[FOM] Concerning Ultrafinitism

Mirco Mannucci mmannucc at cs.gmu.edu
Fri Nov 3 07:44:22 EST 2006

--Hay un concepto que es el corruptor y el desatinador de los otros. No
hablo del Mal, cuyo limitado imperio es la Etica; hablo del infinito--

(There is a concept which corrupts and upsets all others. I refer not to
Evil,whose limited realm is that of Ethics; I refer to the infinite)

Jorge Luis Borges

 ---- Bill Taylor <W.Taylor at math.canterbury.ac.nz> wrote:

> So without necessarily making any approbation or disapprobation of either,
> is it fair to regard ultrafinitism as "fuzzy mathematical logic"?

The answer to your question is simply NO.

Ultrafinitism is not easy to pin down and strait-jacket, as it is still a FOM  Program in
its very infancy (but NOT still-born!), and people who have an interest in it hold
quite different views, as this list has amply borne out......

However, I think there IS a unifying theme:

 ----> Ultrafinitism = Constructivism  MINUS  Potential Infinity

In other words, one could say that Ultrafinitism is any serious  attempt to establish a coherent
fundation of mathematics on constructive basis WITHOUT giving a free ticket to ANY form of infinity,
not excluded POTENTIAL INFINITY (which incarnates in the infamous 1,2, 3 ..... to an ultrafinitist
the dots ARE THE PROBLEM!!!), in all its various forms and disguises (for instance, using unbridled
and unchecked  induction at the meta-level).

This goes as far (see Rupert's excellent reply to you, chiefly focused on Nelson) as to cast doubts
on whether  the long-standing dichotomy FINITE-INFINITE  has an absolute, uncontroversial  character.

To this effect, notions such as feasibility are investigated, involving a degree of "fuzziness".
The Model Theory and, as I suggested in my two postings on TTP, the Proof Theory of Ultrafinitism,
may (I say may, not must) use methods and techniques from Fuzzy Set Theory and other tools
developed to deal with vagueness, in the attempt to create CLASSICAL models of ultrafinitistic
mathematics. This is exactly the same situation as, say, the use of  Kripke models/realizability
models/topos models for standard intuitionism: these are classical mathematics structures that
provide a REPRESENTATION of intuitionism WITHIN the (supposedly solid) framework of classical

My personal goal  is NOT to evangelize anybody on the absence of N (I could not care less);
instead, I intend to provide a concrete  Model Theory and a Proof Theory for Ultrafinitistic
Mathematics that can be understood, studied and played with by ANYONE within the FOM community
(as long as he/she knows some math, and has no irrational apriori prejudices).

----> Do you want to believe in your "crystal-clear jewel"?

I say, no problem at all. Just do not try to suggest that your faith (and
faith it is)  should be necessarily accepted by everybody and do not assume
that you can read other's minds (I can assure you
that if you try to scan mine, you will get a very BIG head-ache!!!).

As for myself, I have other plans: I am striving to show  that it is at least conceivable
to think coherently of a mathematical universe where the very notion of infinite is
blurred, where the entire chain of large cardinals can be reproduced in the finite realm,
where the essential theorems of FOL (completeness, incompleteness, Lowenheim-Skolem,
etc, etc, etc.) can be reproduced at the "below-omega" scale, where....

In other words, my ultimate objective is to show that  there is NO NEED whatever to banish
anyone from Cantor's Paradise, because it can be miniaturized down to the FINITE REALM (this
Program will be the topic of a future posting of mine, titled: Cantorian Nanotechnology).

If such a program can be carried out, even in part, I honestly think that it would be of
some substantial value to the FOM community at large, as it would force everybody to
re-think WHERE the widespread belief in (or presumed  knowledge of)  INFINITY comes FROM.


Mirco A. Mannucci

P.S. On Nelson: Edward Nelson is one of the prominent US mathematicians, with fundamental
contributions to  Analysis, Set Theory, Stochastic Calculus, Foundations of
Quantum Mechanics, and God only knows what else (see  Nelson' s recent
celebration in the book- Diffusion, Quantum Theory, and
Radically Elementary Mathematics- Princeton University Press,
I believe).

In my modest belief though, the single most important thing Nelson ever wrote is not his
magnificient mathematics, not even his predicative arithmetics (not radical enough
for ultrafinitism, by the way), but a small confession (available at
http://www.math.princeton.edu/~nelson/papers/s.pdf), where he candidly tells
the world when and where he lost his faith in N.

I invite you and everyone on this list to read it (incidentally, according to
your unfortunate  definition, he is an "un-natural mathematician", at
least since his apostasy (1976). Well, apparently he can still do excellent
math, in spite of his avowed lack of belief in N. And, last time I checked,
he did not turn into a lawyer or a doctor...)

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