FOM: Finitist prejudices and ontological commitment -- Reply to Simpson and Hayes JoeShipman at
Thu Mar 11 01:37:18 EST 1999

Hayes writes:

> There seem to be a lot of different concepts of 'finite' being used in this
> discussion. Simpson claims that 'finite' means lawlike. However, it is easy
> to describe infinite sets governed by simple laws (the natural numbers
> might be a good candidate, for example); Shipman identifies 'finite' with
> 'having a finite description', which also seems peculiar for a similar
> reason (consider for example the handy Greek letter 'omega'.) Davis wonders
> if the question even makes sense.
The letter "omega" is not a *description* of the set of integers, it is a
*name* for the set of integers.  A pointer is not a description.

> Maybe I'm missing something here, but the meaning of "the universe is
> finite" seems perfectly clear and much more straightforward. It means that
> there is a finite total mass of the universe; that the number of elementary
> particles is finite (maybe indeterminate for quantum reasons, but finite),
> and that the radius of the universe is finite. All of these are currently
> understood to be correct, I believe.

Three possible problems with this: 
1) the universe does not necessarily have a finite total mass-energy if you
count gravitational potential energy (we can't know the answer to this without
a good quantum theory of gravity which we don't have yet)
2) spacetime is modeled as continuous rather than discrete so we still need an
infinite amount of information to describe a finite region of spacetime
3) the radius of the universe considered as a spacetime manifold is not finite
if the universe is "open" (in this case one has the option of regarding this
as a potential rather than an actual infinity if one is prepared to grant a
primordial existence to the concept of "time", though modern physicists would
likely regard such an arbitrary slicing of spacetime as "unphysical"). 

Simpson writes:

> > >(SS) What if we could develop the requisite functional analysis in a
> > > subsystem of second order arithmetic that is conservative over PRA?
> >
> >(JS) ... the functional analysis still has an ontology involving
> > uncountable sets

> Joe, you seem to be saying that by writing down these conservative
> extensions of PRA we are ipso facto committing ourselves to a certain
> ontology involving uncountable sets.  I disagree.  I think you are not
> giving due consideration to Hilbert's ideas in his Hilbert's program
> paper.  My suggestion above is very much in the spirit of Hilbert's
> program.  See also my paper
> <> and my book
> <>.

>Hilbert's point (as I read him) is that it's OK to introduce new
>mathematical entities conservatively over old ones, and this does not
>increase our ontological commitments.  Hilbert views this as a variant
>of a well established mathematical procedure, the `method of ideal
>elements'.  The new entities are not `real'; they are `ideal', i.e.,
>instrumentalities that we introduce for our convenience.
>Conservativity implies that we can freely use the new entities to
>prove theorems about the old entities, and the new entities can always
>be `eliminated' if we want.  A simple example is the conservative
>introduction of the complex numbers as ordered pairs of real numbers.
>These ordered pairs don't have to correspond to anything in reality;
>they are mere instruments, and they can be eliminated.

The problem is that some of the "ideal" elements "conservatively" introduced
in the *mathematical* type theory identified with *physically meaningful
entities* in the physical theory.  If you "eliminate them" your ontology
excludes not only infinite sets but also wave functions, electrons,
gravitational fields, and other elements of reality. I interpret Quine's
point, that we believe our best physical theories to be true and ontological
commitment to these theories requires ontological commitment to the necessary
mathematical entities, to be referring to a commitment not only to the
physical theory but to the objects it is representing.  The alternative is an
entirely instrumentalist metaphysics in which all that "really" exist are dial
settings, pointer readings, and algorithms for translating experimental setups
into observed outputs, and there is no "physical reality".  Such a metaphysics
is untenable because of the complete impenetrability of the algorithms when
they are forcibly rendered in such instrumentalist terms (this is a practical
impossibility anyway, we find we must always actually "do physics", i.e. refer
to entities such as electrons, wave functions, rest masses, gauge fields, or
whatever).  That different but equivalent mathematical descriptions of the
same physical situation can be found (e.g. Schrodinger vs Heisenberg) is of no
consequence because all the descriptions involve complex infinitary
mathematical objects. 

> > (JS) If the coding of the functional analysis into second-order
> > arithmetic is straightforward and preserves meaning (by this I mean
> > that "physically meaningful" entities in the original theory have
> > manageable representations as sets of integers) ...

> (SS) The coding into Z2 and PRA is reasonably nice, but I don't think it
> has the properties that you want.  For instance, real numbers, Banach
> spaces, operators on Banach and Hilbert spaces, etc, would be coded as
> sets of non-negative integers.  This does not seem to be physically
> meaningful.

As I suspected...

> However, that doesn't matter, because the coding in question is merely
> a technical device used to prove the relevant conservation results.

No it's not -- because the "ideal" elements of higher type that have been
introduced are there not just to prove theorems about integers but also to
represent elements of physical reality.

> Here's the situation.  There is a big, many-sorted theory, call it T,
> containing many kinds of entities: natural numbers, integers,
> rationals, reals, complex numbers, separable Hilbert spaces, separable
> Banach spaces, operators on separable Banach spaces, countable
> algebraic structures, etc etc.  Many theorems of functional analysis
> can be proved in T, yet T is conservative over PRA for Pi^0_2
> sentences, via the obvious interpretation of PRA into it.  

> This is a far-reaching partial realization of Hilbert's program.  It
> seems to go a long way toward answering your implicit question.  I
> take your question to be, how can we make mathematical sense out of
> functional analysis used in quantum physics, without ontologically
> committing ouselves to uncountable sets?  I think T shows how to do
> that.

Wrong question -- although we can make "mathematical sense" out of functional
analysis even if we are finitists because of the conservation results, we are
committed not only to the functional analysis used to develop quantum physics
but to quantum physics itself.  The "reduced" theory with "ideal" elements
eliminated may be just as good for proving theorems about integers, but it is
absolutely useless as a model that is supposed to have some connection with
physical reality.

> > What do you think of my proposed test for finitists?  (The test is
> > whether one is more bothered by the unsolvability of Diophantine
> > equations or the undecidability of the Continuum Hypothesis.)

> I don't think it's a good test, because it's too touchy-feely, too
> subjective.  The issue of finitism versus non-finitism is an objective
> scientific issue.  To formulate it as an issue of who is bothered by
> what seems misleading.

A good criticism; but although finitism may have a truth value, we can't see
an objective way to decide the question given what we know now; it is more of
a matter of faith at the moment and my test may help people clarify their own
minds on the issue.

-- JS 

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