This blog post is by Dr Brendan Larvor, University of Hertfordshire, Principal investigator for AHRC Science in Culture Research Network ‘Mathematical Cultures’.
The idea that mathematics can be regarded as a body of culture is not new.
In 1950, the mathematician Raymond Wilder published his paper ‘The cultural basis of mathematics’ (Proc. Int. Congr. Math. Cambridge I, pp. 258) and later developed these thoughts in further publications. The view of mathematics that he advanced gained some adherents, but it did not give rise to a research programme, still less a sub-discipline.
I think this is because Wilder offered a cultural approach as an answer to the ontological question: what is mathematics about? Are mathematical objects timeless Platonic items, or are they ideas in the minds of mathematicians? Both of these options, and other candidate answers such as radical empiricism and the symbol-game view, suffer from serious objections.
The idea that mathematical objects are cultural products promises to escape some of these difficulties. A cultural view explains the conviction that mathematical objects are independent both of the mathematician and of empirical constraints without positing a mysterious Platonic realm and a yet more mysterious intuitive knowledge of it. However, either this ontological view strikes you as an insight, in which case the problem dissolves, or it does not, on which case we are no better off. Either way, there is no motivation here for detailed studies of specific mathematical cultures and practices. Consequently, Wilder’s book and similar efforts did not engender a sub-discipline.
Recent decades have seen interest in mathematical cultures emerging from two sources, both distinct from Wilder’s ontological question: epistemology and pedagogy. In epistemology, curiosity about mathematical cultures results from a recognition that the gapless proofs of logic textbooks are rarely found anywhere else, and certainly not in the texts that mathematicians use to convince themselves and each other of the truth of new theorems. If formal logic does not explain how mathematicians secure new theorems, then what does? This question has led philosophers to attend to mathematical practices, and to call on historians, sociologists and cognitive scientists for help.
Turning to pedagogy: in mathematics education, there has lately been a recognition that the explanation for the differential experiences and success-rates in learning mathematics of both individuals and entire societies may lie in culture rather than in the relative effectiveness of rival teaching methods in recruiting students’ native cognitive endowments. This opens new possibilities for remedial policy. For example, one alarming finding is that most school mathematics teachers have little or no contact with the professional culture of research mathematicians. Snezana Lawrence discussed this in her talk at the first meeting of the AHRC-funded, Mathematical Cultures Research Network.
It’s arguable that this has profound consequences for school mathematics, as Paul Lockhart argues in The Mathematician’s Lament. Lockhart invites his readers to imagine compulsory music lessons that did not involve making music, taught by people who have never played music nor had any contact with those who do. That, he says, is the condition of mathematics teaching in American schools. I just wrote “the professional culture of research mathematicians” but it is doubtful whether there is any such thing. One result of the first meeting in the series funded by the AHRC is that there is considerable variety, both between discipline areas and even between seminars taking place in the same building.
One of the most striking presentations at the first meeting in the AHRC Mathematical Cultures Research Network was Slava Gerovitch’s description of the seminar that Israel Gelfand led at Moscow University.
In Gelfand’s seminar, the usual academic etiquette and hierarchy were abandoned in favour of a ruthless meritocracy. As Gerovitch explains, some Moscow mathematicians thrived in this atmosphere, but others—who were leading professors and productive researchers—found it intolerable. It would be easy to multiply examples of this sort.
The suggestion that schoolteachers and pupils should have some contact with mathematical research culture immediately invites the question: which research culture? On the other hand, there may be values common to all mathematicians that one might try to realise in school classrooms.
For a defence of this last thought see Alan Bishop’s talk from the second meeting of the AHRC Mathematical Cultures Network which took place in London from 17th to 19th September 2013. Alan Bishop is the author of Mathematical Enculturation: a cultural perspective on mathematics education (1991). His talk argues for the existence of universal mathematical values.
This is one of a series of guest blog posts written by AHRC Science in Culture award holders. The Science in Culture Theme is a key area of AHRC funding and supports projects committed to developing reciprocal relationships between scientists and arts and humanities researchers. More information about ‘The Mathematical Cultures’ Research Network’ can be found here.
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