Effective Computations and Foundations of Science

It is clear that by the mid-1980s, the philosophical community was increasingly aware and appreciative of the importance of the topics investigated by Computational/ Informational Sciences, and of the value of its methodologies and theories. Concepts or processes like algorithm, automatic control, complexity, computation, distributed network, dynamic system,

The Oxford Handbook of Philosophy of Science, edited by Paul Humphreys, 2016

This chapter deals with those fields that study computing systems. Among these computational sciences are computer science, computational cognitive science, computational neuroscience, and artificial intelligence. In the first part of the chapter, it is shown that there are varieties of computation, such as human computation, algorithmic machine computation, and physical computation. There are even varieties of versions of the Church-Turing thesis. The conclusion is that different computational sciences are often about different kinds of computation. The second part of the chapter discusses three specific philosophical issues. One is whether computers are natural kinds. Another issue is the nature of computational theories and explanations. The last section of the chapter relates remarkable results in computational complexity theory to problems of verification and confirmation.

Proceedings of the 1993 ACM conference on Computer science - CSC '93, 1993

Computer science inherited its present conceptual foundations from a branch of pure mathematics that, historically, had been exploring the fundamental nature of mathematical computation since before the turn of the century. It is argued that the conceptual concerns of computer science are different from the conceptual concerns of mathematics, and that this mathematical legacy, in particular the notion of the algorithm, has been largely ineffective as a paradigm for computer science. It is fi rst necessary to understand the role of the algorithm in mathematics. The notion of the algorithm is fundamental to mathematics. To understand the signifi cance of the algorithm to mathematics, it is necessary to understand the history of its development. The term derives from the name of an important ninth-century Persian mathematician, Mohammed ibn Musa al-Khowarizmi, who in about ad 825 wrote a small book describing how to calculate with a new ten-symbol, positional value number system developed in India. It described simple procedures for carrying out addition, subtraction, © ACM, 1993. This chapter is a minor revision of my work: K. Fant, "A Critical review of the notion of the algorithm in computer science,"

International Journal for Research in Applied Science & Engineering Technology (IJRASET), 2023

Mathematics (The QUEEN's mother of all Sciences), is the foundation of Computer Science. Mathematics can be perceived in our garden or park from symmetry of leaves, flowers, fruits etc. and by so many examples of Geometry and symmetry can be seen in nature. Scientists and researchers cannot ideally accomplish their work without the inclusion of mathematics. Mathematics is sociable for analytical skills needed in Computer Disciplines like; Concepts of binary number system, Boolean algebra, Calculus, Discrete mathematics, linear algebra, number theory, and graph theory are the most applicable to the subject of computer science with the accessional emergence of new concepts like machine learning, artificial intelligence, virtual reality and augmented reality make the future of mathematics grow endless. Mathematics has been an important intellectual preoccupation of man for a long time. Computer Science as a formal discipline is about seven decades young. Is the almost spontaneous use of computing? In this article, this paper convey to the frontage the many close connections and parallels between the Mother and daughter sciences. The paper underscores the strong interplay and interactions by looking at some exciting contemporary results from number theory and combinatorial mathematics and algorithms of computer science.

Academia Letters, 2021

Effective computation refers to a calculation method that demands no insight, intuition, or ingenuity for producing a result. This is, a step-by-step recipe executable by an automaton, or, in a broad sense, an algorithm. But since algorithm is nowadays so tightly tied with electronic computing, effective computation fits better with the content of this note. The search for effectiveness reached its peak at the beginning of the twentieth century when faced with contradictions, inconsistencies, and paradoxes, some of the finest mathematicians of that time focused their attention on the foundations of the discipline. The three volumes of Principia Mathematica written by Bertrand Russel and Alfred N. Whitehead between 1910 and 1913, was, according to Russell, an attempt to "show that all pure mathematics follows from purely logical premises and uses only concepts definable in logical terms." For Russel and Whitehead consistency was a consequence of the selection of the right logic postulates. David Hilbert went a step further. In 1921 he presented the so-called Hilbert's program, which aimed at embedding all mathematics in a formal axiom-system endowed with a precise language and manipulation rules. That system should also exhibit the next three fundamental properties. a) Consistency, that is no contradiction would arise within the formalism. b) Completeness, meaning that all true mathematical statements could be proved within the formalism. c) Decidability, meaning that the truth value of a statement could be decided with an effective method. Property c) implied that syntax not meaning was the crucial thing in a mathematical proof. For good or ill, Kurt Gödel proved in 1931, in two landmark theorems, that Hilbert's program