[Xmca-l] Re: Objectivity of mathematics

[Martin John Packer]

Hi Ed,

I was simply borrowing a dictionary definition; I'm not sure that I'd define the objectivity of mathematics in precisely those terms.

There is (of course!) a literature on this. For instance:

Goodman, N. D. (1979). Mathematics as an objective science. American Mathematical Monthly, 540-551.

...and a PBS video:

<https://www.youtube.com/watch?v=TbNymweHW4E>

And I've enjoyed Brian Rotman's books:

Rotman, B. (1993). Ad infinitum...: The ghost in Turing’s machine; taking god out of mathematics and putting the body back in; an essay in corporeal semiotics. Palo Alto: Stanford University Press. 

and

Rotman, B. (1993). Signifying nothing: The semiotics of zero. Stanford, CA: Stanford University Press. 

Rotman engages in a kind of deconstruction of the *ways*, the practices, whereby mathematicians construct formalisms by means of their informal discourse, and he diagnoses a kind of collective fantasy of disembodied infinitude.

And I'm searching unsuccessfully for my copy of:

Lachterman, D. R. (1989). The ethics of geometry: A genealogy of modernity. New York: Routledge. 

Lachterman, as the title suggests, traces the history of mathematics and identifies a rupture in that history with the invention, in which Descartes was an important figure, of the notion that mathematicians "construct proofs."  Here, Lachterman concludes, was a change in the ontological status of mathematical entities.

I can't pretend to be more than an interested amateur in this area, but ten years ago I wrote a paper, unpublished, with a student, Jenny Hwang, that we titled "Learning mathematics as ontological change," where we tried to build on Rotman and Lachterman. The abstract reads:

"If mathematics is a sociocultural activity, learning math involves “socialization.” We analyze a fifth grade math lesson in fractional equivalents, and show that the lesson is not about constructing knowledge so much as about producing and acting on new species of mathematical object: “fractions.” As the children learn to recognize and act appropriately on these objects they too, we propose, are ontologically changed. We suggest that classroom academic tasks have an embedded cultural task. While school uses social interaction to achieve academic ends, at the same time academic tasks are used to achieve cultural ends. The children are not just learning math, they are learning to be a particular kind of person."

I haven't looked at it in a long time and I've probably changed my mind on various things, but I'll attach in it case the literature review might be of interest.


Martin

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