Abstract
We use filters of open sets to provide a semantics justifying the use of infinity in informal limit calculations in calculus, and in the same kind of calculations in computer algebra. We compare the behavior of these filters to the way Mathematica behaves when calculating with infinity.
We stress the need to have a proper semantics for computer algebra expressions, especially if one wants to use results and methods from computer algebra in theorem provers. The computer algebra method under discussion in this paper is the use of rewrite rules to evaluate limits involving infinity.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Beeson, M., Mathpert Calculus Assistant. This software program (now known as MathXpert) was published in July, 1997 by Mathpert Systems, Santa Clara, CA, and is commercially available from <http://www.mathxpert.com/>.
Beeson, M., Design Principles of Mathpert: Software to support education in algebra and calculus, in: Kajler, N. (ed.) Computer-Human Interaction in Symbolic Computation, Texts and Monographs in Symbolic Computation, Springer-Verlag, Berlin/Heidelberg/New York (1998), pp. 89–115.
Beeson, M. MathXpert: un logiciel pour aider les élèves à apprendre les mathématiques par láction, to appear in Sciences et Techniques Educatives. An English translation of this article under the title MathXpert: learning mathematics in the twenty-first century is available at <http://www.mathcs.sjsu.edu/faculty/beeson/Pubs/pubs.html>.
Beeson, M., Using nonstandard analysis to verify the correctness of computations, International Journal of Foundations of Computer Science, 6(3) (1995), pp. 299–338.
K. Kuratowski. Topology, volume 1. Academic Press, New York, London, 1966.
M. Monagan, K. Geddes, K. Heal, G. Labahn, and S. Vorkoetter. Maple V Programming Guide for Release 5. Springer-Verlag, Berlin/Heidelberg, 1997.
B. Sims. Fundamentals of Topology. MacMillan, New York, 1976.
Stewart. Calculus, 3rd edition, Brooks-Cole, Pacific Grove, CA 1995.
S. Wolfram. The Mathematica book. Cambridge University Press, Cambridge, 1996.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Beeson, M., Wiedijk, F. (2002). The Meaning of Infinity in Calculus and Computer Algebra Systems. In: Calmet, J., Benhamou, B., Caprotti, O., Henocque, L., Sorge, V. (eds) Artificial Intelligence, Automated Reasoning, and Symbolic Computation. AISC Calculemus 2002 2002. Lecture Notes in Computer Science(), vol 2385. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45470-5_23
Download citation
DOI: https://doi.org/10.1007/3-540-45470-5_23
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43865-6
Online ISBN: 978-3-540-45470-0
eBook Packages: Springer Book Archive