NetNews Usenet Archive 1992 #30/NN_1992_30.iso/spool/sci/math/16922

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Subject: Re: Numbers and sets
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Date: Tue, 15 Dec 92 01:31:28 GMT

Please consider an amateur's comments on numbers, drawn from
practical experience.  For some people, all philosophy should be
practical (pragmatic), as philosophy is the love of wisdom, and
wisdom is the practical side of knowledge.  Anyway, here goes:
    1.  Numbers are not simply abstracts:  they arise from the
interaction of the brain and its environment.  Following Goedel, we
know arithmetic, like all other thought systems (including logic),
cannot stand alone.
    2.  Humans (and other animals) first needed ordinals, not
cardinals:  the pecking order (and the mating order) were more
important to their social structure than any need to count things.
    3.  How do children learn to count?  They first learn a totally
arbitrary sequence (an ordinal operation), and only then can they
perform the one-to-one matching that allows them to assign a
number in that sequence to the cardinal enumeration of the set being
counted.  (The ideas of "one", "two", and "many" might indeed arise

independently of a learned sequence.  These are, in fact, the only
numbers found in certain less numerate cultures.  However, perhaps
the ideas and language of ordinality are very primitive in those
societies.  I wonder...)
    4.  Measurements, even those of trained scientists, are in the
most fundamental sense ordinal.  We never read a scale and know
that a length or a needle exactly matches a grid marker, only that it
points near, above, or below it, at least in the analog world we seem
to inhabit.  An interesting case is given by elevators:  they don't stop

exactly "at" floors (as we sometimes realize to our chagrin), but
only satisfy the condition of stopping above one switch and below
another.  Heisenberg's indeterminacy principle, even though it's
based on the quantum nature of things, might make us think that the
ultimate calculus of the world is based more firmly on inequalities
(with order) than on equalities (with perfect "digital"
correspondence).
    5.  Mathematicians will probably always be revising their
ideas of what their basic elements mean, as will scientists, and
after the magnificent contributions of Russell, Whitehead, Goedel,
and Popper, I would hope we can be a bit tolerant of different
perspectives as we all play our "Magic Bead Game".

Yours sincerely,
        Steven Kilston    (semi-learned astronomer)
        Lockheed Research Labs
        Palo Alto
        [e-mail]