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A very short history of ultrafinitism

Published online by Cambridge University Press:  07 October 2011

By
Rose M. Cherubin  and
Mirco A. Mannucci
Edited by
Juliette Kennedy  and
Roman Kossak
Juliette Kennedy
Affiliation:
University of Helsinki
Roman Kossak
Affiliation:
City University of New York
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Summary

To the memory of our unforgettable friend Stanley Tennenbaum (1927-2005), Mathematician, Educator, Free Spirit.

In this first of a series of papers on ultrafinitistic themes, we offer a short history and a conceptual pre-history of ultrafinistism. While the ancient Greeks did not have a theory of the ultrafinite, they did have two words, murios and apeiron, that express an awareness of crucial and often underemphasized features of the ultrafinite, viz. feasibility, and transcendence of limits within a context. We trace the flowering of these insights in the work of Van Dantzig, Parikh, Nelson and others, concluding with a summary of requirements which we think a satisfactory general theory of the ultrafinite should satisfy.

First papers often tend to take on the character of manifestos, road maps, or both, and this one is no exception. It is the revised version of an invited conference talk, and was aimed at a general audience of philosophers, logicians, computer scientists, and mathematicians. It is therefore not meant to be a detailed investigation. Rather, some proposals are advanced, and questions raised, which will be explored in subsequent works of the series.

Our chief hope is that readers will find the overall flavor somewhat “Tennenbaumian”.

§1. Introduction: The radical Wing of constructivism. In their Constructivism in Mathematics, A. Troelstra and D. Van Dalen dedicate only a small section to Ultrafinitism (UF in the following). This is no accident: as they themselves explain therein, there is no consistent model theory for ultrafinitistic mathematics.

Type
Chapter
Information
Set Theory, Arithmetic, and Foundations of Mathematics
Theorems, Philosophies
 , pp. 180 - 199
Publisher: Cambridge University Press
Print publication year: 2011

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