Bertrand’s paradox is a famous problem of probability theory, pointing to a possible inconsistency in Laplace’s principle of insufficient reason. In this article we show that Bertrand’s paradox contains two different problems: an «easy» problem and a «hard» problem. The easy problem can be solved by formulating Bertrand’s question in sufficiently precise terms, so allowing for a non ambiguous modelization of the entity subjected to the randomization. We then show that once the easy problem is settled, also the hard problem becomes solvable, provided Laplace’s principle of insufficient reason is applied not to the outcomes of the experiment, but to the different possible «ways of selecting» an interaction between the entity under investigation and that producing the randomization.

Just as there are frictional losses associated with moving masses on a surface, what if there were frictional losses associated with moving information on a substrate? Indeed, many methods of communication suffer from such frictional losses. We propose to model these losses as proportional to «bit-meters,» i.e., the product of mass of information (i.e., the number of bits) and the distance of information transport. We use this «information-friction» model to understand fundamental energy requirements on encoding and decoding in communication circuitry. First, for communication across a binary input AWGN channel, we arrive at limits on bit-meters (and thus energy consumption) for decoding implementations that have a predetermined input-independent lengths of messages.

Starting from a short review of the «classical» space problem in the sense of the 19th century (Helmholtz — Lie — Klein) it is discussed how the challenges posed by special and general relativity to the classical analysis were taken up by Hermann Weyl and Elie Cartan. Both mathematicians reconsidered the space problem from the point of view of transformations operating in the infinitesimal neighbourhoods of a manifold (spacetime). In a short outlook we survey further developments in mathematics and physics of the second half of the 20th century, in which core ideas of Weyl’s and/or Cartan’s analysis of the space problem were further investigated (mathematics) or incorporated into basic theories (physics).

Aperiodic tilings are non-periodic tilings defined by local rules. They are widely used to model quasicrystals, and a central question is to understand which of the non-periodic tilings are actually aperiodic. Among tilings, those by rhombi can be easily seen as approximations of surfaces in higher dimensional spaces. In particular, those which approximate irrational planes are non-periodic. But which ones are also aperiodic? This paper introduces the notion of subperiod, which links algebraic properties of a plane with geometric properties of the tilings that approximate it. A necessary and sufficient condition is obtained for tilings that can be seen in the four dimensional Euclidean space. This result is then applied to some examples in higher codimensions, notably tilings with n-fold rotational symmetry

It has recently been shown within a formal axiomatic framework using a definition of four-momentum based on the St\»uckelberg-Feynman-Sudarshan «switching principle» that Einstein’s relativistic dynamics is logically consistent with the existence of interacting faster-than-light inertial particles. Our results here show, using only basic natural assumptions on dynamics, that this definition is the only possible way to get a consistent theory of such particles moving within the geometry of Minkowskian spacetime.
We present a strictly formal proof from a streamlined axiom system that given any slow or fast inertial particle, all inertial observers agree on the value of (m . sqrt{|1-v^2|}), where m is the particle’s relativistic mass (or energy) and v its speed. This confirms formally the widely held belief that the relativistic mass and momentum of a positive-mass particle must decrease as its speed increases.

In this essay I develop quantum contextuality as a potential candidate for Wheeler’s universal regulating principle, arguing — \textit{contrary} to Wheeler — that this ultimately implies that `bit’ comes from `it.’ In the process I develop a formal definition of physical determinism in the languages of domain theory and category theory.

We seek to elucidate the philosophical context in which one of the most important conceptual transformations of modern mathematics took place, namely the so-called revolution in rigor in infinitesimal calculus and mathematical analysis. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind, and Weierstrass. The dominant current of philosophy in Germany at the time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our main thesis is that Marburg neo-Kantian philosophy formulated a sophisticated position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the «great triumvirate» of Cantor, Dedekind, and Weierstrass that declared infinitesimals conceptus nongrati in mathematical discourse. Rather, following Cohen’s lead, the Marburg philosophers sought to clarify Leibniz’s principle of continuity, and to exploit it in making sense of infinitesimals and related concepts.

We introduce a general and systematic theoretical framework for Operational Dynamic Modeling (ODM) by combining a kinematic description of a model with the evolution of the dynamical average values. The kinematics includes the algebra of the observables and their defined averages. The evolution of the average values is drawn in the form of Ehrenfest-like theorems. We show that ODM is capable of encompassing wide ranging dynamics from classical non-relativistic mechanics to quantum field theory. The generality of ODM should provide a basis for formulating novel theories.

In 1926, Dirac stated that quantum mechanics can be obtained from classical theory through a change in the only rule. In his view, classical mechanics is formulated through commutative quantities (c-numbers) while quantum mechanics requires noncommutative one (q-numbers). The rest of theory can be unchanged. In this paper we critically review Dirac’s proposition. We provide a natural formulation of classical mechanics through noncommutative quantities with a non-zero Planck constant. This is done with the help of the nilpotent unit, which squares to zero. Thus, the crucial r\^ole in quantum theory shall be attributed to the usage of complex numbers.

A self-contained introduction is presented of the notion of the (abstract) differentiable manifold and its tangent vector fields. The way in which elementary topological ideas stimulated the passage from Euclidean (vector) spaces and linear maps to abstract spaces (manifolds) and diffeomorphisms is emphasized. Necessary topological ideas are introduced at the beginning in order to keep the text as self-contained as possible. Connectedness is presupposed in the definition of the manifold. Definitions and statements are laid rigorously, lots of examples and figures are scattered to develop the intuitive understanding and exercises of various degree of difficulty are given in order to stimulate the pedagogical character of the manuscript. The text can be used for self-study or as part of the lecture notes of an advanced undergraduate or beginning graduate course, for students of mathematics, physics or engineering.