David Corfield
objections and observations

Doesn’t the analogy between mathematical and physical constitutive change break down too soon?

“I am drawing an analogy, and by nature of analogy, you can, if you want, push it to the point where it breaks.” (Urs)

Certainly, the ‘observations’ in mathematics aren’t generally the results of a group of people training their instruments on the universe, though don’t forget computing for this effect. All the same, Lakatos’s quasi-empiricism may allow us to run the analogy far enough. If we think of the constitutive as allowing new ideas to be thinkable, then profound transformations clearly happen in mathematics.

“A great deal of modern mathematics, by no means just algebraic topology, would quite literally be unthinkable without the language of categories, functors, and natural transformations introduced by Eilenberg and Mac Lane in their 1945 paper. It was perhaps inevitable that some such language would have appeared eventually. It was certainly not inevitable that such an early systematization would have proven so remarkably durable and appropriate; it is hard to imagine that this language will ever be supplanted. It’s introduction heralded the present golden age of mathematics.” (May 2000:11, my emphasis)

Earlier I had put forward the argument: the axioms of category theory are first-order – on the face of it, it can’t be a radically new language. Yet most of the language for Joyce’s Ulysses was there for Jane Austen to use, but such a novel was evidently unthinkable for her. Just because we may rephrase something in set theoretic terms, doesn’t make it thinkable in 1930.

But a few years on and we now have dependent type theory and HoTT! So we do have a revolution in the logic.

We ought to be able to run a parallel account for the changes in mathematics (geometry and logic) which allowed the Einsteinian revolution.

Does Friedman’s account do any better than Quine’s network of beliefs?

Could we not understand in Quinean terms the passage of Cech homology from analytically being a homology to failing to satisfy the axioms? Friedman’s is a layered account; one cannot express Newton’s laws without the calculus, nor measure gravitational force without Newton’s laws, just as one can’t express the Eilenberg-Steenrod axioms (sensibly) without category theory. As suggested by Lakatos and \MacIntyre, let us see who provides the best resources to write a history. Now write a Quinean history of the episode and see what happens.

Quine might say psychology not history is the key to epistemology, but what has psychology achieved in this respect?

See also Friedman and DTT, that the logic of the new revolution sits happier with hierarchical presupposition structures.

Could I invoke the accounts of other historically-minded philosophers: Collingwood, Cassirer, \MacIntyre, Lakatos, Shapere, \McMullin?

Yes, but perhaps there is something special about changes of this magnitude. And it concerns a new mathematical language allowing a new physics, so right in line with Friedman.

If I’m likening mathematical changes to Friedmannian scientific changes, why do we not see meta-mathematical work before Eilenberg-Mac Lane, analogous to the meta-scientific work? (Ralf Krömer).

But we do. See Colin \McLarty’s article on Mac Lane in Göttingen for a partial response. He trained with the phenomenologist Moritz Geiger, as well as drawing on the implicit philosophies of the mathematicians - Noether, Weyl, etc.

Similarly, it’s interesting to note that Per Martin-Löf’s philosophically inspired intensional type theory has been crucial for the development of univalent foundations.

(Hegel and other dialectical philosophy for Lawvere. What for Grothendieck?)

What place would mathematics have in a reconfigured Friedmannian scheme? (Mauricio Suarez)

To follow my line of thought, I think you have to buy into something like Lakatos’s quasi-empiricism about maths. Lakatos was very insistent that science and maths share a similar epistemological structure: ‘observations’ may bring about changes in theory. I took this seriously enough once to try to run a Bayesian account of mathematical inference, not that I think Bayesianism takes us very far in philosophy of science. I think I showed, and here I was only following Polya, that it works rather well in mathematics, and indeed I would argue that in many respects it works better in maths than in science.

Unsurprising then, on seeing Friedman’s work I looked for parallels between his story of scientific change and mathematics. Again I think it works rather well. There’s a shifting about in the status of principles, e.g., from being an observed regularity as definable in one system to becoming part of the defining structure of the next system.

As so briefly described, mathematically, cohomology begins as a method of telling spaces apart (and also aids the study the kinds of structures (algebraic, analytic, holomorphic, etc.) which may be supported on spaces). A bunch of examples of cohomological processes are found, but it’s not very clear how they work. Common features are then extracted and used to define what it is to be a cohomology theory. This uses category theory. What was a bona fide process is no longer. Quickly the idea comes that algebraic entities, such as groups, also can have a cohomology associated with them. And this is followed by arithmetic varieties. Processes which almost satisfy the cohomology axioms are found, and called ‘generalized’. We then get a whole flood of examples of cohomology with little order to them. Recently a framework has emerged, only expressible in a new language of infinity-toposes, which captures the commonality of this zoo of theories.

The question now is how I would want to fit this Friedmannian account of maths into Friedman’s own account of mathematical physics. For him, the maths entered merely at the highest level of a stage of physics as a means to be able to express the constitutive principles, e.g., calculus and 3D Euclidean space to allow the expression of the first law and F= ma, which allow for the expression of the Law of gravitation; tensor calculus and 4D Riemannian manifold to allow the expression of the equivalence principle, etc., which allows for the expression of Einstein’s field equations.

Certainly, there’s a wonderfully rich story to tell of the dance of mathematical and physical thinking through the ages. I am always keen to stress that there’s an important relationship between physical and mathematical intuition. Sometimes this is only seen in one direction, where the physical interpretation of some mathematics is taken as exhausting whatever is intuitive about mathematics. This is a blind spot which leads a lot of philosophy of physics astray, even Friedman. How to characterise things here is very difficult, since they often meet in the same person. Take Weyl. He has mathematical intuitions of how representation theory works, to allow it to operate in number theory, but of course he also understands the representation theory of Lie groups in relation to particle physics.

So, I could have my Friedmannian account of mathematics merely feeding a conceptual linguistic framework into the highest layer of his account of physics, but I would be concerned that this wouldn’t assist us to see properly the interplay between maths and physics. In my year in HPS at Cambridge I had an excellent MPhil student, Chitra Ramalingam, and we worked on the idea that people such as Norton Wise were too quick to understand Kelvin’s analogical thinking between gravitational potential, water flow, heat flow and electrostatic distribution as mediated solely by physical interpretation, when there ought to have been an allowance for mediation by a common mathematical intuition. (I see I complain of this on p. 142, ch. 6 of my 2003 book.)

So, I don’t know how best to set things up. Looking further into that 6th chapter of my book, I wrote out a case where physical observation was providing evidence for a mathematical conjecture. This would be awkward for a picture of mathematics and physics which just has the mathematics written in at the highest constitutive level.

Were things to work out and homotopy type theory becomes accepted as the right foundational mathematical language in which fundamental physics may be written, and better yet it assisted the development of a quantum gravity, what then?

If meta-scientific work could flourish in the nineteenth century by responding to the problems brought about for the Kantian synthesis of Newton and Leibniz, what happens when one revolution, the Einsteinian, leads to an important, but failed response (logical positivism), and the other, the quantum, to untimely philosophical interventions? You would imagine meta-scientific work in the 20th century suffering - perhaps true.

If then, despite this, a breakthrough occurs, what now for philosophy? We would hope that one with a historical orientation would emerge. Wouldn’t it be wonderful if the philosophical lesson of the higher categorification of mathematics and physics was that rationality can only be conceived via narrative? But could the mathematical framework itself be important for philosophy? Well, on the one hand, look how logical positivism turned to logic to the exclusion of history (“although the logical positivists’ preoccupation with the a priori did indeed thereby preclude them from using the history of science as a philosophical tool”, Friedman 1993). On the other hand, the motivation for Lawvere has been that

It is my belief that in the next decade and in the next century the technical advances forged by category theorists will be of value to dialectical philosophy, lending precise form with disputable mathematical models to ancient philosophical distinctions such as general vs. particular, objective vs. subjective, being vs. becoming, space vs. quantity, equality vs. difference, quantitative vs. qualitative etc. In turn the explicit attention by mathematicians to such philosophical questions is necessary to achieve the goal of making mathematics (and hence other sciences) more widely learnable and useable. Of course this will require that philosophers learn mathematics and that mathematicians learn philosophy. (1992, p. 16)

Or do we follow less universalising twentieth century Hegelianism? The truth is the whole — yet this whole cannot be presented all at once but must be unfolded progressively by thought in its own autonomous movement and rhythm. It is this unfolding which constitutes the being and essence of science. The element of thought, in which science is and lives, is consequently fulfilled and made intelligible only through the movement of its becoming. (Cassirer [1929] 1957, xiv)

Weren’t many of the changes in the nineteenth century made by radically rejecting Kant, some seeing him as an obstacle? (Donald Gillies)

This needs investigating. There is a story of Gauss doing so, and of Riemann learning more from Herbart.

Friedman argues that Kant had done so well at reconciling Newton’s physics with certain of Leibniz’s philosophical ideas that the march forward from Newton takes its impetus from struggles with and reactions to Kant.

Apropos of the idea that philosophical interventions for quantum mechanics were untimely, perhaps this was because it was a deeper revolution, so harder to think out from an earlier starting point. Perhaps it could only be discovered in an ad hoc fashion.

Concerning quantum physics Feynman writes

…compared to this discovery that Newton’s laws of motion were quite wrong in atoms, the theory of relativity was only a minor modification. (Feynman, QED, p. 5)

Or is that space needed a radical rethinking first for quantum gauge field theory? Does QM as 0+10+1d QFT mean that we need the latter’s new space to understand it properly?

Perhaps relevant here is Dyson’s divorce between mathematics and physics. Reconciliation comes through geometry - principle/fibre bundles.

Last revised on September 22, 2021 at 03:02:28. See the history of this page for a list of all contributions to it.