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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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  • An enjoyable read which leads me to ask why in geometry, that is plane old fashioned Euclidian geometry, straight edge and compass, is it not allowed to take a chord or an arc with a second compass and transport this dimension?

    If the physical world twists and pulls and stretches surely the rules of geometry should also be more flexible.

  • Mike Will

    An enjoyable read, as always.

    This principle may extend beyond scientific inquiry to nature. In DNA replication, very slightly less than perfect precision does not halt the process, and evolution results. The scientist is tasked with determining ‘appropriate’ in theory and practice, natural selection decides it elsewhere.

  • Approximation is lethal in positive feedback. Economics, meteorology, psychology, climatology… will never globally work. Beware a few small men altering what many clever men provide into sparse nonsense administrators can broadly inflict.

    Approximation requires corrections be applied at local small scale. After failure is planted, it is watered by approximations. When something fails, change it. Space-X slew NASA by lacking Divine Right management. Physics’ triumphs are discovered not perturbated. Newton was changed not patched.

  • Adrian

    There is a difference between approximation and tolerance and I think you mixed between the two.
    The distance between two cities is 100 miles is a good approximation because the size of each is about 10 miles , and I need it to calculate the ETA assuming I drive up to 60 miles /hour( sometimes 20 following a large truck).
    In engineering , be it electronics or carpentry we use the term tolerance which is based on the working tools for measurement, the tools for processing and the actualo needs.
    That’s why you will see on a design that a piece of wood is 5 inch +/- 1/10 or on a resistor that is 5 ohm +/- .1%.
    The smallest the tolerance, the higher the price of the product (more expensive tools required).
    Mathematics ( including geometry) are exact branches as long as you do not involve “practice”. Pi is Pi for any math problem . When you need actual calculations you need 3.14159…. depending on the tolerance you allow for the result in your practical provlem.

  • Approximation bridging science with knowledge :

    Approximation bridging probability – based (science) with possibility – based (knowledgeable science), whereas science currently shifted to the right, from probability – based toward possibility – based domain. The border between science and pseudo science seems becoming blurred (follow the link http://mobeeknowledge.ning.com/forum/topics/important-considerations-why-the-limit-of-science-should-shifted- “Important considerations why the limit of Science should shifted to the right”). Have a look at the Attachment

  • Xezlec

    I tend to agree with Jacques Futrelle, but with the understanding that he is talking about a specific aspect of science. I don’t think he means to say that scientists do not use approximations when performing calculations, only that one must use rigor and make one’s meaning absolutely clear when reasoning. When he says that a mistake causes the structure to topple, I think he’s exactly right, once you consider that he means a mistake in the reasoning by which an idea is derived. He is trying to explain to a layman why it is that scientists are so cautious about checking and verifying things, and why they resort to extremes of cold logic when determining what they can say is “true”, even in ways that may seem absurdly pedantic to an outsider (i.e. 5 sigmas for “discovery”, everything has to be reproduced multiple times to be believed, Monte Carlo methods aren’t considered good enough to say anything “definitive”, etc.). When I look at the way science is done in the real world, I see exactly what he is describing: just look at how that loose cable toppled the entire OPERA result.

  • jfb2252

    Unless you are calculating the _hypothesis_? of a triangle with side of 3 and 4 units,

  • You may like to read my article “Telling lies to describe truth: Do we emphasize the importance of “the art of approximations” to the students?” available at http://wp.me/p1tphj-7j

    I appreciate your feedback.

    Best wishes,
    Dr. M.Jagadesh Kumar, FNAE, FNASc, FIETE
    NXP(Philips)Chair Professor
    Dept. of Electrical Engineering
    Indian Institute of Technology, New Delhi 110016
    Email:[email protected]