FOM: Infinite sequences in the physical world: reply to Richman

[Joe Shipman]

Shipman:
>> >> In particular, if our best theories of physics lead us to conclude

>> >> that a certain reproducible experimental procedure will generate a

>> >> sequence that is mathematically definable but not computable, then

>> >> CT-phys will be false,

Richman:
>> >I'm not sure what this means. "Reproducible" seems to refer to an
>> >actual procedure that can be carried out. Although we are constantly

>> >going back and forth between our mathematical models and the real
>> >world, isn't it some sort of categorical mistake to talk about a
>> >procedure of this sort generating an infinite sequence of positive
>> >integers?

Shipman:
>> Not at all.  The sequence doesn't have to be infinite to be
>> metamathematically significant.

Richman:
>Maybe there is some metamathematics going on in CT-physics that I've
>missed. It seems to me that we are dealing with mathematical models
>and with the real world. Perhaps the phrase "mathematically definable"
>is supposed to take us into the realm of metamathematics. Does the
>sequence have to be infinite to be mathematically significant, or
>physically significant? What is the point of the qualifier
>"metamathematically"?

"Mathematically definable" is not mysterious.  It is just an ordinary
sequence that can be defined with reference to math only and not the
outside world.  For example, the sequence f(n) which equals 1 when the
nth Turing machine computes a total function and 0 otherwise is
definable but not recursive.  If you want to be picky, then
"ZFC-provably first-order definable in the language of set theory" will
do.


Shipman:
>> The idea is that the reproducible
>> experimental procedure, will, in theory, generate integers until you
>> run out of raw materials or die of old age or the sun turns into a
red
>> giant or whatever, which will at any given time represent a finite
>> initial segment of a mathematically definable nonrecursive sequence
f.

Richman:
>Of course we may only get a couple of dozen of those integers by the
>time the sun turns into a red giant. Suppose, for example, that the
>sequence is given by the digits in the decimal expansion of some
>dimensionless physical constant. What kind of experiment can we run to
>distinguish whether that infinite sequence of digits is recursive or
>not? In fact, we may be able to compute the first hundred values of f
>even if f is not recursive.

The EXPERIMENT doesn't tell you the sequence f(n) is recursive or
nonrecursive.  The PHYSICAL THEORY tells you a mathematical definition
of the sequence and asserts that the experiment will produce the
sequence.  MATHEMATICS can then prove the sequence recursive, prove the
sequence nonrecursive, or fail to answer the question of the sequence's
recursivity.

After the experiment is run, the following situation occurs: if the
sequence is recursive, then one can calculate whether experiment matched
theory and thus confirm or disconfirm the theory.  Even if the sequence
is not recursive, it may be the case that the particular terms of the
sequence you have experimentally generated can be "computed" by proving
in ZFC that the values of those terms are certain particular integers.
In this case, you also an say that the experiment either confirms or
disconfirms the theory depending on whether or not it produced those
numbers.

However, the metamathematically interesting case is where the defined
sequence is not  recursive and you cannot establish in ZFC what the
particular values you have experimentally generated "ought" to be
according to the theory.  In this case it no longer makes sense to say
you are running experiments to test the theory.  Rather, if the theory
has passed all other tests, you are assuming the truth of the theory as
an axiom and are deriving NEW mathematical statements by resorting to
experiment.  The only epistemological difference between this and
computer-proved theorems is that we understand the physics that goes
into computers so well that we believe we could simulate them by hand,
so that we believe from the computer result not only that the theorem is
true but also that an ordinary proof of it exists.  (A "computer proof"
of theorem T convinces us not only that T is true but that T is provable
in the ordinary way, but the type of experimental derivation of a
mathematical statement that we are contemplating here could only
convince us of the statement's truth and not of its ordinary
provability.)


Shipman:
>> Even if you never have the completed infinite sequence, at some point

>> you have the value x of f(N) for some N large enough that ZFC does
not
>> settle the truth value of the mathematical assertion "f(N)=x".
>> (Because if contrariwise ZFC did settle all such questions then f
>> would be recursive after all.)

Richman:
>"At some point"? Only if the universe is very accommodating. In any
>event, we will never know when we have reached that point.

We will be able to know, if the nonrecursive sequence is any of the
natural known ones, to which the halting problem is reducible.  Because
then we just have to get enough of the sequence to settle whether the
computation looking for an inconsistency in ZFC halts, for example.


Richman:
>I think I understand this argument, but the question remains as to
>whether it even makes (experimental) sense to say that the decimal
>expansion of a physical constant is nonrecursive, or even irrational.
>The logical positivists may have gotten a lot of things wrong, but
>their insight that some statements should be rejected because they
>have no meaning still has a lot of appeal.

You can't experimentally verify that a constant is nonrecursive or even
irrational, but that's not the point -- you CAN at least test the
hypothesis that the constant is a particular defined real number as far
as you can compute the number.  Either it diverges from the hypothesized
result at some finite point and the theory is falsified, or it matches
for as far as you can compute the theoretical prediction and the theory
is confirmed.  If you never falsify the theory, but can't calculate the
theoretical prediction at some point, then running the experiment to
that point stops being a test of the theory and starts being a new
source of mathematical knowledge.

-- Joe Shipman