Abstract
Four motives for seeking new proofs of previously established results have already been mentioned in Chapter 1: the desires
-
(1)
to correct errors or fill perceived gaps in earlier arguments;
-
(2)
to eliminate superfluous or controversial hypotheses;
-
(3)
to extend a theorem’s range of validity; and
-
(4)
to make proofs more perspicuous.
Even if we have succeeded in finding a satisfactory solution, we may still be interested in finding another solution. We desire to convince ourselves of the validity of a theoretical result by two different derivations as we desire to perceive a material object through two different senses. Having found a proof, we wish to find another proof as we wish to touch an object after having seen it. — George Pólya, How to SolveIt
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.
- 2.
In its simplest form, the Prime Number Theorem (the subject of Chapter 10 below) states that \(\lim \nolimits _{x\rightarrow \infty }\pi (x)\bigg/\dfrac{x} {\ln x} = 1\), where π(x) denotes the number of primes less than x.
- 3.
- 4.
One example is discussed in Chapter 4
- 5.
Such as using topological arguments to prove results in mathematical logic. Another example is Errett Bishop’s text Foundations of Constructive Analysis (Bishop 1967), which Bishop himself called “a piece of constructivist propaganda,” written to demonstrate how large a part of abstract analysis can be developed within a constructive framework.
- 6.
In the form proved by Hilbert, the theorem states that every ideal in the ring of multivariate polynomials over a field is finitely generated, so that for any set of polynomial equations, there is a finite set of such equations that has the same set of solutions.
- 7.
“Mathematisches Notizbuch” (Cod. ms. D. Hilbert 600), preserved in the Handschriftenabteilung of the Niedersächische Staats- und Universitätsbibliothek, Göttingen.
- 8.
See Thiele and Wos (2002) for further details on the history of the twenty-fourth problem and on results related to it found recently by those working in automated theorem proving.
- 9.
Jon Krakauer, e.g., in his book Into Thin Air, wrote: “Getting to the top of any given mountain was considered much less important than how one got there: prestige was earned by tackling the most unforgiving routes with minimal equipment, in the boldest style imaginable.”
- 10.
To paraphrase a remark C.S. Peirce made in regard to philosophy (Pierce 1868).
References
Bishop, E.: Foundations of Constructive Analysis. McGraw-Hill, New York (1967)
Dehn, M.: The mentality of the mathematician. A characterization. Math. Intelligencer 5, 18–26 (1983)
Hallett, M.: Reflections on the purity of method in Hilbert’s Grundlagen der Geometrie. In Mancosu, P., The Philosophy of Mathematical Practice, pp. 198–255. Oxford U.P., Oxford (2008)
Mancosu, P.: Philosophy of Mathematics and Mathematical Practice. Oxford U.P., New York and Oxford (1996)
Mancosu, P.: Mathematical explanation—problems and prospects. Topoi 20(1), 97–117 (2001)
Pierce, C.S.: Some consequences of four incapacities. J. Speculative Philos. 2, 140–157 (1868)
Thiele, C., Wos, L.: Hilbert’s twenty-fourth problem. J. Automated Reasoning 29(1), 67–89 (2002)
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). Motives for Finding Alternative Proofs. In: Why Prove it Again?. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-17368-9_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-17368-9_2
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)