William had a good razor!

The law of parsimony – sometimes called Ockham’s razor after the English Franciscan monk William of Ockham (circa 1285 – 1347), a pioneer in logic – is a principle common to philosophy and science. In fact, it has been stated before, the oldest occurrence that I have found is due to Aristotle¹Ἀριστοτέλης, Φυσικὴ ἀκρόασις. Available on-line. An English translation by Robin Waterfield: David Bostock (editor), 1999. Physics, Oxford University Press. Available on-line. (384 – 322 BC), who attributes it to Empedocles (circa 490 – about 435 BC). However, Proclus (412 AD – 485) traces it back to Pythagoras (about 580 – about 495 BC)²The only version I know of the work in question is this German edition, which appears to be considered the reference: Manitius, C. (éditor et translator), 1909. Procli Diadochi hypotyposis astronomicarum positionum (Bibliotheca scriptorum Graecorum et Romanorum Teubneriana), Teubner, Leipzig. Reprinted in 1974: Teubner, Stuttgart..

This principle is not always well understood, as it is sometimes used in a manner denoting a misunderstanding about what it actually means. I propose you to see a brief history of this law before commenting on what it means. This article is therefore one of the series on history of science and popularisation I started with a view from here.

No, I have not yet specified what states this law. This is a barely honest process that aims to create an almost unbearable suspense to make you captive of my prose, so that you will read this whole article. However, do not worry: the explanation comes in a few lines!

Ockham’s razor has been sharpened before William of Ockham

As noted above, the first mention of this law that I found is due to Aristotle. Here is how he expressed it in the first volume of his Physics, 4, 188a17:

‘You should used fewer principles and in a limited number, as does Empedocles.’

The law of parsimony is sometimes summed up as indicating that between two equivalent explanations you should select the simplest one. Note that as early as Aristotle’s times, this is not how it is presented. Aristotle does not invite us to look for the simplest explanation, but not to multiply the “principles.” Still, it is necessary to clarify what he means when using the word “principle.”

To explain a phenomenon, whatever it is, one uses a variety of mechanisms. Take for example the falling bodies – I take this example as it is in the next article on history of science on this website. For Aristotle, whose model of the World I have already presented, the natural motion of bodies made of element earth or water is orientated downward, while the speed depends on the ability of the body to cleave the element air, which capacity depends on the weight and body shape. To explain the phenomenon of falling bodies, Aristotle therefore uses several mechanisms, which he calls “principle”: the mechanism of natural movement of elements, the ability to cleave the element air, the weight and the body shape. The law of parsimony, as expressed here, invites us not to multiply mechanisms when explaining a given phenomenon.

This principle will be integrated in scholasticism and therefore in Christian thinking. It is found for example in Thomas Aquinas (1225 – 1274)³Thomas de Aquino, 1265 – 1274. Summa theologiæ. Available on-line. An English translation: Thomas Aquinas, 1981. Summa Theologica, Christian Classics. Available on-line.. In the Summa Theologica, part I, Q. 2, art. 3, obj. 2, the law of parsimony is stated as follows:

‘[…] quod potest compleri per pauciora principia, non fit per plura.’

It can be translated into ‘what can be accomplished by principles in small numbers is not done by many more principles.’ Again, the idea is not to multiply mechanisms to explain a phenomenon.

However, the most famous formulation of this law is due to William of Ockham. He stated it several times, the first that I could identify is in Book II of Quaestiones et decisiones in quatuor libros Sententiarum cum centilogio theologico⁴Gulielmus Occamus, 1319. Quaestiones et decisiones in quatuor libros Sententiarum cum centilogio theologico:

‘Pluralitas non est ponenda sine necessitate.’

Which can be translated as ‘a plurality should not be put with no necessity.’ Again, the idea is not to multiply hypotheses if it is possible to avoid such a multiplication.

Note that the expression “Ockham’s razor” is not due to William of Ockham. The earliest mention of this law under this name that I could find comes from William Hamilton (1788 – 1856)⁵William Hamilton, 1852. Discussion in Philosophy, Literature and Education, Longman, Brown, Green and Longmans, Londres..

The purpose of this article is not to produce an exhaustive list of all the times this law has been formulated: it would be particularly daunting, I am not sure this is possible and it would not be very useful. Simply note that it has been reiterated both in philosophy and science. One of the latest formulations of the parsimony law comes from statistical algorithmic. This formulation is due to Ray Solomonoff (1926 – 2009) in the form of inductive inference theory⁶Ray Solomonoff, 1964. A Formal Theory of Inductive Inference Part I, Information and Control 7 (1), pp. 1 – 22. Doi: 10.1016/S0019-9958(64)90223-2. ⁷Ray Solomonoff, 1964. A Formal Theory of Inductive Inference Part II, Information and Control 7 (2), pp. 224 – 254. Doi: 10.1016/S0019-9958(64)90131-7.. In this context, among all the explanations describing the observations perfectly, those requiring fewer computations are provided with a stronger weight. The inductive inference is used in artificial intelligence, specifically in machine learning.

Let us use our razor without cutting ourselves!

As we have just seen, the law of parsimony has been stated and successfully used several times in history. It invites us, you guessed it, to use parsimony in (not to multiply) assumptions and mechanisms we invoke to explain a phenomenon. Thus, among many explanations of a given phenomenon, provided they are all as satisfactory – therefore, an explanation not describing accurately the phenomenon should be eliminated –, the one requiring the fewest assumptions should be favoured. For example, in my article on the principle of relativity, we saw Nicolaus Copernicus has noted that the heliocentric system matched better the law of parsimony that the geocentric one.

Using it should therefore limit the use of undocumented assumptions, which is a way to strengthen the coherence of an explanation to the phenomena it intends to enlighten. Indeed, adding assumptions that have not been validated by observations increases the risk that one or more of the assumptions underlying the explanation is not consistent with the facts to which it relates. Conversely, with a minimum of undocumented assumptions, the explanation is less likely to be invalidated.

However, the use of this law also has some pitfalls that one needs to avoid.

First, be wary of assumptions that actually hide some other complex assumptions. For example, once it has been established that the light behave like a wave, since all waves hitherto observed were propagated in a medium, it has been postulated that light was spreading in what was then called the luminiferous aether⁸The earliest foreshadowing of what will become the luminiferous aether that I could find comes from: Isaac Newton, 1704. Opticks: or, a treatise of the reflexions, refractions, inflexions and colours of light, Royal Society, Londres. It has been republished several times, for instance: Isaac Newton, 1998. Opticks: or, a treatise of the reflexions, refractions, inflexions and colours of light. Also two treatises of the species and magnitude of curvilinear figures, Octavo, Palo Alto, California..

However, it turned out that the luminiferous aether should have had very special properties. In particular, in order to transmit light from distant stars, it would have to be almost infinitely stiff, but it also had to show no resistance to objects motions. Ultimately, this aether was a complex hypothesis. Thus, when proposing the theory of special relativity, Albert Einstein notices, among other things, that luminiferous aether poses more problems than it solves and draws the conclusion that light travels in emptiness⁹Albert Einstein, 1905. Zur Elektrodynamik bewegter Körper, Annalen der Physik, n° 17, pp. 891 – 921. It can be read on-line. An English version translated by George Barker Jeffery is available on-line.. Rather, the middle of light propagation is emptiness, as recent advances in physics, especially in quantum physics, invite us to reconsider the notion of emptiness – I will not develop this any further here, but tell me in comments if you are interested in me to write an article on this subject. Finally, not to use luminiferous aether induced fewer assumptions, even if no other wave propagating in emptiness had been observed at the time.

One should also be wary of explanations that explain nothing. Indicating that bodies fall because of God’s will – the purpose of this article is not to discuss whether there is a god, it is here considered as an explanation of phenomena –, besides that the divine hypothesis is a complex one, so complex that, in the introduction to the Summa Theologica, Thomas Aquinas made the observation that there is no positive definition of God, it actually gives no explanation: it is equivalent to say that things are what they are because that is the way the world works. Whether God exists or not, the fact remains that it does not clarify the phenomenon and gives nothing to make some previsions.

The great virtue of the law of parsimony is to help avoid ad hoc hypotheses. An ad hoc hypothesis is an assumption that is added to a theory to be able to use it in spite it has been refuted. For example, it is clear that there is no evidence showing the existence of leprechauns, some fairy from the Irish folklore. To continue to defend the existence of leprechauns, simply add the ad hoc assumption that, particularly shy, they are very careful not to be noticed and that, moreover, they are invisible. By dint of ad hoc assumptions, we obtain leprechauns who have neither form nor smell, who are invisible and inaudible, who interact with nothing, leave no trace, are undetectable, but who nevertheless exist …

Of course, few people are likely to defend at all costs the existence of leprechauns. In fact, it is one of the reasons I have chosen this example, as it is very unlikely anyone will blame me having indicated leprechauns does not exist. However, the use of ad hoc hypotheses is common when someone sees an explanation she/he cares about to be rebutted, and it can sometimes be difficult to identify these hypotheses. In this context, I think it is useful to remember that being wrong is never something serious if one is ready to acknowledge and correct errors. It does not question someone existence and, as, among other things, this series on history of science try to illustrate, the brightest minds had been wrong at one time or another. The law of parsimony is not always easy to use, but it can help.