FOM: human well-being; constructivism; anti-foundation

[Stephen G Simpson]

This is a reply to Randall Holmes' message of Tue, 30 May 2000
13:05:25 -0600.  It also comments on messages of Frank, Sazonov,
Hazen, and Mahalanobis.

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1. HUMAN WELL-BEING

As part of the ``new axioms'' discussion, I asked:

   Is this area of research [the role of large cardinal axioms, etc]
   likely to lead to new technology which can be expected to
   participate in improving the human standard of living?

Holmes says such questions may be off-topic for the FOM list.  I
disagree, and I don't understand Holmes' reasoning.  A standard
criterion for evaluating scientific research is its potential to
contribute to human well-being.  Why should f.o.m. research be
exceptional in this respect?

Holmes and Frank may be correct in saying that f.o.m. research has no
(and is not to be evaluated in terms of) *immediate* technological
applications.  However, it seems to me that we can ask for a
long-range contribution, as follows.

Philosophy profoundly influences history.  There are some historical
periods (e.g., the Dark Ages) when the human mind is philosophically
shackled or subjected to an unwholesome preoccupation with otherwordly
concerns.  And there are other historical periods (e.g., the
Enlightenment) when the human mind looks boldly outward, seeking to
understand nature and harness it for human purposes, to the enormous
benefit of the human race.  And mathematics is one of the stages upon
which this great drama plays out.  Thus the philosophy of mathematics
and f.o.m. research may have the capability of contributing to human
well-being over the long run, by discovering and validating a
rational, scientific foundation for a kind of mathematics that is
real-world oriented and applications oriented.

In my view, it is appropriate to examine the existing philosophies of
mathematics (formalism, intuitionism, constructivism, Platonism, etc)
and the existing research directions in mathematical logic (set
theory, proof theory, recursion theory, model theory, etc), in terms
of the historical perspective of the preceding paragraph.  Such issues
may be seldom discussed, but they are obviously very important and
worthy of discussion here on FOM.

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2. CONSTRUCTIVIST PHILOSOPHY

I said that constructivistic mathematics is based on a subjectivistic
philosophy, according to which mathematics consists of mental
constructions in the mind of the mathematician.

In support of my view, Ketland quoted Heyting to show that Brouwer's
intuitionism is based on exactly this kind of subjectivism.  And a
couple of people pointed out that Dummett is also somewhere close to
the subjectivist camp.

Against my view, several people noted that one can ``be interested
in'' or ``work on'' intuitionistic systems, without actually
``believing in'' the underlying philosophical ideas.

I concede this point, but I say that it has nothing to do with the
philosophical/foundational issue.  Intuitionistic systems of
mathematics were originally introduced in service of a Kantian or
subjectivist philosophy.  If these formal systems take on a life of
their own, that does not erase the philosophical issues that gave rise
to them.  In particular, if intuitionistic logic and type theory are
convenient for computer-aided algebra or computer-aided proof systems
such as Nuprl, that has no necessary connection to the philosophical
issue, which remains vital for f.o.m.

Also opposing my view, Mahanalobis cited Bridges/Richman to the effect
that there are alternative ``varieties of constructivism'' (e.g., that
of Bishop) which may not be based on Brouwer's subjectivist
philosophy.

I would point out that the Bridges/Richman book explicitly eschews
philosophical concerns.  ``We are writing for mathematicians rather
than for philosophers or logicians.'' (Page 2).  It is true that
Bridges/Richman distinguish three schools which they call INT, BISH,
RUSS (Brouwer's intuitionism, Bishop-style constructivism, Russian
constructivism).  However, they do so at a purely mathematical,
non-philosophical level:

  BISH = classical mathematics with intuitionistic logic

  INT = BISH + fan theorem + continuity principle

  RUSS = BISH + Church's thesis + Markov's principle

In particular, Bridges/Richman do not comment on *why* one might
choose one system over another, except to say that BISH may be better,
because it assumes less.

Is the philosophy that underlies Bishop-style mathematics really so
very different from Brouwerian subjectivism?  If it is, then I think
there is need for a much fuller explanation.  Bishop's discussion at
the beginning of his book on constructive analysis seems woefully
inadequate.  In particular, it does not answer the obvious objections
that can be made by ultrafinitists, formalists, recursive function
theorists, et al.

Can anyone fill this gap in (my knowledge of?) the constructivist
literature?

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3. ANTI-FOUNDATION 

Frank, Sazonov, Holmes and Hazen all praise AFA (the anti-foundation
axiom) in destructive terms.  According to them, AFA is valuable
because it ``knocks the iterative conception of set off its
metaphysical pedestal''.

I disagree with this point.

First, AFA in no way invalidates the iterative concept of set.  The
intended model V of ZFC (including the axiom of foundation) is a
canonical inner submodel of the intended model V* of 

  ZFC*  =  ZFC - foundation + anti-foundation.

Namely, V is the well founded part of V*.  And all of the
f.o.m. action takes place in V.  And each of V and V* is canonically
recoverable from the other.  (The elements of V* are just the
isomorphism types of directed graphs in V.)  If we refer to the
elements of V as ``sets'' and the elements of V* as ``hypersets''
(Sazonov) or ``schmets'' (Anderson) or whatever, there is no conflict.

Proponents of AFA may want to refer to the elements of V* as ``sets''.
But this entails a massive revision of standard terminology, and I see
no good reason for it.

Let me put my objection this way, by disputing one of Holmes' points.
Holmes tries to undermine the standard iterative concept of set, as
formalized in ZFC, in the following terms:

 > I think that many users of set theory (I hope not any set
 > theorists!) actually think on some unreflective level that the set
 > is somehow "made up" of its elements [...]

I guess I am one of those users, because I certainly do think that a
set is made up of its elements.  What is wrong with this view?  Why
does Holmes say that it is not based on reflection?

 > though they learn to avoid in practice the technical mistakes which
 > such a view would imply.

What technical mistakes?  I am not aware of any technical mistakes
which are implied by my (standard) view of sets.

I think that, in order to understand Holmes' remarks, we have to
understand where Holmes is coming from.  Holmes is an advocate of an
alternative set theory known as NFU.  And NF/NFU is an attempt to
formalize a concept of set which many people regard as unclear, or at
least not nearly so coherent and compelling as the standard iterative
concept of set, on which ZFC is based.  Therefore, it is
understandable why Holmes might want to denigrate the standard
iterative concept of set and ``knock it off its pedestal'', by denying
that sets are made of elements.

Note also that the AFA concept of set has nothing in common with the
NF/NFU concept of set.  The only connection is that both of them are
alternatives to the standard iterative concept of set.  What we seem
to have here is a fraternity of mutually antagonistic opponents of the
status quo.

For background, see the FOM discussion of NF/NFU and other alternative
foundational schemes in February-March 2000, especially Holmes'
message of Wed, 8 Mar 2000 13:44:18 -0700.

-- Steve