[Xmca-l] Re: Objectivity of mathematics

[Martin John Packer]

Hola Anna,

Well, here's our summary of your position at the time. Did we get it right?

            Sfard (1998) also gives an account of mathematics that can be called postmodern. She recounts how the search for the elusive referents of mathematical discourse motivated reconceptualizations of this relationship – the move from realism to constructivism, and then the abandonment of the classic dichotomy of symbol/referent in favor of interactionist views of symbols and meaning, such as the semiotics of Saussure and Peirce. Sfard builds on “Foucault’s central claim that the objects ‘referred to’ by symbols, far from being primary to signs and speech acts, are an added value (or the emergent phenomenon) of the discursive activity. This is particularly true for the evanescent objects of mathematics” (p. 14). The “central theme” of her paper is “[t]he process through which the objects ‘represented’ by the symbols come into being retroactively” (p. 15). She suggests that discourse about mathematical referents is “Virtual Reality discourse” rather than “Actual Reality discourse,” a metaphor that “conveys a message as to the particular rights and obligations the mathematical discourse confers upon the participants.... Those who really wish to communicate, not being able to help themselves with their senses, have to use all their mental faculties in an attempt to reconstruct for themselves the realm within which the moves of their interloctors make sense” (p. 2). The task that faces us when we seek to understand mathematics, as she sees it, “consists of not – or no longer – treating discourses as groups of signs (signifying elements referring to contents or representations) but as practices that systematically form the objects of which they speak” (Foucault, 1969/1992, p. 40, emphasis added by Sfard).

Martin

On Nov 7, 2014, at 6:50 AM, anna sfard <sfard@netvision.net.il> wrote:

> Ahoy Martin,
> 
> How nice: so Rotman, Lachterman and the writer of these lines are
> mathematical figures? Positive or negative? :-)
> 
> And while all the other folks you mention indeed view discourse as the heart
> of mathematics, I view it as more than that. Indeed, not just the heart -
> discourse includes all the other parts as well. In mathematical symbols,
> mathematics = [special kind of] discourse (and this is what Huw regards as
> reductionism, perhaps because he equates discourse with languaging?).
> 
> Must reread your paper. It's been awhile... 
> 
> anna 
> 
> -----Original Message-----
> From: xmca-l-bounces@mailman.ucsd.edu
> [mailto:xmca-l-bounces@mailman.ucsd.edu] On Behalf Of Martin John Packer
> Sent: Friday, November 07, 2014 1:29 PM
> To: eXtended Mind, Culture, Activity
> Subject: [Xmca-l] Re: Objectivity of mathematics
> 
> Huw & Anna,
> 
> I had forgotten, until I read the paper again, that Jenny and I based our
> analysis of the fractions class on three main figures: Rotman, Lachterman,
> and Sfard! All three see discourse at the heart of mathematics.
> 
> Martin
> 
> 
> On Nov 7, 2014, at 5:57 AM, anna sfard <sfard@netvision.net.il> wrote:
> 
>> Hi Huw,
>> 
>> Thanks for your thoughts. I agree with much of what you say. I would like
> to know more, though, about why you think that if you talked about problem
> solving in discursive terms, "you'd quickly end up with linguists reducing
> it to wording, and various kinds of "acquisitionists" thinking that this is
> where you're going." I do think about these processes in discursive terms
> and feel, on the contrary, that this is what guards me against
> objectification and acquisitionism. So why?
>> 
>> And on this occasion, to the other debate, the one about "objective". If
> you assume the discursive stance, this word becomes an oxymoron.  Objective,
> as I understand it, means "mind independent", bound have a given form
> independently of one's tastes, values and judgments. But this adjective
> ("objective") refers to narratives, to what people say/think ("facts" are
> subcategory of narratives). So...
>> 
>> anna
>> 
>> -----Original Message-----
>> From: xmca-l-bounces@mailman.ucsd.edu 
>> [mailto:xmca-l-bounces@mailman.ucsd.edu] On Behalf Of Huw Lloyd
>> Sent: Friday, November 07, 2014 3:24 AM
>> To: eXtended Mind, Culture, Activity
>> Subject: [Xmca-l] Re: Objectivity of mathematics
>> 
>> Hi Anna,
>> 
>> Perhaps you could also assert that quantitative choices, predicated upon
> social commitments, offer a means to go beyond those tentative bonds formed
> in numerical rituals.
>> 
>> Commitments, such as commitment to a task that makes it a problem, seem to
> be important.  Also, it seems to me that problem solving (mental searching
> etc) is something that should have a first class status in a theory about
> mathematics. The problem I'd have with referring to these processes as
> discourse is that I think you'd quickly end up with linguists reducing it to
> wording, and various kinds of "acquisitionists" thinking that this is where
> you're going.
>> 
>> A second problem, for me, with fusing communication and cognition is the
> distinct role that communication has in mediating actions, rather than
> comprising the fabric of actions.  For me, the act of exercising that
> fabric, whether mentally or in relation to a present object, induces
> transformations.
>> 
>> I don't think these issues conflict with your account, but perhaps there's
> quite a bit that is skimmed over (such as the bit about individualized
> discourse, perhaps).
>> 
>> I enjoyed your paper.  :)
>> 
>> Best,
>> Huw
>> 
>>