Related papers: The Wild Number Problem: math or fiction?
The article attempts to demonstrate the rich history of one truly remarkable problem situated at the confluence of probability theory and theory of numbers - finding the probability of co-primality of two randomly selected natural numbers.…
Do scientific theories limit human knowledge? In other words, are there physical variables hidden by essence forever? We argue for negative answers and illustrate our point on chaotic classical dynamical systems. We emphasize parallels with…
What is the largest number accessible to the human imagination? The question is neither entirely mathematical nor entirely philosophical. Mathematical formulations of the problem fall into two classes: those that fail to fully capture the…
In this paper we obtain bounds for integer solutions of quadratic polynomials in two variables that represent a natural number. Also we get some results on twin prime numbers. In addition, we use linear functionals to prove some results of…
Since the mathematicians of ancient Greece until Fermat, since Gauss until today; the way how the primes along the numerical straight line are distributed has become perhaps the most difficult math problem; many people believe that their…
I argue for a full mathematisation of the physical theory, including its axioms, which must contain no physical primitives. In provocative words: "physics from no physics". Although this may seem an oxymoron, it is the royal road to keep…
Prime numbers are fascinating by the way they appear in the set of natural numbers. Despite several results enlighting us about their repartition, the set of prime numbers is often informally qualified as misterious. In the present paper,…
The recently developed proof of Fermat's Last Theorem is very lengthy and difficult, so much so as to be beyond all but a small body of specialists. While certainly of value in the developments that resulted, that proof could not be, nor…
I discuss some problems related to extreme mathematical realism, focusing on a recently proposed "shut-up-and-calculate" approach to physics (arXiv:0704.0646, arXiv:0709.4024). I offer arguments for a moderate alternative, the essence of…
We usually construct mathematical objects that are accessible, on which we can put our hands, but a huge part of the mathematical existing is actually wild. Here we explore part of the wild world: its inhabitants are knots that are…
I'll discuss how Goedel's paradox "This statement is false/unprovable" yields his famous result on the limits of axiomatic reasoning. I'll contrast that with my work, which is based on the paradox of "The first uninteresting positive whole…
This article discusses epistemological problems in the philosophy of mathematics and issues concerning the reliability of the mathematical literature.
This paper argues for the status of formal probability theory as a mathematical, rather than a scientific, theory. David Freedman and Philip Stark's concept of model based probabilities is examined and is used as a bridge between the formal…
The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the…
In considering the reliability of numerical programs, it is normal to "limit our study to the semantics dealing with numerical precision" (Martel, 2005). On the other hand, there is a great deal of work on the reliability of programs that…
Fermat's Last Theorem is proved by using the philosophical and mathematical knowledge of 1637 when the French mathematician Pierre de Fermat claimed to have a truly marvelous proof of his conjecture. Our approach consists of setting three…
In our work we give the examples using Fermat's Last Theorem for solving some problems from algebra and number theory.
While complex numbers are essential in mathematics, they are not needed to describe physical experiments, expressed in terms of probabilities, hence real numbers. Physics however aims to explain, rather than describe, experiments through…
It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can't contain more…
An alternative form of Fermats equation[1] is proposed. It represents a portion of the identity that includes three terms of Fermats original equation. This alternative form permits an elementary and compact proof of the first case of…