[mcalus3]

"We will assume no background knowledge on behalf of the student, starting from scratch on both the programming and mathematics."

This is a fantastic "side effect" of the fact that category theory isn't built on any other mathematical knowledge. You don't even need even any arithmetics for that.

[spirographer]

At a meta level, Category Theory requires some comfort with abstraction, which really only comes with a mathematical education. So while it may stand apart from much math, it relies on your strong mathematical foundations.

[dkarl]

As so many undergraduate math textbooks say, "No background is assumed beyond sufficient mathematical maturity."

[inopinatus]

Tautological sufficiency/necessity relativities for an absolute measure are a pet peeve:

“How much salt should I add?”

“Oh, not too much.”

Practically guarantees a withering glare from me.

[dkarl]

That's a really great analogy, actually. You can't tell a brand-new cook "salt to taste," but you can tell a cook with an intermediate level of experience "salt to taste" even if it's a recipe they've never made before. You can't impart the experience in words, pictures, or symbols. You can't add a chapter zero that gets them there. But the right kind of experience will get you there pretty quickly.

I admit it's kind of frustrating that it can't be boiled down to a list of discrete things you need to know, but if I had to explain the difference between me before I had "sufficient mathematical maturity" and me after, I would explain it in terms of habit and confidence and other squishy things that aren't mathematical at all.

[richthegeek]

Giving a precise value for this doesn't work in any case. "Salt to taste" is inherently qualitative.

Most pizza dough recipes quote around 5g of salt for 250g flour. Personally I prefer double that amount, and this is the case with many recipes.

Perhaps specifically with salt there is an insane amount of paranoia about blood pressure. In many ways it feels as irrational as Korean worries about fan death.

On the other hand, giving a "5g of salt" value is a good starting point, to avoid undersalting due to paranoia or oversalting due to inexperience.

[dkarl]

What they really need is someone to tell them to add less than they probably need, then taste, add more, taste, add more, and expect to over-season and under-season some dishes as they get experience. Salt quantities in recipes have to be short, sometimes way short, so you're stuck learning the process with or without the quantities. I've never seen this explained in a cookbook, at least not in a way that made an impression on me.

Edit: A piece of advice that has stuck with me for a long time, long after I forgot where it came from, is it's a mistake to look for the state in between "not salty enough" and "too salty." That state doesn't really exist, especially when you're feeling nervous! Instead you should look for the overlap where you can perceive the dish as alternately "not salty enough" and "too salty," like that ambiguous drawing that your brain can resolve to either an old woman or a young woman[0]. That advice really works for me, but my wife, who seasons like a pro, thinks it's nonsense, which I think illustrates how subjective the process is and how it isn't information you can impart but rather experience you have to guide someone toward.

[0] https://en.wikipedia.org/wiki/My_Wife_and_My_Mother-in-Law#/...

[pizza]

Which is kind of ironic, because students take classes exactly because they feel 'immature' with respect to that subject.. Honestly, most of my smoothest educational experiences with hard topics assumed some immaturity on my part, and that that was OK

[dan-robertson]

People don’t generally take category theory because they don’t really understand how proofs work or how to read definitions. The maturity required is about being able to cope with proving things and following proofs based on definitions which will probably seem somewhat bizarre at first and unmotivated at first. The immaturity you seem to talk about is people taking category theory because they don’t know category theory but that’s different and not what is meant by mathematical maturity.

That said, a background in mathematics helps with category theory. Things like group theory, topology (particularly algebraic topology), Galois theory and set theory can be useful in motivating a lot of category theory. I’m yet to see much of a strong motivation from programming (where is there a functor that isn’t an endofunctor?)

[beezee]

When I first went from CT as an implementation detail of typed FP to studying CT on its own, I was very aggravated by this point: "where is there a functor that isn’t an endofunctor?"

Over time it has fallen away in two directions.

In one direction is that though you are always stuck in a category with types as objects and functions as arrows, that doesn't stop you from subdividing. This became more intuitive to me as I encountered more and more categories that are basically just Set with extra structure (just as the one we program in is basically Set with bottom). If you start putting extra structure on your arrows (that is something that carries around a function along with some extra proof-relevant structures, like say a way to show that some zero element in the domain under the function equals the zero element in the codomain and that multiplication is preserved under the function, now you have monoid-homomorphisms) then you end up in the general case with something that looks a lot like a binatural transformation of profunctors, which pretty cleanly encodes "exo"-functors between different sub-categories of the category with all types as objects and functions as arrows.

In the other direction is realizing that even imprecisely, there's quantifiable value in observing functors to and from the category of types and functions, say one between whiteboard diagrams and code, or between a specific problem domain and code, etc etc. If you have some other space with composable relationships that you want to preserve when you write code to correspond to it, or you have some other representation you want to produce based off some given codebase while preserving the structure of the code taken as input, you can gain a ton of conceptual leverage out of identifying a functor.

[exdsq]

Isn’t an endofunctor a morphism from/to the same category? So a functor would be any morphine from/to different categories, and that wouldn’t be an endofunctor?

Endofunctor: A -> A, for category A

Functor: A -> B, for categories A and B

[Twisol]

More or less. "Morphism" is defined by the category it lives in, so a functor is a morphism between categories in the category of (small) categories. (Insert technicalities about size concerns and Russell's paradox.)

In particular, a map between categories that does not preserve composition is not a functor. It is important that F(f;g) = F(f);F(g).

[dan-robertson]

In the typical functional programmers view of category theory, there is only one category of interest so all functors are endofunctors. My main issue is that category theory the thing being studied is categories plural and so I feel like the language of category theory is being used to only ever talk about a single category.

It feels like doing “group theory” entirely with the symmetric group of the integers. While it’s true that the group is very general and has interesting properties, the focus of group theory isn’t single groups but rather the relationships between groups.

Compare this to something like algebraic topology or Galois theory where you have a Galois correspondence (a functor) between objects you’re interested in and groups.

[mcguire]

"Sufficient mathematical maturity": https://c8.alamy.com/comp/HY1PD9/elderly-math-teacher-HY1PD9...

[gnufx]

I think I got more appreciation of abstraction from physics and programming than what mathematical education I had; I'm weak at maths -- experimental physics doesn't need much :-/ -- but I do understand that weakness, and am quite comfortable with abstraction.

[picklenerd]

This is almost every upper division undergrad math class. It was always fun watching people squirm when they pulled out some useful fact from their past 14 years of math education and then got told they had to prove it before they could use it.

[Someone]

If only it were limited to facts learned from math education. For example, there’s the Jordan curve theorem (https://en.wikipedia.org/wiki/Jordan_curve_theorem), which I guess most four-year olds ‘know to be true’ from their experience with coloring books.

[Tainnor]

Yeah, but from that same Wikipedia article:

> It is easy to establish this result for polygons, but the problem came in generalizing it to all kinds of badly behaved curves, which include nowhere differentiable curves, such as the Koch snowflake and other fractal curves, or even a Jordan curve of positive area constructed by Osgood (1903).

So to some extent, the reason why such an "obvious" statement requires a complicated proof is because our everyday notions of what a "closed curve" is are much more restricted than what we consider in mathematics. This is kind of common in maths, especially in fields with a lot of visual intuition.

[qppo]

to get super meta, one could say that the opposite is true: arithmetic requires categories

[dan-robertson]

Except this isn’t really true in the common sense meaning (schoolchildren do arithmetic fine without knowing about category theory) or in the formal sense you’re trying to get at (there are formal axiomatic foundations which arithmetic can be based on which do not need category theory. A simple proof is by causality: arithmetic was successfully formalised before category theory was invented)

[housecarpenter]

You should read the famous book by Linderholm, Mathematics Made Difficult