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In article <1161923879.394890.236700@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, "Bill
Taylor" <w.taylor@xxxxxxxxxxxxxxxxxxxxx> wrote:
!! 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"?
my biggest exposure to finitists of various persuasions
comes from philosophy departments
that is an indicator of the type of arguments i have seen
the approaches i have seen
might be better called "ultra-engineer"
the claim behind many approaches
is that mathematics does not model mathematics very accurately
that mathematicians are finite manipulators
that the number of steps they will ever take is finite
and the number of symbols manipulated also finite
and so it is not meaningful
at least in modelling mathematics
to speak of infinite processes
they are counterfactual
and in fact anti-factual
can never be factual
now of course this can be formalised in various ways
and one that is particularly well done
can be found at
http://arxiv.org/abs/quant-ph/0108121
the point of ultrafinitists
is that this changes the types of objects
that are allowable in the model's ontology
esenin-volpin
describes some of the structural limitations
one must place on the models of natural numbers
in a modal logical representation
but that is only a small illustration
of the many constraints finite models obey
ebbinghaus has a classic book on finite model theory
ultrafinitsm takes a conceptual step beyond finitism
by stressing that
not only is mathematics a finite process
but there exist hard limits
there is no potential infinity
or legitimate means to assume
a process can be continued indefinitely
in any derivation
it is necessary to question
for any process specification
whether that process can complete
"within the limits of resources"
available to mathematics
because
they insist
mathematics is a physical process
and one day too may suffer the entropic decay
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i have not seen this done in "fuzzifying" the truth characteristic
(by
for instance
making it a more general subobject classifier)
but you could possibly develop such a theory
if you study the formalisation of ultrafinitsm
and follow the debates on appropriate model constraints
papers like
"characterising finite kripke structures
in propositional temporal logic"
by browne, clarke, and grumberg
start standing out
!! Christian rationality:
!! A god killing himself to save his own creation from his own wrath.
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galathaea: prankster, fablist, magician, liar
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