What is the status of the Hilbert 6th problem?

Question

As you know, the Hilbert sixth problem was to axiomatize physics. According to the Wikipedia article, there is some partial succes in this field. For example, Classical mechanics, I believe, can be treated now as an axiomatized discipline since it is properly formalized in the modern theories of Lagrangian and Hamiltonian mechanics (and as a corollary, one can consider it as an extension by definition of the theory of real numbers, which in its turn is an extension by definition of the axiomatic set theory -- and eventually all these theories can be treated as first order theories).

I am not a specialist, but the Theory of relativity seems to be formalized as well in terms of differential geometry (is that correct?). On the other hand, as we understood not long ago, the Quantum mechanics is not axiomatized. I often discuss this with my colleagues and students, so I believe this is an important question, it deserves specification and clarification, the Wikipedia article is too short, it must be detailed. I believe there are people here who can explain clearly

which parts of physics are axiomatized now, and which are not.

For the educational purposes (and I am sure, this will be interesting for specialists as well) it would be good to have a list of these disciplines. In the discussions at the web, for example, in the discussion at PhysicsStackExchange, I don't see this list. If it is possible, I ask specialists to share their knowledge. There is no necessity to list all the disciplines in one answer, each answer can be devoted to one discipline, I only ask people to give the necessary references and to provide some elementary explanations (of course, detailed answers are better) so that non-specialists (and students) could understand.

Thank you.

EDIT. From Ben Crowell's answer I deduce that there is some discrepancy in understanding of what is meant by the 6th Hilbert problem: some people interpret it as a suggestion to construct a first order theory of a given physical discipline (this means, to build a "completely new theory", formally independent from the other mathematics, with its own language of logic -- like in the modern set theories, ZF, NBG, MK, which are the standard examples of the first order theories), while others interpret this wider, as a possibility to give any system of axioms, not necessarily first order (this can be understood as building an extension by definition of another, already constructed first order theory -- and the standard examples are Hilbert's axioms of Euclidean geometry, Kolmogorov's axioms of probability theory, etc.). As to me I don't see serious reasons to restrict ourselves on first order theories: the "axioms as definitions" are quite satisfactory, moreover, I would say they are preferable, since they do not deflet attention from the essential part of the problem (to the details with the new language of logic, etc.). Actually, if we look at the problem from this "practical" point of view, I do not see the difference between "axiomatization" and "formalization", that is why I referred to the books by V.I.Arnold and A.L.Besse from the very beginning.

References:

V.I.Arnol'd, Mathematical methods of classical mechanics

A.L.Besse, Einstein Manifolds

Answer 1

which parts of physics are axiomatized now, and which are not.

As an example, some fairly big fragments of classical physics have been formalized.

A significant part of Newton's Principia has been formalized by Fleuriot and Paulson using the proof assistant Isabelle.

General relativity is pretty trivial to formalize, basically because GR isn't so much a theory per se as a framework that we can plug other theories in to. If you want a concise formalization, it's not much more than this: spacetime is a Hausdorff differentiable manifold and it comes equipped with a nondegenerate metric (usually assumed to have signature +---). You can add in the Einstein field equations, but in a purely mathematical formulation restricted to GR itself, they are vacuous or work just as a definition of the stress-energy tensor, because GR itself says absolutely nothing about the stress-energy tensor or matter fields. You may want to add some subsidiary conditions about the integrability and differentiability of the metric, as discussed in Hawking and Ellis, p. 58. Hawking and Ellis also give some other axioms on pp. 60 (causality) and 61 (local conservation of energy-momentum), but again, these postulates become vacuous unless you plug in a separate theory for the matter fields. For a different approach to formalizing special and general relativity, see Andreka.

What is the status of the Hilbert 6th problem?

I think these two examples can help to clarify why Hilbert's sixth problem is not really very interesting in a modern context. Hilbert was operating in a Victorian era of optimism about physics, when it was believed that we were very close to understanding all the fundamental laws of nature, and all that was necessary was a little more cleaning up and investigation of the details. Totally wrong. Physics turned out to be a lot more complicated than he thought, and all we possess today is a bunch of pieces of a theory of everything (ToE). These pieces sometimes interlock nicely, but in other cases they don't, as in the case of quantum gravity. Since the Planck scale is inaccessible to any foreseeable human technology, it is likely that, even centuries from now, we will never have a ToE that can be tested by experiment. Only once we had a ToE would it even make sense to worry about Hilbert's sixth problem (and at that point I suspect it would be both trivial and of little interest to go ahead and carry it out).

Hilbert was searching for absolute certainty in the universe, and he was working before Godel. An interesting secondary question is whether, given a ToE and its formalization, we could make some sort of decision procedure for physics. The answer is not very clear, because the theory we're talking about might actually not describe enough of arithmetic for Godel's theorems to apply. After all, GR is basically a kind of geometry, and Tarski wrote down a decision procedure for Euclidean geometry. In any case, we wouldn't have absolute certainty, because the ToE itself would be based on experiments and therefore subject to revision.

References

Andreka et al., "On logical analysis of relativity theories," pdf, Hungarian Philosophical Review, 2010/4, pp.204-222

Fleuriot and Paulson, "A combination of nonstandard analysis and geometry theorem proving, with application to Newton’s Principia," pdf, Lecture Notes in Computer Science 1421 (2006) 3.

Hawking and Ellis, "The large scale structure of space-time"

Answer 2

The following might be useful:

Corry, Leo David Hilbert and the axiomatization of physics (1898–1918). From Grundlagen der Geometrie to Grundlagen der Physik. Archimedes: New Studies in the History and Philosophy of Science and Technology, 10. Kluwer Academic Publishers, Dordrecht, 2004. xvii+513 pp. ISBN: 1-4020-2777-X.

The following is the review of the paper taken from mathscinet:

This is a massive scholarly book on the work of David Hilbert on physics. In more than 400 pages the author provides us with an in-depth analysis of the fundamental contributions of this mathematician-born scientist to many branches of physics, from radiation theory to thermodynamics, from mechanics to relativity.

The crucial methodological method identified by the author as the trait d'union of all of Hilbert's work in physics is the idea of "axiomatization'', of being able to deduce the multiplicity of facts and mathematical/physical consequences of a physical theory from a limited but sufficient and consistent set of basic propositions and assumptions. From this point of view, as the author himself suggests, Hilbert's work in physics is the natural continuation of his previous and contemporary achievements in pure mathematics, above all in the foundations of geometry. Underlying this methodological preference is the struggle for clarity, conciseness, mathematical beauty and coherence that was a characterizing feature of Hilbert as a mathematician, and a sign of his genius.

It is in the development of relativity that Hilbert played a most important and influential role, and where Hilbert's ideas on the foundations of physics can be seen most clearly, as far as his contributions to physics are concerned. Consequently, the core of the book is represented by the final 3 chapters dealing with Hilbert's work on gravitation theory and his contribution to Einstein's General Relativity, starting from the influence of both Einstein and Mie's work on him, through a detailed account of Hilbert's own formulation of the theory of gravitation, following his interaction with Einstein (much has been written on this interaction and the author analyses the historical evidence of this with great care), to finish with the last work by Hilbert on General Relativity dating from 1918.

This is a complex but also a fascinating subject, given that what the author beautifully pictures is one of the best minds in the history of science, a towering figure in both physics and, even more, mathematics, struggling to gain understanding of one of the most elusive aspects of nature: gravitation, finally (even if only provisionally, as always in science) obtained thanks to the formulation of one of the most beautiful physical theories that we, as humans, have been able to produce: the General Theory of Relativity. As is to be expected, given the complexity of the subject, this book is no easy read, as the author tries to cover all the historical development of Hilbert's work, while at the same time trying to explain the mathematics and physics involved in this development, alongside the work of other illustrious scientists who either contributed significantly to the area of research under consideration, or influenced Hilbert's work on it: in particular, Minkowski, Born, Mie and Einstein.

The author does not refrain from dwelling on any little aspect of the story, be it mathematical, physical, historical, personal or social, and, if this contributes to the difficulties the reader may face in going through this book, at the same time this contributes also to the pleasure the same reader may find in discovering the many facets that the development of Hilbert's work presented, the same facets that the slow and difficult progress of science shows in real life. As every scientist knows too well, intuition, genius, mistaken judgements, prejudices, social environment, scientific trends, conversations and fights with fellow scientists, hard solitary work, help of others, passion, understanding and misunderstandings, ambition, precise calculations and sketchy ideas, all contribute to the development of science, and nowhere is this amazing complexity of such a very human enterprise more clear than in the careful analysis of the history of any research achievement, as taught to us by a generation of philosophers of science, from Popper to Kunh, from Lakatos to Feyerabend, and as revealed once more by the present book.

If a criticism has to be brought forward to this book, this is that the discussion of the physical and mathematical concepts and results is not always accurate, ranging at times from being a bit confusing or puzzling to being, in a very few occasions, plainly wrong. However, the historical reconstruction, which was the main aim of the author, is, on the contrary, done with extreme competence and care, and it is, I think, of the highest quality.

Answer 3

The work of John Baez about Physics and Category Theory, see

Categories, Quantization, and Much More

Answer 4

I did some work that was quite successful at axiomatis-ing (pardon grammatical abuse!) the principal Hamiltonian operators for all finite quantum theories. My co-worker was able to extend the methodology to continuous operator groups. The basic conclusion was to find a quantum brachistochrone equation- essentially the Heisenberg equation for the Hamiltonian + constraint- and show that all physical matrix groups were given as solutions. So to that extent, I would say that the Hilbert 6th problem has been addressed for quantum mechanics, as all the observed Hamiltonian matrices on SU(2), SU(3),..., SU(N) for N finite come out as natural solutions via this method, and coupled with the infinite dimensional case, all angles are covered and closed out.

The essential link was to state that the path of flow on the complex projective manifold was defined by a Lagrangian that implied a least time principle under energetic constraints.

Quantum mechanics can seem quite esoteric, and believe me when I say that my professor was not a massive fan of the first step being the derivation of the Hamiltonian operator from first principles. Most quantum physics books just assume that it has one form or another. Personally I find it better to derive than to assume.

If there is any interest in exploring this topic, I'm happy to upload some references to some papers and continue the discussion.

Answer 5

Topological Quantum Field Theory -Atiyah

The Definition of Conformal Field Theory-Segal

These both describe the structure of assigning vector spaces to time slices and linear maps for the cobordism of the spacetime. There are distinctions for the structures on collars and cobordisms.