FOM: role of formalization in f.o.m.

[Robert Black]

There is much to agree with in Steve's account of what is involved in
formalization, and (as Steve is probably well aware) much of what he says
bears a striking resemblance to Hilbert's well-known 1917 lecture
'Axiomatisches Denken'.  For example (the translation from Hilbert is mine,
and not particularly careful):

Hilbert:
'When we assemble together the facts from a some field of knowledge,
extensive or not, we soon realize that these facts can be organized.  This
organization always takes place with the aid of a certain *framework of
concepts* so that to a particular object of the field of knowledge
corresponds a concept from this framework and to every fact from the field
corresponds a logical relation between the concepts.  The framework of
concepts is nothing other than the *theory* of the field of knowledge. ...
If we examine a particular theory more closely, we always see that a few
distinguished propositions of the field of knowledge underlie the
construction of the framework of concepts, and that these propositions
alone suffice to build up the whole framework from logical principles. ...
I believe that everything which can be the subject-matter of scientific
thought, submits to the axiomatic method, and thus indirectly to
mathematics, once it is ripe for the formation of a theory. ... Under the
banner of the axiomatic method, mathematics appears called to a leading
role in science generally.'

Steve:
'When a scientific discipline or field of study reaches a certain degree of
maturity, it is desirable to reify or codify the existing knowledge in
terms of basic concepts and principles. ... If we set out to perform and
study the reification of subjects in a highly systematic manner, we are led
to questions such as: Which concepts and propositions are to be taken as
basic for a given subject?  How are the non-basic concepts and propositions
to be derived from the basic ones?  How can we be sure that nothing has
been omitted? ...
In principle, none of this has anything in particular to do with
mathematics.  In principle, the above general scheme ... is applicable to
any scientific subject whatsoever.  But to date it has been worked out only
in mathematics.  In this sense, f.o.m. serves as a model or example of what
can be hoped for in the future in the way of rigorous or formal foundations
of other scientific subjects.'

The striking difference between Steve and Hilbert, however, is Steve's
insistence that the tool of formalization be first-order logic.  The reason
is, of course, that Steve knows, as Hilbert in 1917 could not, how the
completeness of first-order logic contrasts with the incompleteness of
other systems. (Hilbert cites Russell as having brought to completion the
'axiomatization of logic' initially prepared by Frege; it's obvious that
there's no restriction to first-order intended.)

Now I don't really want to start the arguments about second-order logic up
all over again, and I'm certainly no opponent of first-order logic as the
most indispensable tool of formal investigations, because of Steve's points
(ii) and (iii) (topic-neutrality and completeness).  But on at least one
natural reading, it seems to me that his points (i) and (iv) (expressive
power and the possibility of complete axiomatization) claim too much.
Steve quotes as the 'outstanding examples' of formalization 'first-order
arithmetic, Tarski's elementary geometry, second-order arithmetic (i.e.
arithmetic plus geometry), ZFC'.  These four are indeed outstanding
examples of formalization, but of course as we all know, only the second of
them can be be given a *complete* axiomatization in first order logic (or
any other recursively axiomatized formal system).

Hilbert wanted formalizations such that the axioms 'alone suffice to build
up the whole framework from logical principles' and for Steve a requirement
of formalization is that 'nothing has been omitted' and 'all the known
propositions of the given subject are logical consequences of the axioms'.
But surely the greatest result of twentieth-century f.o.m. research is that
in general this *can't be done*. Goedel didn't stop with his completeness
theorem!

[Incidentally, in just what sense is second-order arithmetic (as a
two-sorted first-order theory, of course) 'arithmetic plus geometry'? Or is
this a typo?]

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

tel. 0115-951 5845