FOM: Does Mathematics Need New Axioms? (response to Matthew Frank)

V. Sazonov V.Sazonov at
Sat May 27 09:33:48 EDT 2000

Stephen G Simpson wrote:

> This is in response to Matthew Frank's posting of Mon, 22 May 2000
> 19:31:59 -0500 (CDT).
> 1. Frank says that the anti-foundation axiom might be a new axiom that
> we need to look at in a set-theoretic context.  I am not sure why.
> Could Frank please explain?
> It seems to me that set theorists have good reasons for insisting on
> the axiom of foundation.  Namely, by restricting attention to
> well-founded sets, we simplify our picture of the set-theoretic
> universe, and we lose very little, because the orthodox set-theoretic
> foundational setup (natural numbers as finite von Neumann ordinals,
> reals as Dedekind cuts, etc etc) takes place entirely within the
> well-founded sets.
> My impression is that the anti-foundation axiom has been of interest
> not so much for set-theoretic f.o.m. but in somewhat different
> contexts (Aczel's Frege structures, etc).

Anti-Foundation Axiom (AFA) gives a new view on the concept of
sets and extends it in an interesting way. (Hyper)sets are not a
result of collecting something, but rather the result of abstraction
from directed graphs. In particular, for (finite) graphs with labelled
edges considerted as so called semi- or un-structured databases
(or Web-like databases - WDB, because edges are intuitively
treated as hyperlinks between data by which data may be accessed)
corresponding hyper-set-theoretic approach may be reasonably
considered. Think that an edge leads from a hyperset to its element
(also hyperset, etc.). Actually, informationally equivalent vertices
(URLs, in terms of WWW) are identified and become hypersets.
Moreover, a query language to WDB may be defined in pure set
theoretic terms having also coherent operational semantics in terms
of (WDB-) graphs. (Cf. also my homepage below.)

Of course, AFA may be interpreted in the ordinary set theory.
But it looks very interesting in itself. We can still care about
foundations of "core mathematics", but it seems that other
(more applied) branches of mathematics in computer science
(concurrent processes, database theory, etc.) need it more
urgently. At least, this is also a kind of f.o.m.

> 4. Regarding constructivism, Frank says:
>  > I recommend against using the word "constructivist" to describe
>  > people, as Simpson does. ...
> We need a term to describe a person who endorses a constructivist
> philosophy of mathematics.  I think *constructivist* is the right
> word, but Frank may want to introduce another term.  Of course Frank
> is right in pointing out that there are people who ``work on''
> constructivism (e.g., by proving metatheorems about
> constructivistically-inspired formal systems) without endorsing the
> underlying constructivist philosophy.  On the other hand, there are
> also full-fledged constructivists like Bishop, Bridges, Richman, etc.
> We could call them ``philosophical constructivists'', but that seems
> redundant, since constructivism is nothing but a particular philosophy
> of mathematics.
>  > and certainly not of the subjectivist philosophy which Simpson
>  > describes.
> If subjectivism is not the essence of the constructivist philosophy,
> what is?

By the way, if another philosophy(?) called formalism really exists
(which allegedly asserts that mathematics deals only with formalisms
without any meaning), who are full-fledged formalists, personally?
Or such a formalism is a philosophy without any real ``philosophical
formalists''? Could somebody comment?

Vladimir Sazonov

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