I was tentatively excited about this series, but I have to be honest: I am dismayed by this post.
So now you understand objects and morphisms
… I really, really don’t.
You are (unless I’m grossly misunderstanding) analogizing category theory to grammar. Your analogy starts with some examples of sentences; then provides an intuitive, common-sense explanation of the common parts of speech, in non-technical terms; and also provides the technical terms. This is perfectly sensible, and is easy to follow.
Then, to make use of the analogy, you introduce the mathematical analogues… but this time, you don’t provide any examples, nor any intuitive generalizations of the examples (because there are none to generalize)… you simply introduce the technical terms, assert that they analogize, and declare that understanding has been conveyed. But that doesn’t work at all!
To elaborate:
It turns out that mathematics is pretty much just nouns and verbs at its simplest—just like how, if you read between the lines a bit, any English sentence can be boiled down to its nouns and verbs.
What are some examples of this? What are some things in mathematics which are “nouns” and “verbs”? I don’t have any intuition for this (as I certainly do for English sentences, which clearly deal with things, and actions that people take, etc.).
In mathematics, a noun is called an object.
So we’re talking about… what, exactly? Numbers? Digits? Variables? Functions? Expressions? Equations? Operators? Symbols? All of the above? None of the above? Some of the above?
A verb is called a morphism or arrow.
Again… what are examples of “morphisms” or “arrows”? Like, actual examples, not “examples” by analogy to English sentences?
p:A→B
Ditto. If this is the generalization, what are some specific examples?
I very much hope that you can address these troubles… otherwise, if I can’t understand even the very basic first concepts, there doesn’t seem to be much hope of understanding anything else!
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.
I was tentatively excited about this series, but I have to be honest: I am dismayed by this post.
… I really, really don’t.
You are (unless I’m grossly misunderstanding) analogizing category theory to grammar. Your analogy starts with some examples of sentences; then provides an intuitive, common-sense explanation of the common parts of speech, in non-technical terms; and also provides the technical terms. This is perfectly sensible, and is easy to follow.
Then, to make use of the analogy, you introduce the mathematical analogues… but this time, you don’t provide any examples, nor any intuitive generalizations of the examples (because there are none to generalize)… you simply introduce the technical terms, assert that they analogize, and declare that understanding has been conveyed. But that doesn’t work at all!
To elaborate:
What are some examples of this? What are some things in mathematics which are “nouns” and “verbs”? I don’t have any intuition for this (as I certainly do for English sentences, which clearly deal with things, and actions that people take, etc.).
So we’re talking about… what, exactly? Numbers? Digits? Variables? Functions? Expressions? Equations? Operators? Symbols? All of the above? None of the above? Some of the above?
Again… what are examples of “morphisms” or “arrows”? Like, actual examples, not “examples” by analogy to English sentences?
Ditto. If this is the generalization, what are some specific examples?
I very much hope that you can address these troubles… otherwise, if I can’t understand even the very basic first concepts, there doesn’t seem to be much hope of understanding anything else!
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.
Do you know the monads are like burritos problem? Do you have a plan for how this sequence isn’t going to end up being “mathematics is like burritos”?