FOM: Finitist prejudices -- reply to Simpson

[Joe Shipman]

>  > The standard fundamental theories of physics deal freely with
>  > classes of operators on function spaces
>
> What if we could develop the requisite functional analysis in a
> subsystem of second order arithmetic that is conservative over PRA?
> PRA seems crucial here, because PRA is finitistic: it may commit us to
> potential infinity, but it does not commit us to actual infinity.  See
> also my paper on Hilbert's program
> <http://www.math.psu.edu/simpson/papers/hilbert/> and my book on
> subsystems of second order arithmetic
> <http://www.math.psu.edu/simpson/sosoa/>.

See my later posting (sent 15:42 EST Mar 8).  It doesn't matter if you can
develop this in a restricted system which doesn't allow you to prove as many
facts about integers; the functional analysis still has an ontology involving
uncountable sets even if you can't prove as many of those sets exist in the
weak system.  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) it is possible to maintain an ontology
which admits countably infinite sets but not uncountable sets; but I don't
know that the coding into Z2 is this nice.  In any case, the further coding
into PRA is another matter entirely -- it seems very unlikely there's a way to
do this that represents physically meaningful entities in the original theory
by finite sets, let alone by manageable ones; the restriction of attention
involved in treating such a finitary system as "all there is", and the
impenetrability of such a finitary representation, are so extreme as to be
unjustifiable on any grounds other than an a priori rejection of infinite
sets.

>  > the case (argued by Quine, Putnam, and Maddy) for an ontological
>  > commitment to infinite (even uncountably infinite) sets.
>
> Could you please state Quine's case briefly?  A summary of Putnam's
> and Maddy's case would also be welcome, though I think they have
> changed their minds on this issue.  (Maddy's first book was `Realism
> in Mathematics' and her second was `Naturalism in Mathematics'.)

I'll have to search the FOM archives; this was discussed last year.  Quine
argued that we believe our best theories of physics, implicitly accepting
their ontology which involves several levels of infinite sets, so we should
accept mathematical infinity because to reject it would require rejecting
these theories as nonsense.  Quine went further, accepting Zermelo set theory
on the grounds that it was simpler to formalize the power set operation than
to have several distinct types (I don't know whether the words "simplificatory
rounding out" used to describe this acceptance of Z are Quine's or
Feferman's).

>  > Steve and Martin, are the grounds on which you think the universe is
>  > finite empirical or a priori?
>
> I don't have a final, well-thought-out answer to this, and even my
> tentative answer may appear somewhat strange to you.  First, if I say
> the universe is finite, that doesn't imply that the universe is
> mathematically describable by a finite formula -- I have no evidence
> for such a statement.  What I mean by `finite' in this context is
> something like `definite' or `limited' or `following definite laws'.
> Now, with that proviso, my grounds for believing that the universe is
> finite is, loosely speaking, empirical' because the universe appears
> to be orderly, to follow definite laws.  My grounds are also, loosely
> speaking, a priori, because the idea of an orderly universe is a
> prerequisite for all thought about anything.

Yes, your answer appears very strange.  It's easy to describe a mathematically
infinite universe that follows definite laws (e.g. Newtonian gravitation in
R^3 with point masses and action at a distance according to the inverse square
law; this is already infinite because of the infinite precision needed to
specify the position and velocity of a point, but you could also consistently
have infinitely many points with a bit of care, or even uncountably many
points if you assume a mass density rather than point masses and rigid
3-dimensional bodies).

>  > A seriously finitist ontological position is in fact an atheistic
>  > position;
>
> I'm an atheist, so that's not a problem for me.  By the way, when
> theists say that God is infinite, I think the main implication of that
> is that, according to the theists, God is not limited by the laws of
> nature.

Not exactly, a finite being could also be unlimited by the laws of nature
because if the universe is finite it could have been created by a finite
being.  It is true that God is unlimited by the laws of nature, in the same
way that the author of a play is unlimited by the internal restrictions of the
play's "universe", and could even "write himself into" the play in an
arbitrary way, but the infinity of God is more than this.

>  > unity of knowledge and general intellectual integrity demand that
>  > we at least attempt to reconcile our schizophrenic attitudes.
>  ....
>
>  > It matters professionally whether you "really believe" in finitism.
>
> I thoroughly agree.

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.)

-- Joe Shipman