This is my attempt to modify Michael Friedman’s Dynamics of Reason by redefining the role of mathematics. My 2005 draft makes a start, but I now see that the case study there can be developed to describe a full-blown revolution in mathematical physics.
Friedman draws attention to the retrospective rationality acting through a revolution, casting the old theory as an approximate case of the new theory (e.g., Cartan’s rendition of Newtonian space and time as a Riemannian manifold). He points also to a prospective rationality where meta-scientific work is done on fundamental principles. His paradigm case is Einstein’s work after the metascientific enquiries of Helmholtz, Hertz, Poincaré, following revolutionary changes to geometry by Riemann, Lie, Klein, Hilbert, etc. In the case of quantum mechanics, this metascientific work is taken to have been lacking. Instead we have untimely interventions (DR, 120-121), as in the ad hoc philosophical speculations of Wigner and Schroedinger.
After a revolution, important philosophical work is done, integrating the new science into a novel philosophical framework. E.g., Kant integrating Newtonian science and Leibnizian metaphysics, and the Logical Positivists understanding general relativity.
Friedman draws a distinction between mathematics and physics, which I question in change in status of principles. This is illustrated by cohomology, which plays an important role in modern physics and which is part of higher category theory as revolution. I bring together various themes in diagnosis. Some objections and observations to this line of thinking.
Friedman himself has since developed his views, see ‘Extending the Dynamics of Reason’, his chapter in the Domski and Dixon volume 2010, and in responses to papers in Studies in History and Philosophy of Science 43(1), 2012. Note in his essay in Domski and Dixon, Friedman moves in what I agree is a good direction:
The difficulty arises when one accepts the sharp distinction, emphasized by Schlick, between an uninterpreted axiomatic system and intuitive perceptible experience, and one then views the constitutive principles in question (which, following Reichenbach, I called “coordinating principles” or “axioms of coordination”) as characterizing an abstract function or mapping associating the former with the latter. This picture is deeply problematic, I now believe, in at least two important respects: it assumes an overly simplified “formalistic” account of modern abstract mathematics, and, even worse, it portrays such abstract mathematics as being directly attached to intuitive perceptible experience at one fell swoop. (pp. 697-8)
Our problem, therefore, is not to characterize a purely abstract mapping between an uninterpreted formalism and sensory perceptions, but to understand the concrete historical process by which mathematical structures, physical theories of space, time, and motion, and mechanical constitutive principles organically evolve together so as to issue, successively, in increasingly sophisticated mathematical representations of experience. (p. 698)
In my reconceived version of transcendental philosophy, therefore, integrated intellectual history of both the exact sciences and scientific philosophy (a kind of “synthetic history”) takes over the role of Kant’s original synthetic method; and, in particular, constructive historical investigation of precisely this kind replaces Kant’s original transcendental faculty psychology. (p. 702)