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Byron Jennings | TRIUMF | Canada

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Science: The Art of the Appropriate Approximation

There is this myth that science is exact. It is captured nicely in this quote from an old detective story:

In the sciences we must be exact—not approximately so, but absolutely so. We must know. It isn’t like carpentry. A carpenter may make a trivial mistake in a joint, and it will not weaken his house; but if the scientist makes one mistake the whole structure tumbles down. We must know. Knowledge is progress. We gain knowledge through observation and logic–inevitable logic. And logic tells us that while two and two make four, it is not only sometimes but all the time. – Jacques Futrelle, The Silver Box, 1907

Unless, of course, it is two litres of water and two litres of alcohol, then we get less than four litres. Note also the almost quaint idea that science is certain, not only exact, but certain. We must know. The view expressed in this quote is unfortunately not confined to century-old detective stories, but is part of the modern mythology of science. But in reality, science is much more like carpentry. A trivial mistake does not cause the whole to collapse, but I would not like to live in a house built by that man.

To the best of my knowledge, there has never been an exact calculation in all of physics. In principle, everything in the universe is connected. The earth and everything in it is connected by the gravitational field to the distant quasars. But you say, surely that is negligible, which is precisely the point. It is certainly not exactly zero, but with equal certainty, it is not large enough to be usefully included in any calculation. I know of no terrestrial calculation that includes it. Even closer objects like Jupiter have negligible effect. In the grand scheme, the planets are too far from the earth to have any earthly effect. Actually, it is not the gravitational field itself which is important but the tidal forces which are down an additional factor of the ratio of the radius of the earth to the distance to the planet in question. Hence, one does not expect astrology to be valid. The art of the appropriate approximation tells us so.

Everywhere we turn in science we see the need to make the appropriate approximations. Consider numerical calculations. Unless you are calculating the  hypotenuse of a triangle with side of 3 and 4 units, almost any numerical calculation will involve approximations. Irrational numbers are replaced with rational approximations, derivatives are replaced with finite differences, integrals with sums, and infinite sums with finite sums. Every one of these is an approximation—usually a valid approximation—but never-the-less an approximation. Mathematical constants are replaced by approximate values. Someone once asked me for assistance in debugging a computer program. I noticed that he had pi approximated to only about six digits. I suggested he put it in to fifteen digits (single precision on a CDC computer). That, amazingly enough, fixed the problem. Approximations, even seemingly harmless ones, can bite you.

Even before we start programing and deciding on numerical techniques, it is necessary to make approximations. What effects are important and which can be neglected? Is the four-body force necessary in your nuclear many-body calculation? What about the five-body force? Can we approximate the problem using classical mechanics, or is a full quantum treatment necessary? Thomas Kuhn (1922 – 1996) claimed that classical mechanics is not a valid approximation to relativity because the concept of mass is different. Fortunately, computers do not worry about such details and computationally classical mechanics is frequently a good approximation to relativity. The calculation of the precision of the perihelion of Mercury does not require the full machinery of general relativity, but only the much simpler post-Newtonian limit. And on and on it goes, seeking the appropriate approximation.

Sometimes the whole problem is in finding the appropriate approximation. If we assume nuclear physics can be derived from quantum chromodynamics (QCD), then nuclear physics is reduced to finding the appropriate approximation to the full QCD calculation, which is by no means a simple task. Do we use an approximation to the nuclear force based on power counting, or the old fashioned unitarity and crossing symmetry? (Don’t worry if you do not know what the words mean, they are just jargon and the only important thing is that the approximations lead to very different looking potentials.) Do the results depend on which approach is used, or only the amount work required to get the answer?

Similarly, in materials science, all the work is in identifying the appropriate approximation. The underlying forces are known: electricity and magnetism. The masses and charges of the particles (electrons and atomic nuclei) are known. It only remains to work out the consequences. Only, he says, only. Even in string theory, the current proposed theory of everything, the big question is how to find useful approximations to calculate observables. If that could be done, string theory would be in good shape. Most of science is the art of finding the appropriate approximation. Science may be precise, but it is not exact, and it is in finding the appropriate approximation that we take delight.

Additional posts in this series will appear most Friday afternoons at 3:30 pm Vancouver time. To receive a reminder follow me on Twitter: @musquod.

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