Re: Set Theory: Should You Believe

R. Srinivasan wrote:

<snip>

I would
also suggest to NW not to waste time burying himself too deeply in the
abstruse stuff discussed in the FOM newsgroup; most of the people there
are already committed to this or that viewpoint and will not accept any
questioning of the status quo. In other words, these people are
anything but objective in analysing and questioning the sorry state of
the foundations of logic and mathematics as exists today. However I
will not deny that there are some interesting discussions on that list
from time to time. But if at all anyone successfully questions the
existing foundations and proposes an alternative, it will have to be an
amateur with no stakes in the status quo and with no obligation to
his/her colleagues to justify the status quo; the FOM list does not
meet this criterion.

This is a question from the sociological angle, but I'm curious to know
what you think. Why, in your opinion, is the orthodoxy in set theory
etc. so entrenched? Why is the idea of questioning/rejecting infinity
so threatening? What exactly is the stake which people have in the
status quo? What do they have to lose if the status quo is upset? Why
is the resistance so fierce? (Sorry for so many questions. :-) I think
you can see what I'm driving at.)

Mathemaitcs is about producing proofs
and proofs do require axioms as starting points. To say that
mathematics does not require axioms is the same is saying that
mathematics does not require formal systems or even logic itself. But
that will quickly lead us into contradictions and just plain confusion
as well.

When NW said that "You don't need axioms", I understood him to be
saying something more nuanced -- i.e. that "You don't need set theory,
or the axioms of set theory, to do mathematics."

As he said:
" Whenever discussions about the foundations of mathematics arise, we
pay lip service to the `Axioms' of Zermelo-Fraenkel, but do we every
use them? Hardly ever. With the notable exception of the `Axiom of
Choice', I bet that fewer than 5% of mathematicians have ever employed
even one of these `Axioms' explicitly in their published work. The
average mathematician probably can't even remember the `Axioms'. I
think I am typical---in two weeks time I'll have retired them to their
usual spot in some distant ballpark of my memory, mostly beyond recall.
"

It's very clear that you don't need set theory or the axioms of set
theory to do mathematics. After all, virtually the entire body of
pre-20th century mathematics was developed without set theory, or
axioms of set theory. Even today, mathematicians like NW do productive
work without even knowing set theory, or the axioms of set theory, and
this clearly shows that the whole field is an unnecessary appendix.

.

Relevant Pages

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