This post will be in two, totally unrelated parts. The first part is a bit of a rant, and then the second part will concern more about the Carnap problem I posted about last time.

Random Things

So this might not actually be totally unrelated (cf. the age-old view that everything is like everything else somehow), but it is at least not intended to be related to the next part of the post. One thing I have realised lately is that I should be bringing a big, red, "M" placard to one of my seminars if not for the rest of the quarter then at least for the next several weeks. We are getting in to reading Parsons' Mathematical Thought and its Objects, which, at least at the beginning, is highly metaphysical. I have determined that metaphysics is one of those things that I most commonly fall asleep reading, so this should prove interesting.

With that said, now for something completely different...

More on Carnap - Dedution vs. Description

Important to figuring out whether and how arithmetic and geometry are different for Carnap is determining how he characterises deduction vs. description. First, I will start with geometry and description, and then move back to arithmetic.

The crux of geometry being synthetic for Carnap is that the customary interpretation of the calculus in physical. That is, "we may think of a point as a place in the space of nature; straight lines may be characterized by reference to light rays in a vacuum or to stretched threads... Thus the specific signs of a geometrical calculus are interpreted as descriptive signs."[1, section 21] However, this same thing does not carry over to arithmetical calculi. Carnap thinks that "the specific signs of a mathematical calculus are interpreted as logical signs, even if they occur in descriptive factual sentences stating the results of counting or measuring."[1, section 21] Carnap also sees that the geometrical calculus can be interpreted logically as analytic geometry.

However, physical geometry is still different from mathematical geometry. Physicists, he claims, still ask which (Euclidean or non-Euclidean) geometrical structure space has. This seems like a strange question to me in general. It seems to hold the potential interpretation of the geometrical terms fixed and asks which axiom system makes this interpretation best correspond to observation. Carnap writes, "When an interpretation of the specific signs of is established... then each of the calculi yields a physical geometry as a theory with factual content." [1, section 22] The interpretation itself is up to choice.

It seems strange to consider geometry to be synthetic and arithmetic analytic just on the basis of the choice of customary interpretation. Tomorrow I should probably write about whether arithmetic could be shown to be descriptive in the same way geometry is, or whether there is something special about it such that it can only ever be deductive.

References

[1] Carnap, Rudolf. Foundations of Logic and Mathematics. Foundations of the Unity of Science, vol 1, no 3.