In my previous post I promised to address the idea that “simple arguments are more likely to be right or are inherently superior”. This is unfortunate, because in my first substack post I link a very useful introductory video on the concept of computational complexity classes, which ended with the following two quotes:
“One day I will find the right words, and they will be simple.” ― Jack Kerouac
“Simplicity is the final achievement. After one has played a vast quantity of notes and more notes, it is simplicity that emerges as the crowning reward of art.” ― Frédéric Chopin
The reason those quotes are unfortunate is that, when applied injudiciously, they give an almost entirely misleading idea of both art and science, especially as it pertains to perhaps one of the most frequently abused rules of thumb known to man: Occam’s razor.
Pluralitas non est ponenda sine necessitate: Plurality should not be posited without necessity. This dictum has been frequently restated in various forms throughout the ages, from Aristotle down to the present day, but perhaps the most direct formulation was given by Ludwig Wittgenstein "The procedure of induction consists in accepting as true the simplest law that can be reconciled with our experiences." There is also another class of renditions, that veer closer to Chopin and Kerouac's versions, such as that by Richard Swinburn: "it is an ultimate a priori epistemic principle that simplicity is evidence for truth". It is not difficult to see why this line of thought is enticing. After all, brain tissue is expensive to build and operate, which makes it worthwhile to divide tasks into those that can be quickly solved with limited thought (system 1) and those that require careful and deliberate consideration (system 2), according to Daniel Kahneman's scheme. As I have been re-iterating in previous posts, though, there is no obligation on reality to be simple or readily understandable. Reality is exactly as complicated as it is, no more, no less. Occam's razor, when read partially, is a fallacy. There is no reason whatsoever to expect any connection between the structure of an argument (whether that be simple or complex) and its truth value. In fact, things are considerably worse than that. On the one hand one could consider a universe that is completely. explained by something within that universe, like Douglas Adams' fairy cake. Let's represent this universe as ABA, where B -> ABA (which is to say B maps onto ABA). But notice that your newfound understanding of this magical universe is entirely contingent on the extra information I had to provide to inform you of the mapping. What's worse, since this a self-referential (recursive) mapping, it can never be completed: B -> ABA -> AABAA -> AAABAAA... If the task is to produce a "true" representation of this universe, it cannot be done within the universe itself within finite time (which amounts to the same thing). But that's not the worst of it! If we were to go the other way, and reduce our descriptions as much as possible, there could be (under the simplicity-only reading of the razor) nothing more true than the fundamental laws of thought that forms the basis of logic: 1. The law of identity (If A is true, then so is A) 2. The law of non-contradiction (A cannot be true or false at the same time) 3. The law of the excluded middle (A must be either true or false) In fact, the very idea of something being more or less true is removed by the law of the excluded middle. Furthermore, the simplest of these laws (the one that presupposes the least) is the law of identity. Since the law of identity can be perfectly reconciled with any experience and is the simplest it must be true. And by extension, since nothing is as simple as the law of identity everything else must be false. Swinburn's attempt to resolve the situation by calling greater complexity "evidence of" falsity doesn't work in the domain of logic, where (by most accounts) the law of the excluded middle reigns. As if that were not bad enough, compare a mathematical equation such as E = mc^2 to a poem by Jack Kerouac: How to Meditate -ligts out- fall, hands a-clasped, into instantaneous ecstasy like a shot of heroin or morphine, the gland inside of my brain discharging the good glad fluid (Holy Fluid) as i hap-down and hold all my body parts down to a deadstop trance-Healing all my sicknesses-erasing all-not even the shred of a "I-hope-you" or a Loony Balloon left in it, but the mind blank, serene, thoughtless. When a thought comes a-springing from afar with its held- forth figure of image, you spoof it out, you spuff it off, you fake it, and it fades, and thought never comes-and with joy you realize for the first time "Thinking's just like not thinking- So I don't have to think any more" -Jack Kerouac Or, indeed, what could be regarded as Frédéric Chopin's "simplest" prelude:
Which of these are truly “maximally simple”? Would further simplification add or detract?
Simplicity is indeed a goal to to be striven for, and achieving it is often a crowning glory of art and science, but not if it comes at the cost of the other blade of the razor: Necessity.
In other words, while the command Chekov’s gun should be observed:
If you say in the first chapter that there is a rifle hanging on the wall, in the second or third chapter it absolutely must go off. If it’s not going to be fired, it shouldn’t be hanging there.
…it should not be read as an excuse to introduce a deus ex machina (i.e. simply inserting an undeserved solution into the plot) whenever the script demands. The script, in its fullness, should be exactly as populated by entities as it demands, no more, but also no less.
If anything have a strong, if justifiable, cognitive bias towards making things less complicated than they need to be. Sometimes, as if by magic, that strategy works, and when we encounter explanations or achievements that spark an immediate insight we tend to forget the work that needed to be put in to reach that understanding in the first place. We also tend to forget that most things, most of the time, are, in fact, computationally irreducible. In mathematical terms it is more correct to say that almost nothing can be reduced to simpler terms. We see, and assign importance, to the vanishingly small subset of cases which can be reduced, but at the cost of ignoring practically everything else. But this is just survivorship bias speaking. The world of true things is inhabited virtually exclusively by things far too complex to ever be fully understood.
So for today’s aphorism, let me try a re-framing of Occam’s razor that makes that avoids the fallacy:
Balance what needs to be explained and the explanation required.
Sometimes it is more efficient to reduce what needs to be explained than to try to substitute an overly simplistic but misleading explanation for something that only leads to confusion and extra-work further down the line. While it is never possible to know beforehand exactly what will need to be explained in the future, there are some cases when the goal is indeed clear, such as when an artwork is nearing maturity or when a scientific discovery is ready for use in practical application on a large scale. At that point, and only at that point, it becomes worthwhile to start looking for shortcuts and efficiencies in explanation.
In the end, in explanation, as in all things: