Representing Numbers

Authors

  • Michał Wrocławski Institute of Philosophy, University of Warsaw

DOI:

https://doi.org/10.14394/filnau.2018.0024

Keywords:

representations of numbers, numerals, computable functions, characteristic functions, identity

Abstract

The purpose of this paper is to consider the question of how we can represent numbers (especially natural numbers) and how our choice of a representation affects our ability to compute various functions. In particular, we show the importance of computability of the characteristic function of identity in a representation of numbers. It turns out that it is a very strong assumption that significantly increases the scope of our knowledge about a given representation, including our ability
to tell which functions are computable in this representation.

References

Church A. (1938), "The Constructive Second Number Class," Bulletin of the American Mathematical Society 44, 224-232. https://doi.org/10.1090/S0002-9904-1938-06720-1
Kleene S. (1938), "On Notation for Ordinal Numbers," The Journal of Symbolic Logic 3, 150-155. https://doi.org/10.2307/2267778
Rogers H. Jr. (1967), Theory of Recursive Functions and Effective Computability, Cambridge, MA: MIT.
Shapiro S. (1982), "Acceptable Notation," Notre Dame Journal of Formal Logic 23(1), 14-20. https://doi.org/10.1305/ndjfl/1093883561
Zdanowski K. (2012), On Notation Systems for Natural Numbers and Polynomial Time Computations, Numbers and Truth, Gothenburg (unpublished slides from a conference).

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Published

2018-12-31

How to Cite

Wrocławski, M. (2018). Representing Numbers. The Philosophy of Science, 26(4), 57–73. https://doi.org/10.14394/filnau.2018.0024

Issue

Vol. 26 No. 4 (2018): THE PHILOSOPHY OF SCIENCE

Section

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