Re: Do mathematicians know their axioms?

On 22 Feb 2006 07:29:04 -0800, "abo" <dkfjdklj@xxxxxxxxx> wrote:


matthias@xxxxxxxxxxx wrote:
abo writes:
I don't think the main guns in the field - Friedman or Simpson, etc. -
would call themselves mathematicians,

I can't speak for either of them, but Simpson's web page says ``My full
name is Stephen George Simpson. I am an American mathematician.''
The people I know who are interested in foundations of math either fall
into the mathematical logic community or the philosophy community,
although I am sure there are other people interested in foundations as
well. I believe that the mathematical logicians, like applied
mathematicians, usually consider themselves mathematicians.

My personal opinion is: if it walks like a duck, looks like a duck,
and sounds like a duck...

And isn't one of the characteristics of the duck where they get
published? My impression was their stuff didn't get into math
journals.


OK maybe. Then we get back to the other alternative I suggested:
namely that mathematicians don't actually use the so-called axiomatic
method - churning out theorems from axioms. They're doing something
else - similar, but something else.

I guess it depends on what you mean by ``axiomatic method''. Most
people would say that mathematics does use the axiomatic method, in
contrast to the experimental method used in the physical sciences. You
seem to be saying that because mathematicians don't use formal proofs
they aren't using the axiomatic method, but I think that you are
overdefining the axiomatic method. I agree that nobody writes formal
proofs. Even in proof theory, the proofs are informal...

That's not quite what I would say. Again, it's not a question of
formal vs. informal. My point is more: if a mathematicians doesn't
even know what axioms he is using (formally or informally), then he is
not using the axiomatic method as it is normally conceived.


How is "using the axiomatic method" normally conceived? Maybe you
could give some example?
.

Relevant Pages