\endgroup$
2
$\begingroup$

Euclid's formulation of the axioms is bright and clear, but it doesn't meet the standards of today's axiom systems. In the first place, the main concepts of point, line, angle, circle are borrowed from daily life and the reader is asked to "idealize" them: points have no size, lines have no thickness and have no end. Modern axiom systems (e.g., set theory) do not attempt at defining what its main concepts (set, membership) are about. Instead they describe how the main concept(s) behave. For centuries, mathematicians were stuck to Euclid's "realistic" points and lines, which is probably why it took so long to see that the fifth postulate was just one alternative. E.g., on a curved surface, geodesics (shortest curves) take the role of straight lines. Hence on a sphere (where great circles like meridians on earth are geodesics) there are no parallels. A different problem with Euclid's axioms is that (due to the seemingly realistic description) there are many hidden assumptions. Four specific flaws are described at http://health.uottawa.ca/biomech/laws/euclid.htm

$\endgroup$
1
$\begingroup$

There is nothing wrong with them. The problem is that until the 19th century they were thought to be the only ones possible, giving rise to a single possible geometry (the one called today "Euclidean"). During the 19th century advances in mathematics (Gauss, Bolyai, Lobachevski, Beltrami, Klein but most importantly Riemann and the Italian school of differential geometry) showed that in fact this was just a very particular case of geometry, far from being the single one possible. Today nobody serious does geometry "the old way", i.e. with axioms; today the relay has been taken by differential (and, as a particular case, Riemannian) geometry, which sheds a completely new light on the subject, viewing it from a different point of view, attacking it with totally different tools and asking completely different questions. The geometry that you are referring to belongs, today, to the history of mathematics, nothing more.

(To be fair: a special mention should be given to algebraic geometry which, in turn, asks totally different questions than classical or differential geometry and attacks them with again different tools.)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.