[FOM] a series of grammatical confusions
Gabriel Stolzenberg
gstolzen at math.bu.edu
Thu Feb 16 18:45:30 EST 2006
In his message, "predicative foundations," February 15, Nik Weaver
writes,
> "the very notion of a set ... is based on a series of grammatical
> confusions." This quote is taken from the paper "Grammar and sets"
> by Hartley Slater, to appear in the Australasian Journal of Philosophy.
> Section 2 of that paper contains a thorough, and in my opinion,
> absolutely decisive refutation of the platonic conception of sets.
This is exciting. I spent years (in fact, decades) attempting
(admittedly, not very hard) to share the observation that what we
call "set talk" in mathematics is based on grammatical confusions.
I had zero success. So it will be very interesting to see if Slater
has better luck. I hope so!
If I could ask only one question about his paper, it would be
whether the objects of his critique include singleton sets. If
they don't, have a problem. To me, {1} is pretty much the general
case.
My own approach focused mainly on how set talk is taught.
For example, at the beginning of Paul Halmos's "Naive Set Theory,"
he says something like "A pack of wolves, a flock of pigeons, a bunch
of grapes, all these are examples of sets." But he doesn't say what
property they share that makes them all sets.
That's implicitly left to the reader's intuitions about our use
of words like "bunch," thereby encouraging the assumption that these
non-mathematical examples provide the right intuitions for mathematical
set talk. Not long after I read this, I asked Gerry Sachs whether
bunches of grapes, etc. are the same kind of sets as in mathematics.
He said no.
Sometimes, by way of encouraging a student to believe that she
knows what sets are, she is told that different predicates can define
"the same set." (The same what?) The student is then presented with
simple examples that she is likely to believe she understands because
she is likely to focus on "the same," which she does understand, but
not on "set," which she does not.
I'll leave it at that.
Gabriel Stolzenberg
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