FOM: A critique of pure Kant

[Vaughan R. Pratt]

So far I've been largely in agreement with Joe Shipman, but there's one
little point I'm having trouble with, involving a question of truth in
nature.

>Today, it is also conceivable
>that discoveries in physics could affect FOM.  E.g., if Church's
>thesis is false we could get new math axioms from experiments.
>(And CH or a large cardinal might greatly simplify the physics.)

Are Euclid's Fifth and Church's Thesis the interchangeable propositions
Joe seems to be casting them as?  And for whom?

On the one hand it would appear not: only the former is effectively
testable in nature.

The difficulty arises with measurement.  Even if some model of quantum
mechanics were to yield classically noncomputable reals as values, or
manifest classical noncomputability in some other way, our current
understanding of measurement makes this impossible to test.  All
accounts of measurement to date amount to the reduction of quantum
information to classical.  Since every finite set is computable, any
refutation of Church's thesis must entail some form of contact with a
classically presented infinite set (or an infinite-precision real
depending on how the refutation is presented).  However we have no way
of contacting an infinite quantity of classical information.

Any indirect access to infinitely much information would call for a
completely new notion of measurement, one that would allow us to take
on faith the reports of (say) quantum probes testing whether a given
initial configuration of a computation can reach a halting
configuration.

But this is not the only way to argue the question of testability.
There is nothing in principle that would rule out measurements of this
sort.  Some plausible theory might predict the existence of such
probes, some experimenter succeeds in making one, the probe never fails
in its predictions in those cases that we can confirm (every
terminating program is provably so, and at least *some* though
necessarily not all nonterminating programs are provably so), and
people eventually let down their guard and start using the darn thing
on a routine basis, greatly enhancing the optimizing ability of
compilers for one thing.

The former reasoning is in the style of argument common in philosophy,
particularly of the mathematical kind.  The latter more accurately
reflects the modus operandi of experimental physics.  *No one* has ever
seen a wavefunction, yet wild new theories are proposed, huge
accelerators built to confirm them and look for new physics, and
nuclear power generators are built that work (up to a point that is not
blamed on the theory), all based on the principle that the universe is
one big evolving wavefunction.

While I like the idea of philosophy and mathematics taking
*theoretical* physics as a role model, it seems to me that there is a
fundamental incompatibility between how philosophers and *experimental*
physicists do business.  This brings us to Kant's question of the
legitimacy of pure reason as the basic method of resolving questions.

Which of the above arguments is more consistent with current practice
in either philosophical or mathematical arguments?  If the former,
should either philosophy or mathematics adopt the experimental
physicist's more laid-back view of accepting nature for what they can
make of it, or continue to put their faith in pure reason?  And if the
latter, to what extent are we willing to allow empirically gathered but
widely accepted experimental evidence concerning nature to influence
the shape of mathematics?

However an even more fundamental question than *what* does mathematics
accept is, *does* mathematics accept?  Or does it simply reason from
givens, Peirce's legacy (from his dad)?

The dualistic view of mathematics that I favor makes these the two
complementary halves of mathematics, process and pattern (by
coincidence the title of my wife's master's thesis on the seawead
hormosira banksiae three decades ago).  For the pattern or
propositional half, what mathematics accepts should in my view be
applicable to whatever universe mathematics chooses to focus its
microscope or telescope on.

With regard to nature, mathematicians can indeed help physicists
rationalize their empirical evidence, and they can tell physicists
about some of the consequences of those rationalizations.

But mathematics also has and enjoys its own worlds quite apart from the
world of nature, and even usefully so in the light of experience
relating seemingly remote such worlds to nature many decades later.
These are worlds of pure reason.  Kant's own critique of pure reason is
itself the product of pure reason.  But as an empirical matter, pure
reason has proved its worth time and again.  Kant did not practice what
he preached.

In view of these considerations I see no reason why mathematics should
*found* itself on what currently appears to be true in nature.  By the
same token, what appears to be false in nature should not be
incorporated into the foundations of mathematics either.

Instead foundations should support the main goal of mathematics, which
I would take to be the production of quality mathematics by pure
reason.  Foundations, to the extent that mathematics needs them, should
help rather than hinder this cause.  If tinkering with the foundations
turns out to enhance the flow even a little (as it seems to have done
on occasion), that is a valuable contribution.

That mathematicians can and do make a living reading entrails and
calculating bending moments should not be taken to mean that the
discipline they happen to be serving then is their ultimate raison
d'etre.  Mathematics is its own subject.  When the world says,
"Mathematician, prove thyself", answer that their proof is in the
pudding, ours is but to reason why.

Vaughan