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Based on a variety of philosophical approaches and my own work for decades in pure and applied mathematics teaching and research, I explain my view upon the basic difficulties of acquiring the Mathematical Experience (in the sense of P.J.…
Computer science has grown rapidly since its inception in the 1950s and the pioneers in the field are celebrated annually by the A.M. Turing Award. In this paper, we attempt to shed light on the path to influential computer scientists by…
The Jacobian conjecture is thought to have been proposed by O. H. Keller in 1939. However, we have found that the statement of the conjecture is precisely the main result of a paper published by L. Kraus in 1884. Although the final step of…
This paper describes several classical constructions of thin bases of finite order in additive number theory, and, in particular, gives a complete presentation of a beautiful construction of J. W. S. Cassels of a class of polynomially…
Recent developments of affine algebraic geometry, especially the theory of open algebraic surfaces, provide means to systematically explore geometric and topological properties of polynomials in two variables. Nevertheless, there is one…
We re-evaluate the great Leibniz-Newton calculus debate, exactly three hundred years after it culminated, in 1712. We reflect upon the concept of invention, and to what extent there were indeed two independent inventors of this new…
In February 2014 Time magazine announced to the world that the first quantum computer had been put in use. One key component of this computer is the Josephson-junction, a superconducting device, based on completely different scientific and…
This is the transcript of a lecture given at UMass-Lowell in which I compare and contrast the work of Godel and of Turing and my own work on incompleteness. I also discuss randomness in physics vs randomness in pure mathematics.
We discuss a less known aspect of Feynman's multifaceted scientific work, centered about his interest in molecular biology, which came out around 1959 and lasted for several years. After a quick historical reconstruction about the birth of…
A research problem for undergraduates and graduates is being posed as a cap for the prior antecedent regular discrete mathematics exercises. [Here cap is not necessarily CAP=Competitive Access Provider, though nevertheless ...] The object…
The present paper gives an account for the general mathematical reader of the life and work of Martin Davis. Since two rather comprehensive autobiographical accounts and two long biographical interviews already exist, the present work…
Alan Turing is considered as a founder of current computer science together with Kurt Godel, Alonzo Church and John von Neumann. In this paper multiple new research results are presented. It is demonstrated that there would not be Alan…
A theoretical model of computation is proposed based on Lorentz quantum mechanics. Besides the standard qubits, this model has an additional bit, which we call hyperbolic bit (or hybit in short). A set of basic logical gates are constructed…
This note is a preface to various responses [math.HO/9404229,math.HO/9404236] to an opinion piece by Jaffe and Quinn [math.HO/9307227] on the relationship between mathematics and theoretical physics.
Not only did Turing help found one of the most exciting areas of modern science (computer science), but it may be that his contribution to our understanding of our physical reality is greater than we had hitherto supposed. Here I explore…
The total chromatic number conjecture which has appeared in a few hundred articles and in numerous books thus far is now one of the classic mathematical unsolved problems. It appears that many authors coincidentally have attributed it to…
This paper aims to give a brief account of the mathematical work of the 7th-century Armenian polymath and natural philosopher Anania Shirakatsi. The three sections of Anania's ``Book of Arithmetic'' -- tables of arithmetic operations, a…
One of the variants for systematizing the activities of the historian of mathematics is proposed, as well as a scheme for organizing research and search work in the preparation of scientific articles and reports on the history of science.
These are notes on discrete mathematics for computer scientists. The presentation is somewhat unconventional. Indeed I begin with a discussion of the basic rules of mathematical reasoning and of the notion of proof formalized in a natural…
This article is a short review on the concept of information. We show the strong relation between Information Theory and Physics, beginning by the concept of bit and its representation with classical physical systems, and then going to the…