FOM: Objectivity
John Mayberry
J.P.Mayberry at bristol.ac.uk
Tue Dec 23 06:35:20 EST 1997
Of course our mathematics is permeated with evidence of its
human origins: how could things be otherwise? But is it really possible
to suppose that *all* of our mathematics is "socially constructed"?
What does "socially constructed" mean here? Consider the simple
proposition that seven fives make (a) thirty-five. What this
proposition means is that if you have a finite plurality (set) that has
a partition into seven disjoint fives, then it also has a partition
into three tens and a five, all disjoint. How could this be anything
other than an objective fact, independent of human beings and their
"social constructions"?
But what is this objective fact a fact *about*? Clearly, about
all finite pluralities that have partitions into (a) seven (of)
disjoint fives. Words like "seven", "five", and "thirty-five", should
be seen primarily as names for species in the category of finite,
discrete plurality (number), just as "dog" and "horse" are names for
species in the category of substance. To hold that these species names
have objective content - to hold it to be an objective question whether
certain animals are dogs, or whether certain finite pluralities are
sevens - this is straight-forward Aristotelianism. And the Aristotelian
reading of the proposition that seven fives make (a) thirty-five is
that it is a *general* proposition about *all* finite pluralities
partitionable into a seven of disjoint fives.
The Platonist holds that corresponding to these species names
are Platonic Ideas or Forms: the Form of Dog or the Form of Horse, or,
in the category of finite plurality, the Form of Five or the Form of
Seven. A particular belongs to a species by "imitating" or
"participating in" the Idea or Form corresponding to that species.
Now our modern arithmetical notation - 7 x 5 = 35 - has a kind
of built-in "Platonic" bias, insofar as it suggests that the facts of
arithmetic are facts about "abstract mathematical objects", namely,
natural numbers, of which number words and numerals serve as names. We
have only to identify the natural number seven and Plato's Form of
Seven and we are "Platonists". (In fact there are subtle but important
differences between our natural number seven and Plato's Ideal Seven,
but let's put those aside.)
This assimilation of general propositions to particular
propositions about "abstract mathematical objects" pervades the whole
of mathematics, and this distorts our view of foundations.
Mathematicians speak of "the" group of symmetries of the equilateral
triangle, or of "the" Klein four-group, and thereby disguise *general*
propositions about isomorphism classes as *particular* propositions
about "ideal exemplars" of those classes. Even talk about "the" natural
numbers and "the" real numbers is of this character. What we really
have in those cases are the isomorphism classes of simply infinite
systems and of complete ordered fields, respectively.
Every mathematician will immediately recognise the truth of
what I have just said, but will continue (as I will) to speak,
inaccurately, of "the" Klein four-group and "the" real numbers, if only
to avoid a tedious prolixity in his discourse. And such talk is
harmless - indeed, useful - as long as it doesn't mislead us when we
come to discuss foundations.
Maybe we should say that mathematicians are not genuine
Platonists at all, but natural Aristotelians tempted into "Platonistic"
ways of speaking in order to avoid the onerous necessity of having
always to spell out, explicitly, the level of generality at which they
are working.
John Mayberry
Lecturer in Mathematics
School of Mathematics
University of Bristol
--------------------------
John Mayberry
J.P.Mayberry at bristol.ac.uk
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