Alex Strinka

In the field of linguistics, there are the concepts of descriptivism and prescriptivism. Descriptivism is about describing how language is used. In contrast, prescriptivism is about prescribing how language should be used.

Linguistics, as a science, must necessarily be descriptivist. You don't learn about something by telling it how you think it should work. To learn about the world, you need to observe the world without judgement.

But that doesn't mean there's no place for prescriptivism. In fact, I would argue that prescriptivism is unavoidable, to a degree. When you speak or write, you have to choose what you're going to say. Many of those decisions happen unconsciously, without deliberate intent, but not all of them. Descriptivism alone can't tell you what words to use.

Where prescriptivism goes wrong is in saying that you should speak a certain way because it is the right way. The problem is that there is no single right way of speaking.

Consider, are people who speak a different language wrong? That seems patently absurd to me, and I think most people would agree. But if using different vocabulary and different grammer isn't wrong when you're speaking a different language, why should it be wrong if you're speaking a different dialect?

Furthermore, there's no clear line between a separate language, and a dialect. Typically, two different ways of speaking would be called separate languages if they're not mutually intelligible and dialects if they are. But consider a case where group A can communicate with group B, group B can communicate with group C, but group A can't communicate with group C. This is called a dialect continuum, and is actually fairly common. Do groups A and C speak different languages, or dialects?

And how do you determine which way of speaking is the correct one, anyway? Historically, it was just whichever dialect the monarch or nobility spoke, but why should their opinion matter more than anyone else's today?

Instead of trying to speak correctly, you should try to speak in a way that maximizes your audience's understanding. (At least if you're trying to communicate, which is usually the came when you speak.)

When I was in elementary school, I had a teacher who told us there were fewer even numbers than whole numbers, because half of all whole numbers are even, and the other half are odd. I argued that there must be same amount because there an inifinite amount of whole numbers and an infinite amount of even number, and infinity equals infinity.

Well, it turns out, my conclusion was correct, but my reasoning wasn't.

That probably seems really strange. Surely, if you take the set of all even numbers, and then you add all the odd numbers, it would end up bigger than it started, right?

To answer that, we need to talk about what it means for two sets to be the same size. For finite sets, it's simple; just count how many elements there are. But you can't count to infinity, so that doesn't work for infinite sets.

You can say that if set A contains all the elements of B, but set B doesn't contain all the elements of set A, then set B must be smaller, but that won't work for every pair of sets.

So, mathematicians came up with the concept of cardinality, which works for every set. The way it works is that two sets have the same size if you can come up with a mapping between the two sets, such that every element in each set is associated with exactly one element in the other set.

Here's an example. Let's say you have the two sets {Red, Green, Blue} and {Circle, Square, Triangle}. We can make this mapping:
Red ⇔ Circle
Green ⇔ Square
Blue ⇔ Triangle

Every element in the first set is associated with exaclty one in the second set, and every element in the second set is associated with exactly one in the first, so those two sets have the same cardinality. And, because they're finite, we can also just count them and that works too.

In fact, you could even say that this is actually the same process as counting. You're making a mapping between the set you're counting and the set {1, 2, 3...n}.

And this same concept can be applied to infinite sets. Here's a mapping between all integers and all even integers: For every integer N, N ⇔ 2N

This mapping associated every single integer with exactly one even integer and every single even integer with exactly one integer. Therefore, the two sets have the same cardinality.

You can even make such a mapping between all integers and all positive integers, or between all integers and all rationals, so all of those different sets have the same cardinality too.

Does that mean every infinite set is the same size as every other infinite set?

No!

It turns out that the set of all real numbers is bigger than the set of all integers.

Here's how can show that. First, assume that there is such a mapping, so you can associate every real number with exactly one integer. That means you can put those real numbers in an ordered list, so you can talk about the first number in the list, the second one, and so on.

But there's a real number that's nowhere on the list. To find it, we need to build it, digit by digit, based on the numbers in the list. Here's an example list:
#1: 0.40396...
#2: 0.09714...
#3: 0.73175...
#4: 0.90453...
#5: 0.97491...
⋮

I've bolded some digits, because those are the ones we're looking at. The first digit of the first number, the second digit of the second number, and so on, along the diagonal.

For each digit, we'll add one to it (Or if it's nine, subtract one. What matters is that we get a different number than what's already there.)

By doing that, we'll get a number that starts 0.58262...

This number can't be equal to the first number, because the first digit is different. It can't be equal to the second number, because the second digit is different. And so on, for every single number in the list. And notice that this works for every possible list of real numbers, not just this example.

Since there's a real number that's not in the list, the mapping must not include every single real number, and therefore the sets don't have the same cardinality. This is called Cantor's diagonal argument

So just because two sets are infinite doesn't mean they have the same size.

Spaced repetition is great, and everyone should know about it.

Spaced repetition is a technique for memorizing, well, anything. The basic idea is really simple. You learn a thing, and then you review it. You review it frequently at first and then gradually increase the amount of time between each review. In other words, you review it again and again, with each repetition carefully spaced apart, hence the name.

Now, you might be thinking, "Why worry about memorizing things? I can just look things up online." That's true, but I still think it's worth memorizing things. For one, you can look up information in your brain a lot faster than information online. Furthermore, there are many things that you can't easily look up if you don't already know something about the subject.

You might also be thinking, "It's more important to understand fundamental principles than it is to memorize simple facts." Again, that's true, but I would argue that memorizing facts helps to gain that understanding. A large set of examples can help you internalize the patterns that the principles express.

By now, hopefully you're convinced that it's worth memorizing some things, and you're wondering how to do spaced repetition. Typically it's done with flashcards, but you could do it with anything that gets you to actively recall the thing you're trying to memorize.

You can keep track of the timing using the Leitner system. The way it works is you keep your flashcards in numbered boxes. Every day add a few flashcards to box #1. When you review a card and you get it right, you move it to the next box up. When you review a card and you get it wrong, it goes back to box #1. Each box is reviewed half as frequently as the previous, so you review box #1 every day, box #2 every other day, box #3 every fourth day, and so on.

Day	1	2	3	4	5	6	7	8	9
Box #1	✓	✓	✓	✓	✓	✓	✓	✓	✓
Box #2		✓		✓		✓		✓
Box #3			✓				✓
Box #4					✓
Box #5									✓

This table shows which boxes should be review for the first nine days. To state it mathematically, on day N, you should review box X if N+2^X−2−1 is divisible by 2^X−1.

There's also software to help keep track of all the details for you. Personally, I use Anki, but there are lots of other alternatives you can try.

I strongly encourage you to try it, if there's anything you want to learn. If you want more information, here's a fun interactive article and here's an essay written by a teacher about their experience using spaced repetition software in the classroom.

When kids are taught arithmetic, they're usually given problems that look like "2 + 3 = __", and they're supposed to write "5" in the blank space. So it's not particularly surprising that many kids (and even some adults) don't actually understand what it means. They think it means something like, "Here's the answer" or "Evaluate this expression".

What it actually means is that whatever is on the left of the equals sign is the same as whatever is on the right.

There's some nuance about what exactly it means for something to be the same as something else, which can vary somewhat depending on the context, but generally it just means that the two things have the same value.

So, 2 + 3 = 5 is a true equation because the expression 2 + 3 has the same value as 5.

But that equation could also be written as 5 = 2 + 3. Since thing on the left is the same as the thing on the right, the thing on the right must be the same as the thing on the left too.

You can even write 2 + 3 = 1 + 4. It might not be immediately obvious why you would want to, but being able to manipulate equations like that is the basis of algebra, which is a very useful thing

Sometimes you'll see someone write something like "2 + 3 = 5 + 4 = 9". This is incorrect. What's meant is "2 + 3 = 5 and 5 + 4 = 9", but the correct interpretation is "2 + 3 = 5 + 4 and 5 + 4 = 9". 2 + 3 doesn't have the same value as 5 + 4, so that's false.

Consider this dialogue:

Alice gets a phone call from her friend Bob.
Bob: My car is broken. Can you tell me how to fix it?
Alice: I guess. What's wrong with it?
Bob: I don't know, it doesn't start.
Alice: Well, you could try jumping it.
Bob: And that will definitely fix it?
Alice: It will if the only problem is a dead battery, but there might be something else wrong with it.

You can't give good advice on how to fix a car, if you don't know what the problem with the car is. You can give advice for the most common problems, and you can give advice about how to troubleshoot it, but whether that advice helps depends on whether the problem with the car is something your advice covers. Unless you can give out a detailed technical manual of every part of the car, some problems won't be covered.

Humans are more complicated than cars, and we haven't figured out the fully detailed technical manuals for them yet.

Therefore, all advice is highly situational. Advice that can help one person could hurt someone else.

Which is not to say that advice is bad or worthless. Advice from someone who knows the situation can be very helpful. And even without context, there are many issues that many people share, and it can be helpful to point out common problems. But, any advice that's given without context (for example, self-help books, or random blog posts) should be carefully considered before being put into action.

That especially applies when it's advice that you agree with, advice that makes you say, "Yeah, that's a great idea!". If you feel like that about some piece of advice, then you probably already believe it, and that belief probably already influences your actions. It won't be in the exact same way that the advice says (otherwise your reaction would have been, "Well, duh.") but it's probably more or less in the same direction. Which implies that the advice is less likely to be the bottleneck to improve whatever it is you're trying to improve.