Since I am preparing to write a paper about Carnap and his distinction between the natures of analysis and arithmetic on the one hand and geometry on the other, I thought I would write some preliminary thoughts here.

First, it is important to understand the origin of this distinction. Carnap sees arithmetic and analysis to be primarily deductive in nature. That is, they are used as tools to derive statements from premises. They are entirely L-rules. However, geometry is substantially different. Geometry itself can be broken up into a geometrical calculus and an interpreted system. As far as the calculus is concerned, geometry is just like analysis and arithmetic. However, interpreted geometry is not purely deductive any more, but instead it is descriptive. It is descriptive in the sense that it intends to describe aspects of experience, and thus, its rules become P-rules.

With that as a sketch of the background, let me go into some of my concerns. Carnap seems to think that we are wedded to the standard interpretation of the undefined terms of geometry. That is, "point" as space-time point, "line" as path of a beam of light, etc. Under this interpretation, then, he claims there is a correct geometry of space. Originally, I had not paid attention to the "under the standard interpretation" fact that was lurking in the background. As a result, I was very unhappy about that claim, thinking that the way we interpret "point", "line", and "plane" was a free choice. Therefore, with sufficiently clever choice of those and sufficient willingness to have the math and theory turn out very complex and ugly to work with, we could describe space with any geometry we wished.

With that worry out of the way, things are just peachy, right? Well, not so much. The other worry I have is the implications of this for the rest of mathematics. Clearly, the geometrical and arithmetical calculi, when uninterpreted, are analytic and without content in Carnap's sense. However, interpreting those calculi comes with mixed results. Geometry becomes synthetic, whereas arithmetic stays analytic. This is where the point of confusion comes: what is so special about arithmetic that doesn't force it to have the same fate as geometry?

One possible explanation I see at this point is that interpretations of arithmetic are still not intended to describe. If the purpose of the interpreted enterprise is primary, then this could explain why geometry and arithmetic are so different. Another option is something like any possible interpretation of the arithmetical calculus is essentially the same, lending no information outside of its deductive power. This somehow seems more plausible, but I will have to think about it more.