FOM: reference of mathematical discourse

[Randall Holmes]

This posting is from M. Randall Holmes

In my postings criticizing Hersh, I have stated that for the
applicability of mathematics to make sense, mathematical theories must
refer to some aspect of the real world.  I'm going to try to outline a
hypothesis (NOT original with me) as to how this works.

I think that the simplest hypothesis is the one which I suspect is the
"working hypothesis" of most mathematicians (though they may not
really believe it): mathematical discourse really does refer just as
it appears to on the surface.  There are natural numbers, algebraic
varieties, and so forth, "out there".  This is simple Platonism.

There are two objections.  One is the visceral objection to
non-physical entities; this has very little force for me, and I will
allow others to address it.  One aspect which should be noted is that
the mathematical universe of simple Platonism, if one supposes it to
be or at least include a model of ZFC, is far larger than the physical
universe, even if one supposes the physical universe to be infinite!
The cardinality of the set of all physically relevant objects 
certainly appears to be no more than 2^\omega.

The other, and for me rather more powerful objection is the one posed
by Martin Davis (and even by me in earlier postings): when we talk
about the number 7, it is not clear even in the universe of simple
Platonism what object we are talking about.

My view is that it is easier to start with the reference of
mathematical propositions to the real world; the status of
mathematical objects (and their elusiveness noted by Davis) will come
out in my account of propositions.

I think that all mathematical propositions are "really" of the
following form:

If M is a model of theory T, then P

where P is a statement in the language of theory T about objects in
the model M.

Thus, with regard to one's favorite statement about the number 7 (say
A(7)) the statement usually thought of as A(7)

will be

In any model of arithmetic, A(7)

or, more precisely,

"In any structure with operations including a unary operation S and constant 0 satisfying
<Peano's axioms>, A(S(S(S(S(S(S(S(0))))))))".

Notice that this position handles the elusiveness of 7 just fine:  the
identity of 7 depends on choice of a specific relation S and object 0;
7 is hidden behind an abstract data type interface, as it were.

In the simple Platonist world including a model of ZFC, we can do
arithmetic as soon as we prove the existence of a model of arithmetic.
There are many models of arithmetic; each of them has a different
number 7.  But all questions about 7 _as a natural number_ (all
theorems of arithmetic about 7) will get the same answer in each
model.  There is of course a von Neumann ordinal traditionally called
7 in models of ZFC which has non-arithmetic properties such as 6 \in 7
which do not necessarily hold in all models of arithmetic in ZFC.

There is a serious problem with this view of mathematical propositions as
conditionals.  If the universe is finite, the most obvious interpretation
of the conditional "There is a model of arithmetic" => P , for any P,
is that it is vacuously true because there are no models of arithmetic.

To mkae this approach work, one would then have to view the
conditional as a non-truth-functional operation.  The logic of
counterfactuals is a nasty philosophical issue, which one would like
to avoid.

However, if one supposes enough set theory to be able to prove the
completeness theorem for first-order logic (one does not need ZFC for
this, but one does need infinitely many objects), life becomes much
simpler.  The existence of models of any first-order theory comes to
coincide with the consistency of the theory, so semantics and logic
become easier to relate to one another.

This enormous simplification of the logic of the conditional
interpretations of mathematical statements can be taken to justify the
adoption as a standing hypothesis that there are enough objects to
form a model of (say) second-order arithmetic.

A further point about logic is that the conditionals which I suppose
here to be the true referents of mathematical objects are second-order
assertions: they involve quantification over predicates and
operations, as in the example above "For any operation S and object 0
satisfying <Peano's axioms>, A(S(S(S(S(S(S(S(0)))))))))" It seems
quite natural to admit second-order theories outright, since one has
to do second-order quantification anyway; however, this needs to be
approached warily, since second-order logic does not have a
completeness theorem.

Any mathematical assertion would then be formalized as a conditional
"In any model of theory T (this part is implicit in the context, not
actually expressed in the statement of a theorem in ordinary
mathematical practice), P holds" in the second-order theory of some
domain of entirely uncharacterized objects.  To get the logic to work
nicely (for first-order theories) would require at least a countable
infinity of objects.  Second-order theories would require stronger
assumptions about how many objects there were.

Existential assumptions are required here (there are at least \omega
objects and at least 2^\omega collections of objects in this
framework) but they are all of the form that there are "enough"
objects to build models of a given theory.  Notice that these weakest
assumptions might be on some views be satisfied in the physical world!
It doesn't make sense under this view to refer to particular
mathematical objects except in conditionals of the form "for any model
of T ...", where T is the theory in which the object is defined; every
mathematical object of a particular theory is as it were hiding behind
an ADT interface (which can be implemented by choosing an actual
model of the theory).

Applicability of this to the real world is a further question.  Any
proposition (say, of a scientific theory) which refers to mathematical
objects (natural numbers, real numbers, Hilbert spaces, or whatever)
must on this interpretation include the hypothesis that there is a
model of the mathematical theory in which the mathematical objects in
question are defined; it doesn't make sense, on this interpretation of
mathematical discourse, to refer to a mathematical object outside a
conditional statement about models of the theory in which it is
defined.  What is needed is the principle that a statement not
referring to mathematical objects which can be deduced from the
scientific theory is true.  If models of the mathematical theory
actually exist, this is obvious; if the actual existence of models of
the mathematical theory is doubtful (the existence of models of
second-order ZF might be doubted even by one who regards it as
certainly consistent) then one needs some kind of principle asserting
that supposing the "real world" to be enhanced by the addition of a
model of (for example) second order ZF induces a conservative extension of
the theory of the real world; nothing false of the real world can be
proven using the hypothesized extra objects.  Such a principle would
have to be recognized as a working hypothesis not susceptible of
formal proof.

All of this is pure speculation.  I find it easiest to be a simple Platonist
about ZFC + large cardinals or my favorite strong extension of NFU and
build models of whatever other theories I want to think about there :-)

I do see that there is something dangerous about this proposal; I seem
to have presented a view under which something like second-order
arithmetic (the minimal mathematics required for the proof of the
completeness theorem) is the correct foundation for all mathematics
expressible in first-order theories: work in first-order ZFC is
apparently to be viewed as working out the PA_2 consequences of
Con(ZFC).  But perhaps ZF-istes to whom this view would appeal would
prefer second-order ZF as their foundation.

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
slow-witted and the deliberately obtuse might | holmes at math.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes