Monday, 28 May 2018

Occam's Razor: On the Virtue of Simplicity

As a Franciscan monk, simplicity was at the heart of   William's daily life.
Posted by Keith Tidman

The English philosopher and monk, William of Occam (c. 1287–1347), surely got it about right with his ‘law of parsimony’, which asserts, as a general principle, that when there are two competing explanations or theories, the one with the fewest assumptions (and fewest guesses or variables) more often is to be prefered. As the ‘More than Subtle Doctor’ couched the concept in his Summa Logicae, ‘It is futile to do with more what can be done with fewer’ — itself an example of ‘economy’. William’s law is typically referred to as Occam’s razor — the word ‘razor’ signifying a slicing away of arguably unnecessary postulates. In many instances, Occam’s razor is indeed right; in other examples, well, perhaps not. Let’s explore the ideas further.

Although the law of parsimony has always been most closely associated with William of Occam, (Occam, now called ‘Ockham’, being the village where he was born), he hasn’t been the principle’s only proponent. Just as famously, a millennia and a half earlier, the Greek philosopher Aristotle said something similar in his Posterior Analytics:
‘We may assume the superiority ceteris paribus [other things being equal] of the demonstration which derives from fewer postulates or hypotheses.’
And seven centuries after William, Albert Einstein, perhaps thinking of his own formulation of special relativity, noted that ‘the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible’. Many other philosophers, scientists, and thinkers have also admired the concept.

Science’s favoritism toward the parsimony of Occam’s razor is no more apparent than in the search for a so-called ‘theory of everything’ — an umbrella theory unifying harmoniously all the physical forces of the cosmos, including the two cornerstones of 20th-century physics: the general theory of relativity (describing the macro scale) and quantum theory (describing the micro scale). This holy grail of science has proven an immense but irresistible challenge, its having occupied much of Einstein’s life, as it has the imagination of other physicists. But the appeal to scientists is in a unified (presumed final or all-encompassing) theory being condensed into a single set of equations, or perhaps just one equation, to describe all physical reality. The appeal of the theory’s potential frugality in coherently and irreducibly explaining the universe remains immense.

Certainly, philosophers too, often regard parsimony as a virtue — although there have been exceptions. For clarity, we must first note that parsimony and simplicity are usually, as a practical matter, considered one and the same thing — that is, largely interchangeable. For its part, simplicity comes in at least two variants: one equates to the number and complexity of kinds of things hypothesised, and sometimes referred to as ‘elegance’ or ‘qualitative parsimony’; the second equates to the number and complexity of individual, independent things (entities) hypothesised, and sometimes referred to as ‘quantitative parsimony’. Intuitively, people in their daily lives usually favor simpler hypotheses; so do philosophers and scientists. For example, we assume that Earth’s gravity will always apply rather than its suddenly ceasing — that is, rather than objects falling upward unassisted.
Among the philosophers who weighed in on the principle was Thomas Aquinas, who noted in Summa Theologica in the 13th century, ‘If a thing can be done adequately by means of one, it is superfluous to do it by means of several; for we observe that nature does not employ two instruments where one suffices.’ And the 18th-century German philosopher Immanuel Kant, in the Critique of Pure Reason, similarly observed that ‘rudiments or principles must not be unnecessarily multiplied.’ In this manner, philosophers have sometimes turned to Occam’s razor to criticise broad metaphysical hypotheses that purportedly include the baggage of unnecessary ontological concepts. An example of falling under such criticism via the application of Occam’s razor is Cartesian dualism, which physicalists argue is flawed by an extra category — that is, the notion that the mind is entirely apart from the neuronal and synaptic activity of the brain (the physical and mental purportedly being two separate entities).

Returning to Einstein, his iconic equation, E=mc2, is an example of Occam’s razor. This ‘simple’ mathematical formula, which had more-complex precursors, has only two variables and one constant, relating (via conversion) the amount of energy to the amount of matter (mass) multiplied by the speed of light squared. It allows one to calculate how much energy is tied up in the mass of any given object, such as a chickpea or granite boulder. The result is a perfectly parsimonious snapshot of physical reality. But simplicity isn’t always enough, of course. There must also be consistency with the available data, with the model necessarily accommodating new (better) data as they become available.

Other eminent scientists, like the 17th-century physicist and mathematician Isaac Newton, similarly valued this principle of frugality. The first of Newton’s three ‘rules of reasoning in philosophy’ expressed in his Principia Mathematica offers:
‘We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. . . . Nature is pleased with simplicity, and affects not the pomp of superfluous causes.’
But, as noted above, Occam’s razor doesn’t always lead to truth per se. Nor, importantly, does the notion of ‘simplicity’ necessarily equate to ease of explanation or ease of understanding. Here are two examples where frugality arguably doesn’t win the day. One theory presents a complex cosmological explanation of the Big Bang and the physical evolution of a 13.8-billion-year-old universe. A single, but very-late-on-the-stage thread of that cosmological account is the intricate biological evolution of modern human beings. A second, creationist explanation of the current universe and of human beings — with far fewer assumptions and hypotheses — describes both as having roots in a single event some 6,000 to 10,000 years ago, with the cosmos conveniently made to look older. Available evidence suggests, however, that the first explanation is correct, despite the second explanation’s parsimony.

In broad ways, Occam’s razor has been supported by the empirical successes of theories that proved parsimonious in their explanations: with fewer causes, entities, properties, variables, and processes embedded in fewer assumptions and hypotheses. However, even though people tend instinctively and understandably to be drawn toward simpler accounts of hoped-for reality, simplicity hasn’t always triumphed. For example, the earlier nature-versus-nurture debate posed a simpler, albeit false, either-or dichotomy in trying to understand a person’s development and behaviour on the basis of either the environment — the influence of external factors, such as experience and learning, on an otherwise blank slate or perhaps set of instincts — or genes and heritability — that is, biological pre-wiring. Reality is, of course, a complex mix of both nature and nurture, with one influencing the other.

To avoid such pitfalls, as the English mathematician and philosopher Alfred North Whitehead pointedly (and parsimoniously) suggested:
‘. . . every natural philosopher should seek simplicity and distrust it.


Thomas Scarborough said...

That is a great quote by the philosopher-theologian Alfred Whitehead—a name familiar to me from theological studies. One finds him all over the theological map.

There is something, I think, that many see only as a glimmer today. ‘Frugality arguably doesn’t win the day’ is quite true—yet one needs to ask for what reasons this is so. You point out that over-simplification is a problem. Yes, it is ubiquitous. Or do I over-simplify?

Albert Einstein considered that a unit 'singles out a complex from nature'. Therefore, even if one has simplicity in the form of an equation, this equation hides complexes. Our sin today is that we imagine that simplicity may be perfect. It cannot be perfect. It must always exclude. Thomas Mautner writes, 'All abstraction involves some falsification.'

How do we recognise such falsification in the world? Seldom do we see it in our theories, but in reality, yes. We see the effects of what the equations left out. So deleterious are these effects that Stephen Hawking wrote we are on the point of scoring an 'own goal'. Long before him, the philosophers Wilhelm Kamlah and Paul Lorenzen wrote that science carries with it the 'heavy price' of the unforeseen.

Science will not be complete until it recognises this great limitation. It would be interesting if this was what Whitehead saw.

Martin Cohen said...

Can I take up your fellow Thomas's calim?

'All abstraction involves some falsification.'?

No, I don't accept this. Sure, if we made the calim a bit weaker "Often abstraction involves some falsification'...!

But obviously we only need to find one case to disprove this claim. How about someone who says "All of Keith's posts for Pi are written in English". I'm abstracting, and I'm surely right - point me at the 'falsification...' ?

As per oversimplifying, "Ockham’s influence is the idea that one should always have a bias towards simplicity when constructing a theory. However, he was aware that nature does not always follow the simplest course and was himself prepared to invent new terms – where they really did seem necessary."*

*Quoting Arp & Cohen (forthcoming) with acknowledgements to Tidman

Keith said...

I’d venture, Thomas, that ‘simplification’ can be either boon or bane, depending on circumstances. ‘Boon’ because of how, say, heuristics often, and appropriately, serve as handy mechanisms, especially in the sciences, for arriving at what’s acknowledged as suboptimal approximations of reality, though considered with aforethought as sufficient for immediate purposes. ‘Bane’ because of how those very same suboptimal approximations do indeed risk masking key unknowns — ‘key’ because, if factored in, they may otherwise have heavily affected outcomes in ways that matter to our understanding of reality. That unintended concealment of unknowns is what, I’d propose, potentially weakens, even if only temporarily and until remedied, some of the ‘abstractions’ that the quote by Thomas Mautner refers to.

As to the point by Kamlah and Lorenzen that ‘science carries with it the heavy price of the unforeseen’, I’d be hard put to think of a discipline that’s an exception to shouldering that ‘heavy price’ — though, as an aside, I propose their expression ‘heavy price’ may more encouragingly be translated to the notion of a ‘challenge’. It seems to me that part of every field of study entails the discovery of those very ‘unforeseens’, and figuring out how those former unknowns fit into the larger picture and materially advance understanding. I’d suggest that, no matter how nonlinearly and imperfectly, that’s essentially how fundamental progress (enlightenment) is achieved across fields, moving from the abstract toward the more concrete — one eureka moment indefinitely building upon another eureka moment.

Thomas Scarborough said...

Yes Martin but you are still thinking – in my off-centre mind – in the old-fashioned way. ‘All of Keith's posts for Pi are written in English.’ Yes. What you have written about Keith, however, is an abstraction insofar as it now excludes everything else about Keith. Take something simple like, ‘All motor cars transport humans.’ That excludes information which has been, relatively speaking, the ruination of our world. In fact, that information was unimagined once.

Now about Keith, no matter what you say about him, you exclude everything else about him. Say that extraterrestrials invade our planet. ‘All of Keith's posts for Pi are written in English.’ Therefore they insert a pen in his hand, and enclose him in an extraterrestrial writing booth, which is a vacuum. Strangely, Keith does not write English – nor does he write posts. In fact he doesn't write anything at all. The statement is falsified, though not as we might have expected.

I have been a bit loose. I assume that both the statement about motor cars and the statement about Keith have a bearing on the future of the statement. Both are falsified for reason that the future was not taken into account. That would be the case with statements – above all laws – which are taken to hold for the future. With a 'dead' statement about the past, I am not sure ...

Martin Cohen said...

Methinks you are multiplying entities beyond the strictly necesary, Thomas!

Tessa den Uyl said...

To me it seems that Thomas his statement about falsehood is as simple as the denial of that falsehood. All truth and false is constructed.

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