# Hilbert's 6th Problem: Progress in Physics Axiomatization

• quasar987
In summary, according to the expert, the 6th problem in the list of Hilbert is the axiomatisation of physics. The problem is immense, and has not been fully solved yet. However, progress has been made, and the goal is to find a unified theory regarding the foundation of physics.
quasar987
Science Advisor
Homework Helper
Gold Member
The 6th problem in the list of Hilbert is the axiomatisation of physics.*

My question is simply, how is that coming along? What are the successes, what are the problems, etc? Dissert. *http://aleph0.clarku.edu/~djoyce/hilbert/problems.html

What we need to do I to discover a Unified Theory regarding the foundation of Psysics.The complexity of this task is inimaginable due to the vastness os ares we have to take into account. Electromagnetism, Clasical Mechanics, Statistical Mechanics, Cuantum Mechanics and Relativity. We, the ones searching for an answer to the Sixth should focus on finding the common base for all of those mentioned above.I've recently taken up advanced analisis of Newton's laws. All that is required is knowledge of already accepted facts and logic. Is also have a question for you. How can we link gravity with magnetism ?

quasar987 said:
The 6th problem in the list of Hilbert is the axiomatisation of physics.*

My question is simply, how is that coming along? What are the successes, what are the problems, etc? Dissert.

*http://aleph0.clarku.edu/~djoyce/hilbert/problems.html

It's my understanding that such a thing as 'axiomatisation of physics' is not a viable enterprise. (I haven't read the Hilbert lecture transcript.)

So how come that in some branches there is reference to 'Laws', such as Newton's laws of motion, or the laws of thermodynamics?

Well, those laws are referred to as "laws" for dramatic effect. Maybe not consciously so, but I'm sure that's the case. For instance, Newon's laws are not an actual axiomatisation; they're not exhaustive, they're not rigorous. But it does focus the attention on key issues and key concepts.

If I remember correctly a physicist once tried to axiomatize electromagnetism as codified by Maxwell's equations. According to the description that physicist ended up with six axioms, and it didn't bring more clarity to the field.

In mathematics axiomatisation is achievable, and as I understand it axiomatisation is regarded as a indicator of maturity of a branch of mathematics. In physics what is referred to as 'axioms' is for dramatic effect. I think the 'axioms', 'laws', 'postulates', 'principles' are attempts to capture the essence of what is going on.

Cleonis

Last edited:
From classical to some modern fields of physics, everything is axiomatized. And if it is not, mathematical/theoretical physicists are giving axiodeductive foundations. So, in principal, it is possible. Although, there is some philosophical and realistic problems(will we ever find out?) stopping us.

Remember, when Hilbert made his questions, physics was a lot less mathematically sophisticated.

quasar987 said:
The 6th problem in the list of Hilbert is the axiomatisation of physics.*

My question is simply, how is that coming along? What are the successes, what are the problems, etc? Dissert.

*http://aleph0.clarku.edu/~djoyce/hilbert/problems.html

AFAIK, the most recent activity peaked from the late 1960s to early 1980s. As a result of that work, continuum mechanics and thermodynamics can now be considered nearly axiomatic (nonlinear theories represent the final frontier). I believe electrodynamics has somewhat of an axiomatic foundation, and QM as well, but I am less clear about those.

Hilbert was aware of a long tradition of presenting axiomatic versions of some physical theory or other. I believe Euclid himself may have tried to do that with Optics. Certainly Archimedes did that with Hydrostatics. Newton made some attempt to cast his Principia as if it were axiomatic, but I do not think he had the right personality to really succeed, it is just that the influence of Euclid's style made him imitate it, unsuccessfully.

More importantly were Maxwell, who axiomatised Mechanics in the late 19th century...note well, he didn't really do that for Electromagnetism! (His own field of research), and Hertz, who did the same. Hertz's views on the philosophy of science, and of course he was a great experimentalist too, and an important teacher (Wittgenstein was a pupil) of engineers, were immensely influential in Germany, and Hilbert must have been aware of them.

Unless you wish to criticize Maxwell, Hertz, and Archimedes, one should not *a priori* disparage an axiomatic approach to Physics.

In the 1920's Hilbert ran a seminar in Atomic Physics and all the latest discoveries in Physics and enticed the greatest active scientists, including Einstein, to give talks. Out of this came a paper by HIlbert, von Neumann, and Nordheim which tried to axiomatise Matrix Mechanics, or Quantum MEchanics as we now call it. Hilbert himself had little to do with this paper. Von Neumann was then influenced by Wigner to produce his axiomatisation of Quantum MEchanics a little later, and published it as an influential book.
Independently, Dirac did the same, but without the mathematical rigour of von Neumann. But the physical content of Dirac's axioms are identical. Weyl had published an axiomatisation a year or two earlier, independently of Hilbert's influence, but had not thought of the famous axiom of the reduction of the wave packet, and so had only five axioms, not the usual six we now see in many textbooks.

Wigner has pointed out what, from a Hilbertian standpoint, is the unsatisfactory feature of this axiomatisation, in several famous papers, reprinted in *Symmetries and Reflections*.

It is arguable that since QM encompasses almost all of Physics...wait, see below...that if QM could be axiomatised successfully, then Hilbert's Sixth Problem would be solved. If this is accepted, then only Wigner's criticism (of what is almost his own creation) is the last obstacle to Hilbert's problem

What of QED and General Relativity (GR)? Well...I just said 'arguable', I didn't say 'undeniable'. QED ought to be derivable from QM, so any problems of QED are not genuinely foundational in nature. (One could argue this.) GR by itself has never raised any axiomatic problems, it is straightforward, cut and dried. So if GR and QM are ever unified, one would not expect GR to introduce any foundational difficulties.

From this point of view, then, great progress really has been made. After all, axiomatisation is not supposed to make Physics easy, or intuitive, or dead...

According to the description that physicist ended up with six axioms, and it didn't bring more clarity to the field.http://www.uklv.info/g.php

Last edited by a moderator:
It brought about much more clarity than had been present before. It is not supposed to make Quantum Mechanics conform to our classical intuition! Historically, it had a great influence on our understanding of entanglement. Because von Neumann had axioms to work with, he tried to prove that a hidden variable theory was impossible. Because de Broglie and Bohm succeeded in constructing a hidden variable theory (purely as an axiomatic exercise), J.S. Bell was motivated to examine von Neumann's proof for its assumptions, which were all explicit (that is the virtue of the axiomatic method, it forces you to make all your assumptions explicit, you can't get away with hidden assumptions anymore) and found that one of von Neumann's assumptions was physically unreasonable. This discovery motivated Bell to prove his famous Bell Inequality, which has had a great influence on our understanding of locality, Quantum Probability, and entanglement. In short, it highlighted the assumptions of locality that people used to make unconsciously. There is still more work to be done in axiomatising Quantum Informatin Theory, and researchers such as Lucien Hardy and many others still think axiomatics has some usefulness, sometimes, even in Physics

Hilbert's Sixth Problem now (essentially) solved

A common misconception is that Kolmogoroff solved the part of Hilbert's problem related to probabilities. This misconception was not shared by Kolmogoroff! He well knew that axiomatising the purely mathematical theory of probabilities was merely a useful preliminary: what Hilbert really wanted was to axiomatise the concepts of physical probability. Within physics, is 'probability' a new, primitive, concept to be added to Hertz's list, along with mass and time, or can it be precisely defined in terms of mass, time, etc. ?

Unless grand unification or renormalisation throw up new axiomatic difficulties, then the only two things that were left to do to solve Hilbert's Sixth PRoblem were: a) the problem which Wigner pointed out, about the concept of measurement in QM (Bell analysed http://www.chicuadro.es/BellAgainstMeasurement.pdf the problem the same way Wigner did), and b) the definition of physical probability, i.e., the concept of probability which occurs in QM. Hilbert himself was worried about causality in GR but solved that problem himself. Hilbert pointed to the lack of clarity in the relation between Mechanics and Stat Mech, but Darwin and Fowler solved that in the 1920s.

Many physicists, notably H.S. Green in "Observation in Quantum Mechanics," Nuovo Cimento vol. 9 (1958) no. 5, pp. 880-889, posted by me at http://www.chicuadro.es/Green1958.ps.zip, and now more realistic models by Allahverdyan, Balian, and Nieuwenhuizen arXiv:1003.0453, have pointed to the possiblity of fixing the 'measurement' problem Wigner was worried about: they have analysed the physical behaviour of a measurement apparatus and shown that the measurement axioms of QM follow, approximately, from the wave equation. They do this in a logically circular and sloppy way, but the logic can be fixed.

Physical probability can be defined in QM, and its definition there is parallel to its definition in Classical Mechanics (see "Descriptive statistics as new foundations for probability: A part of Hilbert's sixth problem", Revista Investigaciones Operacionales, to appear): each involves the use of a new kind of thermodynamic limit (in the quantum case http://arxiv.org/abs/quant-ph/0507017, one in which not only does the number of degrees of freedom of the measurement apparatus increase without bound, but Planck's constant goes to zero).

So the people who did the most important work are: Hilbert, Wiener, Weyl, Schroedinger, Darwin, Fowler, Born, Dirac, Kolmogoroff, Wigner, Khintchine, H.S. Green, Bell, Prof. Jan von Plato, and myself, and now it is essentially solved. (Schroedinger could be included twice: he and Debye helped Weyl formulate the first axiomatisation of QM. Later, he influenced H.S. Green in his treatment of measurement as a phase transition.) Of course the solution opens new avenues of research.

## 1. What is Hilbert's 6th Problem?

Hilbert's 6th Problem is one of the 23 problems proposed by German mathematician David Hilbert in 1900. It focuses on the axiomatization of physics and the development of a complete and consistent set of axioms for the physical sciences.

## 2. Why is Hilbert's 6th Problem important?

Hilbert's 6th Problem is important because it addresses the fundamental question of how to establish a solid mathematical foundation for the physical sciences. It highlights the need for a rigorous and axiomatic approach to understanding the laws of physics.

## 3. What progress has been made towards solving Hilbert's 6th Problem?

Significant progress has been made towards solving Hilbert's 6th Problem since it was proposed in 1900. Several theories, such as quantum mechanics and general relativity, have been successfully axiomatized and have provided a solid mathematical framework for understanding the physical world.

## 4. What challenges still remain in solving Hilbert's 6th Problem?

Despite the progress made, Hilbert's 6th Problem still faces challenges, as there is no universally accepted set of axioms for all of physics. Additionally, the reconciliation of quantum mechanics and general relativity, known as the "holy grail" of physics, remains an unresolved issue in the axiomatization of physics.

## 5. How does Hilbert's 6th Problem relate to other areas of mathematics?

Hilbert's 6th Problem has important connections to various areas of mathematics, such as logic, set theory, and topology. It also relates to the foundational issues of mathematics, specifically the quest for a consistent and complete set of axioms for all of mathematics.

Replies
5
Views
1K
Replies
4
Views
1K
Replies
4
Views
2K
Replies
1
Views
1K
Replies
2
Views
1K
Replies
2
Views
2K
Replies
6
Views
816
Replies
0
Views
611
Replies
3
Views
269
Replies
26
Views
2K