[Xmca-l] Re: Objectivity of mathematics

[Rod Parker-Rees]

Doesn't the example from Mikhailov touch on Vygotsky's argument about the inseparability of intellect and affect? Yes, the sun was 'that which we call round' before we came along and noticed its roundness but when we did NOTICE its roundness that noticing was informed by our embodied, affective apprehension of what round feels like - because we have rounded clay with our hands but also because we have held rounded objects and turned them in our hands, rolled them and turned them in the light of the sun! Isn't it the disciplined simplicity of mathematics (the deliberate setting aside of 'real world' knowledge) which makes it so difficult for many people - not the propositions and relationships themselves but the difficulty of finding meaning in ideas which have been deliberately stripped of (much, but not all of) their affective connotations?

As a rather silly example of this (but one which relates to the Yupno response) I like the story of the question, 'If it takes one man one hour to dig a one cubic metre hole, how long would it take two men to dig a hole measuring one metre by half a metre by one metre deep?' If this is seen as a maths question the answer is simple but to anyone with experience of digging holes it becomes more difficult to strip away knowledge of the real world - of how it would be very difficult for two people to dig efficiently in such a small hole. When 'mathematical thinking' is used to 'scale up' educational practices which have been found to be effective in small, specific contexts there can be similar difficulties, even in catering, what is needed to feed 40 people is not likely to be 10 times as much as is needed to feed 4. Multiplying up 'what works' may 'look good on paper' but without knowledge of all the complexity which has to be stripped out to 'do the math', all sorts of difficulties can get in the way.

All the best,

Rod


-----Original Message-----
From: xmca-l-bounces@mailman.ucsd.edu [mailto:xmca-l-bounces@mailman.ucsd.edu] On Behalf Of Luis Radford
Sent: 09 November 2014 01:59
To: eXtended Mind, Culture, Activity
Subject: [Xmca-l] Re: Objectivity of mathematics


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!.?
>>>
>>
>>
>>
>>
>

________________________________
[http://www.plymouth.ac.uk/images/email_footer.gif]<http://www.plymouth.ac.uk/worldclass>

This email and any files with it are confidential and intended solely for the use of the recipient to whom it is addressed. If you are not the intended recipient then copying, distribution or other use of the information contained is strictly prohibited and you should not rely on it. If you have received this email in error please let the sender know immediately and delete it from your system(s). Internet emails are not necessarily secure. While we take every care, Plymouth University accepts no responsibility for viruses and it is your responsibility to scan emails and their attachments. Plymouth University does not accept responsibility for any changes made after it was sent. Nothing in this email or its attachments constitutes an order for goods or services unless accompanied by an official order form.