### Ad hoc approximations

#### by Edward Z. Yang

In his book *Against Method*, Paul Feyerabend writes the following provocative passage about ‘ad hoc approximations’, familiar to anyone whose taken a physics course and thought, “Now where did they get that approximation from...”

The perihelion of Mercury moves along at a rate of about 5600" per century. Of this value, 5026" are geometric, having to do with the movement of the reference system, while 531" are dynamical, due to the perturbations in the solar system. Of these perturbations all but the famous 43" are accounted for by classical mechanics. This is how the situation is usually explained.

The explanation shows that the premise from which we derive 43" is not the general theory of relativity plus suitable initial conditions. The premise contains classical physics

in additionto whatever relativistic assumptions are being made. Furthermore, the relativistic calculation, the so-called ‘Schwarzschild solution’, does not deal with the planetary system as it exists in the real world (i.e. our own asymmetric galaxy); it deals with the entirely fictional case of a central symmetrical universe containing a singularity in the middle and nothing else. What are the reasons for employing such an odd conjunction of premises?The reason, according to the customary reply, is that we are dealing with approximations. The formulae of classical physics do not appear because relativity is incomplete. Nor is the centrally symmetric case used because relativity does not offer anything better. Both schemata flow from the general theory under special circumstances realized in our planetary system

providedwe omit magnitudes too small to be considered. Hence, we are using the theory of relativity throughout, and we are using it in an adequate matter.Note, how this idea of an approximation differs from the legitimate idea. Usually one has a theory, one is able to calculate the particular case one is interested in, one notes that this calculation leads to magnitudes below experimental precision, one omits such magnitudes, and one obtains a vastly simplified formalism. In the present case, making the required approximations would mean calculating the full

n-body problem relativistically (including long-term resonances between different planetary orbits), omitting magnitudes smaller than the precision of observation reached, and showing that the theory thus curtailed coincides with classical celestial mechanics as corrected by Schwarzschild. This procedure has not been used by anyone simply because the relativisticn-body problem has as yet withstood solution. There are not even approximate solutions for important problems such as, for example, the problem of stability (one of the first great stumbling blocks for Newton’s theory). The classical part of the explanans [the premises of that explain our observations], therefore, does not occur just for convenience,it is absolutely necessary. And the approximations made are not a result of relativistic calculations, they are introduced in order to make relativity fit the case. One may properly call themad hoc approximations.

Feyerabend is wonderfully iconoclastic, and I invite the reader to temporarily suspend their gut reaction to the passage. For those thinking, “Of course that’s what physicists do, otherwise we’d never get any work done,” consider the question, *Why do we have any reason to believe that the approximations are justified, that they will not affect the observable results of our calculations, that they actually reflect reality?* One could adopt the viewpoint that such doubts are unproductive and get in the way of doing science, which we know from prior experience to work. But I think this argument does have important implications for *prescriptivists* in all fields—those who would like to say *how things ought to be done* (goodness one sees a lot of that in the software field; even on this blog.) Because, just as the student complains, “There is no way I could have possibly thought up of that approximation” or the mathematician winces and thinks “There is no reason I should believe that approximation should work”, if these approximations do exist and the course of science is to discover them, well, how do you do that?

The ivory tower is not free of the blemishes of real life, it seems.

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Approximations are made because it is

impossibleto calculate things precisely. This is a very practical, and I think reasonable point of view. The downside, of course, is that they may turn out to be wrong, which is actually quite common. Scientific knowledge is build and rebuild with each generation.If anything, I would fault physicists to not appreciate the latter consequence of the approximations they make: most often, they are reluctant to accept that what they thought they knew is wrong.

Sounds like floating point.