[FOM] iterative conception/cumulative hierarchy

d_obrien d_obrien at telus.net
Tue Mar 13 01:39:05 EDT 2012

Nik Weaver wrote:
> d_obrien wrote:
> > A so called  "naive" idea that "sets are extensions of concepts and
> > every concept has an extension" is entirely bogus
> Maybe you just don't understand it?
> What I am getting out of this thread is how invested people are in quasi-
> physical ideas about sets: sets are "formed" in stages, or as you put it,
> plurality can be unified in a totality".  Of course these images are
> incompatible with the idea of sets as atemporal abstract objects (how
> does one go about unifying a plurality?), so we have to say that language
> this is merely a colorful metaphor for some abstract static notion of
> metaphysical dependence.  But this suggestion doesn't pan out.  It turns
> that all of the work is being done by aspects of the iterative conception
> get lost in translation.
> So much effort is put into legitimizing talk of "abstract objects" as if
> such things would be an unqualified good.  The harm is that thinking about
> sets as abstract objects invites us to speak, nonsensically, about
> and "unifying" them as if they were physical objects.
> Concepts do not have to be understood as metaphysical entities.  We are
> really just talking about language, specifically, predicates.  When we
> not to distinguish between two predicates that are tautologically
> we are talking about concepts.  If we go further and decide not to
> between concepts that are intersubstitutable (can replace each other
> without affecting truth values) we are talking about extensions.  There's
> magic about "forming" anything.

"Sets are extensions of concepts and every concept has an extension"
combines two universal affirmations in an assertion that delivers the
conclusion "Every concept determines a set", and I don't think a "naïve" or
beginning student would, left to her own devices, consider either of the
premises a self-evident axiom.  I'm speaking as a liberal-arts student who
gravitated to philosophy, and from there (as a part) to a study of what was
called "logic", but which only covered a rather narrow segment of the
history of studies in logic focussed on developments after Frege, with
references to works in algebra attributed to Boole.   You can't get much
more "naïve" than that.

Consider the following concepts:
dolphin, the species dolphin, boat, mountain, number, mathematical point,
affirmation, negation, nothingness,  unicorn, prophet, metaphor, extension,
comprehension, set, collection, army, dignity, true, good, possible, being,
first intention, second intention, mathematics, meta-mathematics, logic, ...
This variety doesn't lead me, at any rate, to jump to generalizations about
them all. 

Consider the idea 'army', as in Caesar's army.  Such an army is an ordered
collection of individual humans, if it exists.  But no such army exists,
today, since Caesar is long gone.  At any point of time Caesar's army, when
it did exist, was always changing as new soldiers were drafted and cannon
fodder was killed off, but it was always designated "Caesar's army".  We
could always know what we were speaking of.  We can abstract from the
military order  of (ranks within) the army to consider only the individual
human constituents, and for any exact point of time the collection seems
fixed in a sense reminiscent of what mathematicians seem to construe that an
unordered collection of individuals should be fixed, that it be considered
to be a set of elements such that the axioms and rules of set theory may
apply.   This is the same sense as "apples on the table" is fixed in number
at an exact time, but isn't fixed in general.  
Would a person, using language in the most naïve and easily understood
sense, speak of "the collection of apples on the table" when there were no
apples, or only one?  In that case, if there were once an actual
collection, wouldn't someone say "the collection is gone, there's only one
now, or none?"   Who wouldn't laugh if someone showed a butterfly collection
which consisted of only one butterfly, or none?  This is material for comedy
just because the usages are ludicrously incongruous.  So even when we
perform the necessary abstractions, from the order of a collection and to
fix a specific time, a similarity between the naïve term 'collection' and
the mathematical term 'set' fails, since not only does set theory posit a
null set, but it posits the notion 'singleton',  so posits the existence of
a system of nested sets which maps Peano's system of numbers.   And I don't
think this is a bad thing.

Although an army is (by definition) an ordered collection of individuals,
this doesn't mean that the extension of the concept 'army' determines such
an ordered collection.   This difference bears on how we understand the
distinction between concepts of first and second intention.   Likewise,
although a collection is a collection of things somehow described, this
doesn't mean that the concept 'collection' determines a collection.    

Re. the concept 'mountain', consider the sorities paradox:
What does this argument make of the notion that "every concept determines an
extension"?  Doesn't this argument suggest that some concepts, like
'mountain', only determine an extension in a relative sense?

And so on.  The two "naïve" premises you affirm just don't pass muster.

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