RE: [xmca] Math Question

From: Ed Wall (
Date: Wed Jan 03 2007 - 10:15:59 PST


     An interesting question. Let me muddle things with a somewhat
historical answer. In a sense what we call the integers are a result
of solving equations of the form

                    x + a = b (where a and b are counting numbers and a > b)

the rationals are a result of solving equations of the form

                    ax + b = c (where a, b, c are integers - a>0)

etc. For the Greeks, for example, rational number, in the sense we
use it today, had no meaning although they had certainly had an
advanced sense of proportion and like earlier cultures used, in the
market place, a quantity called one-half and some other of which are
termed unit fractions.

    So it would seem that doing number (apologies for changing what
you said here, but 'number' doesn't feel like the right term) is a
process that gets us to an end in view and because of a certain
effectiveness in doing so (O'Halloran talks about this a bit) the
doing - perhaps think tool here - has been applied to processes
outside the particular equation one is working on at the moment. This
application, when it is the everyday, is called by some
mathematizing. Aristotle speaks about this application in somewhat
negative terms.
    A possible example is pi. This comes out of, in a sense, everyday
measurement although it arises out of the doing of equations. In its
denoting the ratio of circumference to diameter it has symbolic value
outside a particular equation. On the other hand, this value, in its
decimal expansion, is not expressible.

I hope this somewhat addressed your question.


>So Ed, would you say that number is a process that gets us to an end in
>view and that it has not symbolic meaning outside of the equation that
>you are working on at the moment? And if the numbers to not get us to a
>rational end in view that we are looking for, then we have to develop
>new numbers as processes?
>-----Original Message-----
>From: []
>On Behalf Of Ed Wall
>Sent: Wednesday, January 03, 2007 12:28 PM
>To: eXtended Mind, Culture, Activity
>Subject: RE: [xmca] Math Question
>Intriguingly Dewey uses such words as valuation, discrimination, etc.
>arguing that number is a rational processes, not a sense (or
>non-sense) fact.
>Ed Wall
>>Quantity (as opposed to Quality). Number is a *much* narrower concept.
>>At 04:50 AM 3/01/2007 -0800, you wrote:
>>> I think that focusing on proofs misses the point about
>>>mathematics since proofs are common to both logic and mathematics
>>>but math has an object about which proofs are developed. That
>>>object is transcendent and can't be reduced to logic (Godel, no?)
>>>And that object is, in all its manifold complexity, Number.
>>> A more fruitful direction for looking at what's going on with
>>>mathematics should relate it back to activity as in Gary Urton's
>>>"The Social Ontology of Numbers".
>>> Paul Dillon
>>>Ed Wall <> wrote:
>>> Michael
>>>I have perhaps slanted this with my talk of proof since this is a
>>>Western notion although I could spoken of the Indian notion of
>>>upapatti instead. I do see very strong indications of this in quite
>>>young children where, in a sense (and this is the best way I have in
>>>speaking about it), they get inside of a piece of mathematics and it
>>>somehow becomes a way of doing/thinking - the tool and the user sort
>>>of blend. The notion of logogenesis re Mattheissen and Halliday seems
>>>to have some bearing here as, I have thought for some time,
>>>Vygotsky's discussion of everyday and scientific concepts in
>>>Development of Scientific Concepts.
>>>The business about creativity may be in the eye of the beholder. I
>>>used to think that some of the drawings of my daughter on the
>>>refrigerator were pretty creative. She, now, being quite a bit older
>>>might disagree.
>>>Ed Wall
>>>>Are we talking about two different mathematics. I have been told
> >>>that mathematics doesn't start getting really creative until you
>>>>stop using numbers. Not being a mathemetician I can't grasp this at
>>>>all - but I have gotten this from two sides - the successful
>>>>mathematician who said to really work on math you have to move
>>>>beyond the use of numbers, and to a fellow who flunked out of the
>>>>Courant Institute (sp?) because he could not get past the use of
>>>>numbers. I think this is true of writing - that really great
>>>>writers are past the use of words as symbols, what they are writing
>>>>is what is happening at the moment for them - the characters takes
> >>>on lives of their own. I think in reading you can always tell who
>>>>has gotten past this point and who hasn't. Some people simply write
>>>>words down on a piece of paper, and for some writers the words are
>>>>only residue - what is left over from the experience. So perhaps
>>>>mathematics and writing are in many ways the same process along
>>>>different trajectories.
>>>>From: on behalf of Cathrene Connery
>>>>Sent: Tue 1/2/2007 9:54 PM
>>>>Subject: [xmca] Math Question
>>>>Hi Ed and everyone,
>>>>What an interesting question. It is true that so many writers and
>>>>artists as well have stated that they felt the ideas they mediate
>>>>cross a line in the creative process where mind and activity and
>>>>object seems to blurr and the work seems to create itself so to
>>>>speak. Michelangelo wrote that his sculptures spoke to him as he
>>>>carved the marble. Sometimes when I am painting, the same
>>>>phenomenon occurs. From a Vygotskian perspective, this experience
>>>>has interesting appeal when considering the inner voice. Vera
>>>>John-Steiner's Notebooks of the Mind and Creative Collaborations
>>>>document this psychological activity.
>>>>To apply it to mathematics is a fascinating question. Being someone
>>>>who can barely balance a checkbook, I am not sure how it would
>>>>apply.......however, I suspect different domains in mathematics
>>>>would reflect variations of this experience as they each depend or
>>>>are derived from various forms of cognitive pluralism. have you
>>>>looked at Reuben Hersh's work?
>>> >M. Cathrene Connery, Ph.D.
>>>>Assistant Professor of Bilingual & TESL Education
>>>>Central Washington University
>>>>>>> Ed Wall 01/02/07 5:06 PM >>>
>>>>Mike and all
>>>> This is not quite on the topic (and, thus, I have held back a
>>>>bit), but given the amount of expertise that people are bringin I ask
>>>>a question I have asked elsewhere (I apologize for how it is phrased,
>>>>but something like this was appropriate in that particular
>>>>> I had a question and wonder if you might point me in a useful
>>>>>direction(s). The situation is such: It has been argued of late that
>>>>>the work mathematicians do - proof and the such - proceeds within
>>>>>mathematics being created. That is, without going into a lot of
>>>>>detail, the mathematics one does is both circumscribed and supported
>>>>>by the mathematics one is doing. This is not exactly a matter of
>>>> >prior knowledge or the hermeneutic circle per se although it might
>>>>>have something to do with being an 'expert.'
>>>>> The reason why I am asking is that, the other day in a somewhat
>>>>>philosophic discussion around a novel, a participant noted that some
>>>>>authors describe the authoring process as open-ended in the sense
>>>>>that what finally takes place may differ from what was originally
>>>>>intended. That is, in a certain sense, the writing writes itself. As
>>>>>this sounded somewhat parallel to the phenomenon I mentioned in
>>>>>mathematics, I was wondering if you knew of someone(s) who makes
>>>>>remarks about a similar phenomenon re writing.
>>>>Ed Wall
>>>>>Hi David--
>>>>>There is a LOT of material on the topic of writing systems.
>>>>>Two interesting places to start are:
>>>>>D. Schmandt-Besserat, Before Writing:. U of Texas Press. 1992 (two
>>>>>R. Harris. The origin of writing. Open Court. 1986.
> >>>>
>>>>>David Olson has written extensively on this topic, primarily from
>>>>>I am unsure of best sources that delve into origins of writing in
>>>>>which were more or less co-incident with
>>>>>events in Euphrates area.
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