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As Thurston says? On using quotations from famous mathematicians to make points about philosophy and education

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Abstract

It is commonplace in the educational literature on mathematical practice to argue for a general conclusion from isolated quotations from famous mathematicians. In this paper, we supply a critique of this mode of inference. We review empirical results that show the diversity and instability of mathematicians’ opinions on mathematical practice. Next, we compare mathematicians’ diverse and conflicting testimony on the nature and purpose of proof. We lay especial emphasis on the diverse responses mathematicians give to the challenges that digital technologies present to older conceptions of mathematical practice. We examine the career of one much cited and anthologised paper, WP Thurston’s ‘On Proof and Progress in Mathematics’ (1994). This paper has been multiply anthologised and cited hundreds of times in educational and philosophical argument. We contrast this paper with the views of other, equally distinguished mathematicians whose use of digital technology in mathematics paints a very different picture of mathematical practice. The interesting question is not whether mathematicians disagree—they are human so of course they do. The question is how homogenous is their mathematical practice. If there are deep differences in practice between mathematicians, then it makes little sense to use isolated quotations as indicators of how mathematics is uniformly or usually done. The paper ends with reflections on the usefulness of quotations from research mathematicians for mathematical education.

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Notes

  1. As one referee pointed out, here we see philosophers (rather than mathematicians or educationalists) assuming that practice is homogenous. It does not matter for our argument who it was who provoked the psychologists we cite to test the homogeneity hypothesis empirically. All that matters for our purpose are the methods and the results of the experiments.

  2. One computer scientist did say that he suspected that the number of proofs from graph theory in this book was evidence of a campaign by graph theorists to advance their sub-discipline, but even he did not complain that the proofs selected are too ugly to deserve selection (private communication).

  3. The locus classicus is Bolzano 1817. In the twentieth century, the view that diagrams have no place in proofs is associated with Bourbaki (see Brown 1999 p. 172). Littlewood, presenting himself as resisting the trend, complains that his students “will not use pictures” and blames this on “heavy warnings” intended to break students from school mathematics (Littlewood 1953 p.36). Moreover, the rejection of diagrams in proofs is a consequence of the view that the inferences in a mathematical proof should be purely logical, that is, make no reference to specific subject matter. An inference that depends crucially on a diagram obviously violates that rule. This view has its origins in the work of Pasch, Frege and Hilbert. For a recent expression, see Hales 2012 p. x.

  4. Manders used the terms ‘exact’ and ‘co-exact’. ‘Metrical’ and ‘non-metrical’ are the terms used by Zhen et al., and by Weber and Mejia-Ramos. Since we discuss their argument, we use their terms rather than Manders’.

  5. Hume (1888), Book I, PART IV. of the sceptical and other systems of philosophy. Section 1. Of scepticism with regard to reason.

  6. Matthew Inglis has argued that this objection is not as reasonable as it sounds, because in shifting from cognitive science to ethnography, we are changing the object of study. “’Mathematical practice’ can be interpreted from an individualist cognitive perspective or an individualist social psychological perspective, or a sociological perspective (roughly speaking I’d say that the difference between the latter two is that the object of study for a social psychologist is the individual in a social setting, whereas the object of study for the sociologist is the social setting itself)… It is like a sociologist saying, ‘cognitive experiments don’t consider social factors’ or a cognitivist saying ‘ethnographies don’t consider cognitive processing’. Both those statements are true, but they’re not sensible objections.” (private communication, 2018).

  7. One reviewer wondered whether, in this section, we are ourselves guilty of argument by isolated quotation. We do not think so, because we do not take these mathematicians to speak for the whole of mathematics, and besides the argument of this section appeals to a change in mathematical practice. The mathematicians we cite here are doing mathematics differently, and that is what breaks up the homogeneity of current practice.

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  1. The University of Toronto, Toronto, Canada

    Gila Hanna

  2. The University of Hertfordshire, Hatfield, UK

    Brendan Larvor

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  1. Gila Hanna
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Hanna, G., Larvor, B. As Thurston says? On using quotations from famous mathematicians to make points about philosophy and education. ZDM Mathematics Education 52, 1137–1147 (2020). https://doi.org/10.1007/s11858-020-01154-w

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