FOM: ultrafinitism again

[Robert Black]

When I characterized the position which I took to be common to Professors
Kanovei and Sazonov as 'ultrafinitist', I did not, of course, intend to
refer to nihilism, or to Dostoyevsky, but to an interesting Russian
tradition in  the philosophy of mathematics with which I associate the name
of Yesenin-Volpin and which I know too little about.

Professor Sazonov seems happy with the appellation, but Professor Kanovei
clearly is not. Let me explain what I take his position to be, and in just
what sense I take it to be 'ultrafinitist'.

1.  The only things which really exist are concrete material objects, and
there are (or for all we can know may be) only finitely many of them.

2.  Therefore, any truth (in the full-blooded, correspondence sense of
truth) is true in virtue of facts about these objects. Further: any set of
axioms which has no finite models thus has, strictly speaking, no models at
all. Sentences of mathematics purporting to quantify over an infinite realm
of abstract objects must be regarded as fictional, for there are no
abstract objects, and there is no infinite realm of anything.

3.  When we say that a sentence of mathematics is 'true', all we mean (or
all we should mean) is that it has a formal proof (in ZFC, say). Such a
formal proof is a finite material object, and thus its existence (or
perhaps its possible existence?) can be a matter of objective truth in the
full-blooded correspondence sense.

Professor Kanovei will no doubt let me know if the above three points are
not a reasonable summary of his position. (I don't myself accept (1), but
(2) seems to follow from (1) all right, and once one has swallowed (1) and
(2), (3) is pretty natural.)

Now: the reason I call the position ultrafinitist is that it is a position
which only allows for the existence of a bounded, finite number of objects.
It is thus much more restrictive than the potential infinity of
intuitionism or of Hilbertian 'finitism'.

Note that on this position incompleteness results come out as trivial. Even
theories which are classically complete will come out as incomplete, since
there will be sentences just short enough to fit into the universe, but
such that neither their proof nor their disproof is short enough to fit
into the universe.

Now I take Kanovei further to be saying that Goedel's results must be seen
as *mathematical* results, e.g. the second incompleteness theorem for ZFC
does not really say that if ZFC is consistent (i.e. if there are no
physical objects fulfilling certain conditions) then Cons(ZFC) is not
provable in ZFC (i.e. there are no physical objects fulfilling certain
other conditions), all it really says is that a certain sentence of
arithmetic, viz. Cons(ZFC) -> not-Prble ([Cons(ZFC]), *is* provable in ZFC.

What I do not really understand about this position is the relation between
the formal result and our expectation (which surely Professor Kanovei
shares) that if we could find a physical object which was a proof of
Cons(ZFC) in ZFC, then we could also find a physical object which was a
proof of 0=1 in ZFC. More generally, I don't understand how Professor
Kanovei thinks formal mathematics gets applied. (Contrast here the
nominalism of Hartry Field, for example, for whom although mathematical
objects are fictions, mathematical theories can be applied to the real
world in virtue of isomorphisms between the fictional realm and the real
realm.)

The other problem which I raised is the possibility that accepted
mathematical proofs might fail to be formalizable in ZFC in the sense that
the full formalization would be too long to fit into the universe. The
obvious reply to this, that one introduces definitions, doesn't seem to me
to work. The reason is that standardly the distinction between a definition
and an axiom is that the former is conservative. But once we have a finite
bound on the lengths of proofs, 'definitions' will no longer be
conservative, and must therefore be regarded as axioms. And what then
justifies them as axioms, once the usual considerations which justified
them as definitions are no longer available?

Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD

tel. 0115-951 5845