Date: Aug 2, 2006 2:42 AM Author: john baez Subject: Re: This Week's Finds in Mathematical Physics (Week 236) In article <12csui1mguecq8a@corp.supernews.com>,

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:

http://en.wikipedia.org/wiki/Constructivism

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:

http://en.wikipedia.org/wiki/Finitism

I can also imagine various sorts of reasonable ultrafinitists:

http://en.wikipedia.org/wiki/Ultrafinitism

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":

http://www.math.princeton.edu/~nelson/papers/faith.pdf

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:

http://www.henryflynt.org/

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":

http://en.wikipedia.org/wiki/Alexander_Esenin-Volpin