If 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 level of correctness of the mathematics in the article as well, which I'm trying to present in sibling comment: https://news.ycombinator.com/item?id=47814213
I 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're more right than you'd think. The whole point of mathematics is precise thinking, yet the article is very inaccurate.
Nobody seems to care or notice. I'm watching in disbelief how nobody is pointing out the article is full of inaccuracies. See my sibling thread for a (very) incomplete list, which should disqualified this as a serious reading: https://news.ycombinator.com/item?id=47814213
My conclusion cannot be other than this ought to be useless for the general practitioner, since even wrong mathematics is appreciated the same as correct mathematics.
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.
Is there a "mind-blowing fact" about category theory? Like the first time I've heard that one can prove there is no analytical solution for a polynomial equation with a degree > 5 with group theory, it was mind-blowing. What's the counterpart of category theory?
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 different rings can have the same modules, for instance.) [0]
Writing a program and proving a theorem are the same act. (Curry–Howard–Lambek.) For well-behaved programs, every program is a proof of something and every proof is a program. The match is exact for simple typed languages and leaks a bit once you add general recursion (an infinite loop “proves” anything in Haskell), but the underlying identity is real. Lambek added the third leg: these are also morphisms in a category. [1]
Algebra and geometry are one thing wearing different costumes. (Stone duality and cousins.) A system of equations and the shape it cuts out aren’t related, they’re the same object seen from opposite sides. Grothendieck rebuilt algebraic geometry on this idea, with schemes (so you can do geometry on the integers themselves) and étale cohomology (topological invariants for shapes with no actual topology). His student Deligne used that machinery to settle the Weil conjectures in 1974. Wiles’s Fermat proof lives in the same world, though it leans on much more than the categorical foundations. [2]
I think that CT is more akin to just a different language for mathematics than a solid set of axioms from which you can prove things. The most fact-y proof I've personally seen was that you can't extend the usual definition of functions in set theory to work with parametric polymorphism (not that just some constructions won't work, but that there isn't one at all).
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?
There 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.
Unless there's some idiosyncratic meaning for the `=>`, the Antisymmetry one basically says `Orange -> Yellow => Yellow -/> Orange`. The diagram is not acurate. The prose is very imprecise. "It also means that no ties are permitted - either I am better than my grandmother at soccer or she is better at it than me." NO. Antisymmetry doesn't exclude `x = y`. Ties are permitted in the equality case. Antisymmetry for a non-strict order says that if both directions hold, the two elements must in fact be the same element. The author is describing strict comparison or total comparability intuition, not antisymmetry.
I don't think they are completely wrong - "=>" is just implication. A hidden assumption in their diagrams is that circles of different colours are assumed to be different elements.
A morphism from orange to yellow means "O <= Y". From this, antisymmetry (and the hidden assumption) implies that "Y not <= O".
Totality is just the other way around (all two distinct elements are comparable in one direction).
If this is meant to be an explainer, that can't be simply implicit. The text actually seems full of imprecise claims, such as:
"All diagrams that look something different than the said chain diagram represent partial orders"
"The different linear orders that make up the partial order are called chains"
The Birkhoff theorem statement, which is materially wrong. A finite distributive lattice is not isomorphic to "the inclusion order of its join-irreducible elements".
It really isn't a long enough section to get lost in.
The 'not accurate' diagram says that orange-less-than-yellow implies yellow-not-less-than-orange. Hard to find fault with.
> NO. Antisymmetry doesn't exclude `x = y`. Ties are permitted in the equality case. Antisymmetry for a non-strict order says that if both directions hold, the two elements must in fact be the same element. The author is describing strict comparison or total comparability intuition, not antisymmetry.
My comment is not long enough either to get lost in.
The prose "It also means that no ties are permitted - either I am better than my grandmother at soccer or she is better at it than me" is inaccurate for describing antisymmetry. In the same short section, you first state the correct condition:
You have x ≤ y and y ≤ x only if x = y
from which it doesn't follow that "It also means that no ties are permitted". The "no ties" idea belongs to a stronger notion such as a strict total order, not to antisymmetry.
I think the last 10% is exactly where the useful part is, at least for programmers.
In a preorder seen as a category, there is at most one arrow between any two objects. So every diagram commutes and uniqueness is basically free. Then products and coproducts stop looking like magic diagrams and become something very familiar: greatest lower bounds and least upper bounds.
Small nit: preorders are thin categories, but posets are the skeletal thin categories. In a preorder you can have distinct a and b with both a <= b and b <= a, which means they are isomorphic, not literally the same. Quotienting by that equivalence gives you the poset.
The software angle is the part I find most useful. This kind of bugs shows up when we force a total order onto something that is only partially ordered, or only preordered. Dependency graphs, versions, permissions, type hierarchies, CRDT states, rule specificity, build steps. A lot of these don’t really want a comparator and a sort. Sometimes they want a quotient, a topological sort, a join, or just the honest answer that two things are not comparable.
That feels like the practical lesson here: category theory is not always adding abstraction. Sometimes it is just a good way to stop pretending two different structures are the same thing.
If 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
})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 level of correctness of the mathematics in the article as well, which I'm trying to present in sibling comment: https://news.ycombinator.com/item?id=47814213
I 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're more right than you'd think. The whole point of mathematics is precise thinking, yet the article is very inaccurate.
Nobody seems to care or notice. I'm watching in disbelief how nobody is pointing out the article is full of inaccuracies. See my sibling thread for a (very) incomplete list, which should disqualified this as a serious reading: https://news.ycombinator.com/item?id=47814213
My conclusion cannot be other than this ought to be useless for the general practitioner, since even wrong mathematics is appreciated the same as correct mathematics.
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.
Is there a "mind-blowing fact" about category theory? Like the first time I've heard that one can prove there is no analytical solution for a polynomial equation with a degree > 5 with group theory, it was mind-blowing. What's the counterpart of category theory?
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 different rings can have the same modules, for instance.) [0]
Writing a program and proving a theorem are the same act. (Curry–Howard–Lambek.) For well-behaved programs, every program is a proof of something and every proof is a program. The match is exact for simple typed languages and leaks a bit once you add general recursion (an infinite loop “proves” anything in Haskell), but the underlying identity is real. Lambek added the third leg: these are also morphisms in a category. [1]
Algebra and geometry are one thing wearing different costumes. (Stone duality and cousins.) A system of equations and the shape it cuts out aren’t related, they’re the same object seen from opposite sides. Grothendieck rebuilt algebraic geometry on this idea, with schemes (so you can do geometry on the integers themselves) and étale cohomology (topological invariants for shapes with no actual topology). His student Deligne used that machinery to settle the Weil conjectures in 1974. Wiles’s Fermat proof lives in the same world, though it leans on much more than the categorical foundations. [2]
[0] https://en.wikipedia.org/wiki/Yoneda_lemma
[1] https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...
[2] https://en.wikipedia.org/wiki/Stone_duality
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.
I think that CT is more akin to just a different language for mathematics than a solid set of axioms from which you can prove things. The most fact-y proof I've personally seen was that you can't extend the usual definition of functions in set theory to work with parametric polymorphism (not that just some constructions won't work, but that there isn't one at all).
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?
Phil?
Just Yoneda Lemma. In fact it feels like the theory just restates Yoneda Lemma over and over in different ways.
And the number of things you can prove using Yoneda lemma just proves how powerful category theory is.
This resource is a really clear breakdown of order relations; visualizing the structure like this makes the abstract concepts much more digestible
There 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.
Unless there's some idiosyncratic meaning for the `=>`, the Antisymmetry one basically says `Orange -> Yellow => Yellow -/> Orange`. The diagram is not acurate. The prose is very imprecise. "It also means that no ties are permitted - either I am better than my grandmother at soccer or she is better at it than me." NO. Antisymmetry doesn't exclude `x = y`. Ties are permitted in the equality case. Antisymmetry for a non-strict order says that if both directions hold, the two elements must in fact be the same element. The author is describing strict comparison or total comparability intuition, not antisymmetry.
I don't think they are completely wrong - "=>" is just implication. A hidden assumption in their diagrams is that circles of different colours are assumed to be different elements.
A morphism from orange to yellow means "O <= Y". From this, antisymmetry (and the hidden assumption) implies that "Y not <= O".
Totality is just the other way around (all two distinct elements are comparable in one direction).
If this is meant to be an explainer, that can't be simply implicit. The text actually seems full of imprecise claims, such as:
"All diagrams that look something different than the said chain diagram represent partial orders"
"The different linear orders that make up the partial order are called chains"
The Birkhoff theorem statement, which is materially wrong. A finite distributive lattice is not isomorphic to "the inclusion order of its join-irreducible elements".
It really isn't a long enough section to get lost in.
The 'not accurate' diagram says that orange-less-than-yellow implies yellow-not-less-than-orange. Hard to find fault with.
> NO. Antisymmetry doesn't exclude `x = y`. Ties are permitted in the equality case. Antisymmetry for a non-strict order says that if both directions hold, the two elements must in fact be the same element. The author is describing strict comparison or total comparability intuition, not antisymmetry.
I like the article's "imprecise prose" better:
My comment is not long enough either to get lost in.
The prose "It also means that no ties are permitted - either I am better than my grandmother at soccer or she is better at it than me" is inaccurate for describing antisymmetry. In the same short section, you first state the correct condition:
You have x ≤ y and y ≤ x only if x = y
from which it doesn't follow that "It also means that no ties are permitted". The "no ties" idea belongs to a stronger notion such as a strict total order, not to antisymmetry.
The first 90% of this is standard set theory.
I'm unclear what the last 10% of 'category theory' gives us.
I think the last 10% is exactly where the useful part is, at least for programmers.
In a preorder seen as a category, there is at most one arrow between any two objects. So every diagram commutes and uniqueness is basically free. Then products and coproducts stop looking like magic diagrams and become something very familiar: greatest lower bounds and least upper bounds.
Small nit: preorders are thin categories, but posets are the skeletal thin categories. In a preorder you can have distinct a and b with both a <= b and b <= a, which means they are isomorphic, not literally the same. Quotienting by that equivalence gives you the poset.
The software angle is the part I find most useful. This kind of bugs shows up when we force a total order onto something that is only partially ordered, or only preordered. Dependency graphs, versions, permissions, type hierarchies, CRDT states, rule specificity, build steps. A lot of these don’t really want a comparator and a sort. Sometimes they want a quotient, a topological sort, a join, or just the honest answer that two things are not comparable.
That feels like the practical lesson here: category theory is not always adding abstraction. Sometimes it is just a good way to stop pretending two different structures are the same thing.