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Date:         Wed, 2 Sep 1992 09:24:43 -0700
              <QUALRS-L@UGA.BITNET>
From:         mhorney@OREGON.UOREGON.EDU
Subject:      Mathematical Invention
Comments: To: QUALRS-L@uga.cc.uga.edu

I would like to make a small aside to Alfred Lang's recent message on
poststructuralism. Towards the end Lang comments:

"Signs conceived in a way that they are forbidden to change meaning by
context are very special cases, they are logical rather than
psychological, such as ABSTRACT NUMBERS, some scentific concepts, and to a
certain extent linguistics signs, etc." [emphasis added]

This suggestion, that numbers are somehow invariant under changes of
context, struck me as requiring some justification, and I've been
assembling such evidence as I can think of that bears on the question.
Please consider the following points:

1. At a surface, and possibly irrelevant, level I think the statement
clearly false. Consider this example. Danial Fader, in his account in
HOOKED ON BOOKS of establishing a reading program in a reform school, cites
the figure that some number, say 15%, of the library had been stolen by the
end of the first year. When told, school authorities expressed concern.
Fader on the other had reframed this occurance miraculous- that these boys
now considered books valuable enough to steal and, as other events showed,
protect. The number in this case, 15, is either a cause for concern or for
rejoyicing depending on a shift in context; when we use numbers we use them
in some context and their meaning shifts as that context shifts. However, I
suspect this is not what Lang was suggesting, which turns more on the
nature of "15ness."

2. In mathematics this question falls under the long standing debate over
whether mathematics is invented or discovered. If invented, then numerical
concepts must shift as contexts shift, particularly shifts in and out of
human contexts. We, being always buried in human culture, can not easily
see such shifts and must await the arrival of some ET anthropologist to
show us the insularities of our mathematical thought.

3. There are differences even in human approaches to numerism. Some
cultures conceive of systems with only 1,2,3, and Many as numerials. How
does the sociologist's "5" translate into "Many." And then, what of "6?"
Closer to home, Greek and Roman numberical systems did not have a symbol
for zero. How does the concept of nothing change when it hasn't a symbol?
And when it later does?

4. Each time mathmaticians have extended the concept of number to include
more complex entities they have caused a stir. One of the most famous
surrounded Cantor's introduction (discovery?) of "transfinite" numbers; the
idea that infinity comes in different sizes. Bitter, horrible disputes
followed, Cantor ended his days in an asylum. Cantor's opponent Kronekker
(spelling???) said "God made the natural numbers [0,1,2,3...]. All else is
the work of man."

5. For sometime mathematicians believe that the axiomatic system as handed
down from antiquity described a "true" picture of physical reality.
However, one of the classical geometric postulates- the statements meant to
be accepted as true without proof, was given in a unique syntactic form,
and there where long efforts to show this postulate could be derived from
the others. If so, it could be reduced to a theorm and the mathematical
system made more "elegant." (Note that elegance is a key if not primal
motivation for mathematical inquiry, See A MATHEMATICIAN'S APPOLOGY by G.H.
Hardy).

The standard approach to deriving the "5th postualate" was to assume its
falsity and attempt to produce a contradiction. Many "monstrous" theorms
were discovered (created?), but no contridictions. By about 1850
mathematicians accepted the new theorms as "Non-Euclidean" Geometry and
generally agreed they has as much mathematical validity as any other
geometry. Einstein later theorized that it was non-euclidean geometry that
actually described reality.

The upshot of all this is that some 140 years ago mathematicians gave up
any pretense they may have had that their work was descriptive of reality.
This opens the door further, in my mind, to the notion that mathematics,
and hence numbers, are human inventions. If numbers are inventions, then I
think their meaning must shift as context shifts.

I appologize for this much too long message on a minor point.



Mark Horney
School of Education
University of Oregon
Eugene, OR 97403
mhorney@oregon.uoregon.edu
(503) 346-2679
Comments: Gated by NETNEWS@AUVM.AMERICAN.EDU