A Kantian view on modern physics

Question

According to the Encyclopedia Britannica article on Immanuel Kant, in the section discussing the Critique of pure reason:

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In the Transcendental Analytic, the most crucial as well as the most difficult part of the book, he maintained that physics is a priori and synthetic because in its ordering of experience it uses concepts of a special sort. These concepts—“categories,” he called them—are not so much read out of experience as read into it and, hence, are a priori, or pure, as opposed to empirical. But they differ from empirical concepts in something more than their origin: their whole role in knowledge is different. For, whereas empirical concepts serve to correlate particular experiences and so to bring out in a detailed way how experience is ordered, the categories have the function of prescribing the general form that this detailed order must take. They belong, as it were, to the very framework of knowledge. But although they are indispensable for objective knowledge, the sole knowledge that the categories can yield is of objects of possible experience; they yield valid and real knowledge only when they are ordering what is given through sense in space and time.

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This view that physics provides the general form of this detailed order seems to be incorrect because physics is now done differently. Before, physics was the development of equations to predict events (classical mechanics). Now, physics is the development of equations to determine probabilities of events (quantum mechanics). Furthermore, spacetime is no longer considered static. So how can physics be a priori then?

Answer 1

Neo-Kantian philosophers, wishing to take into account that physics and mathematics had undergone deep changes in the decades after Kant, proposed a historicized a priori. Fundamental principles frame what we take to be our experience, but these principles are subject to change. Ernst Cassirer's Determinism and Indeterminism in Modern Physics is an important example of such work, which takes up the challenge presented by quantum mechanics.

For current accounts of such a historicized a priori, take a look at Michael Friedman's work, such as The Dynamics of Reason, and his contribution to Discourse on a New Method.

Answer 2

There is a remarkable proposal by William Lawvere, for connecting transcendental philosophy and theoretical physics. Lawvere proposes that the categories in the version in which Hegel presents them in the Science of Logic are faithfully and usefully formalized in categorical logic (a mathematical term! which happens to fit well the use in philosophy) as systems of (co)-reflective subcategories (in the mathematical sense! of category theory) of some ambient topos.

The resulting structure Lawvere called a cohesive topos (following Hegel's discussion of "cohesion" in the Philosophy of Nature), and he indicates how such "gros toposes" may serve as Toposes of laws of motion for physics.

It is possible to refine this a little more to arrive at a concept of cohesive infinity-toposes. In a book-in-progress titled Differential cohomology in a cohesive infinity-topos (web, pdf) I claim to work out how a considerable chunk of modern physics naturally finds its formalization in terms of such categories, see in particular the introductory section 1.2 on Classical field theory via Cohesive homotopy types (web, pdf).

See here for pointers to Lawvere's proposal for formalizing idealistic philosophy in terms of categorical logic.

See here for pointers to Lawvere's work on building a foundation of (classical continuum) physics based on this.

See here for details on how the mathematical formalization of "the categories" according to Hegel's Science of Logic proceeds.

For more background and survey see also the beginning of my lecture slides on Synthetic Quantum Field Theory.

Answer 3

When Kant talked about apriority in the realm of physics, he was talking about background principles being a priori. Science as a whole is like an experiment about the relationship between mathematics and nature, an experiment to test this relationship. (So far, the experiment is going really well!) Experimenters might or might not make or recognize the non-experimental components of their programs---beliefs about inference structures, modality, intuition, etc.---so they may have an appearance of doing science without any question of epistemic apriority. But Kant thought reality as we know it was too orderly for that: there are arguments to be made and decided about some of the background questions, and these can then go on to color the conclusions of our specific arguments about physics.

The notorious case is his doctrine of space and time. His objection, if he would have one, to some of the ways the issue is nowadays framed, would be not so much that we couldn't discursively represent various systems of geometry, applied to spacetime. Remember, this is a man who suggested that time itself being one-dimensional is contingent in an abstract sense. Time as we know it flows along a one-dimensional line (or so it seems), and is subject to the mathematics of a one-dimensional line as such. But we can at least imagine in logical space (but perhaps not in concrete imagination, mind you) a form of time with the mathematics of a two-dimensional structure, or in three dimensions, or whatever.

The next question is whether the above quasi-objection tells against the modern rejection of Euclidean limits on geometry as applied to physics. I don't know that our intuition of space is actually limited in a Euclidean way. I would say that Kant was wrong, not about whether an intuitive spacetime is a fundamental background, here, but about the particulars of that intuition. If physics needs a non-Euclidean geometry because there are some physical experiences of non-Euclidean structures, that seems to me to indicate rather that we do have an intuition of space as such: not as if as humans recently evolved to gain the ability to perceive and visualize non-Euclidean geometry, then!

More broadly, we do have plenty of analogical intuitions of various structures in four- and five-dimensional geometry. We can stereoscopically project or lay out the nets for or show some rotational sequences of geometrical structures whose dimensionality exceeds our direct intuition. This is particular (therefore intuitive, on the Kantian definition of the faculty of intuition) information about such structures, allowing us to differentiate such structures fairly well. But the asymptotic decrease in such analogical projection is such that an increasingly vague boundary between "knowable" and "unknowable" (on the Kantian model) spacetime systems appears: we can provide less and less intuitive descriptions of structures of higher and higher dimensionality, so our possible intuitive evidence for propositions involving those dimensions is reliably decreasing in scope, the more dimensions we assert in our theory. In other words, if all we need to explain can be done in a lower-dimensional model vs. a higher-dimensional one, that is to be preferred. But we take it from experience that what we need to explain might indeed require a few (so to speak) extra dimensions---for space or time (see Itzhak Bar's "2D time theory" for a decent example of the latter case).

Kant doesn't speak of analogical intuitions as such, but they are suggested by his remark about the terms of causation (this is in the section on the analogies of experience):

But in philosophy, analogy is not the equality of two quantitative but of two qualitative relations. In this case, from three given terms, I can give a priori and cognize the relation to a fourth member, but not this fourth term itself, although I certainly possess a rule to guide me in the search for this fourth term in experience, and a mark to assist me in discovering it.

So a 4-dimensional geometrical structure, for example, can be thought of as a fourth term, that by various distinguishing geometrical projections can be related to 3-dimensional space such that the principle of the analogies of experience allows us to "believe in" the 4-dimensional structure, if we need to believe in it (so to speak) in our best mathematical theory (supposing, which is contentious, that there is, after all, such a "best" theory).