Re: A quiet query from a visitor

On Tue, 21 Aug 2007 14:21:34 +0200, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> wrote:

Angus Rodgers wrote:

Hey, leave me out of this! I admit to discomfort with the
usual presentation of ZFC as a foundation for mathematics.

Alas. That alone makes you already kind of a "dissident", in my highly
personal and biased classification. Please, read the disclaimer below.

Disclaimer. The above lists are only an approximation of the reality in
'sci.math' and highly relflect the author's opinion and experience.

I think the main thing wrong with this "approximation"
(questions of individual membership or non-membership
of this or that "school" apart) is that sci.math does
not have more than one "school" of mathematics. Maths,
insofar as it gets done here at all, gets done within
one "school". And then there are people like me who
sit in school, but sometimes stare out of the window,
and forget to pay attention to the lesson; and there
are people outside the school (some of whom want to
burn the school down); and perhaps somewhere there is
a Galois or Ramanujan, who can't get into the school,
but is doing mathematics anyway (while perhaps, in the
case of Galois, also wanting to burn the school down).

However, I haven't seen any evidence of the latter. What
we have is a variety of opinions as to how the school is
run, and whether all its rules make sense; but there is
no other school.

Nor do I think there should be another school. Mathematics
isn't /about/ its foundations. The existence of different
opinions about how mathematics is justified doesn't lead
to the creation of different mathematics. I know this
has to be argued for (especially in view of the claims
of Intuitionism, in particular); I suppose I'm just saying
that whatever the problems with the current orthodoxy in
mathematics, the dragon (or is it angels with flaming
swords?) guards a real treasure; and I for one would like
to be admitted into Cantor's paradise (without necessarily
assuming that it is the whole world of mathematics).

Perhaps the house of mathematics has other mansions too,
but I'd rather see some ... er ... constructive effort
to build those than see Cantor's house being torn down.

I'm sorry about all the mixed metaphors, and I know I
haven't written this at all clearly. I am interested in
problems in the foundations of mathematics, especially
to do with whether everything in mathematics is a set
(which I must say does seem a rather barmy idea), and
how you apply mathematics to the "real world" (especially
how set theory accounts for such applications). But it is
a really tragic mistake to allow one's worries about the
/form/ in which mathematics is presented to deprive one
of contact with its /substance/.

I am not aware of any language other than that of set
theory in which mathematics can be presented. (I am
certainly not against efforts to do so, even if such
efforts must be experimental and toy-like in the early
stages; but I haven't even seen any early constructive
efforts in sci.math. Here I mean "constructive" in the
everyday English sense.)

Apropos of that word that I just used in another sense,
I have a vague question, which perhaps someone here can
answer. Do schools of constructivism differ in respect
of whether they regard non-constructive mathematics as
meaningless or worthless? Do some constructivists want
to tear down the non-constructive edifice, while others
see their work as taking place within the same edifice
as other mathematicians, but limiting attention to the
"constructive" part of it?
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.

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