FOM: Logicism lives...

[M. Randall Holmes]

None of the philosophies which Hersh lumps under "foundationalism" have
necessarily failed in their fundamental goals, even if the founders in
some cases thought that they had.

Intuitionism is the easiest:  it hasn't failed at all, and I don't think that
the leading intuitionists at any time thought they had failed.  The stock
of intuitionistic (constructive) methods is rising, if anything.  The only
sense in which intuitionists might feel that they have failed is that they
have not convinced everyone else to do intuitionistic mathematics.

Formalism might be seen to have failed, because the exact aims which Hilbert
hoped to achieve turned out to be impossible.  Of course, framing a conjecture
clearly enough that it can be decisively disproved is a contribution to
mathematics in itself.  Formalist technique is in use all the time, and I
believe that a good deal of foundational work in mathematics can still
be characterized as carrying on modified versions of the formalist program.

I can speak to logicism better, because I probably am a logicist (this
depends on points of semantics).  Since I am one, I hardly think that it
is a dead program.

First, I'd like to speak to Simpson's remark about NF and its relation to
the question of whether the axiom of infinity is a logical principle.
NF, as originally formulated by Quine, does imply the axiom of infinity.
NF can be viewed as a form of higher order logic -- if one regards it
as logic, then infinity can be proved from logical principles.  I don't
think that this is especially convincing for a couple of reasons.  The
chief argument against this is that the consistency of NF remains an
open question, and the best approximation to NF which is known to be
consistent (the theory NFU of Jensen, in which extensionality is restricted
to nonempty sets, or in which one has many non-sets with no elements) does
not prove the axiom of infinity.  I have argued (as Quine suggested himself
in his remarks accompanyin Jensen's proof of Con(NFU) in Synthese, vol. 19)
that Quine made a mistake by adopting strong extensionality for NF.  (It is
interesting to note further that Marcel Crabbe has shown that NF without
any extensionality axiom at all interprets NFU -- weak extensionality comes
easily).  In reading the rest of this note, one will discover that the
basic logic I accept is in any case not NF.

I don't think that the axiom of infinity is a fatal embarrassment to the 
logicist program.  In the first place (as Russell pointed out) we can
regard ourselves as exploring the logical consequences of the axiom of
infinity when we use it.  It is certainly a hypothesis with logical interest.
In the second place (more daringly) I think that a reasonable case can
be made for the proposition that the axiom of infinity _is_ a logical
principle after all.  Introspection on what we actually do on the meta-level
in logic suggests that the axiom of infinity is indispensible; at any rate,
attempting to do logic with a finitist metatheory is quite demanding (though
the attempt is very interesting and instructive).  For example, it is
quite difficult to make sense of the notions of consistency and completeness
(essential to logic) in a finitist context.

I think that mathematics is the study of abstract structures.  I'm
convinced by well-known arguments (which have been presented on this
list) that there are no specific objects which are the Platonic
"natural numbers" (for example); when we prove theorems about the
natural numbers, we are proving theorems which apply to every
structure with an element "zero" and a relation "successor" which
satisfy Peano's axioms and in particular have the property that every
inductive property (in the familiar sense) belongs to all elements of
the structure.  I don't think that ZFC (or any first order theory) is
the framework in which this is to be understood; I think that
statements about the natural numbers translate to statements of second
order logic with a hypothesis describing the structure to be
considered (note how this harmonizes with Russell's way of reconciling
the axiom of infinity with logicism described above; if mathematics is
a theory of abstract data types, as it were, with every mathematical
statement carrying implicit hypotheses describing the kind of
structure under consideration, then the description of an infinite
structure in this way is no more remarkable than that of a finite
structure -- mod model theoretic considerations discussed below).  

An objection to this is that this would make all statements about
the natural numbers vacuously true if the real world is actually
finite (all structures satisfying PA_2 would then satisfy any sentence
whatsoever); this could be answered by adopting a different kind of
meta-theory of logical connectives, which would probably be
necessary in a finite universe anyway, as I observe above; the
simplifying effect of infinity on logic is considerable.  I doubt that
infinity is disprovable even if the universe is finite in fact, so I'm
not too worried about this.

If second order logic is logic, that makes me a logicist.
Mathematical assertions are logical in character and properly proved
assertions enjoy logical certainty.  Second order logic doesn't enjoy
the nice proof theory of first order logic, but that's tough; we were
never promised a rose garden.  I'm a Platonist with regard to
mathematical truths, but not with regard to mathematical objects; but
facts about how many objects there are (in a general sense) are of
considerable mathematical interest.  To work with first-order ZFC (or
first-order anything) all I need is the usual axiom of infinity (all
first-order theories have countable models).  If I want to adopt
second-order expressible mereological principles (in the style of
David Lewis) I need a real continuum (this is the level at which CH
becomes a meaningful question with an objective answer -- not likely
ever to be known to me!).  If I follow John Mayberry's approach to
foundations and adopt second-order ZFC, I require that the
non-mathematical universe be very large indeed (or I have to be
willing to pretend that it is).

One does not need higher-order logic as a further extension of logic; it
is well-known that higher-order logic can be coded into logic of second
order (with hypotheses about the existence of more objects).

I think further that second order logic should be distinguished from
set theory.  The second-order quantifiers range over properties
(universals); the first-order quantifiers range over objects
(particulars).  When one finds oneself talking about sets, what one is
really doing is associating particulars as "labels" to all universals
on a restricted domain (as in (second order) Zermelo style set theory)
or to a restricted selection of universals on the general domain (as
in Quine style set theory) (or, in other theories, to a restricted
range of universals on a restricted domain).  Properties are not
objects in the same sense that objects are; for example, there is no
reason to include identity criteria for second-order objects.  All of
these considerations are logical in nature (considerations of the
nature of subjects and predicates are certainly logical in a classical
sense).  (a further example of the simplifying effect of assuming infinity
is that I can restrict myself to talking about properties rather than
properties and relations).

Since I'm content with second order logic, I am not demanding a complete
set of logical principles.  The logical principles I accept I regard as
reliable; I don't expect them to be magical.  I fully expect that (as
in the practical world) my ability to express propositions will far outrun
my ability to decide them; that my means of expression outrun my means
of verification even in principle doesn't surprise me.  This does not
mean that I doubt that I can express the things I cannot prove; I know
what CH means, though I don't believe that I can decide it.  I regard my
logical toolkit as essentially open-ended; I could become convinced of
rules of inference that I now do not accept, or even of the utility of
logical constructions that I now eschew.  

I am certain of what I can prove (I sometimes may not be certain that a
proof that I have presented is really a proof, though).  Logicism is still
a living viewpoint.

Sincerely, M. Randall Holmes

holmes at math.idbsu.edu or mrh29 at dpmms.cam.ac.uk
http://diamond.idbsu.edu/~holmes

Boise State University and the University of Cambridge
must be held harmless for any silly thing I may say.

"And God posted an angel with a flaming sword at the gates 
of Cantor's paradise, that the slow-witted 
and the deliberately obtuse might not glimpse 
the wonders therein."  (Holmes, with apologies to Hilbert)