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":
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?