# Mathematics and Computer Science: Coping with Finiteness

@article{Knuth1976MathematicsAC, title={Mathematics and Computer Science: Coping with Finiteness}, author={Donald Ervin Knuth}, journal={Science}, year={1976}, volume={194}, pages={1235 - 1242} }

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…

## 144 Citations

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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.

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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.

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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.

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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.

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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.

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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