FOM: Part I:Ultrafinitism,Naturalism,Vagueness

[Robert Tragesser]

        Sazonov has introduced the topic of
ultrafinitism.  This note means to shed some
circumspective light on the broader logical,
mathematical,  and philosophical issues involved.
I divide it into FOUR Parts (four successive
FOM posting).

        This must be a rather longish note because
the basic concepts of ultrafinitism are so very
poorly understood  [[for example,  it is widely
thought that ultrafinitism is essentially committed
to small,  somehow concrete numbers;  but this
is a _very_ serious error]] and,  not only is
ultrafinitism poorly understood,  but it stands
within a conceptual frame that is rather alien to
"traditional foundations" and f.o.m.: It does
regard mathematical conceptions as being a product
of idealization of the natural world (rather than
abstractions from the natural world and rather than
generalizations based on empirical phenomena). 
But it regards the idealizations culminating in
(say) ZF as not only not the only way of achieving
(a) mathematics; it also regards them as (variously)
deficient,  defective,  unnatural,  ranging far into
the insignificant,  etc.  Ultrafinitism in its more
open-minded form may be regarded as a quest for
a creation of a novel,  more faithfully natural,
and signficant mathematics by seeking for novel
chains of idealization,  beginning with natural
appearances,  and going onto a mathematics that 
is closer to,  and more evenly and thoroughly 
regulated by,  natural appearances/phenomena.
        Exactly: what is then rather different about
ultrafinitism is that while recognizing (with
many "classical" mathematicians,  such as Simpson, 
MacLane) that mathematical concepts significantly orginate
with natural phenomena,  ultrafinitism attends to
_the difference_ between the mathematical concepts
and the natural appearances from which they
orginate and sees (with e.g. Plato} that it is
not just a matter of cutting out the mathematical
ideas from the appearances [not a matter of "abstraction"]
but,  rather,  there is a process bridging
the natural phenomena and the mathematical ideas --
idealization:
        Ultrafinitism goes back to natural
appearances and more self-consciously undertakes
the process of idealization,  but this time 
aiming at mathematical concepts which fit nature
more snugly.  IT IS LOOKING INTO -- EVEN RECOGNIZING --
THIS PROCESS OF IDEALIZATION,  OF thus RE-ORGINATING
MATHEMATICAL CONCEPTS,  but this time with
greater thoughtfulness, with loving care and 
circumspection -- that is so alien to,
or strange to, traditional philosophical, founda-
tional,  and f.o.m. aims,  and the reason that
these "notes" must be so extended.

        Basic literature is located below,
though I will refer to sources before that
location. 
 

PART I: BACKGROUND & OVERVIEW

Ultrafinitism begins with the thought that "classical
mathematics" [crudely: mathematics representable in ZF]
is in some respects monstrous,  degenerate,  largely
insignificant, unnatural [see the quotes from Vopenka
below].  It rather strives for a genuinely natural
mathematics.  It begins with the observation that
"classical mathematical objects",  such as Euclid's
points,  the natural number series, and Cantorian
sets do not occur in nature (do not even "sit lightly"
in nature;  to use Bill Tait's nice locution).
Rather,  natural  phenomena are characteristically
"inexact"[Jack Schwartz: "physics is essentially
an inexact science"] -- what's especially at issue
is of course other than stochastic inexactness [many
examples will be given in Part II].
        This is not to deny that one cannot have
a science without "idealizing";
it is rather to suggest that one look for
idealizations that better reflect nature's manifold
inexactitudes,  a mathematics truer to nature,
a mathematics fully connected to nature,  and
so one incapable of giving birth to the monstrous
but insignificant abstractions,  Balkanizations,
etc.etc. characteristic of current,  classical
pure mathemtics. N.B.,  I mean to be reflecting
-- the melodramatic side of -- the spirit of
ultrafinitism,  and not something I happen to
believe  or worry excessively about.
        Whether one is sympathetic or not to the
spirit of ultrafinitism (as I have just 
melodramatically exposed it),  it nevertheless
presents us with a rich open field for the
creative philosophical and logical and mathematical
imaginations to investigate.  After all,  
there are are least twenty-seven varieties
of natural aggregations which deviate 
strikingly from Cantorianaggregations.  At 
the very least, it is worthwhile
trying to explore their logic and mathematics,
and to do so with an emphasis on the naturalness
of the logic one develops for them to them
[for example, Zadeh's Fuzzy logic is a most
unnatural logic of "vague predicates",  as e.g.,
Rohit Parikh has argued in his papers on
vagueness -- where he also brilliantly argues --
using Goedel's no-finite-matrix theorem -- that
intuitionistic logic is not the logic of vagueness]

        Let me begin with a very brief example
of (I think,  quite failed attempt -- despite
Heyting's and Geiser's generous attempts to find
sense) how one proceeds if one is an ultrafinitist.
In this case,  Esenin-Volpin,  though Esenin-Volpin
is not a representative as Hjelsmlev and Vopenka.

[1] begin with some classical object,  such
as the natural number series, . . n,  n+1, ...

[2] suppress all your habitual distinctions
between the subjective and objective. . .e.g.,
go back to undifferentiated appearances,  but
look for (what you'd normally take to be subjective
or merely apparent traits of the classical object.

[3] For example, Esenin-Volpin [to simplify] noticed
in the process of construction, we can distinguish
(at least) three phaes or phase states of "the
number series": Phase state 1: the
actually constructed,  Phase state 2: constructible
from Phase state 2; Phase State 3: (an ever receding
but never eliminated reservoir of) potential
constructible; Phase state 4: the inaccessible --
beyond the potential.

[4] Give a logical/mathematical representation of
that phase structure, i.e., find that the phase
structure is not a matter of appearance,  but a
hitherto hidden property of the classical
object (the number series).  Essenin-Volpin claimed
to have given a modal-logical representation
of that phase structure in [3].  It is of such
a logical form that it can be mapped onto finte
segments of the classical natural number series.
For example,  it can be mapped onto the segment
{1, 2, 3, 4}, where 4 is then the first inaccessible
number (that is,  from this phase-character point of
view,  the classical series {1, 2, 3, 4} is the
smallest segment of the classical natural numbers
which adequately reflects this ultrafinitist
conception of "natural number series".  N.B.,
that phase structure floats over most finite natural
number series,  but,  for, e.g., {1, 2, 3, 4, 5, 6}
4 would no longer be an inaccessible.  But we do not
have for free that this natural number series shows
all four phases. . .to determine whether it does,
one must understand Essenin-Volpin's modal logic
of the phase structure,  and I do not pretend to do
that.
        Other examples:
        
Example 1.  It is worth observing that Shaughan
Lavine's logically cogent introduction of
a subjective-appearance inspired phase structure
into the natural number
series was discovered by his looking deeply
and (logically) cleverly at an invariant
feature of counting/computation --
having a limited (but variable) reservoir
of "numbers" to draw upon.  Notice how he
followed the script [1]-[4].  (His UNDERSTANDING
INFINITY,  HarvUP 1994 -- now out in paper.)
        
Example 2. Vopenka, by contrast, uses the concept
of a semi-set to logically introduce phases
into the natural number series: The axioms,
there exists a semi-set SS which is a sub-class
of N (the SET of standard natural numbers),  
where I mean by the semi-set SS a proper 
class [sic!] which is a subclass of N. Thus,
the complement of SS in N,  N-SS, 
is also _not_ a set.  Vopenka then finds as
a "natural model" for SS,  any aggregate
generated by a soritically paradoxical
predicate, e.g., "being a small number"
(where,  1 is a small number,  and 
if n is a small number,  then
n+1 is a small number;  and yet some
number of N is decidedly not small. . .
though we need not commit ourselves to
anyone as decidedly not small -- an
important point for developing a
consistent logic of natural semi-sets,
phase structures).
        
        Vopenka's project for producing
a fair amount of servicable mathematics
is decently along; but I don't think there
is as much logical coherence as Lavine 
achieved (unless something has happened
in the last ten years?)

Example 3. Rashevsky proposed yet a fourth
idea for introducing a phase difference in
the natural numnbers,  though I don't know
if it has been worked out.  For _dx_ an
infinitesimal of some sort, the natural
number series goes:
        n, n+(1-dx), n+(1-2dx),  etc.
The idea is that the "discrete" gradually
becomes the "continuous.
 

RECA[ITULATION AND WARNING: There are three well thought
out ultrafinitisms,  those of: J.Hjelmslev, P.Vopenka,
and S.Lavine.  From a logical point of view,  the
last is by far the most successful and convincing,
while the former are more generous about making
clear the naturalistic agenda.  Ultrafinitism
does not mean confining mathematics to a segment
of the natural numbers,  or to a particular
hereditarily finite set,  or to a proper subsystem
of PRA, . . .  For these are all -- let us say --
"classical mathematical objects",  which idea [c.m.o.]
we might IMAGINE to be explained: a classical mathematical
object has ZF as part of the deep structure of 
its concept.  Roughly: if we look deeply enough
into the conceptions, segment of nn, hf set, proper
subsystem of PRA, . . we'd find the moral equivalent
of the conception of set embodied in ZF. [Bill Tait's
deep analysis of the concepts of intuitionistic logic
in "Againast Intuitionism" perhaps exemplifies what
is meant here by "looking deeply enough".]
        Instead,  ultrafinitism looks for nonclassical
objects,  and it looks to nature -- the world of
natural appearances or phenomena -- for them.  For
example,  it is maintained that Cantorian sets do
not occur in nature [=the world of natural appearances],
or occur in such special circumstances as to be negligible.
Rather they are view as "idealizations" which are imposed
on nature or at best [to employ a fine idiom of Bill Tait]
"sit lightly" in nature.
        Be it noted that this naturalism
(which rigorously begins with the world of natural appearances]
is very different from a sort of naturalism which has
a current vogue [which a wag at Brandeis has called
"old Quine in new bottles":  that whatever mathematics
physics calls upon is "natural";  from the ultrafinitist
point of view,  this naturalism is indirect and uninformative,
for it assumes that the mathematics physics happens to employ
is the most "natural",  that is, best fitted to,  optimal for,
truest to,  natural phenomena.
        THE WARNING: Kreisel has done us the service of
drawing attention to the principal pathology of what
he tongue in cheek calls the Heroic Age of foundations:
the obsession with THE.  THE foundations,  THE concept of
computable function,  THE concept of set,  THE definition of. . .
etc.  This is very dramatic in the case of Brouwer.  It didn't
suffice to Brouwer that he should be giving a different
representation of the continuous meeting alternative
criteria of satisfactoriness.  Rather,  he had to surround
his analyses with an ideological body-guard and attack
force (but in his pay,  the gods of refutation).  Ultrafinitists
can either be of the Heroic Age stripe,  needing to make a
case that, e.g.,  the natural number sequence is no-way
as clear as Poincare (say) thought it was.  This THE-attitude
creates confusion and distractions.  It means that a loft
of time is spent having to engage in endless refutation and
refutation.  BETTER: the positive goals of ultrafinitism
as I've outlined them (here,  above and below) have considerable
philosophical and foundational appeal.  The negative
remarks about "classical mathematics" are best reviewed as
a way of clarifying those goals.  Ultrafinitism does not
stand only if classical mathematics falls.  Ultrafinitism
stands,  and has philosophical worth and historical meaning,
only if it succeeds with its novel idealizations;  and
we learn something even if we are not ultrafinitist ideologues.
Lavine's work shows that logical coherent prosecution
of ultrafinitist objectives is indeed feasible.  The
problem is to better appreciate the philosophical meaning
of ultrafinitist objectives and successes,  and to see
what we can learn from them.  For one thing,  we have'live 2,400
with the concept of the process of idealization, but we
know almost nothing about it.  Ultrafinitism is a way of
correcting the deplorable condition:   


I: MOTIVATIONS FOR THINKING ABOUT ULTRAFINITISM:

        Ultrafinitism, properly understood,
is worth pursuing independently of ideological
considerations:
        great light would be shed on, e.g.,

        Naturalism vs.idealism
        the Proof/Truth problem (Tait)
        the relation of the sub-mathematical
                to the mathematical
        the relation of the sub-logical to the
                logical
        the sense in which the natural numbers
                are "perfectly clear" (Poincare...
                Feferman)
        the sense in which Cantor sets are (mot)
                natural entities
        the sense in which the transfinite in
                mathematics has nothing important
                to do with size but essentially
                only with (varieties of) logical
                function (variously: Hilbert, Wittgenstein)
        the question of whether mathematical ideas
                are obtained by (1) generalization
                from the natural-empirical, (2) ab-
                straction from the natural-empirical,
                or (3) idealization of the natural-
                empirical. (1:Mill, 2:Aristotle, 3:Plato;
                2,3:pace Bill Tait).
And much much more, e.g., a terrific opportunity for logical
invention.

II: CORE LITERATURE ON ULTRAFINITISM:

The most characteristic:

Geometry:
        J.Hjelmslev:
                -"Naturliche Geometrie", Abhd a.d.
                Math.Sem.Hamb.Unv. 2(1922),
                -"Geom. der Wirklichkeit" Acta Math.
                        2(1915),        
                -and an inaccessible ultrafinistist
                geometry textbook used
                widely in Danish schools.
Analysis:
        Petr Vopenka, _Mathematics in the Alternative
                Set Theory_ Teubner 1979 (and many
                papers)

Others of considerable importance:

S.Lavine, _Understanding the Infinite_ Harvard 1994

Jan Mycielski, see Lavine biblio

A.S.Yessenin-Volpin, see Lavine biblio [or Buffalo volume,
        Warsaw volume]

Rohit Parikh,  "Existence and Feasibility in Arith."JSL, see
        Saszanov FOM postings.

Greenspan,  _Discrete mathematics_

P.K.Rashevskii, "On the dogma of the natural numbers",
        Russ.Math.Surveys 29 #4 (1975).


CHARACTERISTIC MOTIVATION (Hjelmslev, Vopenka):

[Point A]: [START VOPENKA QUOTE 1]"Set theory 
opened the way to the study of an immense number 
of various structures and to an unprecendented
growth of mathematics. . .a scattering of mathematics
. . .a detrimental growth.. . ."[END VOPENKA QUOTE1]
{quotes from Math.in Alternative set Theory]

[Point B]: [Start Vopenka quote 2] "[After Cantor set
theory] all structures studied by mathematics are
a priori completed and rigid,  and the mathematician's
role is merely that of an observer.   This is why
mathematicians are helpless in grasping
        ESSENTIALLYINEXACT THINGS 
such as realizability,  relation of
continuous and discrete,  and so on. Contemporary
mathematics thus studies a construction whose relation
to the real world is at least problematics.  Moreover,
this construction is not the only possible one and, as
a matter of fact, it is not the most suitable from the
point of view of mathematics itself.  This makes the
role of mathematics as a scientific and useful method rather
questionable.  Mathematics can be degraded to a mere
game played in some specific artificial world.  This is
not a danger for the future but an immediate crisis of
contemporary mathematics.  It manifests itself in the
fact that most quite deep mathematical results
are entirely uninteresting not only for people who are 
not mathematical professionals but even for other mathema-
ticians at present working on problems with differently
situated pieces on the chess board.[End Vopenka quote 2].

[Point C]  Both Hjelmslev and Vopenka seek to restrain mathematics,
to avoid its scattering,  anomalies,  external insignificance.
They both sought to tie it down more closely to the
natural world,  to keep it confined to natural motivations:
to make the constraints natural constraints.

[POINT D] Both Hjelmslev and Vopenka saw (in different ways)
that one could NOT start with elementary "classical" mathematics.
E.g.  Hjelmslev realized that Euclid's geometry already
has implicit it in it the seeds of Cantorian set theory.
That is,  both saw that a _radically different_ starting point
is called for,  one that does not inevitably lead to
all those abstract and finally senseless scatterings of
mathematics,  and the pathologies, as for example geometric 
monstrosities (everywhere continuus nowhere differentiable 
functions), non-measurable sets. . .

[POINT E] INEXACTNESS is characteristic of the natural world.
This is underlined by Hjelsmlev,  Vopenka,  Rashevskii.
Compare also, Jack Schwartz on the inherent/characteristic
inexactness of thinking in physics.  That is,  Cantorian
sets are idealizations;  they are not natural.
        _EVERY PREDICATE_ [SIC: EVERY!} that applies to 
the world about us,directly,  naturally,  is vague,  that is,
it is "soritically deficient"--we can find a "quantity" d for 
which  P(x) -> P(x+d),  and yet for some k, -P(x+kd).
        It is in this sense that nature is on the border of
the sub-logical.
        
[POINT F}:  The core idea of ultrafinitist foundations is
to build a (non-cantorian) mathematics,  one that puts
inexactness at its center.  Of course,  as Hjelmselev and
Vopenka point out,  some idealization is necessary in order
that the mathematics not be a merely empirical subject,  but,
rather,  sustains chains of inferences sufficiently long to 
have a servicable mathematics.

[Point G}:  Where to begin?  With appearances in which
the objective is not yet distinguished from the subjective.
A mathematics of geometric figures will then begin
as a wanna-be mathematics of drawn lines (where,  e.g.,
the triangle inequality can be violated,  where line
tangent to a circle will coincide for an arc,  etc.).
The idea of infinity emerges from the appearance:
(Vopenka) the phenomenon of  infinity is the phenomenon
involved in the observation of large,  incomprehensible
sets.


This end this preliminary PART I.

Part II will present an extended compendium of
natural appearances (e.g., Cantor-deviant aggregations)
,  phenonena of inexactitude occurring in the sciences
and in ordinary life,  which are opportunities
for ultrafinitist idealizations and widely
over-looked by classical mathematics.