Posted by: Alexandre Borovik | May 4, 2010

The “myth of reference”?

I am not that learned in philosophy of mathematics and therefore the following thesis  is quite a new word for me. I cannot classify it as belonging to any established tradition in philosophy of mathematics.
Some time ago, I came across another industrial illustration which relates specifically to school mathematics. The Cockroft Committee of Enquiry into the teaching of mathematics in schools was set up in 1978 as part of James Callghan’s Labour government’s response to widespread criticism of state education, especially from the engineering industry and the writers of the Black Papers (see Kogan, 1978; Lawton, 1980; Salter and Tapper, 1981). The Cockroft Committee was instructed to pay “… particular regard to the mathematics required in further and higher education, employment and adult life generally…” (Cockroft et al. 1982, p. ix).
As part of their response, the Committee set up three research studies. One of these studies was conducted by a team from Bath University. This study focused on the mathematical requirements of the working practices of sixteen-to-nineteen-year-olds. The researchers collected data which they categorized as ‘specific tasks incorporating mathematics ‘ (STIM) and ‘mathematics incorporated in specific tasks’ (MIST). They found that a great many young employees — a ‘vast army of people’ — did not appear to require any formal mathematics, not even counting or recording numbers. Nevertheless, they claimed that ‘all these occupations involve actions which could be described in mathematical terms’ (Bailey et al, 1981; p. 12). The researchers presented a list of these mathematical terms (MIST) as follows:
  • A set, dis-joint sets.
  • Mappings, one-to-one, one-to-many, many-to-one correspondences.
  • Symmetry, bilateral and rotational.
  • Rotation, reflection, translation and combinations of these[.] Tesselation patterns.
  • Logical sequences (if … then … .). (Bailey et al, 1981; p.26)
  • Tasks often have to be carried out in particular orders sometimes requiring simple decisions, but which would not often be verbalised. For example, a creeler in a carpet factory: ‘Is the spool empty? Yes! Replace with another of the same colour’. Awareness of the conseuences of not following the prescribed order may be important. (ibid; pp.25-6)
The terminology used there is very much out of the ‘modern mathematics’ tradition of the era. The tasks (STIM) to which these MIST items corespond are:
  1. (Articles are sorted into separate collections for packing or on an accept/reject basis.
  2. Articles are moved into particular orientations involving moving sideways, turning over or round.
  3. Articles, such as wine glasses[,] are checked for uniformity of shape.
  4. Packed articles form regular patterns.
  5. Assembly tasks can involve matching parts, such as connecting wires to correct terminals.
The researchers are not claiming that creelers need to study formal logic in order to be abe to replace a carpet spool when it’s empty rather than when it’s full. On the contrary, they are clear that such tasks are successfully carried out in the abscence of mathematical knowledge. Yet in making this claim, they are [...] establishing a division between the mathematical-intellectual and the manual and costituting the former as generative of   commentaries upon the latter. It is as if the mathematician casts a knowing gaze upon the non-mathematical world and describes it in mathematical terms. I want to claim that the myth is that the resulting descriptions and commentaries are about that which they appear to describe, that mathematics can refer to something other than itself. I shall refer to this myth as the myth of reference.

[P. C. Dowling,  The Sociology of Mathematics Education: Mathematical Myths/Pedagogic Texts. Routledge, 1997. ISBN-10: 0750707917. ISBN-13: 978-0750707916.]
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  1. When I studied philosophy, I came into peripheral contact with the tradition of ‘marxist analysis’, which according to its followers, could be applied to just about anything involving human actions. This could be expressed in the form “The meaning of mathematical symbols reside in their interconnection with the whole of human practice” (I quote this from memory alone, the reference is long gone now, and will hopefully remain so).

    This means that any abstract notion will only come into being in the course of labour (either physical or mental) and will necessarily have its raison d’etre in the concrete production. I see Dowlings ‘thesis’ more as an anti-marxist comment to the somewhat marxist-inspired discourse of the original research paper.

    This is not a thorough analysis as I have read neither the Dowling nor Cockroft papers/articles/books, it just reminded me of some of the comments my fellow students would make at university.

  2. My first reaction when reading this was

    “The idea that philosophy or pedagogy can refer to something other than themselves might be a potentially dangerous reference myth as well.”

    I believe that most claims that mathematics can refer to a particular aspect of reality, to this or that thought processes, to fundamental structures of action and existence in the universe — are to be justified very thoroughly and precisely, that these links are not granted once and for all, and that most of them have not a clear epistemological status.

    But what this author is expressing is not scepticism or criticism but closer to a form of hate. I would be interested to know how he came to this sentiment.

    It may be by the anti-marxist mechanism suggested by Steen. But if so I find it paradoxical to be associated with a critic of a kind of taylorism (analysis of the intellectual needs of a worker implicitely in order to increase his efficiency) by mathematically minded members of a government comity.

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