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?

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.

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.