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Category Theory Illustrated – Orders

abuseofnotation.github.io

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Re: Category Theory Illustrated – Orders

#21

Post by IsTom »

raincole wrote: post #4
dgan wrote: post #3

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

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?

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).

Re: Category Theory Illustrated – Orders

#22

Post by azan_ »

throw567643u8 wrote: post #13
raincole wrote: post #4

Earlier quoted context omitted.

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?

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.

Re: Category Theory Illustrated – Orders

#23

Post by SkiFire13 »

tux3 wrote: post #6
raincole wrote: post #4

Earlier quoted context omitted.

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?

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?

Re: Category Theory Illustrated – Orders

#24

Post by gobdovan »

mrkeen wrote: post #20
gobdovan wrote: post #7

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 no…

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 comp…

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.

Re: Category Theory Illustrated – Orders

#25

Post by JPC21 »

dgan wrote: post #3

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 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.

Re: Category Theory Illustrated – Orders

#26

Post by [deleted] »

gobdovan wrote: post #7

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 no…

[deleted]

Re: Category Theory Illustrated – Orders

#27

Post by gobdovan »

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

Re: Category Theory Illustrated – Orders

#29

Post by seanhunter »

If 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

Re: Category Theory Illustrated – Orders

#30

Post by scotty79 »

I love how math is like a new language, in a new country, of culture you are not exactly familiar with.

This article is like living there for few months. You see things, some of them you recognize as something similar to what you have at home, then you learn how the locals look at them and call them. And suddenly you can understand what somebody means when they say:

"Each distributive lattice is isomorphic to an inclusion order of its join-irreducible elements."

Having a charitable local (or expat with years there under their belt) that helps you grasp it because they know where you came from, just like the person who wrote this article, is such a treasure.

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