[FOM] "Mathematician in the street"

[Charles Silver]

	What I'm about to say may have some small relevance to the  
"mathematician in the street," (MIS) and his knowledge and opinion of  
AC and CH.
	From speaking to various mathematicians outside of foundations/ 
metamathematics (found/meta), I've started to believe that some MISes  
who are relatively advanced in a specific discipline *outside* set  
theory, *believe* that they ****know*** set theory and also, if they  
can state AC and CH, then they also have opinions about their truth  
value, regardless of their state of knowledge about anything else in  
set theory.  I should add some qualifications.   Those I spoke with do  
*not* belong to a department that has a found/meta area.  Though this  
certainly has some bearing on this issue, it is not unusual.  I'll  
hazard a guess that fewer than 14% ( 1/7) of math departments have  
*any* found/meta area.  [<<--I'm hoping someone can provide a more  
accurate figure.]  (This is not to say that there aren't some isolated  
mathematicians who know some logic & set theory, but I'm supposing few  
of them have any opportunity to teach a course in set theory.  And if  
they do, I think they go only a bit further into the topic and only in  
slightly more detail than can be found in discrete math books.)
	At any rate, I now believe that the MIS has lots of views about AC  
and (if he's heard about it) CH as well.   One particular conversation  
I remember was with a very smart guy whose specialty is listed in his  
department as "Ergodic theory, almost everywhere convergence".  I'd  
been bitching to him that <no one>  knows logic or set theory in the  
math department (actually in <<the entire university>>.   He remarked  
with great surprise, telling me *he* knew set theory, and that AC is  
true.  I asked him a couple of technical questions (which I am not  
knowledgeable about either, all I know are the questions), and he  
quickly settled down and changed the subject.  He's unusual, though.    
Most mathematicians who do not know any set theory (beyond that in  
discrete math) don't know that they don't know it.  But, if they've  
used AC (and knew they were using it) they also have an opinion about  
AC, which is of course that it's true.
	So, in general, I think an MIS will believe in the truth of AC and/or  
CH if he/she has knowingly used that one in a proof or would gladly  
use it if the occasion came up.
	  For those who have only *heard* about AC and CH (even though they  
may have unknowingly used one or the other in a proof--harder to  
believe CH is used without knowing it, though), my notion is that they  
believe them to be true even more strongly than those who know what  
they mean.  I'll emphasize this: the MIS who's only *heard* of AC and/ 
or CH, believes in it (them) even more strongly than those who know  
what they mean--my opinion, anyway.

	By the way, I find it deplorable that very few even excellent schools  
do not have a foundations area--i.e., a college or university in which  
set theory, logic, recursion theory, model theory, proof theory, and  
whatever I've missed
are not taught.   I repeat my question above about the percentage of  
departments in which found/meta is taught.  Does anyone know?  I'd  
also like to hear opinions why the percentage is so low.  I'm biased  
or I wouldn't be on FOM (though I guess many "lurkers" read these  
posts for other reasons).   If mathematics is thought to be taught for  
its "relevance," I think the f. of m. is as justifiable as many  
standard math courses.   Another question:  Has the number of courses  
in found/meta gone up or down?   I will not mention the university  
(which is famous for found/meta), but I was told by someone there that  
nowadays the only subject of interest is pure set theory.  That also  
strikes me as deplorable.
	On a separate matter, it seems curious to me that the only two--or at  
least the *main* two--contenders for foundational status are ZF (or  
maybe ZFC) and category theory.   If I'm sadly mistaken, could someone  
please set me straight about this?   (According to my admittedly  
untutored opinion, neither one is satisfactory, and thus efforts  
should be made to establish <<genuine foundations> for mathematics,  
rather than increasing the number of theorems in either discipline.)

	Well, surely I've stuck my neck out enough to get some responses....

Charlie Silver
P.S.  No loony-tunes need reply; you know who you are.

On Aug 9, 2009, at 1:25 PM, joeshipman at aol.com wrote:

> That can't be right, if you define "mathematician in the streeet" as a
> random person with a Ph.D. in mathematics. Most of the graduate
> cirriculum in pure math (and some of the undergrad curriculum) makes
> regular "unbracketed" use of AC (in the form of Zorn's Lemma, maximum
> principles, constructions involving infinitely many arbitrary choices,
> assuming every vector space has a basis, the Tychonoff theorem, etc.).
>
> I agree that applied mathematicians have no use for AC, but pure
> mathematians are a very important subpopulation of "all  
> mathematicians"
> and they have typically been taught to prove theorems with a toolset
> that relies on AC in many ways.
>
> -- JS
>
>
> -----Original Message-----
> From: T.Forster at dpmms.cam.ac.uk
> To: tchow at alum.mit.edu; Foundations of Mathematics <fom at cs.nyu.edu>
> Cc: fom at cs.nyu.edu
> Sent: Sat, Aug 8, 2009 8:42 pm
> Subject: Re: [FOM] throwing darts at natural numbers (rejoinder to
> Arnon Avron's reply)
>
>
>
>  I think we are in danger of forgetting that not only do
> most mathematicians-in-the-street not believe AC, most of
> them have no intuitions about it and cannot state it even
> roughly, let alone have any idea how to use it.  After all
> most mathematicians are applied mathematicians. Ask yourself:
> how many applied people do i know who can correctly state
> AC?  We are a charmed circle!
>
>       tf
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