Re: Set Theory: Should You Believe

Kevin Karn wrote:
[...]

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.)

Well, your questions have confounded me too, and I don't have clear
answers. I guess it is partly human nature to be intolerant of dissent,
but that doesn't explain everything. I think it is a big mistake made
by the academicians to plunge headlong into set theory and infinitary
mathematics, ignoring the misgivings of geniuses like Poincare, Weyl,
Brouwer and many others in the early part of the twentieth century. I
guess the dominance of great mathematicians like Hilbert ensured that
these objections were ignored. The institutionalization of research
produced many professionals who had no option but to churn out one
paper after another in the mainstream research areas, with virtually
zero support for any significant pursuit of alternative foundations.
All I can say is that this is just plain wrong. Whoever it was that
decided the funding priorities over the last several decades (ever
since the early 1900's) should have had the vision to support and
encourage alternative viewpoints that may have flatly contradicted the
status quo. Why can't these dissenters co-exist with the mainstream
guys and get funding? That would have led to a healthy balance that is
clearly absent today. It is amazing that today research into
foundations is a very low priority for mathematicians and even
logicians. I would recommend that you look at a recent paper by Lee
Smolin on these issues in the context of theoretical physics (I forget
the title of the paper, but you can search his website and get it).

[..]

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."

In the passage that I quoted in my previous post, what NW said was that
mathematics did not need axioms at all, period (whether set theoretical
or otherwise). That viewpoint presumes a certain 'reality' that
mathematcians have to take for granted and simply make definitions to
access that reality, in their proofs. This is not right -- there is no
way for us humans to access any such reality (certainly about
infinitely many objects, such as, natural numbers). This alleged
'reality' is just a figment of our imaginations -- which is what I
would call axiomatic declarations in the human mind. i.e., these
"truths" have their (temporary) existence in the human mind as axioms.
Using the rules of inference in some system of logic that the human
mind accepts, one can deduce theorems from these axioms. Ultimately all
the theorems of such a theory are therefore just declarations made by
the human mind. This is the position that I take in my work on the
logic NAFL, and you can show why infinite sets are not acceptable in
NAFL theories (but infinite proper classes, which are not mathematical
objects (i.e., sets) are acceptable in NAFL theories).

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 Wildberger 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.

How would you define real numbers, for example, without set theory?
Wildberger seems to assert in his paper that as long as we have a
finitely stated rule for generating the n'th term in a Cauchy sequence
of rationals, there is no need to consider a real number as an infinite
object. I don't agree with this. For starters, you need to talk about
an "arbitrary" natural number n to generate this rule -- what is this
n? It only makes sense to consider n as a variable that can take on
*any* of infinitely many possible values; in my view this presumes the
existence of an infinite class of natural numbers. Similarly the range
of the sequence is another infinite class of rationals described by the
rule for generating r_n, the n'th rational in the list. Wildberger
rejects this and says that a function does not need to be considered as
having an infinite domain and an infinite range, as long as there is a
finite rule to generate the function. My own view is that the existence
of these infinite classes themselves is not the problem; it is
quantifying over these classes, i.e., formally referring to infinitely
many such infinite classes in a formula, that constitutes infinitary
reasoning, tacit in set theory or almost any modern mathematical
theory. This is what my logic NAFL avoids and one can still do real
analysis in NAFL as I pointed out in my previous post. Wildberger's
objection to an *arbitrary* real number x (see my previous post) can
now be rationalized as an objection to quantifying over all the
possible real values that x can take, since each real is an infinite
object. Otherwise what precisely is Wildberger's objection to an
arbitrary real number, since he already accepts arbitrary natural
numbers?

Regards, RS

.

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