FOM: The logical order v. the psychological order

[Neil Tennant]

Dean Bruckner asks for arguments against---maybe even just some simply
stated  objections to---the view that learners can grasp the concept of
number before they grasp, or appreciate the truth of, Hume's Principle.
I'd like to provide some such objections here.

But first, let me quickly say that I'm not one of those neo-Fregeans
enthralled by HP. I see no reason to accept the existence, for *every*
property F, of the number of Fs. Yet this extravagant existence claim is a
quick consequence of HP, using standard logic (which is what the
proponents of HP use).  I believe that the constructive content of Frege's
logicist grounding of arithmetic can be extracted from ontologically less
committal assumptions: assumptions strong enough to yield just the natural
numbers, and no more. I prefer instead a constructive logicist account,
based on a free logic, that secures the existence of 0, and thereafter the
existence of a successor of any number that exists. The rules that do this
are given in my book Anti-Realism and Logic, 1987. They suffice for the
derivation of the Peano-Dedekind axioms within intuitionistic relevant
logic. The intuitionist/constructivist can be a logicist about natural
numbers.

The adequacy condition for a fully satisfactory theory of the natural
numbers is that the theory should have as theorems all instances of the
following adequacy schema:

	There are exactly n Fs
		iff
	    #xFx = _n_

where _n_ is the numeral for n (that is, s...s0, with n occurrences of the
successor function sign s). This requires a language containing both the
numerals and the primitive variable-binding term-forming operator
#x(...x...), applicable to sortal predicates. Thus the overall theory of
number will be both "pure" and "applied".

I suppose it is a matter of both intuition and rigorously developed
logical and philosophical analysis, combining to yield what philosophers
like to call a reflective equilibrium, that inclines me to say (whereas,
perhaps, Bruckner would not say) that one needs to acknowledge a grasp of
the truth of all instances of the adequacy schema as a necessary condition
for the attribution of full grasp of the concept of (natural) number. That
is, little Johnny needs, in order to be a fully-fledged master of the
latter concept, to appreciate not only the truth of

	There are no apples in the basket
		iff
	the number of apples in the basket = 0


	There is exactly one apple in the basket
		iff
	the number of apples in the basket = 1 (i.e., s0)


	There are exactly two apples in the basket
		iff
	the number of apples in the basket = 2 (i.e., ss0)

	... etc.


but also to appreciate that this would hold quite generally, for any
sortal predicate Fx in place of "x is an apple in the basket".

Any adult who grasp the concept of natural number appreciates, on
reflection (that is, a priori), that all instances of the adequacy schema
must be true. Why should we demand less of a child, and be prepared to
attribute full mastery of the concept of natural number when in fact one
has at most a few bits of evidence for merely partial, but not total,
grasp?

Today my son, aged 2 years and 11 months, counted to 17 as he descended a
staircase, uttering exactly one numeral for each step, in the right order.
Plainly in view was the fact that on each step was exactly one upright to
the handrail. In all likelihood, if one were to ask him how many such
uprights there were, he would respond by counting them separately (and
"again", as it were).

Proud father though I am, I would hesitate to credit my son with a grasp
of the concept of natural number. Maybe that's his unfortunate lot as the
offspring of a philosopher. I just know that he is at too immature a stage
of intellectual development. Indeed, he displays just the sort of
responses that Bruckner latches on to, in being all too anxious to
attribute a grasp of number. My son distinguishes one foot from his other
foot, and knows that he has two eyes. That is, in response to the question
"How many eyes do you have?", he will answer "Two". But at this stage he
is too happy to count every small finite set in order to reach its
"number", rather than take any shortcut afforded by an obvious one-one
correspondence. Put out five plates, each with exactly one cup-cake on it,
and he will count the cup-cakes and reach 5. Ask him how many plates there
are on the table, and he will re-count, and not be struck by the
coincidence that the number of places is also 5. Every quintet is sui
generis for him at this stage: each one seems to clamor for the right to
be counted "iself", rather than be treated in a sort of second-class,
derivative way, via the quick-and-dirty method of spotting a one-one
correspondence with an already counted quintet.

For as long as this primitive fascination for the re-count continues, I
shall refrain from crediting my son with a full grasp of the concept of
natural number.

And what is wrong with that? Why should the fullest, and deepest, truths
about the matter be the most evident, or the first ones to be mastered? My
son will eventually tumble to the fact that he could have determined the
number of plates by just recalling the number of cup-cakes. And then I
shall count him as a real counter, or number him among the number-minded.
Until such time, he is just playing the necessary language-games on the
way to fluency in the contingent conventions by means of which we make our
counts. ("Eleven, twelve,.." caused some non-plussed looks, until he heard
the refrain of more familiar word-stems in "fourteen, fifteen (just),
sixteen, seventeen,...". "Twenty" will be the next hurdle, and after that
it'll be plain phonological sailing until he hits a "hundred".)

There are also many problems of interpretation to be overcome before
crediting learners with proper grasp of the concept of natural number.
Does the learner understand both of the sortal predicates involved in a
one-one correspondence? Can he see that the correspondence is indeed 
one-one? Does he understand the *point* of counting? Small finite counts
are treated by many a child as just a form of linguistic entertainment,
like reciting a nursery rhyme; when do they "twig" that this so-called
final number that they reach, for the predicate F in question, is not just
a familiar sound, arrived at by a familiar pattern of recitation, but is
actually the *number* of Fs? When do they appreciate that they will reach
that same number, for these Fs, every time they count the Fs, regardless
of the order in which they count them?

So, to summarize my reply to Bruckner, I'd say that he is perhaps
underestimating the strength of the necessary conditions for proper grasp,
and perhaps mislocating the crucial elements of such grasp in the
developmental-psychological order.

And these are the sorts of points to which, for philosophers, the
brief mentions of Aristotle and Frege were highly elliptical pointers.

___________________________________________________________________
Neil W. Tennant
Professor of Philosophy and Adjunct Professor of Cognitive Science

http://www.cohums.ohio-state.edu/philo/people/tennant.html

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