[Xmca-l] Re: Objectivity of mathematics

[Luis Radford]

Here are my two cents to this interesting discussion.

The question of objectivity has been a central question in understanding
what is meant by mathematics. Sociologists of knowledge, even those
inclined to think about the world as culturally formed (e.g., Berger,
Luckmann, and even Karl Mannheim), have usually preferred to stay away
from a sociological understanding of mathematics.
The crux of the problem seems to me to be this: If we do not have a clear
sense of what we mean by mathematics, we won’t be able to tackle the
question of its objectivity. Mathematics objectivity, I think, cannot be
established on the basis that 4x7 = 28, regardless of the culture. What is
often missed in examples of this kind is that the examples are already
embedded in a particular rationality (numbers are treated as
decontextualized quantities and the multiplication operator bears an
abstract, specific meaning). It is “objective” only within a culturally
institutionalized way of thinking and doing about quantities.
My argument does not amount to saying that in another culture 4x7 may be
equal to 20 or something else. What I am trying to say is that the
question of objectivity can only be asked and investigated within what
Foucault used to call a regime of truth (the Western contemporary regimes
of truth include a strong interest in abstraction and in expressing
abstractions in a written manner). When the Jesuits brought Euclid to
China in the 17th century, they did not simply brought Euclidean theorems;
they also brought new ways of thinking about figures where proving things
in a syllogistic manner makes sense. They brought an Euclidean regime of
truth.

In the attached chapter, I refer to a psychological expedition that
Wassmann and Dasen carried out several years ago with Yupno subjects (from
the Madang Province of Papua New Guinea). The example illustrates, from
another angle, the previous idea. Wassmann and Dasen used the following
“Bride price story”: “You want to marry P’s daughter. The bride price was
set at 19 pigs. You have already paid 8 pigs. How many will you have to
pay later?” The answer was: “Friend, I am not rich enough to buy a new
wife. Where would I find 8 pigs? Besides, I am an old man and have no more
strength.” As the interviewers remark, “After this he could not be moved
to tackle the problem again, it having been rejected as preposterous” (W &
D). Although the question seems to make sense because of its “contextual
cultural nature” (pigs, bride price, etc.) the way the question is asked
is already asked within a certain rationality that provides us (no the
Yupno) with a range of expected responses. The rationality has its own
normativity (which is partially explicit, partially implicit) from where a
kind of objectivity can be spoken about.
The Euclidean and the Yupno examples do not explain, however, what can be
understood by “mathematics.” They do not provide any hint about from where
the various forms of mathematics we see in the world may come.
Following Ilyenkov, I see mathematics as idealities. Of course, not
platonic idealities. I see mathematics rather as idealities or generals,
in Hegel’s sense.  I have suggested that Mathematics is crystalized human
labour. More precisely, Mathematics (taken as an ethno-plural noun here)
are (like knowledge in general) an evolving culturally codified synthesis
of doing, thinking, and relating to others and the world.
Mathematics, in this sense, cannot be equated to social practices. They
are syntheses of social practices. Mathematics, as Hegelian generals, are
put into motion, actualized and transformed through social practices.
Within this context, the Yupno mathematics are syntheses of reflection and
action in the form of the Yupno activities.

To come back to the question of objectivity and to Martin’s and Andy’s
reply to Julian’s post, I am reminded of a beautiful passage from
Mikhailov’s “The  Riddle of the Self.” Mikhailov asserts that “People
could see the sun as round only because they rounded clay with their
hands. With their hands they shaped stone, sharpened its borders, gave it
facets” (Mikhailov, 1980, p. 199).
Mikhailov is not saying that we invented ex nihilo the idea of the sun as
something round. Nor that the sun was already round having all the
Euclidean properties of geometric spheres (symmetries, etc.) before we
noticed it for the first time. We qualify the sun as round because we have
experienced and objectify roundness through embodied activity and
recognize (as far as our culturally evolving perceptual systems allow us
to do) a similarity with artifacts we and others before us have shaped to
satisfy some needs.


Luis

On 8/11/2014, 7:18 PM, "Andy Blunden" <ablunden@mira.net> wrote:

>Oh dear! some times I despair of the possibility of communication.
>That the Earth is round is a social convention, but it is not *only* a
>social convention; it has a sound basis in material reality. That is to
>say, Julian, no amount of discoursing and activity can alter the fact
>that the world is round. The roundness of the Earth is also outside
>discourse and activity, even though it is made meaningful and known for
>us only thanks to discourse/activity.
>Driving on the right is subject to discourse/activity. In about 1968
>Sweden changed from left to right. RIght-hand driving is *only* a social
>convention.
>Simple, eh? I would have thought so.
>Andy
>------------------------------------------------------------------------
>*Andy Blunden*
>http://home.pacific.net.au/~andy/
>
>
>Martin John Packer wrote:
>> And also that the earth is round is a convention! Go figure!
>>
>> Martin
>>
>> On Nov 8, 2014, at 5:55 PM, Julian Williams
>><julian.williams@manchester.ac.uk> wrote:
>>
>>   
>>> I'm struggling to keep up here... Surely I didn't hear Andy Blunden
>>>say that 'objectivity' implies stuff that can't be transformed? I'm
>>>sure I must have misremembered that!.?
>>>     
>>
>>
>>
>>   
>

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