Hilbert's Sixth Problem Research Articles

On the Existence of One-Point Time on an Oriented Set

The oriented set notion is the elementary fundamental concept of the theory of changeable sets. In turn, the changeable set theory is closely related to Hilbert's sixth problem. From the formal point of view, any oriented set is a simple relational system with a single reflexive binary relation. Such mathematical structure is the simplest construction, within the framework of which it is possible to give a mathematically strict definition of the time concept. In this regard, the problem of the existence of time with given properties on an oriented set is very interesting. In the present paper, we establish the necessary and sufficient condition for the existence of one-point time on an oriented set. From the intuitive point of view, any one-point time is the time related to the evolution of a system, which consists of a single object (for example, from a single material point). The main result of the paper provides that the one-point time exists on the oriented set if and only if this oriented set is a quasi-chain. Also, using the obtained result, we solve the problem of describing all possible images of linearly ordered sets, which naturally arises in the theory of ordered sets.

Rational extended thermodynamics: a link between kinetic theory and continuum theory

There are three different levels of description for macroscopic physical systems: macroscopic level using thermodynamics and continuum mechanics, mesoscopic level by the kinetic theory, and microscopic level by statistical mechanics of many-particle systems. The search for possible links bridging among these levels is the core part of the Hilbert Sixth Problem. Through a concrete example, we explain the links and also the main idea of Rational Extended Thermodynamics.

Cosmology and Hilbert's sixth problem

There have been tantalizing indications from many quarters of physical cosmology that we are living in the multiverse - a huge set of cosmological domains ("universes"). What is the structure of this larger whole is an entirely open problem on the interface between physics and metaphysics. A goal of the present paper is to draw attention to the connection between this problem and an old and celebrated puzzle in mathematical physics. Among the unresolved problems David Hilbert posed in 1900 as a challenge for the dawning century, none is more philosophically controversial than the Sixth Problem, requiring the axiomatization of physical theories. In the new century and the new millennium, this problem has remained a challenge, usually swept under the rug as "not belonging to mathematics" (as if that impacts its epistemical status) or simply "unresolved". Recent radical ontological/cosmological hypothesis of Max Tegmark, identifying mathematical and physical structures, might shed some new light onto this allegedly antiquated subject: it might be the case that the problem has already been solved, insofar we have formalized mathematical structures! While this can be seen as "cutting the Gordian knot" rather than patiently resolving the issue, we suggest that there are several advantages to taking Tegmark's solution seriously, notably in the domain of (future) physics of the observer.

Hilbert's sixth problem and the failure of the Boltzmann to Euler limit.

This paper addresses the main issue of Hilbert's sixth problem, namely the rigorous passage of solutions to the mesoscopic Boltzmann equation to macroscopic solutions of the Euler equations of compressible gas dynamics. The results of the paper are that (i) in general Hilbert's program will fail because of the appearance of van der Waals-Korteweg capillarity terms in a macroscopic description of motion of a gas, and (ii) the van der Waals-Korteweg theory itself might satisfy Hilbert's quest for a map from the 'atomistic view' to the laws of motion of continua.This article is part of the theme issue 'Hilbert's sixth problem'.

Hilbert's sixth problem: between the foundations of geometry and the axiomatization of physics.

The sixth of Hilbert's famous 1900 list of 23 problems was a programmatic call for the axiomatization of the physical sciences. It was naturally and organically rooted at the core of Hilbert's conception of what axiomatization is all about. In fact, the axiomatic method which he applied at the turn of the twentieth century in his famous work on the foundations of geometry originated in a preoccupation with foundational questions related with empirical science in general. Indeed, far from a purely formal conception, Hilbert counted geometry among the sciences with strong empirical content, closely related to other branches of physics and deserving a treatment similar to that reserved for the latter. In this treatment, the axiomatization project was meant to play, in his view, a crucial role. Curiously, and contrary to a once-prevalent view, from all the problems in the list, the sixth is the only one that continually engaged Hilbet's efforts over a very long period of time, at least between 1894 and 1932.This article is part of the theme issue 'Hilbert's sixth problem'.

Scale matters.

The applicability of Navier-Stokes equations is limited to near-equilibrium flows in which the gradients of density, velocity and energy are small. Here I propose an extension of the Chapman-Enskog approximation in which the velocity probability distribution function (PDF) is averaged in the coordinate phase space as well as the velocity phase space. I derive a PDF that depends on the gradients and represents a first-order generalization of local thermodynamic equilibrium. I then integrate this PDF to derive a hydrodynamic model. I discuss the properties of that model and its relation to the discrete equations of computational fluid dynamics.This article is part of the theme issue 'Hilbert's sixth problem'.

On the emergence of the structure of physics.

We consider Hilbert's problem of the axioms of physics at a qualitative or conceptual level. This is more pressing than ever as we seek to understand how both general relativity and quantum theory could emerge from some deeper theory of quantum gravity, and in this regard I have previously proposed a principle of self-duality or quantum Born reciprocity as a key structure. Here, I outline some of my recent work around the idea of quantum space-time as motivated by this non-standard philosophy, including a new toy model of gravity on a space-time consisting of four points forming a square.This article is part of the theme issue 'Hilbert's sixth problem'.

Aggregation of Markov flows I: theory.

A Markov flow is a stationary measure, with the associated flows and mean first passage times, for a continuous-time regular jump homogeneous semi-Markov process on a discrete state space. Nodes in the state space can be eliminated to produce a smaller Markov flow which is a factor of the original one. Some improvements to the elimination methods of Wales are given. The main contribution of the paper is to present an alternative, namely a method to aggregate groups of nodes to produce a factor. The method can be iterated to make hierarchical aggregation schemes. The potential benefits are efficient computation, including recomputation to take into account local changes, and insights into the macroscopic behaviour.This article is part of the theme issue 'Hilbert's sixth problem'.

The quantum N-body problem in the mean-field and semiclassical regime.

The present work discusses the mean-field limit for the quantum N-body problem in the semiclassical regime. More precisely, we establish a convergence rate for the mean-field limit which is uniform as the ratio of Planck constant to the action of the typical single particle tends to zero. This convergence rate is formulated in terms of a quantum analogue of the quadratic Monge-Kantorovich or Wasserstein distance. This paper is an account of some recent collaboration with C. Mouhot, T. Paul and M. Pulvirenti.This article is part of the themed issue 'Hilbert's sixth problem'.

From axiomatics of quantum probability to modelling geological uncertainty and management of intelligent hydrocarbon reservoirs with the theory of open quantum systems.

As was recently shown by the authors, quantum probability theory can be used for the modelling of the process of decision-making (e.g. probabilistic risk analysis) for macroscopic geophysical structures such as hydrocarbon reservoirs. This approach can be considered as a geophysical realization of Hilbert's programme on axiomatization of statistical models in physics (the famous sixth Hilbert problem). In this conceptual paper, we continue development of this approach to decision-making under uncertainty which is generated by complexity, variability, heterogeneity, anisotropy, as well as the restrictions to accessibility of subsurface structures. The belief state of a geological expert about the potential of exploring a hydrocarbon reservoir is continuously updated by outputs of measurements, and selection of mathematical models and scales of numerical simulation. These outputs can be treated as signals from the information environment E The dynamics of the belief state can be modelled with the aid of the theory of open quantum systems: a quantum state (representing uncertainty in beliefs) is dynamically modified through coupling with E; stabilization to a steady state determines a decision strategy. In this paper, the process of decision-making about hydrocarbon reservoirs (e.g. 'explore or not?'; 'open new well or not?'; 'contaminated by water or not?'; 'double or triple porosity medium?') is modelled by using the Gorini-Kossakowski-Sudarshan-Lindblad equation. In our model, this equation describes the evolution of experts' predictions about a geophysical structure. We proceed with the information approach to quantum theory and the subjective interpretation of quantum probabilities (due to quantum Bayesianism).This article is part of the theme issue 'Hilbert's sixth problem'.

John von Neumann's mathematical “Utopia” in quantum theory

This paper surveys John von Neumann's work on the mathematical foundations of quantum theories in the light of Hilbert's Sixth Problem concerning the geometrical axiomatization of physics. We argue that in von Neumann's view geometry was so tied to logic that he ultimately developed a logical interpretation of quantum probabilities. That motivated his abandonment of Hilbert space in favor of von Neumann algebras, specifically the type II 1 factors, as the proper limit of quantum mechanics in infinite dimensions. Finally, we present the reasons why his axiomatic program remained an “unsolved problem” in mathematical physics. A recent unpublished result by Huzimiro Araki, proving that no algebra with a tracial state defined on it, such as the type II 1 factors, can support any (regular) representation of the canonical commutation relations, is also reviewed and its consequences for von Neumann's projects are discussed.

Problems with the Deductivist Image of Scientific Reasoning

There seem to be some very good reasons for a philosopher of science to be a deductivist about scientific reasoning. Deductivism is apparently connected with a demand for clarity and definiteness in the reconstruction of scientists' reasonings. And some philosophers even think that deductivism is the way around the problem of induction. But the deductivist image is challenged by cases of actual scientific reasoning, in which hard-to-state and thus discursively ill-defined elements of thought nonetheless significantly condition what practitioners accept as cogent argument. And arguably, these problem cases abound. For example, even geometry—for most of its history—was such a problem case, despite its exactness and rigor. It took a tremendous effort on the part of Hilbert and others, to make geometry fit the deductivist image. Looking to the empirical sciences, the problems seem worse. Even the most exact and rigorous of empirical sciences—mechanics—is still the kind of problem case which geometry once was. In order for the deductivist image to fit mechanics, Hilbert's sixth problem (for mechanics) would need to be solved. This is a difficult, and perhaps ultimately impossible task, in which the success so far achieved is very limited. I shall explore some consequences of this for realism as well as for deductivism. Through discussing links between non-monotonicity, skills, meaning, globality in cognition, models, scientific understanding, and the ideal of rational unification, I argue that deductivists can defend their image of scientific reasoning only by trivializing it, and that for the adequate illumination of science, insights from anti-deductivism are needed as much as those which come from deductivism.