While I was preparing for this post, I idly watched some videos. Since I already basically knew that I wanted to continue the previous thread I wasn’t particularly looking for inspiration directly. However, as is so often the case with these things, once you start down a road on a topic little serendipities tend to come along for the ride.
This time, the serendipities were in the form of these two videos. The first is by a YouTuber called Veritassium, who makes some excellent science videos. In this video he talks about analogue engines (computers), a particular interest of mine specifically when it comes to my theory of cognition that I hope to expound and perhaps even expand over the course of this blog series.
The second is by Adam Savage of Mythbusters fame. I used to enjoy the series in my youth, and while I wouldn’t exactly say that he is someone who I look to for scientific inspiration, this particular video really hit the mark in a surprising and entertaining way.
So, what’s the big connection between analogue engines, the science of metrology (measurement) and Occam’s razor? More specifically, what is the connection between the asymmetry of knowledge and cognition?
Simply put, the aphorism I offered last time (“balance what needs to be explained with the explanation required”) is an injunction to find the best digital representation of an analogue world. By analogue here, I mean that the world is what it is, in all its infinite complexity, and by digital I mean the objects that make up the plurality that Occam enjoins in his “Pluralitas non est ponenda sine necessitate”.
Simpler still:
Objects are digital, reality is analogue.
This principle follows quite naturally from how logic works and the laws of thought. If I want to talk about a thing in logical terms, what needs to happen is I need to assign it a name (or a position in a memory bank) and then, by the law of the excluded middle, assign a truth value of either true or false (traditionally rendered as 1 or 0) to it.
Now, there is as lovely principle in logic, which in some ways echoes Occam’s razor, called the deflationary or disquotational principle. Without getting too deeply stuck into the weeds with it, what it essentially says is that the statement
“A”
is equivalent to the statement
“A is true”,
and
“It is the case that A is true”,
as well as
“It is true that it is the case “A is true” is true if and only if it is the case that A is true”.
In other words, naming an object is the same as assigning a truth value to it. This is the sense in which I mean that objects are intrinsically digital.
But now let’s consider a world consisting of only one object, namely “A”. Well, this world is an exact analogue to the logical proposition A. So the question becomes: Just how many objects are there in the real world?
This is a question that really would get us into the weeds, but suffice to say that if we look closely enough, as Adam Savage explains, the question ceases to even be a meaningful one. This is essentially the big insight of the Heisenberg uncertainty principle. It turns out that when you look really closely, there are not just more objects in the world than can be represented digitally, but there are uncountably infinitely many more objects than can be represented. Put another way: “Most decision problems are uncomputable”.
To be clear, “most” in this context means, for intents purposes, essentially “all” but a (literally) vanishingly small subset.
The reason this matters at all is that it means that whenever we describe the world in terms of simple objects — which is to say: digitally — we have to assign values of “false” to any proposition. There is no way to even represent the uncountably infinite complexity of the real world in terms of objects in a way that doesn’t lead to contradiction. There is no sense in which any claim we can make that has any content (which is to say aside from pure logical statements) could ever be true.
But where does this leave science? Many commentators, after all, seem to labor under the completely erroneous impression that science is some sort of search for truth. Well, as it happens, the situation can be somewhat salvaged in a very particular way.
That way involves the recognition of an easily missed tiny little detail in the formulation I offered above: “It is true that it is the case “A is true” is true if and only if it is the case that A is true”. Can you spot it?
The answer is in the “and only if”. Logic, it turns out, is not perfectly symmetrical. Logical rules are set out according to “truth tables”. Logic is essentially a list of outputs that should be given for a specific set of inputs plus connectives. Let’s say we have two propositions A and B. Let’s then connect them with an AND connective. The joined term now has a set of perfectly predictable truth values depending on the values of A and B. Specifically, in this case, “A AND B” is true if (and only if) both A and B are true.
The trick comes in with the IF connective itself. When we say “IF, AND ONLY IF, A, THEN B”, it means that the joined proposition is true if both A and B are either true or are both false. This may seem counter-intuitive, but it follows from the nature of the IF rule.
“IF A THEN B” is only false when A is true, but B is false. When B is true and A is false, in contrast, the joined proposition remains true, because it makes no claims about the truth value of A given B.
This is a bit of a head-scratcher, I will concede, but in it lies the key to understanding why Occam’s razor can work by eliminating objects that are not necessary in an explanation while it fails if simplicity is sought as a goal in itself.
Rather than thinking of science as a an exercise in of trying to develop true digital statements about an analogue world, an exercise we know in advance to be futile, it is better to think of it as an exercise that seeks to develop a series of IF statements. David Hume would call these “constant conjunctions” and taught us that such formulations are the only way that we can actually make causal claims. In other words, when we say that force causes acceleration, what we are actually saying is that every time we observe a force applied, we see acceleration.
IF there is an acceleration THEN there was a force applied.
But there are cases we could observe when a force can be applied where no acceleration would occur! What then? Notice that the conditional (IF...THEN) is still true in this case. Now we can go postulate and then observe a bunch of different hypotheses about this situation, each exquisitely unique to the situation. Essentially positing plurality, which one sided accounts of Occam’s razor tells us to eschew.
However, after having done so we can take up the razor and shave off the unnecessary dross. In the end, what we are left with is a law of nature (and this is a distinction I wish to formally make at this point, in contrast with theory). The law of motion that Isaac Newton developed was not that force causes acceleration, but that:
IF AND ONLY IF there is an acceleration THEN there was a unbalanced force applied.
You cannot accelerate if there is no unbalanced force applied, and if you find acceleration then an unbalanced force was applied.
What has happened here is that we have removed all the infinitely many entities which represent all the different possible arrangements of forces acting on an object and replaced them with the single entity: The unbalanced force.
But hold on one second. All that has really happened here is a redefinition of the term “acceleration” as “unbalanced force”. This is what I meant when I subtitled this piece as “Occam’s pinprick”. We have made a little hole in the endless darkness of reality and seen some light shine through. But neither “acceleration” nor “unbalanced force” actually posits any objects. It is just proposing an identity. It’s not really simplifying anything. It is only after applying this identity to the world of things which interests us, that we can conclude that something was simplified.
To actually be able to use the law we have to go back to the digital world where we see a thing changing its rate of motion in a reference frame. IF object is accelerating THEN an unbalanced force was applied. This statement is always true, regardless of how we define an object and whether we consider the claim that it is accelerating to be true. It is only false when we consider an object to be accelerating and find no unbalanced force. But since we have defined acceleration to be an unbalanced force that can’t happen.
What this tells us, is that we need to postulate exactly as many objects as to identify the unbalanced force, and (at least in terms of this law) we need no others. Thinking about science in this way is thinking about it in terms of identifying objects that satisfies laws. Science cannot be a matter discovering truths (or facts). Truth values are assigned after the facts, which is to say after the objects are identified.
Instead, science is a way to create a universe of objects that satisfy laws. And laws can judged on how well they allow a balance between explanatory power and explanatory objects to be reached.
In conclusion then, it is my firm belief that a lot of controversy in science can be resolved by clarifying the presently rather muddy distinction between law and theory in the following way:
A scientific law is a statement equivalence between two or more concepts that creates useful objects through theory.
The three laws of thought can then be read as (1) simply stating identity, (2) stating an identity between truth and lack of falsity and, (3) stating an identity between truth and falsity and binary numbers. All the rest is then the theory of logic.
It’s not about simplicity, and it’s not about truth. It is about combing ideas in ways that generate the most understanding with the least possible objects.
That means that we don’t have to choose between many objects:
or just a few:
Reality is just what it is. Neither simplicity nor complexity reigns. Explanations must postulate exactly as many objects as they require. No more, and no less.