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

[V. Sazonov]

Stephen G Simpson wrote:

> 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.
>
> ----------
>
> 1. HUMAN WELL-BEING
>

> .... 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.

Completely agree.


> 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.

Completely agree. But if any of these philosophies make a sense
today and have not only some historical value, they should be,
I believe, presented in a clear contemporary (non "orthodox")
terms, in a simple language, as concrete as possible.


----------

>
> 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.

May be they have a different philosophy from the original one
(say, more concrete and probably only implicit)?


> 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.

Cf. my previous note.



> 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.

I would say, that all of the f.o.m. action takes place in
appropriate formalisms of sufficiently fundamental character.
ZFC* in principle has such a character. It gives sets of
a different flavor than ZFC. It is useful in some applications
say in Web-like databases. That is all what I mean.

Actually, I started to work on a set-theoretic approach
to databases (first, in more abstract setting) by using
well-founded heredetarily-finite (HF) sets *represented*
by vertices of finite *acyclic* graphs. It was important
to have some representation to speak on computability,
especially on polynomial-time computability over HF.
PTIME over HF is very sensitive to encoding. (Tree or
bracket encoding/representation/view are crucially different
from graphical; well-known Ackermann encoding of HF by natural
numbers is probably the worst one in this respect.) Already
this gives a new feeling of sets-as-graphs. (When somebody say:
"let us consider or let us take an HF-set", what do he mean,
especially if "he" is a computer?) Later the book of Azcel
was published which gave for me (with my colleagues in Russia)
a good impulse to very natural Anti-Well-Founded case (HFA)
with considering *arbitrary* finite (or even infinite, but
finitely-branching) graphs-as-sets. Whether it is fundamental
to the whole f.o.m. or not, AFA gives a new, interesting,
useful and more general (say, for database people) view on
the nature of sets (even if it is formally reducible to
the well-founded case).

Do these simple notes have a relation to discussed above
"metaphysical pedestal"? I do not know. But this makes
sets to be very concrete things, even if quite unusual
axiom AFA is assumed.


> 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.)

According to AFA we should speak here on bisimulation
rather than on isomorphism.


> 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.

No problem. When you are tired to repeat many times
hyper- or anti-well-founded, or non-well-founded sets
you just begin to use the old term "set" in the new
sense.


> 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.

If I am opponent of the status quo of traditional set
theory, it is not because the iterative concept of set
or because of AFA, but because some its declared absolutism
I sometimes feel behind some discussions on its relation to
f.o.m. I am pluralist (as seems also Matthew Frank is).
I do not reduce mathematics to ZFC or its extensions,
despite the well known fact that almost all *contemporary*
mathematics seems is reducible. Nobody knows what will be
tomorrow. And for me "new axioms for mathematics" means,
e.g. AFA or probably anything not reducible to (extensions
of) ZFC at all. I think that the role of axioms is not only
to give proof-theoretic strength, but to describe new
(fundamental and desirably applicable) concepts. The latter
seems to me even more important, the former being rather a
("pay-less"?) consequence.


Vladimir Sazonov