The paradox of phase transitions raises the problem of how to reconcile the fact that we see phase transitions happen in concrete, finite systems around us, with the fact that our best theories—i.e. statistical-mechanical theories of phase transitions—tell us that phase transitions occur only in infinite systems. In this paper we aim to clarify to which extent this paradox is relative to the mathematical framework which is used in these theories, i.e. classical mathematics. To this aim, we will explore the (...) philosophical consequences of adopting constructive instead of classical mathematics in a statistical-mechanical theory of phase transitions. It will be shown that constructive mathematics forces certain ‘de-idealizations’ of such theories: talk of actually infinite systems is meaningless, there are no discontinuous functions, and—in a sense which will be clarified—constructive real numbers reflect our imperfect methods of determining the values of physical quantities. As such, so it will be argued, constructive mathematics offers a means to gain insight in the idealizations introduced in classical theories and the philosophical issues surrounding them. (shrink)

Theoria, Volume 87, Issue 4, Page 971-985, August 2021.

We analyse the proposition that the spacetime structure is modified at short distances or at high energies due to weakening of classical logic. The logic assigned to the regions of spacetime is intuitionistic logic of some topoi. Several cases of special topoi are considered. The quantum mechanical effects can be generated by such semi-classical spacetimes. The issues of: background independence and general relativity covariance, field theoretic renormalization of divergent expressions, the existence and definition of path integral measures, are briefly discussed (...) in the proposal. The connection with some problems in foundations of mathematics and differential topology are also discussed. (shrink)

Our aim in this paper is to replace the old concept of truth in mathematics, based on the Set Structure provided with idea of true and false characterized by the presence of a characteric function \, by a mathematical structures founded on the idea of Topos, the triple structure \\}\) and the notion of Gradual Truth or Steps from the truth. Our motivations is to understand the mathematical structures underlying the emergence’s mechanism and phenomena. We think that this approach could (...) be useful to better appreciate the subtleties of the notion of truth and gives us a better understanding of the complex and changing phenomena. (shrink)

We propose an operator constraint equation for the wavefunction of the Universe that admits genuine evolution. While the corresponding classical theory is equivalent to the canonical decomposition of General Relativity, the quantum theory contains an evolution equation distinct from standard Wheeler–DeWitt cosmology. Furthermore, the local symmetry principle—and corresponding observables—of the theory have a direct interpretation in terms of a conventional gauge theory, where the gauge symmetry group is that of spatial conformal diffeomorphisms (that preserve the spatial volume of the Universe). (...) The global evolution is in terms of an arbitrary parameter that serves only as an unobservable label for successive states of the Universe. Our proposal follows unambiguously from a suggestion of York whereby the independently specifiable initial data in the action principle of General Relativity is given by a conformal geometry and the spatial average of the York time on the spacelike hypersurfaces that bound the variation. Remarkably, such a variational principle uniquely selects the form of the constraints of the theory so that we can establish a precise notion of both symmetry and evolution in quantum gravity. (shrink)

In this article we present a notion of “logical perfection”. We first describe through examples a notion of logical perfection extracted from the contemporary logical concept of categoricity. Categoricity (in power) has become in the past half century a main driver of ideas in model theory, both mathematically (stability theory may be regarded as a way of approximating categoricity) and philosophically. In the past two decades, categoricity notions have started to overlap with more classical notions of robustness and smoothness. These (...) have been crucial in various parts of mathematics since the nineteenth century. We postulate and present the category of logical perfection. We draw on various notions of perfection from mathematics of the 19th and 20th centuries and then trace the relation to the concept of categoricity in power as a logical notion of what a “mathematically perfect” structure is. (shrink)

Radical Ontic Structural Realism (ROSR) claims that structure exists independently of objects that may instantiate it. Critics of ROSR contend that this claim is conceptually incoherent, insofar as, (i) it entails there can be relations without relata, and (ii) there is a conceptual dependence between relations and relata. In this essay I suggest that (ii) is motivated by a set-theoretic formulation of structure, and that adopting a category-theoretic formulation may provide ROSR with more support. In particular, I consider how a (...) category-theoretic formulation of structure can be developed that denies (ii), and can be made to do work in the context of formulating theories in physics. Keywords: structural realism, category theory, general relativity.. (shrink)

Motivated by examples from general relativity and Newtonian gravitation, this essay attempts to distinguish between the dynamical structure associated with a theory in physics, and its kinematical structure. This enables a distinction to be made between a structural realist interpretation of a theory based on its dynamical structure, and a structural realist interpretation of spacetime, as described by a theory, based on its kinematical structure. I offer category-theoretic formulations of dynamical and kinematical structure and indicate the extent to which such (...) formulations deflect recent criticism of the radical ontic structural realist's conception of structure as "relations devoid of relata". (shrink)

The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is (...) Heisenberg's `quantum-theoretical Umdeutung (reinterpretation) of classical observables', which lies at the basis of quantization theory. Similarly, Bohr's correspondence principle (in somewhat revised form) and Schroedinger's wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from quantum mechanics. On the other hand, no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail. On the assumption that quantum mechanics is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely in the limit h -> 0 of small Planck's constant (in a finite system), in the limit N goes to infinity of a large system with $N$ degrees of freedom (at fixed h), and through decoherence and consistent histories. The first limit is closely related to modern quantization theory and microlocal analysis, whereas the second involves methods of C*-algebras and the concepts of superselection sectors and macroscopic observables. In these limits, the classical world does not emerge as a sharply defined objective reality, but rather as an approximate appearance relative to certain ``classical" states and observables. Decoherence subsequently clarifies the role of such states, in that they are ``einselected", i.e. robust against coupling to the environment. Furthermore, the nature of classical observables is elucidated by the fact that they typically define (approximately) consistent sets of histories. This combination of ideas and techniques does not quite resolve the measurement problem, but it does make the point that classicality results from the elimination of certain states and observables from quantum theory. Thus the classical world is not created by observation (as Heisenberg once claimed), but rather by the lack of it. (shrink)

Special relativity is no longer a new revolutionary theory but a firmly established cornerstone of modern physics. The teaching of special relativity, however, still follows its presentation as it unfolded historically, trying to convince the audience of this teaching that Newtonian physics is natural but incorrect and special relativity is its paradoxical but correct amendment. I argue in this article in favor of logical instead of historical trend in teaching of relativity and that special relativity is neither paradoxical nor correct (...) (in the absolute sense of the nineteenth century) but the most natural and expected description of the real space-time around us valid for all practical purposes. This last circumstance constitutes a profound mystery of modern physics better known as the cosmological constant problem. (shrink)