Algorithms

Ada Lovelace's diagram from "note G", the first published computer algorithm.

Algorithms, in mathematics and computer science, are an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning. In simple words an algorithm is a step-by-step procedure for calculations.

Quotes

• Algorithms are the computational content of proofs.
  • Robert Harper, Benjamin C. Pierce et al. Software Foundations

• Mathematics is what we want to keep for ourselves. When playing games, we stick to the rules (or we are changing the game...), but when doing serious mathematics (not executing algorithms) we make up the rules—definitions, axioms... even logics. ...[I]n arithmetic we find prime numbers, which are a whole new 'game'... [T]o identify mathematics with games would be one of those part-for-whole mistakes (like 'all geometry is projective geometry' or 'arithmetic is just logic' from the nineteenth century)... [M]y separation of game analysis from playing games tells... against the analogy of mathematics to the expert play of the game itself.
  • Robert Spencer David Thomas, "Mathematics is Not a Game But..." (January, 2009) The Mathematical Intelligencer Vol. 31, No. 1, pp. 4-8. Also published in The Best Writing on Mathematics 2010 (2011) pp. 79-88.

• Numerical analysis is very much an experimental science.
  • Peter Wynn. "On some recent developments in the theory and application of continued fractions". Journal of the Society for Industrial and Applied Mathematics: Series B, Numerical Analysis Volume 1, pages 177–197.