Reductionism is a prevalent viewpoint in science according to which all physical phenomena can be understood from fundamental laws of physics. Anderson [Science, 177, 393 (1972)], Laughlin and Pines [PNAS, 97, 28 (2000)], and others have countered this viewpoint and argued in favour hierarchical structure of the universe and laws. In this paper we advance the latter perspective by showing that some of the complex flow properties derived using hydrodynamic equations (macroscopic laws) are very difficult, if not impossible, to describe in microscopic framework—kinetic theory. These properties include Kolmogorov’s theory of turbulence, turbulence dissipation and diffusion, and dynamic pressure. We also provide several other examples of hierarchical description.

In this paper, from the historical point of view, we present short anecdotes about the development of the object that we well known as complex numbers.

The ontology proposed in this paper is aimed at demonstrating that it is possible to understand the counter-intuitive predictions of quantum mechanics while still retaining much of the framework underlying classical physics, the implication being that it is better to avoid wandering into unnecessarily speculative realms without the support of conclusive evidence. In particular, it is argued that it is possible to interpret quantum mechanics as simply describing an external world consisting of familiar physical entities (e.g., particles or fields) residing in classical 3-dimensional space (not configuration space) with Lorentz covariance maintained.

In this article we present a technique for selecting models of set theory that are complete in a model-theoretic sense. Specifically, we will apply Robinson infinite forcing to the collections of models of ZFC obtained by Cohen forcing. This technique will be used to suggest a unified perspective on generic absoluteness principles.

This paper argues against the proposal to draw from current research into a physical theory of quantum gravity the ontological conclusion that spacetime or spatiotemporal relations are not fundamental. As things stand, the status of this proposal is like the one of all the other claims about radical changes in ontology that were made during the development of quantum mechanics and quantum field theory. However, none of these claims held up to scrutiny as a consequence of the physics once the theory was established and a serious discussion about its ontology had begun.

In this paper we undertake to examine how algebra, its tools and its methods, can be used to formulate the mathematics used in applications. We give particular attention to the mathematics used in application to physics. We suggest that methods first proposed by Henry Siggins Leonard are well suited to such an examination.

Let k be a field of characteristic zero with a fixed embedding σ:k↪C into the field of complex numbers. Given a k-variety X, we use the triangulated category of étale motives with rational coefficients on X to construct an abelian category M(X) of perverse mixed motives. We show that over Spec(k) the category obtained is canonically equivalent to the usual category of Nori motives and that the derived categories Db(M(X)) are equipped with the four operations of Grothendieck (for morphisms of quasi-projective k-varieties) as well as nearby and vanishing cycles functors.

In a quantum universe with a strong arrow of time, it is standard to postulate that the initial wave function started in a particular macrostate–the special low-entropy macrostate selected by the Past Hypothesis. Moreover, there is an additional postulate about statistical mechanical probabilities according to which the initial wave function is a “typical” choice in the macrostate (the Statistical Postulate). Together, they support a probabilistic version of the Second Law of Thermodynamics: typical initial wave functions will increase in entropy. Hence, there are two sources of randomness in such a universe: the quantum-mechanical probabilities of the Born rule and the statistical mechanical probabilities of the Statistical Postulate. I propose a new way to understand time’s arrow in a quantum universe. It is based on what I call the Thermodynamic Theories of Quantum Mechanics.

In the summer of 1918, Emmy Noether published the theorem that now bears her name, establishing a profound two-way connection between symmetries and conservation laws. The influence of this insight is pervasive in physics; it underlies all of our theories of the fundamental interactions and gives meaning to conservation laws that elevates them beyond useful empirical rules.

We extend prior results of Cody-Eskew, showing the consistency of GCH with the statement that for all regular cardinals κ≤λ, where κ is the successor of a regular cardinal, there is a rigid saturated ideal on Pκλ. We also show the consistency of some instances of rigid saturated ideals on Pκλ where κ is the successor of a singular cardinal.