Yesterday I talked some about Carnap on the synthetic nature of geometry. To recap, geometry is synthetic because the truth of the sentences of the geometrical calculus under its customary interpretation depends on empirical facts. Arithmetic, on the other hand, is analytic according to Carnap, for similar reasons. The customary interpretation of Peano arithmetic is customarily reducible just to a higher-order logic (second or third order), which in turn in analytic. Therefore, the truth of sentences of arithmetic under the customary interpretation of the calculus only depends on logic, making it analytic.

Reflecting on these differences, it seems there is nothing substantially different between the two cases. It is just a matter of the "choice" of customary interpretation that makes arithmetic analytic and geometry synthetic. If, for instance, a different standard interpretation of geometry in terms of modern analytic geometry could be established, geometry should turn to being analytic instead. Furthermore, if the interpretation of arithmetical sentences rested on apples or some other physical object, it seems that it would be considered synthetic.

There is, however, an important difference between arithmetic and geometry that might lead to certain natural interpretations. Geometry is the study of different classes of structures. There are Euclidean, hyperbolic, and elliptical geometries that are mutually exclusive, but also relatively consistent. Similarly to group theory, the point is to develop something of an if-then-ism about geometrical structures. That is, proofs in geometry seem to work something like "If the space is Euclidean, then the sum of the interior angles of a triangle is 180-degrees." Without that antecedent, the statement "The sum of the interior angles of a triangle is 180-degrees." seems like it should not have a truth value, just like the fact that "There is an element of order 7." in group theory doesn't make sense without an antecedent condition of what group or class of groups is intended to be discussed. The same concerns don't apply to arithmetic. There is only one object under consideration, not many as in geometry or group theory. Therefore, the sentences of arithmetic do not need a proper antecedent to be meaningful.

The standard interpretation of geometry, then, seeks to pinpoint a certain geometrical structure to prepend as an antecedent to statements that one might wish to make. That is, it fixes what the basic terms mean and then tries to decide which type of geometrical space, when added as a hypothesis, best makes geometrical statements coincide with empirical data.

We now seem to be left with the question of why sort of argument from geometry doesn't apply to other branches of mathematics. That is, why isn't there a physical group theory and a mathematical group theory? Why isn't there a physical theory of complex systems and a mathematical one?