[FOM] "Mathematician in the street"
Charles Silver
silver_1 at mindspring.com
Mon Aug 10 13:39:46 EDT 2009
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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