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Which of Euclid's postulates can be relaxed, to take us closer to the laws of physics?

Euclid's 5th postulate states that parallel lines never meet. There are many equivalent ways of stating it, given the other postulates - perhaps most simply "the sum of the internal angles of a triangle is 180 degrees":

$$\sum_\triangle\theta=\pi$$

There is some elegance in the notion, I believe, that the laws of the Universe require no assumptions, except perhaps the assumption of existence. If we therefore start with any theory and relax the assumptions one by one it stands to reason that we may be moving that theory closer to the laws of the Universe.

Starting with the laws of Euclidean geometry, and relaxing the postulates or assumptions one by one, which of Euclid's assumptions can we relax in order to move the laws of geometry inarguably closer to the laws of the Universe (i.e. the laws of physics)?

As a starter for one we can relax the 5th postulate to show the Euclidean geometry in which classical mechanics is expressed is just a subset of non-Euclidean geometry. This includes the geometry of General Relativity and therefore relaxation of the 5th postulate fulfils our objective.

Which of the other postulates can be relaxed, and shown to move us closer to reality?

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    $\begingroup$ The parallel postulate is true in Minkowski spacetime. It is not true in the more general spacetime of GR. $\endgroup$
    – user107153
    Apr 10, 2017 at 15:22
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    $\begingroup$ I am unconvinced that this is a question about physics. FWIW it's hard to see how any of the first four postulates could be discarded and produce anything like a realistic geometry. $\endgroup$
    – John Rennie
    Apr 10, 2017 at 15:28
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    $\begingroup$ Adding to @tfb's comment, finite-dimensional inner product spaces are ubiquitous in every branch of physics, and they all satisfy Euclid's postulates. So what does "discarding a postulate" mean? Does it mean refusing ever to make use of those inner product spaces? Does it mean using them, but ignoring those properties that depend on some postulate or another? Or what? $\endgroup$
    – WillO
    Apr 10, 2017 at 15:56
  • $\begingroup$ @WillO I'm sure you understand the mathematical details better than me but any such space is a subset of another space in which the parallel postulate is not necessarily observed. Examples in which distances are short or motion is slow, are the special cases in relation to relativity, in which the parallel postulate is approximated. $\endgroup$
    – it's a hire car baby
    Apr 10, 2017 at 16:06
  • $\begingroup$ @JohnRennie there are big questions in physics, which bear this kind of thinking. Just because the answers don't yet exist in any textbook, doesn't make them not about physics. Answer the question, and you are writing the textbook. $\endgroup$
    – it's a hire car baby
    Apr 10, 2017 at 16:08

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It will depend to start with what you mean by the Euclid axioms. The Euclid axioms are not actually a sound axiomatization of geometry, as it was discovered in the 19th century, and a few axioms are missing, such as Pasch's axiom and the axiom of continuity, both of which cannot be proven from Euclid's axioms. Better axiomatizations of geometry include Hilbert's axioms and Tarski's axioms.

So let's look at the Euclid axioms :

1 : "Two points determine a line segment"

In terms of modern physics, this roughly means that there is always a geodesic between two points. There isn't any necessity to change the geometry of physics to find counterexamples : any incomplete manifold possesses points that cannot be joined by a geodesic. Also for a variety of manifolds, geodesics between two points may not be unique.

2 : "Any line segment with given endpoints may be continued in either direction."

This is related to geodesic completeness. Again, this breaks down fairly easily with the appearance of a singularity.

3 : "It is possible to construct a circle with any point as its center and with a radius of any length"

Once more, singularities can break down this property. It is perfectly possible to draw a circle with a center that is not part of the manifold, or to have two points that do not define a circle due to this.

Axiom 4 I'm not quite sure how to break but here we go. It's overall not terribly difficult to find counterexample geometries without even removing more important properties of manifolds used to describe space, such as the Hausdorff property, paracompactness, or even removing the manifold structure altogether, such as having spacetime as a lattice (which would remove the axiom of continuity, for instance).

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  • $\begingroup$ I take this answer as pretty strong evidence that the question should be closed. Almost no mathematical object satisfies Euclid's axioms, and many of those mathematical objects arise in physics. You've listed a few, but we could also list the natural numbers, the group $SO(3)$, the Bloch sphere, and Newtonian mechanics (it's not even clear what concepts in Newtonian mechanics you'd want to map the notions of "point" and "line" to, but whatever you choose, they're unlikely to satisfy Euclid's axioms. Do two accelerations determine a force?) .... (CONTINUED) $\endgroup$
    – WillO
    Apr 11, 2017 at 6:51
  • $\begingroup$ (CONTINUED) One might as well write down any set of axioms whatsoever and ask whether there are important objects in physics that do not satisfy these axioms. With only trivial exceptions (such as the empty set of axioms), the answer is always going to be "Obviously yes". $\endgroup$
    – WillO
    Apr 11, 2017 at 6:52
  • $\begingroup$ @WillO I interpret in the contrary. I do find it quite profound that even before our knowledge of black holes, we might have looked at our postulates and realised that by relaxing them we would have predicted and explained things we were hundreds of years away from understanding. And with respect to "one might write down any set of axioms, yes that is exactly correct since every set of axioms, if relaxed one by one, tends to the same place. It is obvious after all, that existence of the Universe itself should depend on no other thing. $\endgroup$
    – it's a hire car baby
    Apr 11, 2017 at 8:10
  • $\begingroup$ @Slereah I think you have unwittingly highlighted another example; namely the discreteness of space which we find at the Planck level implies that the axiom of continuity is untrue. Do you not think that with these axioms we make it easier for ourselves to model the Universe, but as we understand the Universe better we might ultimately relax them one by one. $\endgroup$
    – it's a hire car baby
    Apr 11, 2017 at 8:14

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