Newsgroups: comp.ai.philosophy,sci.physics,alt.consciousness,sci.math
Path: cantaloupe.srv.cs.cmu.edu!das-news2.harvard.edu!news2.near.net!bloom-beacon.mit.edu!galois!baez
From: baez@math.mit.edu (John Baez)
Subject: Re: Penrose in the shadows -- Part 4
Message-ID: <1994Dec24.161430.20690@galois.mit.edu>
Sender: usenet@galois.mit.edu
Nntp-Posting-Host: nevanlinna
Organization: MIT Department of Mathematics
References: <3cqgsp$e85@galaxy.ucr.edu> <D0x9v5.DJs@gpu.utcc.utoronto.ca> <3ctu27$6tl@mp.cs.niu.edu>
Date: Sat, 24 Dec 94 16:14:30 GMT
Lines: 74
Xref: glinda.oz.cs.cmu.edu comp.ai.philosophy:24052 sci.physics:104323 sci.math:90211

In article <3ctu27$6tl@mp.cs.niu.edu>, Neil Rickert <rickert@cs.niu.edu> wrote:
>In <D0x9v5.DJs@gpu.utcc.utoronto.ca> pindor@gpu.utcc.utoronto.ca (Andrzej Pindor) writes:
>>In article <3cqgsp$e85@galaxy.ucr.edu>, john baez <baez@guitar.ucr.edu> wrote:

>>>                                                           (Indeed,
>>>since the wavefunction lies in L^2(R^n), which consists of equivalence
>>>classes of functions, it doesn't really have well-defined values at
>>>points.)

>>Do you mean that a sequence of successive averages over smaller and smaller 
>>volumes around a point does not converge?

Yes, that's one way to put it, or better, "need not converge."

>I should let John answer.  But let me throw in a comment.  The wave
>equation is a second order partial differential equation.  Strictly
>speaking, any solution should be a twice continuously differentiable
>function.  If f and g are twice differentiable functions of one
>variable, then the function u(x,y,z,t) = f(x-ct) + g(x+ct) is a
>particular solution to the wave equation, representing two planar
>waves moving in opposite directions along the x-axis.  But we can
>make sense of u, even when f and g are not required to be
>differentiable.  Mathematical physicists consider the equation as an
>unbounded operator on L^2, and that function space contains functions
>which can be far from continuous and, as John says, don't have well
>defined values at individual points.

I think the folks who proved this much quoted uncomputability result,
Pour-El and Richards, were looking at C^2 solutions of the wave
equation on the line.  They showed something like that you could have
a solution u(x,t) with Cauchy data u(x,0) and udot(x,0) (here udot is
the time derivative of u) that were computable functions, but such
that u(0,T) was not computable for some particular time T > 0.  I
forget the precise statement of their result and it's important to
emphasize that the details matter immensely.  First, there are several
different definitions of computability for real-valued functions.  I
think they were using the most common one, which says basically that F
is computable if the set of rationals a,b,c,d for which F maps the
open interval (a,b) into the open interval (c,d) is recursively
enumerable.  (NOT recursive!)  There are other equivalent ways of
saying this definition.  Secondly, it makes a huge bunch of difference
if they said u(x,0) was computable as a function of x, or u and its
first derivative were.  It's easy to have computable functions with
uncomputable derivatives.  I *think* they just made u and udot be
computable, not any spatial derivatives thereof.

Now I find this all a bit too idealized; of course everything is
idealized, but I don't like C^2 solutions very much; I prefer the
space of finite-energy solutions.  The norm on this space is the
energy squared, which seems a bit more physical than the C^2 norm,
which requires you to know the supremum of the 2nd derivative of u
over (say) some interval of times t.  And one can show that the
finite-energy space is a real Hilbert space, and that time evolution
on it is continuous, orthogonal (i.e. 1-1, onto, and norm-preserving),
and computable (in the most common sense of computability for
continuous functions between separable metric spaces equipped with a
favored dense set of points, which is just a generalization of the
definition for real-valued functions I gave above.)

So anyway, depending on how you make the statement precise, you can
prove either that time evolution for the wave equation is computable,
or is not computable, which suggests that it's a terrible idea to
build a philosophical argument on one theorem or another unless your
philosophy is so sophisticated that the difference between C^2
solutions and finite-energy solutions is really taken into account!

Anyone interested in this stuff should read

Computability in analysis and physics / Marian B. Pour-El, J. Ian Richards.  
   Berlin ; New York : Springer Verlag, c1989.
     Series title:  Perspectives in mathematical logic.



