# Workshop: Hilbert's Sixth Problem

Lead Research Organisation:
University of Leicester

Department Name: Mathematics

### Abstract

In the year 1900 Hilbert presented his problems to the International Congress of Mathematicians (he presented 10 problems at the talk; the full list of 23 problems was published later). The list of Hilbert's 23 problems was very influential for 20th century mathematics. The sixth problem concerns the axiomatization of those parts of physics which are ready for a rigorous mathematical approach.

The original formulation (in English translation) was:

"6. Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics." This is definitely "a programmatic call" for the axiomatization of existent physical theories.

In a further explanation Hilbert proposed two specific problems: (i) axiomatic treatment of probability with limit theorems for the foundation of statistical physics, and (ii) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua":

The Sixth Problem has inspired several waves of research. Its mathematical content changes in time what is very natural for a programmatic call. In the 1930s, the axiomatic foundation of probability seemed to be finalized by Kolmogorov on the basis of measure theory. Nevertheless, in the 1960s Kolmogorov and Solomonoff stimulated new interest in the foundation of probability (Algorithmic Probability).

Hilbert, Chapman and Enskog created asymptotic expansions for the hydrodynamic limit of the Boltzmann equations. The higher terms of the Chapman-Enskog expansion are singular and truncation of this expansion does not have rigorous sense (Bobylev). Golse, Bardos, Levermore and Saint-Raymond proved rigorously the Euler limit of the Boltzmann equation in the scaling limit of very smooth flows, but recently Slemrod used the exact results of Karlin and Gorban and proposed a new, Korteweg asymptotic of the Boltzmann equation.

It seems that Hilbert presumed the kinetic level of description (the "Boltzmann level") as a compulsory intermediate step between the atomistic view and continuum mechanics. Nevertheless, this intermediate description may be omitted. Now, L. Saint-Raymond with co-authors is developing a new approach to the problem "from the atomistic view to the laws of motion of continua" without the intermediate kinetic equations.

Quantum mechanics was invented after the Hilbert problems were stated. The first attempt at formalization of quantum mechanics was performed by von Neumann (who was Hilbert's assistant). Subsequently, the axiomatic approach to the quantum world has been developed by many researchers and there are many versions of its axiomatization. Ideas and methods of quantum computing and quantum cryptography have transformed research in the foundation of quantum mechanics into an applied discipline with a potential for engineering applications. Many new mathematical structures and methods have been invented.

Work on Hilbert's Sixth Problem involves many areas of mathematics: mathematical logic, algebra, functional analysis, differential equations, geometry, probability theory, theory of algorithms, and many others. It remains one of the most seminal areas of interdisciplinary dialog in mathematics and mathematical physics.

The proposed workshop aims to gather top experts in Hilbert's Sixth Problem, to review the current achievements in the solutions of this problem, and to formulate the main mathematical challenges and problems which have arisen from 115 years of work. This renewed programmatic call should be disseminated and explained together with modern achievements to interdisciplinary research community, to a new generation of mathematicians, and to the public so that they can better appreciate the contribution of mathematics to science and technology.

The original formulation (in English translation) was:

"6. Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics." This is definitely "a programmatic call" for the axiomatization of existent physical theories.

In a further explanation Hilbert proposed two specific problems: (i) axiomatic treatment of probability with limit theorems for the foundation of statistical physics, and (ii) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua":

The Sixth Problem has inspired several waves of research. Its mathematical content changes in time what is very natural for a programmatic call. In the 1930s, the axiomatic foundation of probability seemed to be finalized by Kolmogorov on the basis of measure theory. Nevertheless, in the 1960s Kolmogorov and Solomonoff stimulated new interest in the foundation of probability (Algorithmic Probability).

Hilbert, Chapman and Enskog created asymptotic expansions for the hydrodynamic limit of the Boltzmann equations. The higher terms of the Chapman-Enskog expansion are singular and truncation of this expansion does not have rigorous sense (Bobylev). Golse, Bardos, Levermore and Saint-Raymond proved rigorously the Euler limit of the Boltzmann equation in the scaling limit of very smooth flows, but recently Slemrod used the exact results of Karlin and Gorban and proposed a new, Korteweg asymptotic of the Boltzmann equation.

It seems that Hilbert presumed the kinetic level of description (the "Boltzmann level") as a compulsory intermediate step between the atomistic view and continuum mechanics. Nevertheless, this intermediate description may be omitted. Now, L. Saint-Raymond with co-authors is developing a new approach to the problem "from the atomistic view to the laws of motion of continua" without the intermediate kinetic equations.

Quantum mechanics was invented after the Hilbert problems were stated. The first attempt at formalization of quantum mechanics was performed by von Neumann (who was Hilbert's assistant). Subsequently, the axiomatic approach to the quantum world has been developed by many researchers and there are many versions of its axiomatization. Ideas and methods of quantum computing and quantum cryptography have transformed research in the foundation of quantum mechanics into an applied discipline with a potential for engineering applications. Many new mathematical structures and methods have been invented.

Work on Hilbert's Sixth Problem involves many areas of mathematics: mathematical logic, algebra, functional analysis, differential equations, geometry, probability theory, theory of algorithms, and many others. It remains one of the most seminal areas of interdisciplinary dialog in mathematics and mathematical physics.

The proposed workshop aims to gather top experts in Hilbert's Sixth Problem, to review the current achievements in the solutions of this problem, and to formulate the main mathematical challenges and problems which have arisen from 115 years of work. This renewed programmatic call should be disseminated and explained together with modern achievements to interdisciplinary research community, to a new generation of mathematicians, and to the public so that they can better appreciate the contribution of mathematics to science and technology.

### Planned Impact

Without doubt, work on Hilbert's Sixth Problem has had a great impact on a number of strands of pure mathematics. For example, von Neumann's foundation of quantum mechanics was, at the same time, a big step in the development of functional analysis. But what is more surprising is, despite the abstraction of the formulation, aimed towards to the deep foundations of science, Hilbert's Sixth Problem had already great impact not only on pure mathematics but on many applied disciplines as well. Numerous model reduction methods in physics, applied physics, chemical and biochemical kinetics have their origin in the works of Hilbert, Enskog and Chapman regarding solutions of Boltzmann's equation, "which lead from the atomistic view to the laws of motion of continua". Quantum technologies exploit results arising in the fundamental analysis of the logical foundation of quantum mechanics. Axiomatization of probability has had impact on many applications areas, from statistical physics to financial mathematics. Computer sciences gain much from results of algorithmic probability and complexity-based foundations of the notion of randomness. Another direction of impact of Hilbert's Sixth Problem is our general understanding of the world and notions of rationality. This impact is communicated to society via the educational system and popular scientific writers (in particular, by scientific journalism).

As suggested by the preceding narrative, there are three stream of impact of the proposed workshop besides the direct influence on the researchers working with Hilbert's Sixth Problem:

1. Wider impact on the research in related areas of pure and applied mathematics. For this purpose, a focussed discussion about related mathematical problems will be organised, and a list of scientifically significant open problems will be published.

2. Impact on applications of mathematics in science and engineering. Discussion of various applied aspects of the work in Hilbert's Sixth Problem will be specially organised in the round table format with invitation of the experts in applications. It is possible to name two applications immediately: (i) model reduction in science and engineering and (ii) quantum technologies. The possible spectrum of applications may be wider and we expect significant extension to the ones suggested here by the workshop participants.

3. Wide cultural and educational impact. We will organise an open lecture for teachers and students (in collaboration with IMA) and publication of a series of popular articles in various journals and by news agencies.

Model reduction is a central tool in applied mathematics as it looks for dominant behaviours in complex systems, and covers such wide ranging applications as big data analytics and modelling of fluid flow. Such systems arise in all physical, chemical, biological, economic and societal processes, and so the potential beneficiaries related to development of the knowledge base, outside of the academic mathematical community, include physicists, economists, psychologists, and biologists. There is clear national significance of the development of this area of applications.

Quantum technologies form a strategic direction of the technological development of the UK. We plan to organise a discussion about application of results related to Hilbert's Sixth Problem in quantum technologies with close collaboration with the Quantum Technology Hubs.

We will organise special interest groups around specific themes to create new partnerships and develop action plans for new projects.

As suggested by the preceding narrative, there are three stream of impact of the proposed workshop besides the direct influence on the researchers working with Hilbert's Sixth Problem:

1. Wider impact on the research in related areas of pure and applied mathematics. For this purpose, a focussed discussion about related mathematical problems will be organised, and a list of scientifically significant open problems will be published.

2. Impact on applications of mathematics in science and engineering. Discussion of various applied aspects of the work in Hilbert's Sixth Problem will be specially organised in the round table format with invitation of the experts in applications. It is possible to name two applications immediately: (i) model reduction in science and engineering and (ii) quantum technologies. The possible spectrum of applications may be wider and we expect significant extension to the ones suggested here by the workshop participants.

3. Wide cultural and educational impact. We will organise an open lecture for teachers and students (in collaboration with IMA) and publication of a series of popular articles in various journals and by news agencies.

Model reduction is a central tool in applied mathematics as it looks for dominant behaviours in complex systems, and covers such wide ranging applications as big data analytics and modelling of fluid flow. Such systems arise in all physical, chemical, biological, economic and societal processes, and so the potential beneficiaries related to development of the knowledge base, outside of the academic mathematical community, include physicists, economists, psychologists, and biologists. There is clear national significance of the development of this area of applications.

Quantum technologies form a strategic direction of the technological development of the UK. We plan to organise a discussion about application of results related to Hilbert's Sixth Problem in quantum technologies with close collaboration with the Quantum Technology Hubs.

We will organise special interest groups around specific themes to create new partnerships and develop action plans for new projects.

### Organisations

## People |
## ORCID iD |

Alexander Gorban (Principal Investigator) |

### Publications

Alexander N. Gorban
(2019)

*Hilbert's Sixth Problem: the Path to Mathematical Rigour in Science*in LMS Newsletter
Gorban A
(2018)

*Hilbert's Sixth Problem: the endless road to rigour*
Gorban A
(2016)

*Beyond Navier-Stokes equations: capillarity of ideal gas*in Contemporary Physics
Gorban AN
(2018)

*Hilbert's sixth problem: the endless road to rigour.*in Philosophical transactions. Series A, Mathematical, physical, and engineering sciencesDescription | The main aims of the workshop were 1. To facilitate interdisciplinary discussion across key mathematical and physical disciplines involved in solution of Hilbert's sixth problem about the state of the art. DONE 2. To synthetize an integral interdisciplinary point of view on Hilbert's sixth problem and renew the programmatic call in the light of the latest achievements. DONE Sixth Hilbert's problem was analysed as the programmatic call, the current state-of-the art in this direction was reviewed and the modern revised version of the Sixth Problem was presented in the series of publications. 3. To provide guidance to early career researchers via an indication of future research directions in Hilbert's sixth problem. DONE 4. To disseminate the modern achievements and renewed programmatic call in a series of review publication DONE (to be continued) |

Exploitation Route | Dissemination of the results is organised in the series of the publications. New fluid mechanics equations for highly non-equilibrium flows are derived and published. New approaches to correction of mistakes of legacy AI systems is developed. Non-classical applications of non-classical probabilities are developed and published. |

Sectors | Aerospace Defence and Marine Digital/Communication/Information Technologies (including Software) |

URL | https://royalsocietypublishing.org/toc/rsta/2018/376/2118 |

Description | There are three stream of impact of the proposed workshop besides the direct influence on the researchers working with Hilbert's Sixth Problem: 1. Wider impact on the research in related areas of pure and applied mathematics. For this purpose, a focussed discussion about related mathematical problems is organised. A list of scientifically significant open problems will be published. This academic impact is developing, step by step. The special lectures "Derivation of hydrodynamic equations from kinetic theory" for UK-based PhD students and researchers were delivered at the Imperial College, CDT in Fluid Dynamics across Scales: Summer School 11-15 July 2016. The series of publications is under preparation 2. Impact on applications of mathematics in science and engineering. Model reduction in science and engineering (including quantum technologies) are deeply discussed. The possible spectrum of applications may be wider and we expect significant extension to the ones suggested by the workshop participants. 3. Wide cultural and educational impact. We organised an open lecture for teachers and students (in collaboration with IMA) and publication of a series of popular articles in various journals will follow. The first big introductory review paper is published in Contemporary physics. More popular publications are on the way to publications. |

First Year Of Impact | 2016 |

Sector | Education |

Impact Types | Cultural |

Description | New capillarity effects in ideal gases solve an old mathematical mystery |

Form Of Engagement Activity | A press release, press conference or response to a media enquiry/interview |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Media (as a channel to the public) |

Results and Impact | Attention to the problem was attracted worldwide with media reaction. Students and PhD students contacted the organiser with questions. Media asked for further explanations. |

Year(s) Of Engagement Activity | 2016 |

URL | https://www.sciencedaily.com/releases/2016/11/161124100046.htm |

Description | Unsolved after 116 years: International academics to discuss Hilbert's sixth problem |

Form Of Engagement Activity | A press release, press conference or response to a media enquiry/interview |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Media (as a channel to the public) |

Results and Impact | Press-release with media reaction and requests for participation in such events in the future. |

Year(s) Of Engagement Activity | 2016 |

URL | https://www.eurekalert.org/pub_releases/2016-04/uol-ua1041116.php |