Understanding History
The Principle of Simplicity

William of Occam

Two principles mentioned in these notes are old enough to have Latin formulations. One is associated with William of Occam (variously spelled). In one version, it runs thus:

Entia non sunt multiplicanda praeter necessitatem.

Or, "entities should not be multiplied beyond what is necessary." A historical explanation should not be more complicated than it needs to be, to cover the observed facts. It is the first of the Rules of Reasoning in Philosophy which begins the third section of Isaac Newton's Principia Mathematica. Newton's statement is:

Rule 1. We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes.

The simpler theory is better that the more complicated theory. All this has its intuitive appeal. A difficulty arises when we try to be precise about what we mean by "more complicated," and thus what we mean by "simpler."

The Copernican conception can be used as an example. Ptolemaic astronomy posited circular orbits for the planets, because the circle was considered the most perfect shape, and the universe was considered to be a perfect creation. Differences between these theoretical circular orbits and the observed motions of the planets led to the positing of epicycles, additional circular orbits superimposed on the basic one. One set of epicycles did not quite suffice to account for the actual motions, so other epicycles were added to the system. Eventually the agreement with observation was fairly good, though never perfect. Copernicus's contribution was to notice that the assumption of an elliptical orbit accounted at once for the actual observed motion. He was able to get rid of the epicycles. A single equation, not a nest of equations, governed planetary motion.

Someone might object: This violates the principle of simplicity, because the equation of a circle:

x² + y² = a²

is simpler than the equation of an ellipse:

x²/a² + y²/b² = 1

That judgement would be wrong. Simplicity is not to be judged by the elements of the system, but by the complexity of the system. An orbit plus modifications is more complicated than an orbit by itself. Since the (elliptical) orbit by itself will account for observations, the modifications of the (circular) orbit are unnecessary. They are excrescences. Occam's razor will accordingly shear them off.

Laplace's masterpiece Mécanique Céleste (1799-1825) applied Newtonian principles to all celestial motions. He made few and simple assumptions, among them that Newtonian gravity held throughout the system. Napoleon, to whom he presented an abridged account (Exposition du Système du Monde, 1796), famously remarked, "You have written this huge book on the system of the world without once mentioning the author of the universe." Laplace's equally famous reply took a certain amount of nerve:

Sire, je n'avais pas besoin de cette hypothèse

The assumption of universal gravitation sufficed. He did not need the God hypothesis.


Occam's razor is often wrongly applied. The key is the the phrase "praeter necessitatem" in the Latin formula. Not all complications are forbidden. On the contrary, complications necessary to cover the facts are not only allowed but required. However simple or even beautiful a theory may be, if it does not cover the facts, it is not viable. A complex theory may be a sign that the thing it is trying to explain is not yet deeply understood. But it may also be the correct explanation of a genuinely complex phenomenon.

To many, "simple" means nothing more than "familiar." A familiar theory is simple because it requires no mental effort. This is not the sort of simplicity envisioned by the principle. Historians should never make things simpler than they are, either for their own convenience or for that of their students. Those who are incapable of making a further effort of thought, those who cannot update their familiar notions, are disqualified from having opinions in the first place, let alone retailing those opinions to others.

The fallacy of a too simple theory was understood by the ancients.

You can't talk of ocean with a frog in a well: he is confined to his own space. You can't talk of ice with a summer insect: he is limited to his own season. You can't talk of the Way to a quiddling scholar: he is bound by his own knowledge. (Jwangdz 17:1a)

This is merely using a tube to peep at the sky, or an awl to probe the depth of the earth. Are they not too small for the purpose? (Jwangdz 17:4)

As for the dangers of not recognizing what work is being done by the seemingly unused parts of a theory (or a physical situation), we have this:

The earth is nothing if not wide and vast. The amount of it a man uses is no more than what he actually puts his foot on. But if you dug away the earth from around his foot, all the way down to the Yellow Springs, would the man still be able to use it? (Jwangdz 26:7)

First, one must be judicious in including enough data, and in choosing an analysis adequate to the range of the data. Next, one must be subtle in assessing the functions of the elements in the resulting explanation. Once one has met these basic requirements, useless elements may safely be eliminated by Occam's razor.

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