Date: Aug 2, 2006 2:42 AM
Author: john baez
Subject: Re: This Week's Finds in Mathematical Physics (Week 236)

In article <>,
Jim Heckman <weu_rznvy-hfrarg@lnubb.pbz.invalid> wrote:

>OK, but I'd be interested to know which ZF axioms your "imagine[d]
>reasonable people" don't believe. Or is their problem with
>mathematical logic?

I can imagine all sorts of reasonable people who believe all sorts
of things. And, I even know some of them.

For example, I can imagine various sorts of reasonable constructivists:

and my former student Toby Bartels (who just got his PhD) is one.
Most such people don't believe in the law of excluded middle, so
ZF is right out. And, I believe most of them don't believe you
can well-order uncountable sets, because I've never heard of any
way to "construct" a well-ordered uncountable set, in the technical
sense of "construct".

I can also imagine various sorts of reasonable finitists:

I can also imagine various sorts of reasonable ultrafinitists:

meaning people who don't believe in unbelievably large finite numbers.
Unfortunately, it seems hard to develop good axioms formalizing this
view, perhaps because the normal concept of proof allows arbitrarily
long proofs. I know Christer Hennix and Alexander Esenin-Volpin have
tried, but I don't know how far they've gotten. Edward Nelson hasn't
worked much on ultrafinitism, but he has expressed sympathetic views in
his book "Predicative Arithmetic". In his article "Mathematics and Faith":

he writes:

I must relate how I lost my faith in Pythagorean numbers. One
morning at the 1976 Summer Meeting of the American Mathematical
Society in Toronto, I woke early. As I lay meditating about numbers,
I felt the momentary overwhelming presence of one who convicted me
of arrogance for my belief in the real existence of an infinite
world of numbers, leaving me like an infant in a crib reduced to
counting on my fingers. Now I believe in a world where there are no
numbers save that human beings on occasion construct.

Personally I don't advocate any of these positions, and like Tom
Leinster I am happy that you can do mathematics without "believing
in" any specific axiom system.

Personal stuff:

Edward Nelson is a mathematical physicist at Princeton who like me
was a student of Irving Segal. I never discussed logic with him,
though he read and critiqued my senior thesis when I was an undergrad,
and this thesis was on applications of recursive function theory to
quantum mechanics.

I used to argue heatedly with Christer Hennix, because he regarded
all mathematics using infinity as a sham. I should have spent my
time asking him how Esenin-Volpin's alternative system was supposed
to work. But our discussions weren't a total waste, because I met
my wife through a friend of his - Henry Flynt:

known as the inventor of "concept art", musician, and cognitive
nihilist. I'm not sure Henry Flynt would want to be characterized
as a "reasonable person".

I only met Esenin-Volpin a couple of times. Besides being the son of
the famous Russian poet Sergey Yesenin and the main proponent of
ultra-intuitionism, he is known for being a topologist, a dissident
during the Soviet era, and a political prisoner who spent a total
of 14 years in jail and was exiled to Kazakhstan for 5. His
imprisonments were supposedly for psychiatric reasons, but Vladimir
Bukovsky has been quoted as saying that Volpin's diagnosis was
"pathological honesty":