As Thurston says? On using quotations from famous mathematicians to make points about philosophy and education
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ORIGINAL ARTICLE
As Thurston says? On using quotations from famous mathematicians to make points about philosophy and education Gila Hanna1 · Brendan Larvor2 Accepted: 24 March 2020 © FIZ Karlsruhe 2020
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.
1 Introduction The reflections and reminiscences of mathematicians are a rich source of data for mathematics education and the philosophy of mathematical practice. Mathematical research practice is hard to study, because it takes place in private and semi-private places (offices, seminar rooms) and in the minds of mathematicians. In addition, it is technically difficult. For these reasons, we need the reflective testimony of mathematicians to help us to understand what they are doing. On the other hand, the usual reservations about practitioner-testimony apply to mathematics. Adepts in any practice can fail to understand what they are doing, how they are doing it and what conditions make it possible. Moreover, they can disagree among themselves about the meaning and We are grateful to Paul Dawkins and three anonymous referees for extensive comments on drafts of this paper. * Brendan Larvor [email protected] 1
The University of Toronto, Toronto, Canada
The University of Hertfordshire, Hatfield, UK
2
nature of the practice, in which case the testimony of any one practitioner cannot be taken as a reliable guide to the whole. In this paper, we first review empirical results in cognitive psychology that show the diversity and instability of mathematicians’ spontaneous opinions on selected aspects of mathematica
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