Introduction

The mathematical structure of a theory is a very abstract collection of assumptions about the nature of the sphere of phenomena the theory studies. Given the great cultural gap which has opened between mathematics and physics, it is all too easy for these assumptions to become unconscious.

General Relativity (GR) is a classical theory. Its mathematical foundation is a smooth manifold with a pseudometric on it. This entails the following assumptions.

1. (i) Spacetime contains a continuously infinite set of pointlike events which is independent of the observer.

2. (ii) Arbitrarily small intervals and durations are well defined quantities. They are either simultaneously measurable or must be treated as existing in principle, even if unmeasurable.

3. (iii) At very short distances, special relativity becomes extremely accurate, because spacetime is nearly flat.

4. (iv) Physical effects from the infinite set of past events can all affect an event in their future, consequently they must all be integrated over.

The problem of the infinities in Quantum General Relativity is intimately connected to the consequences of these assumptions.

In my experience, most relativists do not actually believe these assumptions to be reasonable. Nevertheless, any attempt to quantize relativity which begins with a metric on a three or four dimensional manifold, a connection on a manifold, or strings moving in a geometric background metric on a manifold, is in effect making them.

Philosophically, the concept of a continuum of points is an idealization of the principles of classical physics applied to the spacetime location of events.