# [FOM] Question about theoretical physics

Timothy Y. Chow tchow at alum.mit.edu
Thu Feb 28 19:29:31 EST 2013

```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
```