[Xmca-l] Re: Objectivity of mathematics

[Ed Wall]

Martin

     I read Lachterman's a number of years ago (and I know where my copy is - smile) and found it to be quite a lovely book. I'm in the process of reading Hopkins' The Origin of the Logic of Symbolic Mathematics which nicely takes up Jacob Klein's work. 
     Anyway, as a person who does, teaches, teaches others to teach mathematics, I admit to being a little unsure of the word 'objective' and, to be honest, I'm not always sure what  people mean by 'fact.' I do more or less agree with your conclusion about "becoming a particular kind of person" and I have heard children clearly express their awareness of such 'becoming.' My initial reason for bringing up some of this on the list was that I've been wondering how to help teachers notice some aspects of this 'becoming.' 

     Thanks for the paper!

Ed

On Nov 6, 2014, at  12:59 PM, Martin John Packer wrote:

> 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
> 
> <Packer&Hwang_draft.pdf>
> 
> On Nov 6, 2014, at 11:50 AM, Ed Wall <ewall@umich.edu> wrote:
> 
>> Martin
>> 
>>   Nicely put and I have been wondering about this also. I'm sure I have contributed to the confusion as I have been struggling to find words that capture all this in school classrooms. Anyway, so for you objective means "actual matters of fact?" That seems reasonably like Andy's definition?
>> 
>> Ed
>> 
>> On Nov 6, 2014, at  9:50 AM, Martin John Packer wrote:
>> 
>>> I don't want to hijack this thread, but I've been puzzled by the supposed need to decide whether mathematics is 'objective' or whether it is a 'social convention.'  One of my pet peeves about the way that people - both lay and academics - talk about 'social construction' is that it is assumed that what has been socially constructed is insubstantial, maleable. 'Oh, gender is just a social construction!' as though this implies that we can change it tomorrow if we want to. Introducing the term 'convention' seems to me to make matters worse, because we all 'know' that a 'convention' is, well, merely conventional.
>>> 
>>> Surely we live in a social world in which there are many things that have been socially constructed - I would prefer to say socially constituted - and that as a consequence are objective, that is to say actual, matters of fact. It is a fact that Obama is US president, but this is equally a social construction. Want to change that fact? A whole lot of work - social work - is needed to do so.
>>> 
>>> The same with mathematics, I would suggest. Math is a social construction. And as a result it is objective. Math has a history: it has been constructed in different ways at different times. At each point is has been a matter of fact. Different cultures have invented different mathematics, each of which has been a matter of fact for that culture. 
>>> 
>>> It seems to me that we need to stop opposing what is 'social' to what is 'objective.' 
>>> 
>>> Martin
>>> 
>>> On Nov 6, 2014, at 1:10 AM, anna sfard <sfard@netvision.net.il> wrote:
>>> 
>>>> Hi,
>>>> 
>>>> I have not been aware of this super-interesting (for me) thread, and now, when I eventually noticed  it, I cannot chime in properly. So I am doing this improperly, simply by attaching my own paper. Those who are interested enough to open the attachment will see the relevance of its theme to the present conversation. And although I mention Davydov only in an endnote, he is very much present. The theses I'm arguing for seem to substantiate his request for taking the quantitative discourse, rather than the numerical, as a point of departure for the process of  developing child's mathematical thinking (we cannot help it, but in our society, these two discourses appear in the child's life separately and more or less in parallel, with the quantitative discourse free from numbers and the numerical one innocent of any connection to quantities; at a certain point, these two discourses coalescence, thus giving rise to the incipient mathematical discourse; but at the pre-mathematical stage, quantitative discourse is meaningful to the child on its own, as it supports the activity of choosing, whereas numerical discourse is but a way to bond with grownups).
>>>> 
>>>> anna
>>>> 
>>>> -----Original Message-----
>>>> From: xmca-l-bounces@mailman.ucsd.edu [mailto:xmca-l-bounces@mailman.ucsd.edu] On Behalf Of HENRY SHONERD
>>>> Sent: Thursday, November 06, 2014 3:11 AM
>>>> To: eXtended Mind, Culture, Activity
>>>> Subject: [Xmca-l] Re: Objectivity of mathematics
>>>> 
>>>> Ed and Andy,
>>>> Just a little while ago, while I was finishing the Moxhay paper, which seems to have produced an AHA! moment‰ regarding object-mediated action for Andy, I had my own AHA! moment, and it is this:
>>>> Some years ago, after teaching Intro to Linguistics many times, I decided that the most important property of human language that clearly sets it apart from what we know about other species‚ ability to communicate is what is called DISPLACEMENT: the ability to use language to refer to things removed from the here and now, including imaginary happenings or things. The Davydov tasks in the Moxhay article give children the same problem of displacement by requiring that they figure a way to compare two objects removed from one another in space, and, effectively, in time. And I am wondering if this touches on the other threads I have been following: L2 and the Blommmaert/Silverstein. Does the need for standardization in measurement of the objects in the world today find its way into L2 teaching and language policy? The blending of qualitative and quantitative research methods come to mind, to my mind at least. Moxhay‚s article ended with a comparison of  Classroom A and B that  certainly was a blend of the two methods, though the ways in which the dialog broke down in Classroom B (a qualitative issue, I would think) was only hinted at. That would have required a narrative. So, the interplay of narrative and dialog, objects mentioned by David K. I know I have bitten off more than I can chew. 
>>>> Henry
>>>> 
>>>> 
>>>>> On Nov 3, 2014, at 10:51 PM, Ed Wall <ewall@umich.edu> wrote:
>>>>> 
>>>>> Andy
>>>>> 
>>>>> What you say here fits somewhat with some of the thinking I've been doing, but, in part, it is at the point of symbol manipulation that things seem get complicated for me. Also, I find myself wondering whether teaching mathematics, in effect, as mathematics or even Davydov-style is just the things you list. There seems to be more that is needed (and I could be wrong about this) and I have yet to factor in something  like those pre-concepts you mentioned earlier. So I need to do a little reading/rereading on the symbolic question, think a bit more about the space the teacher opens up for studying mathematics, and factor in those 'pre-concepts' before I can reply reasonably to what you are saying here. 
>>>>> I admit that I tend to complicate things too much (smile), but that may come from thinking about them too much. 
>>>>> 
>>>>> Thanks
>>>>> 
>>>>> Ed
>>>>> 
>>>>> On Nov 3, 2014, at  10:45 PM, Andy Blunden wrote:
>>>>> 
>>>>>> Particularly after reading Peter Moxhays' paper, it is clear to me that teaching mathematics, Davydov-style, is orchestrating concept-formation in a particular domain of activity, and that what the children are doing in forming a concept is a system of artefact-mediated actions: "For Davydov," he says, "a theoretical concept is itself a /general method of acting/ - a method for solving an entire class of problems - and is related to a whole system of object-oriented actions." Pure Vygotsky, and also equally pure Activity Theory except that here the object becomes a "theoretical concept," which is characteristically Vygotsky, the point of difference between ANL and LSV! Just as in all those dual stimulation experiments of Vygotsky, the teacher introduces a symbol which the student can use to solve the task they are working on.
>>>>>> So the unit of learning mathematics is *an artefact-mediated action*. The artefact is introduced by the teacher who also sets up the task. At first the symbols is a means of solving the material task, but later, the symbol is manipulated for its own sake, and the material task remains in the background. This is what is special about mathematics I think, that the symbolic operation begins as means and becomes the object. C.f. Capital: the unit is initially C-C' becomes C-M-C' and then from this arises M-C-M' - the unit of capital.
>>>>>> 
>>>>>> Andy
>>>>>> 
>>>>>> ---------------------------------------------------------------------
>>>>>> ---
>>>>>> *Andy Blunden*
>>>>>> http://home.pacific.net.au/~andy/
>>>>>> 
>>>>>> 
>>>>>> mike cole wrote:
>>>>>>> That is really a great addition to Andy's example, Ed. Being a total duffer here i am assuming that the invert v is a sign for "power of" ?
>>>>>>> 
>>>>>>> You, collectively, are making thinking about "simple" mathematical questions unusually interesting. 
>>>>>>> The word problem problem is really interesting too.
>>>>>>> 
>>>>>>> mike
>>>>>>> 
>>>>>>> PS - I assume that when you type:  There is, one might say, a necessity within the integers is that 5 x -1 = -5.   you mean a SUCH not is?
>>>>>>> mike**2
>>>>>>> :-)
>>>>>>> 
>>>>>>> 
>>>>>> 
>>>>> 
>>>>> 
>>>> 
>>>> 
>>>> <Nantes - Sep 13 - final.pdf>
>>> 
>>> 
>> 
>> 
>