[FOM] "Mathematician in the street" on AC
Charles Silver
silver_1 at mindspring.com
Fri Aug 14 10:08:14 EDT 2009
On Aug 12, 2009, at 4:57 AM, Sara L. Uckelman wrote:
> Henrik Nordmark wrote:
>> A very large majority claimed that AC was generally accepted as true.
>
> Do you mean by this that a large majority claim that they themselves
> think it is true, or that a large majority claim that they think most
> other people think it's true, though they themselves may have a
> different opinion? If the former, I find this result pretty amazing.
> Since this thread began, I've started quizzing some of my non-
> logician-
> but-still-mathematician friends about their knowledge and beliefs of
> AC,
> to get some informal data of my own. Most of them can state AC or one
> of its equivalents (usually Zorn's Lemma) without having to look it
> up (or feel guilty that they can't because they think they should be
> able to), but when asked whether they think it's true, they take a
> much
> more pragamatic approach, saying things like "I'm not sure that it
> even
> makes sense to ask whether it's true or false, but I have no qualms
> about using it in a proof" and "I'm not sure that "true" is the right
> term. I think it's useful, and that doing without it would be
> frustrating and annoying (unless you were making a career of doing
> without)."
>
> Did the mathematicians you spoke with who think AC is true say
> anything
> about what grounds their belief that it is?
Your results are much more interesting. Mine were very vague, and
perhaps those I asked felt defensive and just went along. You're
right about using Zorn's Lemma, and I think you're right that they use
AC in this form. But I think they just use it as one of the elements
in their bag of proof tricks. I'm not sure they're conscious of using
it, yet they seem to still have the notion that perhaps they do use it
and therefore think they better say it's OK.
I shouldn't have put CH in the same class with AC, except that it
seemed to me that many think c must come immediately after
Aleph_naught, and may even think the proof that the cardinality of the
reals is greater than that of the naturals establishes that. I don't
know; I'm not at all sure of this and didn't want to press them.
To Joe Shipman: I do not think that AC's status as an axiom matters,
since mathematicians in the large aren't interested in set theoretic
axioms anyway--and there's a good chance they're not at all familiar
with set theory beyond unions and intersections, except extensionality
and the proof that the def'n of ordered pairs passes the test.
However, I did notice one time when I passed an empty classroom that
the question of CH was posed in an odd way (which possibly was
incorrect; I couldn't tell from the scribbling). There were lots of
mistakes on the board about G's theorem and his CH result. I got the
impression that the material was not really part of the course but was
just lagniappe.
The views I expressed depend on my interpretation of what the
mathematicians I spoke to <<really thought>>, since their remarks were
so--I think deliberately-murky. Hence, I admit I could well have
misinterpreted them. (Understanding what they thought amidst the fog
of their utterances presented a challenge.)
Please try it out yourselves. At a university that gives no
foundations/metamathematics courses, which I already mentioned, seem
to dominate, ask about AC and CH. Of course, I'm addressing FOMers
here, so probably most of you are not at such schools (or you wouldn't
be able to teach metalogic, set theory, etc.)
My guess is that Sara L. Uckelman is not at a place where foundations
are excluded. Am I wrong?
Charlie Silver
> --
> Sara L. Uckelman
> Institute for Logic, Language, & Computation
> Universiteit van Amsterdam
> http://staff.science.uva.nl/~suckelma/
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