binary relations defining order are more nuanced than they seem; a linear order isn't just about ranking, it's about the structure of the relationships themselves.
Category Theory Illustrated – Orders
31–40 of 72 posts
Re: Category Theory Illustrated – Orders
#32[dead]
Re: Category Theory Illustrated – Orders
#33[dead]
Re: Category Theory Illustrated – Orders
#34If you want to learn category theory in a way that is more orthodox, a lot of people recommend Tom Leinster’s Basic Category Theory, which is free[1]. I’m going to be working through it soon, but the bit I’ve skimmed through looks really good if more “mathsy” than things like TFA. It also does a better job (imo) of justifying the existence of category theory as a field of study. [1] https://arxiv.org/pdf/1612.09375
Disclaimer for the book, and for category theory in general: most books are optimized for people who already master mathematics at an undergraduate level. If you're not familiar with algebraic structures, linear algebra, or topology, be prepared to learn them along the way from different resources.
Category theory is also not that impressive unless you already understand some of the semantics it is trying to unify. In this regards, the book itself presents, for example, the initial property as trivial at first hand, unless you notice that it does not simply hold for arbitrary structures.
Re: Category Theory Illustrated – Orders
#35Earlier quoted context omitted.
A thing is its relationships. (Yoneda lemma.) Keep track of how an object connects to everything else, and you’ve recovered the object itself, up to isomorphism. It’s why mathematicians study things by probing them: a group by its actions, a space by the maps into it, a scheme in algebraic geometry defined as the rule for what maps into it look like. (You do need the full pattern of connections, not just a list — two…
We should call it “relationship lemma”. That way its function is contained within its name. And would not require the definition step every time. We should strive to name all things by their function not by their inventor or discoverer IMO. But people like their ribbons.
In my study, it's basically never that the person names the thing after themselves. My theory goes: Often a discovery is presented in a paper by someone(s), who gives it a usually only barely passable name. For a time, only a handful of experts in the field know about it and none of them care to write general explainers for the layman. So they call it what's easy. "[Name] [concept]" because they're used to talking in names all the time. Academic experts have a large library of people's names tied to the concepts in their papers, i know my PI certainly did, every query was met with a name that had solved it to go look up.
Anyways, the discussion begins with these people. Who all use the name to reference the paper which contains the result. As the discussion expand, it remains centered on this group and you have to talk _with_ them and not at them so you use the name they do. This usage slowly expands, until eventually it gets written in a textbook, taught to grad students, then to undergrads, and it becomes hopeless to change the name.
I share the frustration with naming, we can come up with such better names for things now. But until we give stipend bonuses for good naming, the experts will never care to do so. But i wholeheartedly disagree that the problem as a whole can be reduced to "people like their ribbons". Naming something after yourself is so gauche and would not be tolerated in my field at least. The other professors would create a better name simply out of spite for your greed.
Re: Category Theory Illustrated – Orders
#36Earlier quoted context omitted.
Sure, category theory can't prove the unsolvability of the quintic. But did you know that a monad is really just a monoid object in the monoidal category of endofunctors on the category of types of your favorite language?
Isn't that just the definition?
I think they're making a joke
Re: Category Theory Illustrated – Orders
#37If someone does not want to check the mathematics line by line and prefers to give the article the benefit of the doubt, note that it also presents this JavaScript: [1, 3, 2].sort((a, b) => { if (a > b) { return true } else { return false } }) This is not a valid comparator. It returns bools where the API expects a negative, zero or positive result, on my Chrome instance it returns `[1, 3, 2]`. That is roughly the le…
Why assume it is javascript? The article doesn't indicate the language anywhere that I can see.
Re: Category Theory Illustrated – Orders
#38I think it is pretty obvious that at the challenge with all abstract mathematics in general and the category theory in particular isnt the fact that people dont understand what a "linear order" is, but the fact it is so distant from daily routine that it seems completely pointless. It's like pouring water over pefectly smooth glass
You say pretty obvious, but it took me 2 years during my PhD to be consciously aware of this. And once I did, I immediately knew I wanted to leave my field as soon as I would finish.
I'm just curious. Do you play computer games?
Re: Category Theory Illustrated – Orders
#39There is a way to frame category theory such that it's all just arrows -- by associating the identity arrow (which all objects have by definition) with the object itself. In a sense, the object is syntactic sugar.
This is obvious within about 3 seconds of opening the article, noticing it's filled with coloured M&M's, and closing it again.
Re: Category Theory Illustrated – Orders
#40If someone does not want to check the mathematics line by line and prefers to give the article the benefit of the doubt, note that it also presents this JavaScript: [1, 3, 2].sort((a, b) => { if (a > b) { return true } else { return false } }) This is not a valid comparator. It returns bools where the API expects a negative, zero or positive result, on my Chrome instance it returns `[1, 3, 2]`. That is roughly the le…
Why assume it is javascript? The article doesn't indicate the language anywhere that I can see.
Ok, let's say that it is not JS, but an untyped, closure-based programming language with a strikingly similar array and sort API to JS. Sadly, this comparator is still wrong for any sorting API that expects a general three-way comparison, because it does not handle equality as a separate case.
And to tie it down to the mathematics: if a sorting algorithm asks for a full comparison between a and b, and your function returns only a bool, you are conflating the "no" (a is before b) with the "no" (a is the same as b). This fails to represent equality as a separate case, which is exactly the kind of imprecision the author should be trying to teach against.