Splitting Concepts

post by Hazard · 2019-06-21T16:03:11.177Z · LW · GW · 0 comments

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For anyone who's gone through some form of primary education, I'd expect numbers to be a tight neural category. Numbers feel like things, they are pretty much the same, and there's little ambiguity over if something is a number or not.

So it might come to one's surprise when they learn about the ordinal number omega which has the interesting property: $$\omega <\omega +1|\omega |=|\omega +1|$$ Where "<" means "less than" and "|x|" means "the size of the number x"

You're telling me there's a number that's less than another number, but still the same size as another number? Bullshit, quit pulling my leg.

A mathematician will quickly tell you that I got the meaning the above symbols wrong. "<" talks about order and "|x|" talks about cardinality. Ordinality refers to somethings position in a list relative to other elements. You can come before another element or after another element.

Cardinality refers to being "bigger" or "smaller".

Here's another fun fact: $$\omega +1\ne 1+\omega$$ Okay, now you're just shitting me. This might as well not be a number!

This annoy someone for a few reasons:

• In the numbers you're used to, "comes before" and "is smaller than" always meant the same thing.
• You don't even remember the phrase "communative" and the second piece of math just looks fake.
• In general, though these squiggly things look like the numbers you're used to, they sure don't act like them.

You are being told to split a concept. This is always a bit difficult, and a source of mental resistance. You have to literally rewire your brain to prevent certain inferences being made. Rewiring = work = experience of effort.

If numbers never felt like yours to begin with, this will be easier. If you don't care about numbers that much you may resent your teacher for "adding" another cognitive stumbling block, but you will proceed.

If you feel like you "know what a number is", this will be harder. Even if you learn the math of ordinals, you will wail and gnash your teeth every time someone refers to them as numbers. This is the mistake of the student. Fine, naturals and ordinals are different, but we can't use the word number for both of them, because what if I was asked this question on a test and wasn't able to give the right answer?!?

Mathematicians redefine words all the time. In this way, they are like poets. They can get away with this because this is what they do. In school you will be told that if you intuition on what a mathematical word means clashes with how it is defined, well then it sucks to be you. Yes, the word still triggers an old neural category and you make incorrect inferences. Learn to swim or drop out. This attitude may or may not be the most helpful.

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