FOM: Objectivity

[John Mayberry]

	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

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John Mayberry
J.P.Mayberry at bristol.ac.uk
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