Michael Friedman in his Dynamics of Reason is intrigued by the physical theory of a time being composed of a mathematical language, coordinating principles,and empirical laws and regularities, and how their status changes through a revolution. E.g., what is an empirical regularity for Newton that gravitational and empirical mass are identical becomes a consequence of a more fundamental principle - the equivalence principle - for Einstein. What was constitutive for Newton, that space is flat, becomes an approximately correct observation under Einstein. Now the Riemannian structure is constitutive, but the particular metric is empirical.
So there is both promotion and demotion through a revolution, according to Friedman's schema and the observations of a period may relate to findings to be written into the principles or evidence that existing principles are merely approximate truths.
Friedman claims that this lack of reversal is where mathematics differs principally from the natural sciences (see Friedman 2001: 96-99).
The main disanalogy with pure mathematics, however, is not simply that the retrospective containment is here only approximate. The crucial point, rather, is that the later constitutive framework employs essentially different constitutive principles.” (p. 98)
In pure mathematics, however, there is a very clear sense in which an earlier conceptual framework (such as classical Euclidean geometry) is always translatable into a later one (such as the Riemannian theory of manifolds). In the case of coordinating principles in mathematical physics, however, the situation is quite different. To move to a new set of coordinating principles in a new constitutive framework (given by the principle of equivalence, for example): what counted as coordinating principles in the old framework now hold only (and approximately) as empirical laws, and the old constitutive framework, for precisely this reason, cannot be recovered as such. By embedding the old constitutive framework within a new expanded space of possibilities it has, at the same time, entirely lost its constitutive (possibly defining) role. (p. 99)
Let’s look for evidence against this claim, i.e., evidence that mathematics changes its constitutive principles. First, note a possible objection. Friedman knows about profound changes happening in mathematics, so then why would he choose not to count these as replacement of constitutive principles? But then, if he believes that mathematical change is not about changes to constitutive principles, why when indicating the possibility of a major change in mathematics would he point to the adoption of quantum logic? Why not an existing change, such as category theory? Is it that to count as a major change there must be an increase in expressibility, and that, as a first-order theory, anything category theoretic is to be counted as ‘expressible’ within set theory and classical logic, and so no great change?
Urs: While these two principles are about very different things, they share one common property:
while they were/are both regarded as being a major guiding principle while the theory was/is developed, after a while the principle became/will become so tautological as to be essentially empty.
Because, if you think about it, what does the “equivalence principle” in general relativity really say, from the modern perspective of differential geometry? It really just says “there are tangent spaces to a manifold”. That used to be an important insight while humanity was still figuring out that GR is about differential geometry. But once you accept this, the “equivalence principle” is not a big deal.
Same here: while higher category theory is still in its adolescence, we go on about the meaning of equivalence. But in 100 years people will find it hard to imagine how anyone could have ever made an “evil” statement in the first place.
Mike: I thought the “equivalence principle” in GR referred to the equivalence of inertial and gravitational mass. Which is an empirical observation, not a logical tautology.
Urs: There are many “equivalent” ways to state the equivalence principle, all of them informal of course. To make full sense of the statement as you quote it, one needs to say what “inertial mass” is, and this is done with respect to momentum / acceleration in flat space.
A formulation of the statement that more directly goes to this point is (I am quoting from the list given in the Wikipedia entry):
The local effects of motion in a curved space (gravitation) are indistinguishable from those of an accelerated observer in flat space, without exception.
That flat space appearing in that sentence is the tangent space to the spacetime manifold at the given locus. The statement says: a manifold looks locally like its tangent space at that point.
But, of course, initially this is an empirical observation, true. It is not logically necessary (well, as far as we know, we are working on it… ;-) that gravity is described by pseudo-Riemannian manifolds. But once you have established this, the original fact that brought you to this insight sits so deep at the root of the very formalism that describes it all, that in retrospect it is hard to see it as a deep statement. There are tangent spaces.
(Well, of course you may disagree. I am drawing an analogy, and by nature of analogy, you can, if you want, push it to the point where it breaks. So if you feel that point is too close, that’s okay with me, but it’s maybe not worth discussing further. This just to prevent us from spiraling into another philosophical argument not related to formal mathematics. ;-)
Mike: Okay, I think now I get it. The fact that mathematics is basically all isomorphism-invariant is (or at least, could be said to be) also an empirical observation. So in both cases we start with an empirical observation, but then that observation gets built into our formal framework at an extremely basic level, so that when expressed in terms of the formal framework the original empirical observation looks like an obvious tautology. Thanks for explaining!
In his chapter for Circles Disturbed, Barry Mazur writes
The mathematical visions that I am currently fascinated by are those that begin with the mission of explaining something precise, and then—because of their extreme success—expand as a template refashioned and reshaped to explain, and to unify, larger and larger constellations of mathematical or scientific issues—this refashioning done by whole generations of mathematicians or scientists, as if a single orchestra. Things become particularly interesting, not when these templates fit perfectly, but rather when they don’t quite fit, and yet despite this, their explanatory force, their unifying force, is so intense that we are impelled to reorganize the very constellation they are supposed to explain, so as to make them fit. A clean example of such a vision is conservation of energy in Physics, where the clarity of such a principle is so unifying a template that one perfectly happily has the instinct of preservation of conservation laws by simply expecting, and possibly positing, new, as yet unconsidered, agents—if it comes to that—to balance the books, and thereby retain the principle of conservation of energy. Such visions become organizing principles, so useful in determining the phenomena to be explained, and at the same time in shaping what it means to explain the phenomena. There is a curious non-falsifiable element to such principles, for they get to organize our thoughts-about-explanation on a level higher than the notion of falsifiability can reach.
Both Collingwood and Cassirer were interested in such principles, the latter recognising that such things exist in mathematics. This drawing of an analogy between mathematics and physics encouraged me to ask the following of Mazur in an interview
Corfield: Friedman’s notion of retrospection, is definitely the telling of a story. And it’s more the telling of the sort of story it sounds like you want to tell, rather than nitty gritty history. He talks about constitutive principles. Consider the passage from the Newtonian paradigm to the Einsteinian. In the Newtonian one you need a mathematical language like the calculus that provides you with a way in which you can state various constitutive physical principles such as the laws of motion. And then within that framework you’ll have various facts. And one of the facts is the equivalence of inertial mass to gravitational mass. As time goes by, there will be some tensions and difficulties in that setup. But as it stands, for a reasonable amount of time at least, it can’t be falsified. They are not all on the same level.
Mazur: Yes absolutely.
Corfield: You can falsify the facts and you’ll try to adjust various principles to keep the main principles intact.
Mazur: That’s right.
Corfield: Right, which is rather like what you are saying. But it will come to a point when something isn’t fitting anymore and then a new mathematics is needed, like the tensor calculus, which will allow the expression of new principles, general relativity say, and then you have this lovely turn around where what were facts before can become principles now. Mass, inertial mass, gravitational mass. And what were laws before can become now maybe just approximate facts. He goes on to say in that book that you don’t get this in mathematics. The difference between mathematics and physics is you don’t get that change, from something being a fact to something being a principle.
Mazur: But I do.
Corfield: I know exactly. You are right. I have an example. You had some axioms of topology and algebra and a bunch of homology theories emerged, Cech, singular, etc. Later on, with a new constitutive language, if you like, category theory, you can set up the Eilenberg-Steenrod axioms and then what was a fact, that Cech homology was a homology is now no longer a fact. I think it’s the one that fails, isn’t it?
Mazur: Yes, but it has made its comeback in étale cohomology.
Rather than homology, let’s take cohomology as an example of an idea which changes status from something ‘empirical’ to some deeper constitutive principle.