A very short history of ultrafinitism

Rose M. Cherubin; Mirco A. Mannucci

doi:10.1017/CBO9780511910616.010

A very short history of ultrafinitism

Published online by Cambridge University Press: 07 October 2011

Rose M. Cherubin and

Mirco A. Mannucci

Edited by

Juliette Kennedy and

Roman Kossak

Show author details

Juliette Kennedy: Affiliation:
University of Helsinki
Roman Kossak: Affiliation:
City University of New York

Book contents

Get access

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

DOI: https://doi.org/10.1017/CBO9780511910616.010 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] P. V., Andreev and E. I., Gordon, A theory of hyperfinite sets, Annals of Pure and Applied Logic, vol. 143 (2006), no. 1-3, pp. 3–19.Google Scholar

[2] Archimedes, , The Sand Reckoner, available at http://www.math.uwaterloo.ca/navigation/ideas/reckoner.shtml.

[3] Aristotle, , Physica, edited by W.D., Ross. Oxford Classical Texts, Oxford University Press, Oxford, 1950.Google Scholar

[4] Scott, Aronson, Who Can Name the Bigger Number?, available at http://www.scottaaronson.com/writings/bignumbers.html.

[5] Jon M., Beck, Simplicial sets and the foundations of analysis, Applications of Sheaves (Proc. Res. Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal., Univ. Durham, Durham, 1977), Lecture Notes in Mathematics, vol. 753, Springer, Berlin, 1979, pp. 113–124.Google Scholar

[6] Jon M., Beck, On the relationship between algebra and analysis, Journal of Pure and Applied Algebra, vol. 19 (1980), pp. 43–60.Google Scholar

[7] Emil, Borel, Les Nombres Inaccesibles, avec une note de M. Daniel Dugué, Gauthier-Villars, Paris, 1952.Google Scholar

[8] A., Carbone and S., Semmes, Looking from the inside and from the outside, Synthese. An International Journal for Epistemology, Methodology and Philosophy of Science, vol. 125 (2000), no. 3, pp. 385–416.Google Scholar

[9] Rose, Cherubin, Inquiry and what is: Eleatics and monisms, Epoché, vol. 8 (2003), no. 1, pp. 1–26.Google Scholar

[10] Rose, Cherubin, Why matter? aristotle, the eleatics, and the possibility of explanation, Graduate Faculty Philosophy Journal, vol. 26 (2005), no. 2, pp. 1–29.Google Scholar

[11] Van, Dantzig, Is 101010 a natural number, Dialectica, vol. 9 (1956), pp. 273–277.Google Scholar

[12] Jean-Yves, Girard, Les Fondements des Mathématiques, 2003, available at http://iml.univ-mrs.fr/∼girard/Articles.html.

[13] David, Isles, Remarks on the notion of standard nonisomorphic natural number series, Constructive Mathematics (Las Cruces, N.M., 1980) (F., Richman, editor), Lecture Notes in Mathematics, vol. 873, Springer, Berlin, 1981, pp. 111–134.Google Scholar

[14] Charles H., Kahn, Anaximander and the Origins of Greek Cosmology, Columbia University Press, Columbia, 1960, corrected edition, Centrum Press, 1985; reprint Hackett, 1994.Google Scholar

[15] Shaughan, Lavine, Understanding the Infinite, second ed., Harvard University Press, Cambridge, MA, 1998.Google Scholar

[16] Mirco A., Mannucci, Feasible and Utterable Numbers, FOM posting, at http://cs.nyu.edu/pipermail/fom/2006-July/010657.html.

[17] Mirco A., Mannucci, Feasible Consistency-Truth Transfer Policy, 2003, FOM posting, at http://www.cs.nyu.edu/pipermail/fom/2006-October/011016.html.

[18] Per, Martin-Löf, On the meanings of the logical constants and the justifications of the logical laws, Nordic Journal of Philosophical Logic, vol. 1 (1996), no. 1, pp. 11–60 (electronic), Lecture notes to a short course at Università degli Studi di Siena, April 1983.Google Scholar

[19] Richard D., McKirahan Jr., Philosophy Before Socrates, Hackett, Indianapolis, IN, 1994.Google Scholar

[20] Jan, Mycielski, Analysis without actual infinity, The Journal of Symbolic Logic, vol. 46 (1981), no. 3, pp. 625–633.Google Scholar

[21] Gerard, Naddaf, The Greek Concept of Nature, Suny Series in Greek Philosophy, SUNY Press, Albany, NY, 2007.Google Scholar

[22] Ed, Nelson, Faith and Mathematics, available at http://www.math.princeton.edu/∼nelson/papers/s.pdf.

[23] Ed, Nelson, Predicative Arithmetics, available at http://www.math.princeton.edu/∼nelson/books/pa.pdf.

[24] Rohit, Parikh, Existence and feasibility in arithmetic, The Journal of Symbolic Logic, vol. 36 (1971), no. 3, pp. 494–508.Google Scholar

[25] Vladimir, Sazonov, On feasible numbers, Logic and Computational Complexity (Indianapolis, IN, 1994), Lecture Notes in Computer Science, vol. 960, Springer, Berlin, 1995, pp. 30–51.Google Scholar

[26] Simplicius, , In Aristotelis Physicorum. Commentaria in Aristotelem Graeca IX, edited by Hermann, Diels, Reimer, Berlin, 1882.Google Scholar

[27] Anne S., Troelstra and Dirk, van Dalen, Constructivism in Mathematics. Vols. I & II, Studies in Logic and the Foundations of Mathematics, vol. 123, North-Holland, Amsterdam, 1988.Google Scholar

[28] Jouko, Väänänen, A Short Course on Finite Model Theory, available at http://www.math.helsinki.fi/logic/people/jouko.vaananen/jvaaftp.html.

[29] Jouko, Väänänen, Pseudo-finite model theory, Matemática Contemporânea, vol. 24 (2003), pp. 169–183, 8th Workshop on Logic, Language, Informations and Computation—WoLLIC'2001 (Brasília).Google Scholar

[30] Jean Paul, Van Bendegem, Classical arithmetic is quite unnatural, Logic and Logical Philosophy, (2003), no. 11-12, pp. 231–249, Flemish-Polish Workshops II–IV and varia.Google Scholar

[31] Petr, Vopênka, Mathematics in the Alternative Set Theory, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1979, Teubner-Texte zur Mathematik. [Teubner Texts in Mathematics], With German, French and Russian summaries.Google Scholar

[32] A. S., Yessenin-Volpin, The ultra-intuitionistic criticism and the antitraditional program for foundations of mathematics, Intuitionism and Proof Theory (Proc. Conf., Buffalo, NY, 1968), North-Holland, Amsterdam, 1970, pp. 3–45.Google Scholar

Book contents

A very short history of ultrafinitism

Summary

Access options

References

Save book to Kindle

Save book to Dropbox

Save book to Google Drive