The Argument of Mathematics pp 199-237 | Cite as

# Analogical Arguments in Mathematics

## Abstract

This chapter proposes an analysis of analogical arguments in mathematics. Such arguments are used to show that mathematical conjectures are plausible. The core idea of the chapter is that there is a strong link between good analogies and fruitful generalization. Specifically, a good analogical argument in mathematics is one that articulates a clear proof that is ‘fit for imitation’. This idea is first stated as a simple test, and then refined and deepened through a set of models for relationships of similarity that are important in mathematical analogies. The chapter concludes by addressing the philosophical basis for analogical reasoning and the use of analogies in extended mathematical research programs.

### Keywords

analogy analogical arguments heuristics mathematical analogy plausible reasoning.## Notes

### Acknowledgements

This chapter is based largely on material in my book (Bartha, 2010), with some additions and clarifications. Sections of that book are reproduced here by kind permission of Oxford University Press. Many issues that are neglected or only briefly mentioned here are discussed in the book. I also wish to acknowledge the helpful contributions of the editors.

### References

- Bartha, P. (2010).
*By parallel reasoning: The construction and evaluation of analogical arguments*. New York: Oxford University Press.CrossRefGoogle Scholar - Campbell, N. (1957).
*Foundations of science*. New York: Dover.Google Scholar - Davies, T., & Russell, S. (1987). A logical approach to reasoning by analogy. In J. McDermott (Ed.),
*IJCAI 87: Proceedings of the tenth international joint conference on artificial intelligence*(pp. 264–270). Los Altos, CA: Morgan Kaufmann.Google Scholar - Descartes, R. (1954 [1637]).
*The geometry of René Descartes*(D. E. Smith & M. L. Latham, Trans.). New York: Dover.Google Scholar - Gentner, D. (1983). Structure-mapping: A theoretical framework for analogy.
*Cognitive Science, 7*, 155–170.CrossRefGoogle Scholar - Hadamard, J. (1949).
*An essay on the psychology of invention in the mathematical field*. Princeton, NJ: Princeton University Press.Google Scholar - Herstein, I. (1975).
*Topics in algebra*(2nd ed.). New York: Wiley.Google Scholar - Hesse, M. (1966).
*Models and analogies in science*. Notre Dame, IN: University of Notre Dame Press.Google Scholar - Hume, D. (1947 [1779]).
*Dialogues concerning natural religion*. Indianapolis, IN: Bobbs-Merrill.Google Scholar - Kline, M. (1972).
*Mathematical thought from ancient to modern times*. New York: Oxford University Press.Google Scholar - Kokinov, B., Holyoak, K., & Gentner, D. (Eds.) (2009).
*Proceedings of the second international conference on analogy (Analogy-2009)*. Sofia: New Bulgarian University Press.Google Scholar - Lakatos, I. (1976).
*Proofs and refutations: The logic of mathematical discovery*(edited by J. Worrall & E. Zahar). Cambridge: Cambridge University Press.Google Scholar - Macintyre, A. (1986). Twenty years of p-adic model theory. In J. Paris, A. Wilkie, & G. Wilmers (Eds.),
*Logic colloquium 1984*(pp. 121–153). Amsterdam: North-Holland.Google Scholar - Mill, J. S. (1930 [1843]).
*A system of logic, ratiocinative and inductive, being a connected view of the principles of evidence and the methods of scientific investigation*. London: Longmans, Green and Co.Google Scholar - Munkres, J. (1984).
*Elements of algebraic topology*. Menlo Park, CA: Addison-Wesley.Google Scholar - Playfair, J. (1778). On the arithmetic of impossible quantities.
*Philosophical Transactions of The Royal Society, 68*, 318–343.Google Scholar - Poincaré, H. (1952).
*Science and hypothesis*(W. J. Greenstreet, Trans.). New York: Dover.Google Scholar - Pólya, G. (1954).
*Mathematics and plausible reasoning*(2 Vols.). Princeton, NJ: Princeton University Press.Google Scholar - Russell, S. (1986).
*Analogical and inductive reasoning*. PhD thesis, Department of Computer Science, Stanford University.Google Scholar - Schlimm, D. (2008). Two ways of analogy: Extending the study of analogies to mathematical domains.
*Philosophy of Science, 75*(2), 178–200.CrossRefGoogle Scholar - Snyder, L. (2006).
*Reforming philosophy: A Victorian debate on science and society*. Chicago, IL: University of Chicago Press.CrossRefGoogle Scholar - Van Fraassen, B. (1989).
*Laws and symmetry*. Oxford: Clarendon.CrossRefGoogle Scholar