Math and Reality

[Purple Rabbit]

That mathematical concepts don't follow reality is demonstrated by the history of Euclid's Geometry (EG).

Standard or Euclidean geometry is based on five basic postulates:
1. Any two points can be connected by a straight line.
2. A finite line may be extended indefinitely in a straight line.
3. A circle may be drawn with any given center and any given radius.
4. All right angles are equal to one another.
5. That, if a straight line falling on two straight lines makes 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.

From these five postulates all of Euclidean geometry can be deduced. And Euclid's attempt at it is quite convincing. Around 300 BC he put it in the 13 collected books of Euclid. For centuries mathematicians believed the oddly phrased out of place fifth postulate could be deduced from the preceding four. Over the ages many would announce a derivation of the parallel postulate (fifth postulate) that would later be disproved.

But then Girolamo Saccheri (1667-1733) decided to deny the fifth postulate and see where subsequent deductions would lead him. He was led to the outrageous result that the sum of angles of a triangle could be less than 180 degrees. It convinced him the parallel postulate should be true. As hard as was tried by him and others no logical contradictions were ever found. Non-Euclidean geometry was self-consistent. What then are we to think of the question whether Euclidean geometry is true? It has no meaning.

In the 1800s when elliptic and hyperbolic geometry were developed the foundations of mathematics were shaken. This was a time when mathematics flew off the landscape of reality into a far more expansive plane. Before there was a confidence that Euclidean geometry was absolute and somehow real, but now the validity of one geometry over another could only be verified from experiment.

They were all true in the sense that they were all logically deduced and could be applied to certain circumstances. No one was any more insightful than another, but each provided very rich and exotic structures that could be analyzed rigorously. With Einstein’s general relativity (1915), a vastly new universe was revealed exposing Euclidean geometry for what it was: a simplification of reality. In the neighbourhood of mass the sum of angles of a triangle could indeed be less than 180 degrees.