Sets and Functions 2019-10-11T05:06:23.711Z
Examples of Categories 2019-10-10T01:25:32.374Z
Categories: models of models 2019-10-09T02:45:29.617Z
The sentence structure of mathematics 2019-10-07T18:58:46.630Z
Introduction to Introduction to Category Theory 2019-10-06T14:43:46.977Z
What is category theory? 2019-10-06T14:33:50.455Z


Comment by countedblessings on Pricing externalities is not necessarily economically efficient · 2019-11-10T02:54:55.833Z · LW · GW

Here's a category-theoretic perspective. (Check out the rest of the lectures and the associated free textbook.)

Comment by countedblessings on Sets and Functions · 2019-10-15T17:44:43.829Z · LW · GW

My next series of posts will be directly about the Yoneda lemma, which basically tells us that everything you could want to know about an object is contained in the morphisms going into/out of the object. Moreover, we get this knowledge in a "natural" way that makes life really easy. It's a pretty cool theorem.

Comment by countedblessings on Sets and Functions · 2019-10-15T15:53:16.592Z · LW · GW

In the end, we don't really care about sets at all. They're just bags with stuff in them. Who cares about bags? But we do care about functions—we want those to be rule-based. We need functions to go "from" somewhere and "to" somewhere. Let's call those things sets. Then we need these "sets" to be rule-based.

I'm grateful for your comments. They're very useful, and you raise good points. I've got most of a post already about how functions give meaning to the elements of sets. As for how functions is a verb, think of properties as existing in verbs. So to know something, you need to observe it in some way, which means it has to affect your sensory devices, such as your ears, eyes, thermometers, whatever. You know dogs, for example, by the way they bark, by the way they lick, they way they look, etc. So properties exist in the verbs. "Legs" are a noun, but all of your knowledge about them has to come from verbs. Does that make sense?

Comment by countedblessings on Sets and Functions · 2019-10-15T15:47:49.668Z · LW · GW

You raise a good point. Think of category theory as a language for expressing, in this case, the logic of sets and functions. You still need to know what that logic is. Then you can use category theory to work efficiently with that logic owing to its general-abstract nature.

Comment by countedblessings on Sets and Functions · 2019-10-15T13:23:52.630Z · LW · GW

I agree. I'm shifting gears to work on something basically aimed at the idea that the intelligent layperson can grasp Yoneda lemma and adjunction if it's explained.

Comment by countedblessings on Sets and Functions · 2019-10-15T13:22:43.567Z · LW · GW

It's older terminology. Everyone says image now.

Comment by countedblessings on Sets and Functions · 2019-10-15T13:21:36.810Z · LW · GW

I thought I had it right, and then mixed it up in my head myself.

Comment by countedblessings on Sets and Functions · 2019-10-14T01:12:48.815Z · LW · GW

That was my intention. Thanks for pointing it out. One of the mistakes of this series was the naive belief that simplicity comes from vagueness, when it actually comes from precision. Dumb of me.

Comment by countedblessings on Sets and Functions · 2019-10-12T02:17:31.981Z · LW · GW

Steam is run out of. This was poorly conceived to begin with, arrogant in its inherent design, and even I don't have the patience for it anyway. I'll do a series about adjunction directly and Yoneda as well.

Comment by countedblessings on Categories: models of models · 2019-10-10T01:43:21.227Z · LW · GW

Honestly my real justification would be "adjoint functors awesome, and you need categories to do adjoint functors, so use categories." More long as it's free to create a category out of whatever you're studying, there's clearly no harm. The question is whether anything's lost by treating the subject as a category, and while I fully expect that there are entire universes of mathematics and reality out there where categories are harmful, I don't think we live in one like that. Categories may not capture everything you can think of, but they can capture so much that I'd be stunned if they didn't yield amazing fruit eventually. I'd acknowledge that novel, groundbreaking theorems are still forthcoming.

Comment by countedblessings on Categories: models of models · 2019-10-10T01:38:32.805Z · LW · GW

Thank you for the positive feedback. (A very underrated thing in terms of encouraging free content production.) I can go back to each post and add a link to the next one. I am concerned that I may want to add, rearrange, or even delete individual posts at some point, but I suppose that's no reason not to add in the links right now for convenience's sake.

Comment by countedblessings on Categories: models of models · 2019-10-10T01:36:34.763Z · LW · GW

One of the reasons for my own interest in category theory is my interest in the question you raise. I'm hoping that we'll explore the idea that universal properties offer an "objective" way of defining "subjective" categories.

Maybe a more direct answer is that in the very next post in the series, we'll see that sets can be considered the objects of the category of sets and functions, and also the objects of the category of sets and binary relations. Functions are binary relations, so that's not a perfect answer, but yes, you can think of an individual category as a context of sorts through which you view the objects, like how you can view a tomato as a fruit or vegetable depending on the context.

Comment by countedblessings on The sentence structure of mathematics · 2019-10-08T01:16:07.602Z · LW · GW

You have thought about the language analogy much harder than I did. I will think about how to avoid this issue better in the future, so thank you. In any case, don't stress it too much—all that this post seeks to establish is that category theory is a mathematics of "stuff taking action on stuff"—moreover, it does so in a logical, intuitive way that you are already familiar with, even if you don't know higher maths. Judging by Said's comment, I also should have clarified that specific branches of mathematics fill in particular things for "stuff" and "taking action." E.g., you get set theory when you fill in "sets" for stuff and "functions" for taking action.

Comment by countedblessings on The sentence structure of mathematics · 2019-10-08T01:11:21.779Z · LW · GW

Thank you very much for your reaction to this post. As it happens, I find myself in agreement with you. I leaned too hard in the direction of avoiding any discussion of mathematics. The next post is already written to clarify that sentences are all about nouns and verbs because we use sentences to model reality, and reality seems to consist of nouns and verbs. (Cats, drinking, milk, etc., are all part of reality. Even adjectives like "blue" are broken down by our physics into nouns and verbs.) We use various specific kinds of mathematics to model various specific parts of reality, and so various specific kinds of mathematics themselves boil down to nouns and verbs. So when you do a "mathematics of math" it ends up being a mathematics that is analogous to a mathematics of nouns and verbs, which get called objects and morphisms respectively. (We probably can't carry this analogy forever—I don't know that there's a real-world language analogy to n-categories. But that won't come up anyway.) I'll very much look forward to your reaction to the next post, which motivates category theory as a general description of how you'd want to model pretty much anything in a universe of cause-and-effect, which correspondingly generalizes, almost as a byproduct, the mathematics any human is likely to invent.

There are many options for being clearer about objects and morphisms in this post, and I will consider them...I will also take pains to ensure it is not necessary to reconsider future posts for this particular mistake, thanks to you.

Comment by countedblessings on Introduction to Introduction to Category Theory · 2019-10-07T19:27:53.761Z · LW · GW

I'm motivated by the essays themselves. I believe this material should exist. It's also good for my own understanding to write them.

Comment by countedblessings on Introduction to Introduction to Category Theory · 2019-10-07T19:26:47.466Z · LW · GW

When we get to adjunction, I'm hoping the utility of the series will start to become clear.

Comment by countedblessings on What do the baby eaters tell us about ethics? · 2019-10-07T00:16:33.720Z · LW · GW

You're playing chess. You're white, so it's your move first. It's a big board, and there's a lot of pieces, so you're not quite sure what to do.

Luckily for you, there definitely exists a rule that tells you the best possible move to play for every given configuration of pieces—the rule that tells you the move that maximizes the probability of victory (or since draws exist and may be acceptable, the move that minimizes the probability of defeat. Or maximizes the points you gain, 1 for a win and 0.5 for a draw, over an infinite number of games against the opponent in question—whatever.).

But many of these configurations of pieces will have more than one possible move to play, so it's not like this rule is just a given. You have to figure it out.

So what is a rule? It's something that tells you what move to play in any and every given position. Two rules are equal when, for every possible position, they tell you to play the same move.

When two rules are equal, we just merge them into one rule—they're literally the same, after all. So let's consider the list of all unequal rules—rules that differ in at least one move recommendation for at least one position from every other rule.

How many of these rules are there? Mathematically speaking, the answer is "a super-huge amount that would literally cause your mind to explode if you tried to hold them all in your head." After all, there is a universe-eating number of possible chess positions (remember, this is a rule that works for all possible chess positions, even ones that would never happen in a real game). And in every chess position, the number of possible ways that rules can be distinguished is equal to the number of possible moves in that position.

So imagine each rule as a black ball, each attached to this really big wall in this vast infinite array. Out of this huge infinity of black balls, one of these balls gets a little pink dot placed on its backside, so you can't see that it's there.

Now, out of this huge infinite array, you have to find the one ball with the little pink dot on it. That is the challenge of finding the rule that tells you the best chess move for each position.

Is ethics just as hard? No! Ethics is insanely harder. Because ethics tells you a most ethical move possible for each chess position, which includes illegal moves and therefore there are more black balls corresponding to just the subset of ethical choices for chess positions!

Now consider that also Go exists, and checkers, and, Fortnite, and situations that aren't even games at all, like most of everything in the universe.

There's a ball for each way a rule can be distinguished from all other rules over the full list of all possible situations, including but hardly limited to chess situations.

One of them has a little pink dot on its backside. Go find it.

That is the challenge of ethics.

Yes, people disagree about which ball has the little pink dot on it! Yes, you can search your heart and soul and still not know which ball has the little pink dot on it! That does not mean there is not a ball with a little pink dot on it!

The pink-dotted ball exists!

Alas, the Babyeaters were looking for the ball with the little red dot on it, and the Super Happy people looking for the ball with the little blue dot on it. Looking for different colored dots, or disagreeing about which ball has the pink/red/blue dot on it, is the stuff that wars and trade are made of.

Comment by countedblessings on What do the baby eaters tell us about ethics? · 2019-10-06T23:47:59.264Z · LW · GW

I'm tempted to put it like this: ethics is a rule for producing something called a "total order" that tells you what to do in any and every given situation. Basically, you have a list of all the things that could happen, and then ethics puts them in an order so you have the most ethical conceivable thing at the top and the least ethical conceivable thing at the bottom.

From there, you go to the top and then start chopping off things that you can't do. For example, maybe your ethics has "give everyone an immortality pill" really high up on the order. But you can't do that, so you chop it off and keep going down. Eventually you run into something that you can do, so you do it because it's the most ethical thing remaining.

What the humans, Babyeaters, and Super Happy people all find out when they meet in literature-space is that you can define an ethical rule for producing any total order from the unordered list of all things that could happen. Say that list is really small, just A, B, and C. Well, A < B < C is clearly one valid order. But so is C < B < A. And B < A < C. Etc.

The humans are following one rule, the Babyeaters another, and the Super Happy people yet a third. Because of the way algorithms feel from the inside, they all perceive each other as monstrous.

Ethical nihilism is an easy mistake, I think. Label the rule producing the order A < B < C as "moral." Then it is an objective fact to say that the rule producing C < B < A is "not moral." It's also possible that you live in a big universe with lots of stuff in it like chess, genocide, and chocolate, and so your ethics rule is really complicated and so you might not have the order it gives you fully derived. Thus you might find yourself asking questions like "Is it moral to do [insert action here]?" Still, the order that is "moral" is the order that is "moral."

That's my initial guess after skimming the set of links given by riceissa. We'll be discussing orders (albeit usually partial, not total) in my category theory series of posts, so if this interests you, follow along....

Comment by countedblessings on What is category theory? · 2019-10-06T18:17:35.788Z · LW · GW

I'd think so, although I've heard mixed reports from programmers about this. But give a shot. Why not?

Comment by countedblessings on What is category theory? · 2019-10-06T15:04:55.861Z · LW · GW

Category theory is useful for understanding lambda calculus—I feel like anyone who studies the latter will certainly encounter the former soon enough.

Comment by countedblessings on Introduction to Introduction to Category Theory · 2019-10-06T14:45:29.220Z · LW · GW

I apologize for spamming with two posts on the same day about the same content, but I actually wrote this post first. It was waiting to be moderated for several days, so I wrote the other one and submitted it, and to my surprise it was immediately published. I guessed this one must have gotten stuck in a queue or something, so I rewrote it and published it. My bad, we'll definitely go one at a time from now on.

But in a way it's fitting, as having two introductory posts should definitely indicate just how slow the pace of this series is going to be.

Comment by countedblessings on Open & Welcome Thread - October 2019 · 2019-10-04T19:01:23.111Z · LW · GW

There are a bunch of really great introductions to category theory, and this is definitely one of them. There's also Youtube videos of his lectures going over the same material.

My plan is to go very slowly, and to assume everything needs to be explained in as much detail as possible. This will make for a tediously long but hopefully very readable series.

Comment by countedblessings on LW Team Updates - October 2019 · 2019-10-03T00:40:40.444Z · LW · GW

What's the typical timeline for a post to be approved by the moderators? I'd just like to know so I can plan accordingly.

Comment by countedblessings on Human instincts, symbol grounding, and the blank-slate neocortex · 2019-10-02T19:15:32.339Z · LW · GW

Any thoughts on the relevance of category theory as the same kind of "universal modeling system" for mathematics as the brain might well be for real life?

Comment by countedblessings on Open & Welcome Thread - October 2019 · 2019-10-02T02:21:24.397Z · LW · GW

Would people be interested in a series of posts about category theory? There's a lot of great introductions to the subject out there, but I'd like to fill a particular niche—I don't want to assume my audience knows topology yet. I think you can still get a lot of value out of category theory at the high school senior level.