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Even though Plato's philosophy in ancient times was always closely associated with mathematics, modern Platonic scholarship, during the last five centuries, has moved steadily toward de-mathematization. The present work aims to outline a…
This paper gives a counterexample to the impossibility, by G\"odel's second incompleteness theorem, of proving a formula expressing the consistency of arithmetic in a fragment of arithmetic on the assumption that the latter is consistent.…
This article critically reappraises arguments in support of Cantor's theory of transfinite numbers. The following results are reported: i) Cantor's proofs of nondenumerability are refuted by analyzing the logical inconsistencies in…
We investigate some examples of quantum Zeno dynamics, when a system undergoes very frequent (projective) measurements that ascertain whether it is within a given spatial region. In agreement with previously obtained results, the evolution…
Prevention of a quantum system's time evolution by repetitive, frequent measurements of the system's state has been called the quantum Zeno effect (or paradox). Here we investigate theoretically and numerically the effect of repeated…
The main aim of this article is to defend the thesis that Plato apprehended the structure of incommensurable magnitudes in a way that these magnitudes correspond in a unique and well defined manner to the modern concept of the "Dedekind…
We investigate the structure common to causal theories that attempt to explain a (part of) the world. Causality implies conservation of identity, itself a far from simple notion. It imposes strong demands on the universalizing power of the…
A geometric illustration of the Lorentz transformations is given. According to similarity between space and time and correspondence between a ruler and a clock, like the division number in a moving ruler, the tick number of a moving clock…
The problem of absolute motion in the context of the twin paradox is discussed. It is shown that the various versions of the clock paradox feature some aspects which Mach might have been appreciated. However, the ultimate cause of the…
Pythagoras' theorem lies at the heart of physics as well as mathematics, yet its historical origins are obscure. We highlight a purely pictorial, gestalt-like proof that may have originated during the Zhou Dynasty. Generalizations of the…
Set theoretical paradoxes have a common root -- lack of understanding of why some multitudes are not sets. Why some multitudes of objects of thought cannot themselves be objects of thought? Moreover, it is a logical truth that such…
Some recent philosophical debate about persistence has focussed on an argument against perdurantism that discusses rotating perfectly homogeneous discs (the `rotating discs argument'; RDA). The argument has been mostly discussed by…
In two articles ([Brisson-Ofman1, 2]), we have analyzed the so-called 'mathematical passage' of Plato's Theaetetus, the first dialogue of a trilogy including the Sophist and the Statesman. In the present article, we study an important point…
In this paper, we present a paradox arising from the acceptance of the Law of Excluded Middle (LEM) within classical mathematics. Specifically, we construct a nonzero analytic function on a connected open subset of the complex plane whose…
This article describes a Turing machine which can solve for $\beta^{'}$ which is RE-complete. RE-complete problems are proven to be undecidable by Turing's accepted proof on the Entscheidungsproblem. Thus, constructing a machine which…
The overarching theme of the following pages is that mathematical logic -- centered around the incompleteness theorems -- is first and foremost an investigation of $\textit{computation}$, not arithmetic. Guided by this intuition we will…
Using a graph representation of classical logic, the paper shows that the liar or Yablo pattern occurs in every semantic paradox. The core graph theoretic result generalizes theorem of Richardson, showing solvability of finite graphs…
It is shown that the ``retrodiction paradox'' recently introduced by Peres arises not because of the fallacy of the time-symmetric approach as he claimed, but due to an inappropriate usage of retrodiction.
We motivate and construct a mathematical theory for the separation of space and time in general relativity. The formalism only requires a single observer and an optional choice of reference frame at each instant. As the splitting is done…
A universal schema for diagonalization was popularized by N. S. Yanofsky (2003) in which the existence of a (diagonolized-out and contradictory) object implies the existence of a fixed-point for a certain function. It was shown that many…