Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- Acknowledgments
- Introduction
- I Perspectives on Infinity from History
- II Perspectives on Infinity from Mathematics
- 2 The Mathematical Infinity
- 3 Warning Signs of a Possible Collapse of Contemporary Mathematics
- III Technical Perspectives on Infinity from Advanced Mathematics
- IV Perspectives on Infinity from Physics and Cosmology
- V Perspectives on Infinity from Philosophy and Theology
- Index
- References
2 - The Mathematical Infinity
Published online by Cambridge University Press: 07 June 2011
Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- Acknowledgments
- Introduction
- I Perspectives on Infinity from History
- II Perspectives on Infinity from Mathematics
- 2 The Mathematical Infinity
- 3 Warning Signs of a Possible Collapse of Contemporary Mathematics
- III Technical Perspectives on Infinity from Advanced Mathematics
- IV Perspectives on Infinity from Physics and Cosmology
- V Perspectives on Infinity from Philosophy and Theology
- Index
- References
Summary
Early History
In my youth I read a popular book, written by the famous physicist Gamow (1947), aimed at guiding the reader to a glimpse of modern science, with special emphasis on the microcosm of the atom, the macrocosm of the galaxies all the way back to the Big Bang, and Einstein's theory of relativity along the way. It was a fascinating book indeed. The title was One, Two, Three…Infinity, in reference to how counting may have started in primitive tribes as “One, two, three…many.”
We may smile, thinking proudly of how far ahead we have gone in our understanding of counting, but in a certain sense we have not made much progress beyond this. Studies have shown that the average person, when shown a multitude of objects, is not able to recollect more than seven of them with accuracy, if not even less. So our inner way of counting may still be today, “One, two, three,…seven…many.” On the other hand, there are ways of understanding the very large and the incredibly small, and mathematics provides the tools to do so.
What is infinity? Is it the inaccessible, the uncountable, the unmeasurable? Or should we consider infinity as the ultimate, complete, perfect entity? Can mathematics, the science of measuring, deal with infinity? Is infinity a number, or can it be treated as such?
The concept of infinity plays a fundamental, positive role in today's mathematics, but it was not always so positive in antiquity.
Information
- Type
- Chapter
- Information
- InfinityNew Research Frontiers, pp. 55 - 75Publisher: Cambridge University PressPrint publication year: 2011
References
, and 1965. Diophantine problems over local fields. I. American Journal of Mathematics 87: 605–30.CrossRefGoogle Scholar
, and 2004. Variazioni sul problema dei buoi di Archimede, ovvero, alla ricerca di soluzioni “possibili”…http://www.cartesio-episteme.net/mat/cattle-engl.htm.
2001. Johannes de Tinemue's Redaction of Euclid's Elements, the So-called Adelard III version. vol. 1. Stuttgart: Franz Steiner Verlag.Google Scholar
1969. Decision procedures for real and p-adic fields. Communications on Pure and Applied Mathematics 22: 131–51.CrossRefGoogle Scholar
1971. The complexity of theorem proving procedures. In Proceedings, Third Annual ACM Symposium on the Theory of Computing, ACM, New York, pp. 151–58.Google Scholar
Correspondence d'Hermite et de Stieltjes. 1905. T.II. Paris: Gauthier-Villars.
. 1966–1967. La Commedia secondo l'antica vulgata a cura di Giorgio Petrocchi. Edizione Nazionale a cura della Società Dantesca Italiana. Milano: Arnoldo Mondadori Editore.Google Scholar
(ed.) 1965. The Undecidable. Hewlett, NY: Raven Press. http://cs.nyu.edu/pipermail/fom/2007-October/012156.html. http://forum.wolframscience.com/showthread.php?s=&threadid=1472.Google Scholar
1947. One, Two, Three…Infinity: Facts and Speculations of Science. Illustrated by author. Rev. ed. 1961. London: Macmillan.Google Scholar
1975. Mathematical Carnival. New York: Borzoi Books, Alfred A. Knopf. p. 264. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians.Google Scholar
, , and 2001. A new reading of Method Proposition 14: Preliminary evidence from the Archimedes palimpsest. I. Source and Commentaries in Exact Sciences 2: 9–29.Google Scholar
2001. A simple solution to Archimedes' cattle problem. University of Oulu, Linnanmaa, Oulu, Finland. Acta Universitatis, Ouluensis, Scientiae Rerum Naturalium. http://herkules.oulu.fi/isbn9514259327/index.html?tulosta=yes.
, and 1977. A mathematical incompleteness in Peano arithmetic. In Proof Theory and Constructive Mathematics, with contributions by C. Smorynski, Helmut Schwichtenberg, Richard Statman, et al. Handbook of Mathematical Logic, part D, pp. 819–1142. Studies in Logic and the Foundations of Math, vol. 90. Amsterdam: North-Holland.Google Scholar
, and 1965. Computer studies of Turing machine problems. Journal of the Association of Computing Machinery 12: 196–212.CrossRefGoogle Scholar
1994. Algorithms for quantum computation: Discrete logarithms and factoring. In 35th Annual Symposium on Foundations of Computer Science, Santa Fe, New Mexico, pp. 124–34. Los Alamitos, CA: IEEE Computer Society Press.Google Scholar
1966. Un contre-example à une conjecture d'Artin. Compte Rendus de L'Académie des Sciences Paris Séries A-B 262: A612.Google Scholar
and (1977). From Mathematical Reviews, MR0491063 (58 #10343)
Accessibility standard: Unknown
Why this information is here
This section outlines the accessibility features of this content - including support for screen readers, full keyboard navigation and high-contrast display options. This may not be relevant for you.Accessibility Information
Accessibility compliance for the PDF of this chapter is currently unknown
and may be updated in the future.
- 1
- Cited by