[FOM] Question about theoretical physics

[Timothy Y. Chow]

I can't, of course, answer Joe Shipman's more technical questions. 
However, I can say which of his demands I believe are reasonable.

On Sat, 6 Jul 2013, joeshipman at aol.com wrote:
> (1) Can the result of Laporta and Remiddi be regarded as a rigorous 
> theorem of mathematics?

This doesn't seem relevant to me.  They have written down a well-defined 
number and have officially proposed it as a theoretical prediction.  That 
should be enough.

> (2) considering this experiment only (anomalous magnetic moment of the 
> electron), does the fourth-order coefficient that has been calculated 
> come with a rigorous upper and lower numerical bound?

Here I agree that this is a reasonable demand, although I would replace 
"rigorous" with "explicit".

> (3) Is there any reason to believe that an analytic expression of the 
> type that Laporta and Remiddi established for the third-order 
> coefficient does not exist for higher-order coefficients?

A reasonable question, though somewhat tangential in my opinion; question 
(4) below is more to the point.

> (4) Is there a method in principle to calculate arbitrarily precise 
> bounds for the fourth-order and higher coefficients, which we may regard 
> as algorithmic by applying the Church-Turing thesis to the mind of 
> either Professor Neumaier, or of some more expert specialist in the 
> field whom Professor Neumaier could name?

This is a reasonable question in the sense that the answer might be yes, 
in which case Joe Shipman's objections should be satisfactorily answered. 
I'm not convinced, though, that the answer *needs* to be yes.  Suppose, 
for example, that the calculation is "open-ended" in the following sense. 
There is an outline of a procedure to be followed, whose initial steps are 
clearcut.  At some point, however, one may need to verify that certain 
calculations do indeed produce answers of a certain form that we expect 
them to have.  If they do, then all is well and we can continue.  But if 
they don't, then it may not be quite clear how to proceed, depending on 
how badly the intermediate calculation violates expectations.

In this scenario, there is strictly speaking no "algorithm."  At best we 
have a "conjectural algorithm" where some steps are only conjectured to 
terminate.  In my mind this would still be good enough as long as the 
details of the fourth-order case have been worked out in enough detail to 
produce sufficient precision (note: not "arbitrary precision") to make a 
meaningful theoretical prediction.  The important thing is that the 
calculation that has been done already is verifiable by someone else who 
is willing to study the subject in detail, and it's not so important that 
the procedure be defined in a way that guarantees the ability to 
extrapolate the results indefinitely.

For a more mathematical analogy, one could imagine some calculation that 
requires computing the ranks of certain elliptic curves over Q, where one 
conjectures, but cannot prove, that the elliptic curves that come up have 
ranks that are computable using known methods.

Tim