FOM: Brouwer & Beyond ZF

[Robert Tragesser]

[Abstract:  Here I present a reasonable
alternative to the Byzantine (and I think
wrong) Dummett/Tennant "rescue"
of Brouwer.  It rests on the argument that
the foundational
problems dropping out of FINITE/INFINITE
were not as central to Brouwer as those
dropping out of DISCRETE(orSET)/
CONTINUA.
        I here elaborate
the sense in which Brouwer's
thinking raises foundational problems
(alternative to FINITE/INFINITE problems
with which Brouwer is misleadingly
identified!) that dogmatic ZFer's can not 
handle --
        that what Brouwer reveals is that
SET and CONTINUA are two primitive
but absolutely inseparable mathematical
FORMS. CANTORIAN SET attempts (according 
to Brouwer,  impossibly) to eliminate 
CONTINUA.  The failure to settle THE CH
bitterly proves Brouwer right.
   One can understand ZF as peculiarly suited
to the investigation of the foundational
problems surrounding FINITE/INFINITE.
But it is wholly unsuited to exploring
the foundational problems isolated
by Brouwer surrounding DISCRETE/
CONTINUOUS,  between SET and "Geometry".
It is here that Brouwer pioneered;
it is THE deepest (and most intractable)
foundational problem facing us now. [Here
put Friedman's vitriolic
remarks about the unprofessional
character of this assertion,  so
that I need not be treated to them
again_____]] END ABSTRACT]]
        

        It is an easy enough mistake,  but
nevertheless a serious mistake,  and for 
Simpson a very convenient mistake, to take
Brouwer's "subjectivist" assertions at
face value.  There is an alternative and more
certain reading of Brouwer that makes
better sense of Brouwer and wholly evades
the Byzantine and (to me) unconvincing recon-
structive approach of the Dummettians.


[Some References beyond Brouwer's Collected
Papers; vn Stigt,  Brouwer's Intuitionism
(1990), with Koestier's essay in Handbook
of the History of Topology I [1997] as 
an important corrective to van Stigt's
characterization of the interelationships
among Brouwer's Diss,  his topological work,
and his "constructivism"; the essays by
Brouwer and Weyl in Mancosu's anthology
From Brouwer to Hilbert(1997); it is
important to notice there Brouwer's
corrections to Weyl 1921, esp. the
fifth remark on p.119 ("From my...")
and its elaboration on p.120 ln.-9 
"By this I understand..."]    

[1] As van Stigt elaborately points out,
Brouwer sought to evade the anomalies
and aporia he perceived as being generated
out of -- for simplicity let me say -- 
Cantorian modelling of coninua.

[2] Brouwer's over all strategy was to
begin from scratch (with hindsight),
by identifying,  and reanalyzing,
the fundamental forms of mathematics
given in intellectual vision.  In
his Dissertation,  Brouwer identifies
as the basic form:
        [long quote from Brouwer
Collected Works I, p.[[17]], Diss.;
boldface emphases mine]:
        __"the substratum,
divested of all quality,  of any
perception of change, and unity of
continuity and discreteness, a
possibility of thinking together
several entities. connected by a 
'between',  which is never exhausted
by the insertion of of new entities.
Since continuity and discreteness
occur as INSEPARABLE COMPLEMENTS...
it is impossible to avoid one of the,m
as a primitive entity...it is impossible
to consider [either] as self-sufficient.  
Having recognized that the intuition
[=intellectual vision/perception!] of
continuity,  of "fluidity",  is as
primitive as that of several things
conceived as forming a unit,  the latter
being AT THE BASIS OF EVERY MATHEMATICAL
CONSTRUCTION,  we are able to sate
properties of THE CONTINUUM AS A MATRIX OF
POINTS TO BE THOUGHT OF AS A WHOLE".__
{end Brouwer quote]
        
        In Heyting's commentary on this
passage,  he remarks [Brouwer I, p.[[565]]],
"In his later work,  Brouwer abandoned the
notion of the continuum as the basic notion
of mathematics...".
        But a careful reading reveals that
 Brouwer did not think of the notion
of the continuum as THE basic notion of
mathematics. As Brouwer remarks,  it is
nonindependent of the notion of discreteness
-- indeed,  they are nonindependent of
one another -- we can't develop one "self-
sufficiently" without developing the other.
Cantorian set theory assumes that one can
(independently) develop the notion of discreteness,
that is,  the notion "several things
conceived as forming a unit",  independently of
the notion/form "fluidity". -- There is
good reason to believe that Brouwer would
view the present failure to decide Cantor's Continuum
Hypothesis as bitterly proving Brouwer
right.  Remember that Cantor's CH is formulated
with reference to "the discrete" only -- again,
remember that Brouwer's language for the "discrete"
is just that of Cantor's for a set -- a unity
of a manifold of items.
        That is,  Brouwer would have thought that
discrete measures of continua (like the CH tries to
invoke) could not be satisfactory unless they were
essentially responding to,  and drawing on, the
irreducible form of "fluidity".  (As Brouwer's
own "solution" to his analogue of CH ILLUSTRATES --
the crucial word here is "illustrates" what it would
be to take that something more into account.--I do not
want to suggest that I think Brouwer did it in
a fully satisfactory way.)
        
        One can see very clearly here how the
dogmatism [see next para. ]of Harvey Friedman 
(which we and I have recently experienced on FOM --
HF's two Feb 26 postings),
that all mathematical notions absolutely can be 
reduced to,  and not distorted
by,  ZF would preclude him,  and other flag
waving Cantorians,  from taking seriously
the very thought (that Brouwer teaches us 
how to take seriously)
that the highly problematic character of
CH may indicate that we must take up again the
conceptual problem of seeing what that
fundamental form,  CONTINUITY/fluidity might
contain that we have overlooked.  This,  I
think,  should be the fourth conceptual problem
that bugs Sol Feferman.
        [HF's "Dogmatism" means:insisting on a thesis
without a formulation of a thoroughly reasoned
defense of it and the principles on which
it is based...see Friedman's two postings on
Feb. 26,  in which he insists,  without argument,
but simply declaration, that Boolos' Conjecture
is not conjecture at all but an absolute truth.
Friedman has his own reasons for my not being
willing to go along with him;  but at the
very least I don't imagine that I could until
CH is no longer an open problem.]

[3]  Now,  as Kreisel has elaborately
pointed out in papers in the 1980's
(I would give thorough references if
need be),  figures in what he calls
the Heroic Age,  sought THE
foundations,  THE definition,
THE explanatory proof,  and so on.
Indeed, Brouwer sought THE mathematics
of continua.  It is clear that
he would need definitions & proofs which
could not be assimilated by the
Cantorians/Formalists (again, I must
abbreviate), --in short a method that
would reserve/sustain an abiding place
for "fluidity" as an ineliminable
form.
        He found this needed novel
way of reasoning in the
Solution to the Third Antinomy of
Kant's Critique of Pure Reason --
"this" being the constructions, reasoning,
behind, e.g.,  the Fan Theorem.
        I do not have direct evidence
for this.  But here are my reasons for
claiming it.  (N.B.,  I am not a Kantian
and am not held in thrall by the very
idea of a mathematician getting an idea
from Kant.)  My reasoning is this: (1)
Brouwer read the first Critique
carefully;  (2) the reasoning in the
Solution to the Third Antinomy is novel &
unique -- no one had ever seen its like
before,
(3) it bears an obvious analogy to
the reasoning underlying the Fan Theorem,
(4) anyone developing a mathematical
argument by close analogy with Kant's would
be burdened with the freight of having
to think in terms of (albeit ideal) sub-
jective processes. (Only with the work of
Kleene, Kreisel,  Troelstra. . .Scott,
 J.R.Moschkovakis does one begin to see the
objective content of Brouwer's constructions.)
        Stripped down to its essentials,
Kant's reasoning is this. 

THE CORE OF KANT'S SOLUTION TO THE THIRD
ANTINOMY AND ITS _ANALOGY_ WITH REASONING
/METHOD UNDERLYING FAN THEOREM.
.

[K-A] How does one reconcile physical
determinism with the practically
successful human institution of assigning
responsibility (of assuming a measure
of free will)? [As to the latter,  notice
that Kant takes a specific example a brief 
analysis of the assumptions underlying/
underwriting, prosecution of a criminal.]

[K-B} Kant has argued that the principle
that every event has a cause is synthetic
a priori.  But what does this mean?  One
looks to how this principle is used.
First,  it is a fundamental principle that
one uses to pull knowledge of physical
nature out of appearance.  Second,
the principle is confined to the course of
possible experience.  Third,  how is it
applied?  Here is the rule of application:
begin with phenomena,  and research the
causes.  That is: assume there is a cause,
and look for it.  But there is absolutely
no necessity that one will go from
phenomenon to unique cause to unique cause.
It is rather that "physical nature" will be
manifest only to the extent that you
succeed.  Kant would quite allow Hume's
"skepticism" here,  that one never knows
that one has gotten to "the unique"
preceding cause.  It is rather that,
for Kant, pursuing causes MAKES SENSE,
for our understanding provides the
relevant pure concepts,  even if there
is never any guarantee that one has gotten
down to THE explanatory cause.
        Notice that there is considerable
(transcendental) logical freedom here:
One gets a matephysics of the natural
sciences BY CONFINING ONESSELF to the
"sythetic a priori" rule(s) -- for
in the end that's what they are -- that
elicit deterministic physical nature out
of Appearance.  But we are quite free to
apply other rules to appearance if they
make sense.  

[K-C]  It is exactly because Kant
can argue that,  since we only go finitely
in our quest for causes,  and because we
can never know that we have gotten to
the immediate antecedent cause of any event,
that we can consistently interprolate
"spontaneity" (acts of free will) into appearance
as "numena" without fear of our researches
in deterministic science ever contradicting
us in some definitive way.  Briefly:
our quest for "the causes" can only ever
go fiitely far without our ever reaching
a "certain" (necessary) end (and being able to
know that we have reached such an end.  So
that all that our experience forces,  it
can never force "There is no spontaneity".
(no free will).  Therefore we can
CONSISTENTLY interpolate as much "free will"
or "spontaneity" as we like into the noumenal.
INDEED,  FROM A KANTIAN POINT OF VIEW,
SINKING THE DETERMINISTIC _AND_ THE SPONTANEOUS
INTO THE NOUMENAL _IS_ MAKING DISCOVERIES ABOUT
THE NOUMENAL.
        This reasoning is very much analogous
to Brouwer's deployment of the method of
free choice sequences to (N.B. two points of view
here) super saturate the intuitive form of
the continuum with points (so that all functions
at this first level are continuous) or to 
justify the assertion that "points" are locally
tangled or order confused.
        What Brouwer interpolates into the
continuum (= what correponds in our analogy to
Kant's noumena) is e.g. local failure of "less than"
etc. (where this local failure of "less than"
corresponds in the analogy to "spontaneity").
END DISCUSSION OF KANT & BROUWER

[4] One can understand why Brouwer would 
then resist,  e.g.,  logical foundations for 
his proof of the Fan and Bar Theorem(s);
and why therefore he would want to bolster
at great cost (namely by a perfectly silly
and incoherent "subjectivism",  his 
emergent mathematics of continua which he
was determined to make "the" mathematics
of continua.
        At the same time,  I don't think
for a moment that things are this simple.
I think that there was also the factor
of Brouwer NOT FULLY KNOWING HOW to keep
live the form CONTINUA as
a pregiven "whole" [see his holistic
thesis in what I quoted from Brouwer above]
with mathematical success.

[5]  I read the current problem with the CH
as suggesting that Brouwer's surely
"professional" insight had something to it:
that set (the DISCRETE) and the continuous
are deeply co-dependent.  I see Brouwer's
attempt to this co-dependence live to be
heroic,  but so far it has failed.  
        The greatest foundational tension
in mathematics NOW is between CANTORIAN/ZF
SET and Geometric Intuition,  and not so much
any more between the Finite and the Infinite.
        ZF --as Sol Feferman has elaborated -- 
is a good place to begin to explore the 
foundational problems arising in
the finite/infinite. It is not,  as Brouwer
saw that it was not,  a good place from
which to explore the relation between
SET  and GEOMETRY or DISCRETE
and CONTINUOUS.
        Brouwer was then a pioneer
in the new foundational task that lies before 
us.  But no one seems to have a clue how to begin.
        Certainly not the dogmatic ZFer's!!!
(Who'd rather pretend the problem didn't
exist.) 

Postscript on Brouwer/Wittgenstein:  The
big issue deviding them had to do with
Brouwer's pet project of achieving world peace
by (a forced) reforming of all of
human language!  (This is makes the Dummett/
Tennant line(s) a bit ironic.)

Robert Tragesser