[FOM] Concerning Ultrafinitism.

[V.Sazonov@csc.liv.ac.uk]

Quoting Bill Taylor <W.Taylor at math.canterbury.ac.nz> Wed, 01 Nov 2006:

> I'm still kind of wondering where Yessenin-Volpin, Edward Nelson,
> and other ultrafinitists are coming from.

One of the first papers of Yessenin-Volpin on ultrafinitism was called 
Analysis of Potential Feasibility (Logical Investigations, Moscow, AN 
USSR, 1959, 218-262 (in Russian). I think, the title says for itself.

>
> They purport to find, or rather take the public stance of finding,
> that the concept of "all the naturals" is confusing and vague,
> whereas it is indeed *crystal-clear* to the rest of us. I'm sure
> it was once crystal clear to them too.

Until they have not tried to analyse.

It is NOT necessarily
> crystal clear, initially, to the non-mathematician


> But by and large, it might almost be considered a criterion to
> be a "natural mathematician",

...appropriately educated one [all of us are somewhat "damaged" by the 
education system] and knowing (either explicitly or implicitly just by 
the training provided by the "school") which formal rules and axioms to 
use. When you have learned how to reason and have no further questions 
then, of course, you will start to believe

that the idea of N, the naturals,
> is crystal clear.

see also below about ideas and fantasies and mathematics.

(As opposed to, say, an intelligent doctor
> or lawyer who may have doubts about it.)

or an educated mathematician who starts asking some "stupid" questions.


>
> Now, given this, what are we to make of ultrafinitists,
> who purport to find vagueness or ambiguity in this basic
> crystalline abstract jewell of ours,

a "Sacred cow"?

> Some time ago, back in the late seventies to early eighties,
> there was a brief flurry of interest from fringe mathematicians
> in "fuzzy math".  It was never quite clear what this was, but it
> still has a small amount of library shelf space, though perhaps
> little or no presence in math departments in academia.
> It seemed to be (AFAICT), basically, that joke that
> used to go around about "Generalized Mathematics" -

an awful idea on mathematics, by my opinion.


> *  "In Orthodox math we derive true results by valid means;
> *   in Generalized math both these restrictions are dropped!"

Mathematics, "by definition", deals with our (inevitably vague) 
imagination and fantasies governed/restricted/strengthened by FORMAL 
axioms and rules. This definition is very general, but it does not 
change the nature of mathematics. It only changes some traditional 
angle of view on mathematics - a wrong angle assuming beliefs in 
fictions having no scientific grounds. Of course this definition is not 
about true results (which ever truth if we are dealing with fantasies 
and imaginations; recall, e.g. Imaginary Geometry of Lobatshevsky). But 
what you call valid methods is, in fact, formal axioms and rules 
(governing our fantasies).

>
> Anyway, one can hardly say that Fuzzy math even died - it was
> practically still-born...  math departments gave it very short shrift.

I am not an advocate of Fuzzy math, but as I know Peter Hajek presented 
a quite nice mathematical (rigorous) approach to fuzzy logic. Any 
problems with this? Or any pretensions to probability theory, etc? The 
ideas may be as arbitrary as our fantasies. The only problem is whether 
they are governed by formal rules. Then it is just a normal mathematics.

> So finally, my question is this:-  is it a fair point of view
> to regard ultrafinitism as essentially, fuzzy mathematical logic?

I think not. The point of ultrafinitism, as I understand it, is the 
analysis of potential feasibility. Fuzzy math has some other starting 
point. They might be related in some way, but this is a different story 
which does not exist yet.

See also some citations from Troelstra and van Dalen presented, e.g., 
in http://cs.nyu.edu/pipermail/fom/2003-June/006716.html

The point is that constructivism requires that existence should mean 
the possibility to construct an object. The question is whether we can 
guarantee that the construction is real(istic).

Thus, the question is how the ordinary abstract natural numbers are 
related with "real", or feasible numbers (some initial part of N). 
Either we reduce everything to complexity theory based mainly on 
asymptotical considerations (which, being based on classical 
mathematics, also ignores or has no non-asymptotic approaches to the 
"real" feasibility) or we could try to do something on the level of 
foundations of mathematics trying to find a reasonable formal approach 
to the very concept of feasible numbers.

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

As I told, I do not believe so. The "set" of feasible numbers looks a 
vague or fuzzy one, but the question is rather about how, starting with 
naively understood feasible numbers we arrive to highly idealised 
(still vague AS ANY OTHER IDEA at all) of mathematical numbers. Can we 
approach to all of this rigorously, as it is required in mathematics so 
that the formalisation will include the idea of feasibility in an 
appropriate way, or the only thing we could do is dully speculations?  
This could extend mathematics by new concept of feasible number, but 
the main feature of mathematics - its rigorous character - will remain 
at least at the same level if not higher. Thus, as I understand, the 
question is not in introducing in mathematics something foreign to its 
nature. Mathematical rigour is not intended to be relaxed in any way! 
It is about extension of its ability to formalise new kind of ideas (or 
fantasies; see the definition above).


Quoting Gabriel Stolzenberg <gstolzen at math.bu.edu> Thu, 02 Nov 2006:

>   Alik Volpin's "ultra-finitist" program had the aim of proving
> the consistency of mathematics.

I remember his further paper with this "attempt", but, by my opinion, 
this was not a real proof and too far to generate a proof. To prove 
anything from "ultra-finitism" it was necessary to have it first 
formalised. I like the general ideas and fantasies of Yesenin-Volpin as 
stimulating ones, but I have not seen anything written by him what 
could be considered as a sound formalisation of *these* his ideas. More 
precisely, I was never been able to understand his writings on this 
subject (even when he used some formulas) as truely mathematical. (See 
again the above definition of mathematics.) Is anybody able to 
understand his "proofs" concerning ultrafinitism? As I know, the first 
person who indeed formalised something similar was Rohit Parikh.


Vladimir Sazonov


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