Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-29T09:24:20.988Z Has data issue: false hasContentIssue false
  • English
  • Français

Developing new picture proofs that the sums of the first n odd integers are squares

Published online by Cambridge University Press:  03 July 2023

Chris Sangwin  and
Fenner Stanley Tanswell
Chris Sangwin
Affiliation:
School of Mathematics, University of Edinburgh, EH9 3FD e-mail: C.J.Sangwin@ed.ac.uk
Fenner Stanley Tanswell
Affiliation:
Centre for Logic and Philosophy of Science, Vrije Universiteit, Brussels, Belgium e-mail: fenner.tanswell@vub.be
Get access Rights & Permissions [Opens in a new window]

Extract

What is it that makes us judge two proofs of the same theorem to be the same or different? This is not an idle question: one central aspect of judging mathematics is the novelty of the mathematics presented. This is important everywhere, from the peer-review system, to assigning international prestige, to funding agencies’ grant decisions. It even matters to some extent in examinations, to avoid accusations of collusion. Surprisingly, philosophers of mathematics have not paid the question of novelty much attention. In this Article, we will consider the appealing conjecture that the main ideas that make up the proof, the essence of a proof, can indeed be identified and that very different styles of proofs can share common main ideas. Further, that a particular theorem can be proved using quite different, independent main ideas. As a means of exploring whether this is plausible, we will present a number of novel proofs of the following theorem.

Type
Articles
Information
The Mathematical Gazette , Volume 107 , Issue 569 , July 2023 , pp. 249 - 262
Check for updates
Copyright
© The Authors, 2023. Published by Cambridge University Press on behalf of The Mathematical Association

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Polya, G., Mathematics and plausible reasoning. Vol.1: Induction and analogy in mathematics. Vol.2: Patterns of plausible inference. Princeton University Press (1954).Google Scholar
Nardi, E. and Steward, S., Is mathematics T.I.R.E.D? a profile of quiet disaffection in the secondary mathematics classroom, British Educational Research Journal 29 (2003) pp. 345–366.CrossRefGoogle Scholar
Hanna, G., Some pedagogical aspects of proof, Interchange 21(1) (1990) pp. 6–13.CrossRefGoogle Scholar
Buckley, S. M. and MacHale, D., Variations on a theme: Rings satisfying x 3 = x are commutative, Amer. Math. Monthly 120(5) (2013) pp. 430–440.CrossRefGoogle Scholar
Kalman, D., Six ways to sum a series, College Maths J., 24(5) (1993) pp. 402–421.CrossRefGoogle Scholar
Ording, P., 99 Variations on a proof, Princeton University Press (2019).Google Scholar
Duchêne, L. and LeBlanc, A., Rationnel mon Q: 65 exercices de styles, Hermann (2010).Google Scholar
Loomis, E., The Pythagorean Proposition, National Council of Teachers of Mathematics (1968).Google Scholar
Sangwin, C. J., Sums of the first odd integers, Math. Gaz., 107 (March 2023) pp. 10–24.CrossRefGoogle Scholar
Nelsen, R. B., Proofs Without Words: Exercises in visual thinking, Mathematical Association of America (1993).Google Scholar
Nelsen, R. B., Proofs Without Words II: Exercises in visual thinking, Mathematical Association of America (2000).Google Scholar
Nelsen, R. B., Proofs Without Words III: Further exercises in visual thinking, Mathematical Association of America (2016).Google Scholar
Leversha, G., What makes a good proof without words? Math. Gaz. 105 (July 2021) pp. 271–281.CrossRefGoogle Scholar
Hayes, B., Gauss’s day of reckoning, American Scientist 94(3), (2006).CrossRefGoogle Scholar