[FOM] Concerning Ultrafinitism.

Bill Taylor W.Taylor at math.canterbury.ac.nz
Wed Nov 1 00:32:46 EST 2006

[Note to moderator:  I sent this several days ago, and have seen/had 
                     no reply.  Is it still in the pipeline? ]

Is the following a reasonable point of view, do people think?

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

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.  It is NOT necessarily
crystal clear, initially, to the non-mathematician - sometimes
I've had CS or business students (amazingly) wonder
"do the numbers go on for ever", though they are
always happy with the simple answer "yes".

But by and large, it might almost be considered a criterion to
be a "natural mathematician", that the idea of N, the naturals,
is crystal clear.  (As opposed to, say, an intelligent doctor
or lawyer who may have doubts about it.) 

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, but who nevertheless seem
to be reputable mathematicians.  At least it seems so, judging
from the fact that they get quite a bit of air time.

My take is this, and I wonder if it is a reasonable view?


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" -

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

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


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

They insist on keeping a fuzzy view on what is the largest
feasible number, and similarly with the largest feasible
derivation; indeed feasible anything - the very concept
of feasibility seems to be the ultimate in fuzzy concepts. 
This viewpoint is *not necessarily* a negative one,
I must point out.  It may be that (unknown to me)
there IS a lot of value in fuzzy math, whatever FM may be. 
This being so, there could easily be value in ultrafinitist
math logic, also.

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


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