Near-circular definitions

Question

I've been playing around with mathematical definitions for a naive (non-relativistic) sense of space and time. Now, the reason I'm posting on the math forum instead of the physics forum is because I don't yet care to argue the definitions themselves. Instead, I'm interested in the logical structure of these definitions.

In particular, I want to define them both as parameters with particular qualities, but the only way I can make their mathematical definitions agree with their intuitive properties is to define each of the parameters strictly for cases in which the other is held constant.

For instance, space is a parameter that distinguishes two events that occur at the same time. Now, this feature doesn't depend on the definition of time, so it's not circular in the sense that I recall circularity, but it does depend on the value of time. My definition of time, similarly depends on space being constant, but not on how space is defined.

This is dangerously reminiscent of a circular argument, but avoids strictly defining one in terms of the definition of the other. My question is: is this valid?

Answer 1

Your question is too vague to be concretely answered, but here are two general points:

1. You cannot define a value using another value and then the second value using the first. Consider what happens if you say:

   Let $t,x$ be real numbers such that $t = x^2+1$ and $x = t^2+1$.

   It is simply invalid.

2. You can define axioms and investigate whether or not there are any structures satisfying them. So you can consider a system whose axioms include:

   $t = x^2+1$.

   $x = t^2+1$.

   It may be the case that these together are inconsistent with the other axiom, in which case there is no model of your system. The point is that it is not a circular definition since we are not asserting that there is a model, but are simply defining what axioms a model must satisfy.