David Corfield
Vienna

Contents

Contents

Two talks at the University of Vienna, December 2019.

The Dynamics of Mathematical Reason

5 December 16:45, FoNTI Invited Talks Series

(draft slides, needs much paring down)

A return to old interests of mine, see here.

Recognizing the dialectic between philosophy, science, history, and mathematics, our task is to locate those issues that bind these disciplines together so that we can make the most informed decisions about our own methods and concepts as we attempt to move forward. (Domski & Dixon 2010, p. 17)

Friedman directs his attention to mathematical physics, and in particular to two key episodes. Rather than a Kuhnian irrationalism.

What distinguishes Friedman’s approach, then, is a re-examination of the role of philosophy in scientific change, restoring philosophy to its place as a field for the rational discussion of theoretical alternatives. (DiSalle 526-7)

To call this process a conceptual analysis, indeed, is to understate the force of the argument: it is typically a dialectical argument from the prevailing definition to a new one, revealing the hidden presuppositions of the old conception, and exhibiting the internal difficulties that must be resolved by the new. Indeed, such an argument deserves to be called a transcendental argument. If a certain way of defining a concept is shown to be a condition of the possibility of employing that concept at all, and thereby a condition of the possibility of the scientific reasoning on which that concept depends, then it can hardly be seen as an arbitrary coordination. Nor, therefore, can the argument for it be seen as an a posteriori appeal to the convenience or simplicity of the framework that the definition constitutes. The argument, rather, reveals the new conception in its transcendental role, as uniquely making possible the synthesis of experience under formal scientific principles (DiSalle, 538)

Friedman’s tale is for mathematical science, physics in particular. Involves ineluctably history of philosophy. Kant for Newton, Vienna Circle for Einstein. Sympathy for Cassirer, Husserl.

When we bring such an approach to bear on the thesis expounded by Michael Friedman in his 2001 book, The Dynamics of Reason, we find we can construct a satisfying new case study, which involves developments in the mathematics of the past 75 years of importance to physicists.

When I first sketched this, I was hoping higher category theory would play this role. At the same time we needed a dynamic account of large swaths of modern maths. Imagine continuing the story to today. A change in the foundations of mathematics (modal HoTT) allows for the natural expression of cocycles in differential cohomology, which in turn allow for the expression of higher fields in physics, and so the expression of M-theory as equivariant differential cohomotopy.

In the limit of D=11 supergravity, the covariant phase space of M-theory must consist of torsion constraints in supertorsion-free super-orbi 10,1|32\mathbb{R}^{10,1\vert \mathbf{32}}-folds equipped with a suitable higher gauge field: the C-field. The first ingredient of a non-perturbative quantization of this phase space must be a choice of Dirac charge quantization-condition for C-field.

Hypothesis H: The C-field 4-flux and 7-flux forms in M-theory are subject to charge quantization in J-twisted Cohomotopy cohomology theory in that they are in the image of the non-abelian Chern character map from J-twisted Cohomotopy theory.

This is better than the Einstein case where the foundations (Frege-Peano-Hilbert) were less influential, just requiring geometry of Minkowski and Riemann, Christoffel, etc. rather than Frege. Yet Hilbert’s work fed into an alternative version leading to the Einstein-Hilbert action.

Discussing the two lines of work on space, time and motion (Mach) and on the foundations of geometry (Helmholtz, Poincaré), Friedman writes:

two continuously evolving traditions of thought in mathematics and scientific philosophy at first developed independently of one another, but then unexpectedly intersected with one another at the level of scientific theory itself–and this, in my view, was the source of the genuine conceptual discontinuity that does in fact arise at this level. The upshot, however, is that, although Einstein’s theory remains conceptually incommensurable or nonintertranslatable with the Newtonian theory it replaced, the way in which it was inextricably entangled with the previous traditions in question explains how it was nevertheless rational–for all parties to the earlier debates–to accept Einstein’s radical expansion of the Newtonian space of conceptual possibilities as both mathematically and physically possible (as, in Kantian terminology, really possible).(2010A, 500)

6 December, 16.45, Logik-Cafe

(draft slides)

Modal variants of the new foundational language, homotopy type theory, are now been intensively studied. In this talk I aim to show how philosophy can prosper from adopting modal homotopy type theory as its formal language of choice. I do this by unpealing its layers to explain: Why types? Why dependent types? Why homotopy types? And, why modal types? In the process we shall see the importance of these constructions for philosophy of language, philosophy of logic and metaphysics.

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