Abstract
The history of mathematics provides evidence that proofs let mathematicians distinguish between true results and merely plausible ones; that the careful formulation of arguments allows them to see how individual mathematical results relate to broader mathematical ideas; and that the process of proving teaches logical reasoning. This paper presents key historical examples, including the origin of logical proof in Greek geometry, Aristotle’s proof-based model for science, the classical uses of visual demonstration, the developing power of abstraction and symbolism, the role of the principles of optimisation and symmetry, the changing standards of rigour between the algorithmic calculus of the eighteenth century and the proof-based version of the nineteenth, the discovery of non-Euclidean geometry, the modern reciprocal influences between philosophy and proof-based mathematics, and the current importance to society of understanding logic. We conclude that observing and teaching this history also helps us teach proof and proving.
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Notes
- 1.
Since the Babylonians used fractions with a base of 60, a practice which, incidentally, is the origin of our base-60 divisions of time into minutes and seconds, it would be more accurate to say that the area calculation given is equivalent to approximating π as 3 + 7/60 + 30/602.
- 2.
None of the present discussion is meant to rule out the possibility that there were already steps towards forming axioms and logical proofs in Egyptian and Babylonian mathematics. No evidence of this is known to me at the present time, but scholarship on ancient mathematics continues, and one should keep an open mind. Even should such evidence be found, though, we would still need to explain why the Greeks chose to make logical proofs, using the smallest possible set of axioms, so central to their mathematics.
- 3.
I have used the phrase ‘Greek proofs by contradiction’. In philosophy, proof by contradiction exists in cultures independent of the Greek, notably in China (Leslie 1964; Siu 2009). But as far as I know, only the Greeks and their mathematical heirs used it within mathematics; Professor Siu (2009) says that he does not know of an example from China before the coming of the Jesuits in the seventeenth century.
- 4.
Vieta, and many mathematicians in the century following him, thought of expressions like x3 as volumes, x2 as areas, and x as lines, and so they would not write an expression like ax2 + bx + c. Vieta also designated his unknowns by upper-case vowels and his constants by upper-case consonants; the use of the lower-case letters x and y for the principal unknowns was introduced later by Descartes. But these details do not affect the main point here.
- 5.
‘If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two straight lines, if produced indefinitely, meet on that side where the angles are less than two right angles’ (Euclid 1956, Postulate 5, p. 155; Fauvel and Gray 1987, p. 101).
- 6.
If one wants to read only one history of mathematics, I recommend Katz (2009). For further readings, Katz’s bibliography will take one as far as one would like. Another good scholarly general history is Boyer and Merzbach (1989).
For a collection of original sources in the history of mathematics from the period between 1200 and 1800, with excellent commentary, I recommend Struik (1969). Another fine collection of original sources, with shorter excerpts and commentary but covering the entire period from antiquity to the twentieth century, is Fauvel and Gray (1987). Original sources from Egypt, Babylon, China, India and the Islamic world may be found in Katz (2007), and, in classical analysis, in Birkhoff (1973). In this paper, I have given references to these four source books whenever relevant materials are accessible there. Finally, I highly recommend the Mathematical Association of America’s online ‘magazine’ Convergence (http://mathdl.maa.org/mathDL/46/) dedicated to the history of mathematics and its use in teaching. Besides articles, Convergence includes book reviews, translations of original sources, quotations about mathematics, portraits, ‘mathematics in the news’, and a great deal more.
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Grabiner, J.V. (2012). Why Proof? A Historian’s Perspective. In: Hanna, G., de Villiers, M. (eds) Proof and Proving in Mathematics Education. New ICMI Study Series, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2129-6_6
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