Abstract
Before proceeding to consider the questions posed in the Preface, it is necessary to clarify some logical issues. Paramount among them is the question: What is a proof?
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
That may well change soon, however, given that computer proofs of such major results as the Four-color Theorem, the Prime Number Theorem, and the Jordan Curve Theorem have now been obtained.
- 2.
Nevertheless, as a colleague has rightly noted, if T is a statement whose proof has long been sought and the implication S ⇒ T has already been established, one who proves S is often said to have proved T. For example, Andrew Wiles proved the Taniyama conjecture, but is often said to have proved Fermat’s Last Theorem.
- 3.
It seems that only very recently has such a proof of the Law of Cosines been given. See http://www.cut-the-knot.org/pythagoras/CosLawMolokach.shtml (discussed further in Chapter 5 below).
- 4.
Quoted from Avigad (2006), an article whose concerns overlap to some extent with those of the present text. Avigad suggests that a more fruitful model for analyzing broader aspects of proofs that occur in mathematical practice may be that employed by workers in the field of automated deduction.
- 5.
It is interesting to note, however, how many arguments later deemed to be ‘faulty’ have yielded correct results — have contained a ‘germ’ of truth, so to speak. In some cases, the methods originally used to prove such results have been discarded and the theorems reestablished by other, quite different means, while in other instances the original approaches have subsequently been revalidated in light of more sophisticated analyses. (One example is Laurent Schwartz’s theory of distributions, which provided a rigorous foundation for arguments based on Dirac’s ‘δ-function.’ Another is Abraham Robinson’s creation of non-standard analysis, in terms of which the Newtonian concept of infinitesimal was made comprehensible and arguments based upon it were seen to be correct.)
- 6.
Proofs may, for example, be crafted to serve the needs of a particular segment within or outside of the mathematical community (students, for example, or lay persons with an interest in mathematics).
References
Aigner, M., Ziegler, G.M.: Proofs from the Book, 2nd ed. Springer, Berlin (2000)
Avigad, J.: Mathematical method and proof. Synthese 193(1), 105–159 (2006)
Barwise, J.: Mathematical proofs of computer system correctness. Notices Amer. Math. Soc. 36(7), 844–851 (1989)
Hoffman, K., Kunze, R.: Linear Algebra. Prentice-Hall, Englewood Cliffs, N.J. (1961)
Rav, Y.: Why do we prove theorems? Philosophia Math.(III) 7, 5–41 (1999)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Dawson, J.W. (2015). Proofs in Mathematical Practice. In: Why Prove it Again?. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-17368-9_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-17368-9_1
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-17367-2
Online ISBN: 978-3-319-17368-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)