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The Binomial Theorem has long been essential in mathematics. In one form or another it was known to the ancients and, in the hands of Leibniz, Newton, Euler, Galois, and others, it became an essential tool in both algebra and analysis.…
Working in point-free topology under the constraints of geometric logic, we prove the Fundamental Theorem of Calculus, and apply it to prove the usual rules for the derivatives of $x^\alpha$, $\gamma^x$, and $\log_\gamma x$.
There are the longstanding differences in the continuity of continuum among mathematicians. Starting from studies on a mathematical model of contact, we construct a set that is in contact everywhere by using the original idea of Dedekind's…
The assumptions needed to prove Cox's Theorem are discussed and examined. Various sets of assumptions under which a Cox-style theorem can be proved are provided, although all are rather strong and, arguably, not natural.
A thought experiment is described and the probability of a particular type of results is predicted according to the quantum formalism. Then, the assumption is made that there exists a particle that travels from the source to one of the…
Although the suspicion that quantum mechanics is emergent has been lingering for a long time, only now we begin to understand how a bridge between classical and quantum mechanics might be squared with Bell's inequalities and other…
We propose a novel foundation for calculus that focuses on the notion of approximations while avoiding the use of limits altogether. Continuity is defined as approximation at a point, while differentiability is defined as approximation with…
My purpose is to examine some concepts of mathematical logic, which have been studied by Carlo Cellucci. Today the aim of classical mathematical logic is not to guarantee the certainty of mathematics, but I will argue that logic can help us…
This is a brief review of the experimental and theoretical quantum computing. The hopes for eventually building a useful quantum computer rely entirely on the so-called "threshold theorem". In turn, this theorem is based on a number of…
This essay considers the special character of mathematical reasoning, and draws on observations from interactive theorem proving and the history of mathematics to clarify the nature of formal and informal mathematical language. It proposes…
In this note, we discuss how possible expansion histories of the universe can be inferred in a simple way, for arbitrary energy contents. No new physical results are obtained, but the goal is rather to discuss an alternative way of writing…
We discuss the motivation for pursuing research on the foundations of quantum theory.
If the history of science has taught us anything, it's that persistence and creativity makes the once impossible possible. It has long been thought experimental tests of quantum gravity are impossible. But during the last decade, several…
Mathematical conception of infinite quantities forms a cornerstone of many disciplines of modern mathematics --- from differential calculus to set theory. In fact, it could be argued that the most significant revolutions in mathematics in…
A sketch of some of the fundamental notions related to the nature of knowledge is offered, with special focus on the role of mathematics and my own opinions. No single idea exposed here is entirely original; indeed, this topic has been…
We formulate physically-motivated axioms for a physical theory which for systems with a finite number of degrees of freedom uniquely lead to Quantum Mechanics as the only nontrivial consistent theory. Complex numbers and the existence of…
The hypothesis concerning the off-site continuum existence is investigated from the point of view of the mathematical theory of sets. The principles and methods of the mathematical description of the physical objects from different off-site…
The notion of the quantum angle is introduced. The quantum angle turns out to be a metric on the set of physical states of a quantum system. Its kinematics and dynamics is studied. The certainty principle for quantum systems is formulated…
We cast aspects of consciousness in axiomatic mathematical terms, using the graphical calculus of general process theories (a.k.a symmetric monoidal categories and Frobenius algebras therein). This calculus exploits the ontological…
Bayesian probability theory is used as a framework to develop a formalism for the scientific method based on principles of inductive reasoning. The formalism allows for precise definitions of the key concepts in theories of physics and also…