[FOM] "Mathematician in the street" on AC

[Charles Silver]

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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