[FOM] Question about theoretical physics

[Joe Shipman]

Regarding (1), the paper of Laporta and Remiddi certainly feels like a mathematical proof--they appear to be relying on common assumptions about a numerical procedure that they are deriving a shortcut to the answer for (because evaluating the analytic expression requires an amount of work polynomial in the number of digits, while the underlying procedure it is supposed to represent required exponential time). If they were asked to state their theorem rigorously, they might say that a certain limit of finite approximations exists and has the value they state, where the limit can be rigorously defined; we should rule that possibility out before giving up on the idea of their having proved a purely mathematical theorem.

-- JS

Sent from my iPhone

On Jul 7, 2013, at 11:41 PM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:

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
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