So today's post is going to be quite short, but it is connected to the posts I made late last week. In general, I want to ponder what happens in the case of a failure of physical theory.

Let us consider the following thought experiment: Physicists are testing some hypothesis about the structure of space. Presumably this is possible, though I don't actually know. The current best physical theory suggests space is Lobachevskian (that is, is non-Euclidean with constant negative curvature -- from what I recall this is what general relativity suggests, though again, I am not an expert; at any rate, it shouldn't matter). However, the test does not conform to predictions. Furthermore, suppose many other tests also fail to conform to the theory. Now what should happen?

According to Quine, the whole of scientific knowledge goes through the "tribunal of experience" as a whole. As such, it is surely possible to say that Lobachevskian geometry, or some other part of mathematics, is false and should be revised. This seems profoundly strange. Another option is to say that the mathematical model is a bad model of the theory of space; it is not true or false, just performs poorly. Perhaps this way is open to Quine. It seems natural that Quine would suggest that the physical theory of science should be altered to indicate that space is not Lobachevskian. Is that the same as just saying the model was bad? Was there really any chance of mathematics being altered based on physical experience?

It is somewhat harder to imagine what Carnap would say in the same situation. At the least, he would say that the customary interpretation of the geometrical calculus was not a true interpretation. As I believe I have outlined before, this seems strange in itself. I think it is better to say that the geometrical calculi are interpreted analytically, and that it is the correspondence map from the world to the interpretation that is no good. (By correspondence map I mean something like the map(s) that allow for the counting of objects despite the fact that the arithmetical calculus is interpreted analytically within second/third(?)-order logic.) That is, trying to make paths of beams of light correspond to lines, etc. is not a good model of the universe within the geometrical calculus being considered. If it is desirable that the path of a beam of light indeed correspond to a line, then perhaps an interpretation of a different geometrical calculus should be used instead. On the other hand, if it is desirable to continue considering Lobachevskian geometry, a different correspondence should be considered. Granted, this is not really what Carnap thinks, but it seems to me to provide a more consistent theory within which to work. All of mathematics stays analytic L-rules, whereas the correspondence maps and other aspects of modeling are synthetic P-rules.