Generalization as a Means of Intelligence Amplification

pedagogy, abstraction

I realized recently that my blog posts might seem a little further off the wall than I usually intend them to. This seems like there is more inferential distance anticipated, so I thought it might be a good opportunity to take a step back and see if we can’t try to close the priors gap a little bit.

Today I want to talk about the most valuable skill I possess. It’s the reason I’ve been able to make big changes in my life over the last year; it’s the only reason that Facebook pays me silly amounts of money to work for the (surprise: it’s not being able to program); and it’s my only claim to fame in the intelligence department. Being the nice guy that I am, I’m going to share it with you today, but first, a little backstory.

I

A new friend of mine posted on Facebook the other day: “everything is hard until someone makes it easy”. This is a sentiment which resonates with me, and (because I had previously been thinking about this blog post), I realized that a big part of making things easy is having concepts and words which allow you to succinctly reason about it.

Visualize with me for a second, the true story of a friend of mine, an amateur musician with a knack for gnarly guitar solos. Unfortunately, he grew up in a metal kind of counterculture and seems to have a disdain for the musical establishment. He avoids any kind of classical training that he can, preferring to generate it for himself. While this works to an extent (he comes up with some sick riffs), my friend is plagued by the fact that he can’t read or write musical notation. Whenever he comes up with a particularly cool lick, his only option is to record it or to practice it enough times that he won’t forget it. Instead of, you know, just like, writing it down.

Musical theory exists for a reason. While it’s certainly an error to only study musical theory without actually playing music, it’s also a mistake to go the other direction and avoid it altogether. Knowing the theory means you can listen to it and save yourself some time, or explicitly go against the traditional and do something different. And it also gives you a convenient notation for being able to encode your ideas.

But I don’t want to write about musical theory. I want to talk about encoding ideas in convenient forms.

Your brain does this all the time. Like literally, always. There is infinitely too much information in the universe to soak in, so instead you gloss over some of it, looking only at what is most relevant to the circumstances. For instance, you know that except in extreme circumstances, you don’t really need to ever think about the last conversation your coworker had with his mom. It’s information you know must exist if you stop to think about it, but it’s really not all that interesting to you. Furthermore, you probably have a mental image of what your best friend looks like, but likely couldn’t accurately draw her nose without looking, even if you were good at drawing.

The point I’m trying to impress is that we gloss over specifics all the time. We know that they’re there, and we know where to find them if they ever become relevant, but for most intents and purposes, we don’t really care about them.

Along similar lines, our minds are naturally prone to generalize. “Humans have 10 fingers.” This is broadly true, but I’m sure you can bring to mind someone with fewer fingers. If you try, you can probably think of someone with more than 10 fingers too. So why do we say that “humans have 10 fingers” if it’s not ostensibly true? Because it’s a useful abstraction, and it’s very concise.

Imagine instead having the belief that “humans have 10 fingers, except for the ones who have 9, or 11. Oh and don’t forget, there are people who have lost two or more fingers! Well I guess it’s safe to say that humans have fingers, even if we can’t put an actual number on them. Unless of course they were exposed to thalidomide as a child.” That’s a mouthful, and it’s still not even entirely true.

So instead, our brains (through what is known as a need for closure) happily cut their losses and retire to the fact that they’re not going to have correct beliefs, and instead might as well have mostly-correct beliefs.

This is the process of abstraction. It’s discarding inane, surface-level inconsequential details, and instead focusing on the form behind what it is that you’re looking at. It’s the process of realizing the underlying features that make this thing what it is.

As a quick exercise, create a list of 5 features about vehicles that are mostly-correct.

Aside: the classical philosophy that you have likely been exposed to in your lifetime gets this task wrong. Not only does it get it wrong, but it butchers all of the usefulness out of it. Remember, you’re trying to find a list of features that seem to uniquely identify a vehicle from a non-vehicle. What you are not trying to do is impose lots of words defining what a vehicle is, and then write a treatise on it and fight tooth and nail to tell other people why what they call a vehicle is wrong. No! We just want to know what kinds of things a vehicle is or does.

II

This ability to generalize from specifics is the key to intelligence. Not just like, book-smarts, this is the characteristic that separates us from animals. (Which is not to say that humans are Aristotle’s “rational animal”). This ability to abstract is what gave humanity agriculture, industry, science, politics and technology. We’re able to stop and look at the useful parts of something, and separate the wheat from the chaff.

Please take a second to realize the gravitas of this. It’s an incredibly useful skill, and it turns out that, like any other skill (can you see me generalizing here?) it can be practiced and trained. Think about that – this is an easy technique you can practice to become smarter. It has nothing to do with your IQ, but instead it’s your ability to see patterns and make connections.

Making connections is good, because if you have a generalization of a problem, and you come up with a solution to this generalization, the solution will usually work for all of the specific problems that you began with. Instead of solving one problem at a time, you’ve just gained the ability to solve thousands of problems simultaneously. In my books, that counts as being 1000x smarter.

However, making connections is hard. Perfectly finding patterns turns out to not just be hard, but computationally impossible. But don’t let that get you down, remember what my friend on Facebook said? “Everything is hard until someone makes it easy.” And remember, that the key to making things easy is to have concepts that allow you to reason about them.

It turns out then, if our goal is intelligence amplification, we should focus on developing concepts which allow us to think more concisely about generalizations.

III

As it turns out, some nice people have gone ahead and done a whole bunch of work on this topic for us. You’re probably going to hate me when I tell you who, but please bear with me for a second and let me make my argument before dismissing it. Promise?

Those people are mathematicians.

And the concepts are mathematical ideas. If you have misgivings about math, cast them out of your mind right now. Mathematics isn’t the dry symbol manipulation without real-life applications that you were taught in grade school. It’s really, really and truly not. Your teachers were just shitty, and they were teaching by rote and didn’t actually know what mathematics were either.

I used to be of the same mindset as you, and I didn’t know what mathematics were either, despite majoring in the faculty of math at a prestigious university for three years. If you’re interested in what mathematics is really like, from the eyes of a mathematician, I would highly recommend A Mathematician’s Lament. It changed my mind.

Anyway, the point to which I’m laboriously trying to get is that the entire study of mathematics is this never-ending explosion of abstractions. Unlike our earlier examples of generalizing over people, it turns out that in mathematics you can generalize your generalizations, and then again. My favorite quote of all time is from E. T. Jayne’s Probability Theory: “We will not prove this result in general here, as we will find out later that it is in fact a special-case of a more general rule still.”

Let’s look at a quick, simple example. Numbers.

If I were to ask you to start counting, how would you start? “one… two… three..”. This sequence is known as the natural numbers, and they’re really the only numbers that actually exist in the universe1. When you are counting apples, these are the only numbers you will use.

Except that, if you think for a second, you can come up with another number. Zero. Oh yeah! That one! While it requires a little bit of philosophizing to argue that you can point out zero apples, you immediately recognize this to be a useful concept. If we include zero with the natural numbers we come up with the whole numbers, and the whole numbers let us describe the count of things, or the absense of things.

Aside: don’t worry about remembering what these sets of numbers are called. If you’re not a mathematician it’s not going to make any difference to you in ten minutes. Instead focus your attention on what happens as we climb up the abstraction tree. Feel your mind expanding as we think about these bigger and bigger structures.

The next obvious thought is, well, if we can have positive numbers, why can’t we also have negative numbers? With whole numbers we were able to express addition, but we couldn’t talk about being able to remove objects from a count. The set of negative, zero, and positive numbers is known as the integers. All of a sudden we can now think about bank accounts and business transactions and eating pieces of pie. The things we can apply numbers to has just become bigger again.

There is a notable problem with our current number system though. We can’t express parts of things. We can’t talk about half of an apple, nor can we talk about dollars and cents. Clearly our model of what numbers are is lacking, and so we invent the real numbers – numbers which can be split and divided indefinitely. Now we can also talk about distances and time durations. Cool! For reasons that will become evident in the near future, we will also call this set of numbers R1 – short for real.

Surely now we’re done, right? What other kinds of numbers could there be?

Well, if we have a set known as R1, does it make any sense to talk about R2? What would that mean? A complete stab in the dark would estimate that it’s twice as big as R1, which would mean we’d need two R1s. You can think about R1 as a line (remember, it allows us to talk about distance?), so it might make sense if we position these two lines in such a way that they form a corner, and this is exactly what R2 is. You can think of R2 as a piece of paper. It’s flat, but has two dimensions, width and height. Now that we have R2, instead of just talking about distances we can now talk about areas! You can now impress people at parties by telling them the floor area of your house, and it’s all thanks to R2.

And again, if we have R2, what about R3? What would that be? By analogous construction of R2, if we take three lines and position them so they all form a single corner, this means one of them is going to need to stick straight out of the piece of paper that we had in R2. You could also call R3 “3-D”. R3 allows us to discuss physical spaces. We can now talk about the locations of airplanes and we can gossip about our friends on the Atkins diet.

Again, I’m going to ask you to take a second and realize what we’ve done. We just generalized a number from something you can count with to something you can use you judge your fat friends. Again, none of this should be new to you – it’s not a surprise that you have a weight, but hopefully this way of thinking how to get there from here is. Numbers are all of the things we have talked about, and they are way more things that I’m not going to get into here.

It all depends at how you look at them, and how willing you are to let your mind be flexible on the topic.

Next week we’re going to generalize better and harder and (most importantly) more usefully. It will be less mathy, I promise. Get hype.


  1. One could make an argument that truly no numbers exist, and they’re just generalizations of structures in the universe, but this wouldn’t be a useful exercise in “trying to find abstractions”, now would it? Don’t be that guy.