FOM: RE: Connections between mathematics, physics and FOM

[Matt Insall]

I very much enjoyed reading your posting of Friday, January 28, 2000 9:23
AM.  No need to apologize for the length.  I have some comments and
questions.  You (Professor Ketland) said:

> (ii) Our theories must be recursively axiomatizable in order for them
> to be usable by the presumably finite human mind.
>

I am not entirely convinced of this, for I am not sure

(a) what you mean by the human mind,

and, if you mean by ``mind'' something more than the physical human brain,

(b) what you mean by ``finite''.

If we assume, as you do, that space-time includes infintely many points,
then might it not be the case that, although a particular human mind may
occupy a finite volume of space-time, that volume would, in general, include
infinitely many (aye, even uncountably many) points of space-time?  Thus,
although finite in volume, a human mind may be yet infinite in other ways,
and thus be capable of ``using'' a ``non-recursively axiomatizable'' theory.


Now, for more questions.  :-)

Professor Ketland said:


> (i)+(ii) Mathematics is the science of patterns and structures.



I am glad you included structures in your description of Mathematics, for I
am unconvinced that Mathematics is ``the science of patterns'', as some have
wont to say.  Basically, I see that a large part of Mathematics is an
attempt to accurately describe some portion of reality, whether this reality
is physical, cognitive, ideal, utilitarian, or philosophical.  Its role in
this activity of accurately portraying reality is frequently in the form of
analysis of structures, which may or may not exhibit ``patterns''.  There is
no a priori assumption that a pattern exists, in such pursuits.  (Consider,
for example, the current widespread use of the Axiom of Choice in much of
Western Mathematics, which guarantees the existence of a function for which,
in many cases, no known ``pattern'' is available to produce this function
without a strong axiom like the axiom of choice.)



Professor Ketland said:


>The
> physical universe is, as Martin Gardner and zillions of others have
> put it, mathematically structured. (Of course, it is a contingent and
> not a priori fact that the physical universe exemplifies one structure
> rather than another, but it is still a fact).


I agree with you (and Martin Gardner and zillions of others, apparently)
that ``the physical universe'' is ``mathematically structured'', but in
light of my comments above, my view is that this is an a priori fact,
because Mathematics, or at least some large part of Mathematics, is
specifically intended to accurately describe the physical universe.  As you
say, however, the fact that some particular structure describes the physical
universe (or Mathematics, or its foundations, IMHO) is not an a priori fact.


Professor Ketland said:


> It is also implausible (but an open research problem) that
> theoretical physics can get by either without mathematics at all
> (Field thinks it can) or with just constructive or predicative
> mathematics (Feferman thinks it can).


I can understand why this is an open research problem, and it is hard for me
to agree with either Field or Feferman on this point.  The reason I can
understand that it is an open research problem is that I realize there are
many who do not agree with me about what constitutes Mathematics.  You see,
even if the Formalist argument is [or should I say ``were'' :-)] correct,
and if Field were ``correct'' about what he sees now as ``Mathematics''
being superfluous in theoretical Physics, whatever he would replace it with
would, of a necessity, be a study of structure, at the very least, which is,
IMHO, (a significant portion of) Mathematics.  It's harder to argue with
Feferman on these grounds, I believe, so I maintain, to a great extent a
``wait-and-see'' stance on the necessity of impredicative or
non-constructive Mathematics in theoretical Physics.  (By ``wait-and-see'',
I do not mean I am entirely passive about it.  I am quite interested in some
of the same questions you raise later about the usefulness of concepts such
as CH, etc, in predicting the existence of some new particle or particles,
or some equally important result in theoretical physics.)



Professor Ketland said:


>If that is right, it follows that
> some scientific theories probably require mathematics which only
> has a platonistic interpretation (completed infinite sets, sets
> thereof, and so on). In a nutshell, (if Field and Feferman are wrong),
> we have: SCIENCE ENTAILS PLATONISM.


In fact, based on what I said above, we can, ``in a nutshell'' conclude
that, since I contend that Field is a priori wrong (if he is in fact,
claiming that theoretical physics can do without Mathematics, as you say),
it follows, a priori, that SCIENCE ENTAILS (some part of) MATHEMATICS, even
if it did not entail Platonism.



Professor Ketland said:


> I suspect that there is no predicting what pieces of pure
> mathematics might turn out to be useful in describing the physical
> universe.


I'll make a prediction (what the heck, I can't get into trouble over this
one can I):  All pieces of pure mathematics will turn out to be useful in
describing the physical universe.  :-)



Professor Ketland said:

>
> (i) The physical universe is infinite (i.e., contains infinitely many
> space-time points) and exemplifies a structure which contains at
> least one standard model of the natural numbers (one of our best
> physical theories, GR, does say this: it says spacetime is a
> differentiable manifold, and within this theory one can define space-
> time regions which are omega-sequences of space-time points)
>


Precisely!  Even if the old euclidean dream died with the introduction of
GR, the idealization introduced by GR (which appears to be a very good
description of reality) continues to suggest the existence of the actual
infinite.  Thus if we believe GR, we need not ASSUME the physical universe
is infinite.  In fact, it is not clear to me that ANY form of GR even makes
sense on a finite domain, although I am willing to be shown that some form
of it does make sense in a finite reality.


Professor Ketland said:


> First make two assumptions:
>
> (i) <snip>

>
> (ii) Our theories must be recursively axiomatizable in order for them
> to be usable by the presumably finite human mind.
>
> It follows from Godel's theorem that some of our physical theories
> will be deductively incomplete with respect to the set of physical
> statements.

Yes.  If you assume this ``strong form'' of Church's thesis, this will
follow.  (Actually, it says more than you have said here, IMHO, but I
believe it says what you intended, namely that even though our theories will
be incomplete, of necessity because we will never, in finite time, find
their deductive closure, that [Platonistic] deductive closure itself is
incomplete.)


>For some of our physical theories, there will be godel
> sentences which cannot be proved within the theory, but can be
> proved by some set-theoretical extension. An example would be
> the consistency of a precise formalization of GR. The consitency
> statement Con(GR) could be coded, using a reference omega
> sequence of spacetime points, as a statement about spacetime.
> Assuming that GR is true (and thus consistent), this new
> statement would be another true sentence about spacetime, but it
> would not be provable in GR. However, presumably, the
> consistency of GR *is* provable in some suitable set theory.
> Stewart Shapiro discusses something like this (in very careful
> detail) in a paper in Journal of Philosophy 1983, called
> "Conservativenesss and Incompleteness", criticizing Field's idea
> that adding maths to a nominalistic theory (e.g., a synthetic theory
> of space-time) must be a conservative extension.


I hope to look some of these things up soon, but I am very unclear about the
precise nature of the notion of a ``nominalistic theory''.  Can you point me
to some (more basic) literature on this particular topic?  The only place
I've seen much about ``nominalism'' is in philosophy, and I've read precious
little philosophy.  :-)  (I am, however, quite comfortable, in general, with
the notion of a conservative extension.)


Professor Ketland said:

> A: Could the addition of presently undecided highly abstract *set-
> theoretical* principles lead to the prediction of new *physical
> phenomena*? (I.e., could adding CH or a large cardinal axiom lead
> to the prediction of a new particle field, or fix the dimensionality of
> space-time, or something amazing like that).
>


At this point, I don't see how CH or a large cardinal axiom could lead to a
determination of the dimensionality of space-time, but I imagine that such
things could predict the existence of a new particle, or some other
phenomenon.  Since I have now seen some of the results of assuming CH (or
its negation), especially those in analysis and probability, I can imagine
that some interesting predictions will probably be made in the near future.


Professor Ketland said:

> B: Could the physical universe contain non-algorithmic processes
> which might be harnessed by humans to "compute" non-recursive
> functions (such as the characteristic function of the set of truths in
> arithmetic)?
>


Similar ideas are being developed already at the interface of quantum
physics, computer science and mathematics:  Witness the recent meeting of
the AMS, at which a short course was held on ``quantum computing''.  They
are not quite looking at actual ``non-recursive'' computation, but they do
seem to believe that their method of using quantum processes to compute will
make some exponential-time problems tractable.  The problem I see with
``computing'' non-recursive functions is that the problem of tractability is
overshot quite a bit in this case.  Now if you mean we could find some way
of describing a ``non-recursive'' set of axioms for arithmetic (for
example), I always hold out that hope, in complete defiance of the
Church/Turing thesis.  :-)


 Name: Matt Insall
 Position: Associate Professor of Mathematics
 Institution: University of Missouri - Rolla
 Research interest: Foundations of Mathematics
 More information: http://www.umr.edu/~insall