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C00002 00002	Margaret Boden's %2Jean Piaget%1, Viking Press 1980
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Margaret Boden's %2Jean Piaget%1, Viking Press 1980

47 I'm not convinced that it is wrong to teach a child to
count without imparting a notion of set and 1-1 
correspondence.  From a mathematical logical point of view,
defining numbers as equivalence classes of equinumerous sets is just
one way of doing it, and not even the most popular among
logicians.  Psychologically, the ability to recite numbers
and generate as much of an infinite set as may be wanted
is a worthwhile bit of understanding and quite independent
of any set theory.  Children like to exercise their ability
to count; objecting is like objecting to a child waving its
feet before it is ready to walk.  I am less certain about
the examples Boden cites - learning that "two twos are four"
and learning the sequence of English kings.  However, children
often like to memorize, and the sequence of kings gives
something to tie other knowledge to.

Consider Piaget's point that a small child takes the longer
object to be bigger and also the fatter object to be bigger
and is confused when the Plasticine snake is reshaped.

normally ∧ longer(x,y) ⊃ bigger(x,y)

and

normally ∧ fatter(x,y) ⊃ bigger(x,y)

incite bits of non-monotonic reasoning (which don't fit very well
into previous schemes).  The more sophisticated rule specifically
excepts changes produced by rearranging an object - appealing
to conservation.  By the way, would a child, who found an offered
flexible piece of candy too small, be appeased by an offer to
stretch it?  If not, this would indicate that the concept of
conservation exists but is compartmentalized.  In general, it would
seem that many concepts arise earlier than Piaget thinks but
are compartmentalized to parts of experience not touched by
his experiments.

We can more generally try on the idea that the mental weaknesses
of children consist of compartmentalization and premature
circumscription.