I posted the following comment on The n-Category Cafe blog, discussing Smolin’s book:
Victor Grauer writes (in response to a preceding comment of mine):What is really at stake here is NOT the question of what space and time really really are in the most fundamental sense, but the even more fundamental question of how it is possible to represent them. In other words, we are dealing with not only epistemology but semiotics, the “science” of representation.Man, I hope this is not considered off-topic. I don’t want to get deleted again. I just want to stress that while Smolin’s thesis, the trouble with physics, is ultimately our inability to unify the discrete and continuous theories, he asserts that there is more than this reflected in the string theory controversy. It’s as if the string controversy is the center piece of the table, focusing our attention on the state of physics as a whole, not just the latest innovation, which might, or might not, have outlived it’s usefulness.
If string theory is justified on mathematical grounds, it’s not just because it is “beautiful mathematics,” but because it’s beautiful mathematics that, to some extent, unifies the discrete and continuous theories. The fact that it has no contact with experiment and can’t predict anything right now, is overridden, in the minds of many, by the apparent achievement of a consistent unification of the discrete and continuous, in a very compelling manner.
The details of how the development of string theory has led to the current prospect of “the end of a science” and consideration of the serious question “What comes next?” are not so important at this point. What is important is gaining a clear understanding of the method of thinking that led us to this point, and without a doubt that thinking is best characterized as the history of developments in the science of mathematics.
String theory must live in a minimum of ten dimensions, but what bothers Glashow, as quoted by Smolin, should bother all of us:…Worst of all, superstring theory does not follow as a logical consequence of some appealing set of hypotheses about nature. Why, you may ask, do the string theorists insist that space is nine-dimensional? Simply because string theory doesn’t make sense in other kind of space.In other words, string theory is not inductive science, it’s inventive science, and the comments of Einstein, regarding the significance of epistemology in science, rise to the top of our thoughts, like the cream in unhomogenized milk.
However, and this question must be asked, if string theory is basically an exercise in mathematics, and it is inventive science, then doesn’t that imply that Glashow’s criticism applies to the science of mathematics represented by string theory?
I believe the conclusion that it does is just inescapable. Writing about this in a historical context, Hestenes sees the development of mathematics as the centuries-long effort to unify discrete numbers with continous physcial magnitudes, which Euclid deliberately kept separate, proving theorems first with line segments and then with numbers.
Clearly, though, this history is a history of inventive science, not inductive science. String theory mathematics is simply a continuation to unify discrete numbers with continuous magnitudes. Briefly, the three properties of physical magnitudes versus natural numbers are:
- Continuous vs. discrete quantity
- Two directions vs. one direction
- Limited vs. unlimited dimensions
In the development of our inventive mathematical science over centuries, the ad hoc invention of the real numbers addresses number 1; the ad hoc invention of the imaginary numbers addresses number 2; and the ad hoc invention of “compactified dimensions” adresses number 3.
Nevertheless, in the spirit of Glashow’s complaint, wouldn’t we rather have a numerical theory that “follows as a logical consequence of some appealing set of hypotheses about nature?” Is this even possible in mathematics, or is formalism the only “game in town?”
Of course, I couldn’t give an answer to the question on Baez & company’s blog, but I can here. The answer is clear: Inventive science is no longer the “only game in town.” Inductive science has returned. As Larson explained, in his “Principal Address to the Third Annual NSA Conference,” in 1978, inductive science inevitably falls behind the available empirical information of experiment, and thus makes it possible for inventive science to fill the void in the effort to explain natural phenomena. However, ultimately, the number of ad hoc inventions becomes so great that the human spirit recoils. We intuitively know that the physical laws of nature are not like the tax laws, designed by government bureaucracy, but most men are not as wise, nor as patient, as Newton, who insisted that basic concepts and laws of nature can only be derived from experience.
The new inductive science is possible, because the new information that Larson discovered, that space is the reciprocal of time, makes all the difference in the world.