[FOM] Question about theoretical physics

[Timothy Y. Chow]

On Thu, 28 Feb 2013, Kreinovich, Vladik wrote:
> If the results are close but not that accurate, they try second 
> approximation.
>
> For QED, we get correct result with I think 10 digits or so, very 
> accurate, by using the appropriate approximation, enough to explain most 
> experiments

Again, allowing me to caricature the situation for simplicity, I'd say 
that the objection is this.  If the sequence of approximations is not 
believed to converge, then this looks like "cheating" to an outsider.  I 
compute the first approximation, and it's not so good.  So I compute the 
second approximation, and it's better, but still not great.  I compute the 
third approximation, and wow!  It matches to 10 digits.  I collect my 
Nobel Prize and conveniently forget to mention that if I had computed the 
fourth approximation, it would have matched only 5 digits.

Sort of like tossing a needle a multiple of 213 times so that after 3408 
trials one can "estimate" pi to 7 digits.

I think I know what the correct rejoinder is.  The situation is not like 
Buffon's needle because even if physicists have a whole array of different 
theoretical calculations that they could try, there's no reason a priori 
to expect *any* of the methods to agree with experiment to 10 digits. 
Admittedly, because the physicists can't precisely map out the space of 
possible theoretical calculations ahead of time, they can't make a precise 
quantitative statement about just how remarkable the agreement with 
experiment is.  But that is generally the case with scientific predictions 
anyway---we don't have any quantitative estimate of "how remarkable" 
Einstein's calculation of the perihelion precession of Mercury is. 
Though QED isn't mathematically rigorous, that doesn't mean it's 
infinitely malleable, and it's still possible to have an intuitive sense 
that there's something very remarkable about a particular non-rigorous 
calculation agreeing with experiment to that extent.

Having said that, I think that popular accounts do sometimes give the 
impression that the large number of digits of agreement makes this the 
most remarkable agreement between theory and experiment of all time, and 
maybe that is overstating the case?  It's not like every digit of 
agreement exponentially increases our confidence in the correctness of the 
theory?

Tim