[FOM] PoV on Ultrafinitism

Bill Taylor W.Taylor at math.canterbury.ac.nz
Wed Oct 25 02:41:42 EDT 2006

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

I'm still kind of wondering where Sazonov 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 am sure it was once crystal clear to them too.  It is NOT necessarily
crystal clear, initially, to the non-mathematician - sometimes I have had
CS/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 for
being 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
jewel of ours, but who nevertheless seem to be reputable mathematicians.
At least it seems so, judging from the fact that they get air time here.

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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