Mathematics and Computer Science: Coping with Finiteness

  title={Mathematics and Computer Science: Coping with Finiteness},
  author={Donald Ervin Knuth},
  pages={1235 - 1242}
  • D. Knuth
  • Published 17 December 1976
  • Philosophy
  • Science
By presenting these examples, I have tried to illustrate four main points. 1) Finite numbers can be really enormous, and the known universe is very small. Therefore the distinction between finite and infinite is not as relevant as the distinction between realistic and unrealistic. 2) In many cases there are subtle ways to solve very large problems quickly, in spite of the fact that they appear at first to require examination of too many possibilities. 3) There are also cases where we can prove… 
Unsolved Problems in Computational Science III: A Special Self-Avoiding Walk
  • Shanzhen Gao, Keh-Hsun Chen
  • Mathematics, Computer Science
    2016 International Conference on Computational Science and Computational Intelligence (CSCI)
  • 2016
This paper will present prudent walks, a sequence of moves on a lattice not visiting the same point more than once, which differ from most sub-classes of SAWs that have been counted so far in that they can wind around their starting point.
Introduction to Projective Arithmetics
Science and mathematics help people to better understand world, eliminating many inconsistencies, fallacies and misconceptions. One of such misconceptions is related to arithmetic of natural numbers,
Reflections on the role of statements on statements in mathematics
Mathematicians reason about abstract objects, or at least they seem to do so. In the course of doing so, they routinely make statements on mathematical statements. In fact, every inference, such as
Beyond Knuth's Notation for Unimaginable Numbers within Computational Number Theory
This topic is developed by determining in some cases an explicit recursive algorithm for the number of steps required to reach zero, as well as an effective bound for it using Knuth’s notation.
Experiencing Mathematics: What Do We Do, When We Do Mathematics?
Most mathematicians, when asked about the nature and meaning of mathematics, vacillate between the two unrealistic poles of Platonism and formalism. By looking carefully at what mathematicians really
Diophantine and Non-Diophantine Arithmetics: Operations with Numbers in Science and Everyday Life
Science and mathematics help people better to understand world, eliminating different fallacies and misconceptions. One of such misconception is related to arithmetic, which is so important both for
On the Arithmetic of Knuth's Powers and Some Computational Results About Their Density
The object of the paper is the so-called “unimaginable numbers”, and some arithmetic and computational aspects of the Knuth’s powers notation are dealt with and some first steps into the investigation of their density are moved into.
The notion of "Unimaginable Numbers" in computational number theory
This work gives an axiomatic set for this topic, and tries to find on one hand other ways to represent unimaginable numbers, as well as applications to computer science, where the algorithmic nature of representations and the increased computation capabilities of computers give the perfect field to develop further the topic.
New Approaches to Basic Calculus: An Experimentation via Numerical Computation
The experimental project aims to explore the teaching potential offered by non-classical approaches to calculus jointly with the so-called “unimaginable numbers” and employed the computational method recently proposed by Y.D. Sergeyev.
Proof identity for mere mortals
A new perspective on the proof identity problem is outlined, which employs the concepts and tools of automated theorem proving and complements the rather more theoretical perspectives coming from pure proof theory.


A note on two problems in connexion with graphs
  • E. Dijkstra
  • Mathematics, Computer Science
    Numerische Mathematik
  • 1959
A tree is a graph with one and only one path between every two nodes, where at least one path exists between any two nodes and the length of each branch is given.
A method of factoring and the factorization of
The continued fraction method for factoring integers, which was introduced by D. H. Lehmer and R. E. Powers, is discussed along with its computer implementation. The power of the method is
The complexity of decision problems in automata theory and logic.
This paper is intended to demonstrate the efforts of the Massachusetts Institute of Technology’s graduate students to improve the quality of their teaching and research.
The 11 possibilities for bar 8 are all identical, and Mozart gave only two distinct possibilities for bar 16, so the total number of waltzes is 2 x 1114 rather than 1116
    A determinant formula that specifies the number of spanning trees in a particular graph was discovered by
      See also A
      • The Design and Analysis ofComputer Algorithms
      • 1972
      Musikalisches Wdirfelspiel (Schott, Mainz, 1956), K 516 f Anh
      • C 30.01; this was first published in
      • 1793