User:Marc Goossens/Hilbert's 6th problem

Introduction
In stating his sixth problem, Hilbert outlines the task of bringing theoretical physics in axiomatic form, starting with mechanics and statistical mechanics. This is one of the original 10 mathematical problems listed by David Hilbert in his address held on August 8, 1900, to the International Congress of Mathematicians in Paris (see

).

The fact that Hilbert highlights this domain in the form of a separately stated problem indicates that he saw it as a potentially rewarding field of research. The broadening of its scope to “the whole of physics” is a natural extension of the original formulation of the problem, and appears to be generally accepted

.

Being somehow close to physics, this issue is sometimes regarded as non-mathematical, and the odd one out in Hilbert’s list. Yet referring to geometry as an example, Hilbert himself noticed that a physical theory may be cast into a mathematically rigorous, axiomatic form. Any physical theory formulated in such a way constitutes a fully fledged and “pure” mathematical theory in its own right.

Hilbert’s explicit mention of axiomatic investigation of physical theories brings in the aspect of interpretation and foundation of these theories from the formal point of view, with possible links to its more firmly established “foundations of mathematics” sibling. By the same token, it enters territory of controversy (interpretation of physical theories) and competition (different attitudes and schools of thought regarding axiomatics and foundations of physics).

This article provides an overview of several elements that furnish partial answers and solutions to the sixth problem. A rich axiomatic development exists for classical continuum mechanics and thermodynamics. Important axiomatic research results are also available for General Relativity, Quantum Mechanics, Relativistic Quantum Mechanics and the kinetic theory of gases. In addition to axiomatics of individual theories or domains, significant advances have been made in the direction of generic formal metatheory (or “theory of theories”) for physics. Even combined, these do not present a complete answer to what is probably an open-ended question.

HF Hilbert’s original formulation
Hilbert refers in general to the mathematical treatment of the axioms of physics (“mathematische Behandlung der Axiome der Physik”). To him, the example of the foundations of geometry suggests to certainly handle these physical disciplines in which mathematics already in his days plays an “excellent” role, in a similar axiomatic fashion. He goes on to mention probability theory in mathematical physics (especially as used in the kinetic theory of gases) and mechanics. Hilbert cites the writings of Mach and Hertz on mechanics. In his view, it is desirable for mathematicians to take up the investigation of the foundations of this discipline. He is further intrigued by Boltzmann’s work on deriving bulk behaviour of continua from atomic behaviour, by statistical means. To Hilbert, a rigorous mathematical underpinning of this is called for. The other way around, one should check if rigid body motion may be shown to be a limiting case of a deformable continuous medium. To what extent certain of these axiom systems might be equivalent, appears to him a highly interesting question. Still according to the original formulation of the 6th problem, a lean set of general axioms encompassing a broad class of physical phenomena may be specialised by adjoining more axioms, of which the consistency is to be checked. As is the case with Lie groups, Hilbert expects this may lead to classification schemes for various such theories. From a mathematical point of view, the theories may also be generalized beyond those strictly modelling physical reality, and all consequences ensuing from the axioms ought to be explored.

Scope of the problem
(something here)


 * A physical theory, when formulated in a mathematically rigorous, axiomatic form, provides a mathematical theory.
 * The mathematical theories thus found or constructed, may not be those typically explored by pure mathematicians, without the spur of physical problem solving or physical intuition. So at least, the formalization of physical theories is a likely source of new (variants of) mathematical theories.
 * By the same token, theoretical physics may state new mathematical problems, as well as provide instruments for solving known ones.

Apart from possible different motivations, tricks and insights, the aspects listed above, once formalized, remain strictly mathematical. To this is added an additional slant, which – while formal – goes beyond mathematics.


 * How and to what extent, if at all, can the physical interpretation of such a mathematical theory, itself be formalized?
 * Can one specify a useful meta-theoretical scheme for this kind of formalization?

From the original statement of the problem, it is not clear whether Hilbert had also this “foundations of physics” aspect in mind. Still, methodologies and (partial) solutions in this direction are truly Hilbertian in spirit and aim and are therefore included in this overview.

At a practical level, many theoretical physicists have endeavoured to tackle mathematical issues, and many mathematicians have directed their efforts towards contributing to theoretical physics. Beyond that, many scientists from both domains have sought to clarify the physical content and meaning of the general theory of relativity and quantum mechanics in particular, by means of axiomatic formulations. By stating his 6th problem, Hilbert deemed this rich field of interaction between mathematics and physics worthy of systematic research.

Caveat on foundational aspects of physics
In pure mathematics, “foundations” and meta-theoretical research are an established discipline. In physics, this is much less the case. Yet, even when looked upon restrictively as a strictly mathematical enquiry, the 6th problem probes into these foundations of theoretical physics.

Indeed, when one asks about the physical content of the axioms of a physical theory, Hilbert’s question necessarily touches upon the interpretation of these theories. For many if not all of these, their interpretation remains a matter of passionate debate and controversy, some of which inevitably rubs off on the scope as well as the subject matter of this article.

Apart from that, there are several approaches for, or schools of, foundational theoretical physics. While complementary, these lines of thought are also to some degree in competition with one another.

Finally, axiomatics as such is regarded with some suspicion by many physicists, if not considered as pointless and sterile. The challenge of showing how axiomatic methods may be helpful and bring insight and fruitful new ideas into theoretical physics arguably adds one more dimension to this problem posed by Hilbert more than one century ago.

An open-ended problem
In contrast to many other of Hilbert’s problems, this 6th one is open-ended - at least to the extent that physics itself is not a finalized and closed endeavour.

If a hypothetical physical “theory of everything” would exist and be known and brought into axiomatic form, the task set by Hilbert could in principle be considered as completed.

Whether any such theory or catch-all model exists, is a matter of conjecture and controversy. Today, no such theory is known. Some of the theory-of-theories concepts considered in this article, suggest that (theoretical) physics is essentially and by its nature formed of a network of interrelated theories.

From this perspective, any suitable generic methodological scheme (of which there may be more than one!) for the formalization of “every” physical theory may be viewed as the most encompassing single answer to the question posed by Hilbert. Even then, such a scheme still needs to be properly applied to each specific physical theory, whether old or new. So in this scenario too, the challenge posed by Hilbert ultimately remains a permanent one.

GCA General considerations on axiomatics in physical theories

 * constructive axiomatics: physical interpretation is built into the theory explicitly in a bottom-up fashion; in many presentations of physical theory, the physical meaning of certain mathematical objects present in the model are derived or explored “a posteriori” (e.g. Hawking-Ellis derive the implications of space-time curvature in terms of its “effect” on geodesics, themselves interpreted as the world lines of particles or photons).


 * per theory formal system


 * general axiomatization schemes stronger than set theory

A2 Classical mechanics and thermodynamics of continua
A comprehensive axiomatic formulation was given by W. Noll. Truesdell regarded this result as a major contribution towards the solution of Hilbert’s sixth problem.

Further generalization of this scheme have since been proposed and explored. Still, Noll’s axioms in essence solve the axiomatization of (classical) mechanics as a mathematical (“rational”) theory, which Hilbert had called for.

This framework has also laid to an significant body of work and ongoing research in this field, with important applications in materials science and engineering.

The mathematical model adopted by the theory also takes in a powerful and general formulation of classical thermodynamics of continuous media, including irreversible processes, memory effects,

B5 Ludwig’s Theory of Theories
Pro: -	ordinary logic suffices -	standard set theory suffices -	in this sense lean (occam) relative to its scope (all of physics) -	a common interpretation scheme for all theories -	clarification of what distinguishes the physics from math -	scheme for relating physical theories

classification of axioms as physical hypotheses, …

drawback -	in its complete form, highly formal -	not widely known -

C demarcation from other rigorous mathematics in theoretical physics
Dynamical systems; symplectic geometry; functional analysis and operator algebra’s; gauge field theory (fibre bundles) and other geometrical stuff (ashtekhar) … … qualifies as sound mathematics and mathematical physics. While generally rigorous and mathematically advanced, the initial mathematical structures (axioms) chosen are the ones naturally adopted from the mathematical point of view. This is in contrast with the search for alternative and complementary axiomatic formulations, aiming to deepen the physical understanding. This distinction is not always very marked, and is certainly one of style, priorities and aims, not one of quality, importance and rigor.