8 ms·

Knowing that a ring (in abstract algebra) is equivalently defined as "a monoid object in the category of abelian groups" is not going to help you with your firs

by joppy 8y ago

Knowing that a ring (in abstract algebra) is equivalently defined as "a monoid object in the category of abelian groups" is not going to help you with your first abstract algebra course. It makes things that are glaringly obvious from the usual perspective totally opaque, and means you have to define things like the tensor product of abelian groups properly before you can even define a ring. I don't know of a professor who would even mention this perspective during a first abstract algebra course. You get much a much better intuition for what rings are by just using them. Mathematicians who work in category theory have often seen many, many examples of algebraic structures before they get abstracted away to "a monoid object in the category of blah", and many mathematicians still find it an exercise to unpack these compact definitions back to something more intuitive, or more familiar. And even so, most of the time identifying your favourite object as some abstract category thing doesn't bring anything new or useful to the table for the study of that object. (It can help spot patterns between objects though). I say this for any aspiring Haskell programmers: don't feel like understanding this statement will make you a better programmer. The best way to get better at using monads is to just use monads more, and play around with them. Tracking down all the definitions in the statement is a neat exercise, though.

nerdponx 8y ago
On the flipside, knowing that a ring is "like a group (which we already covered), with another operation that follows these axioms" is useful, and still mostly rigorous. It's better than "eh just use it a lot and you'll figure it out."
- throwawaymath 8y ago
  True, but that likeness is normally taught in abstract algebra without requiring category theory. For example this: https://i.redd.it/yqnbigi032i11.png https://i.redd.it/yqnbigi032i11.png doesn’t require any category to understand.
  - mlevental 8y ago
    that's definitely a neat diagram but the examples (the blue) almost entail already know the definitively.
  - nerdponx 8y ago
    Not my point. For education, there is a happy medium between providing theoretical context and hand-waving over technicalities and learning by example. Very nice diagram though.
    - throwawaymath 8y ago
      To give credit where it's due, that diagram is from a /r/math thread: https://www.reddit.com/r/math/comments/99y547/my_digram_of_abstract_algebra_and_other_stuff/ https://www.reddit.com/r/math/comments/99y547/my_digram_of_a...
- zozbot123 8y ago
  And "these axioms" are indeed the axioms of a monoid operation (for multiplication), plus some sensible notion of "consistency" with the underlying abelian-group structure. Which is all that the "monoid object" characterization is saying, actually. So, I disagree that the perspective is inherently unintuitive.
  - throwawaymath 8y ago
    I think you may have a different definition of intuitive than the top level comment, or else you two are speaking to different things. Category theory is certainly "elegant" and provides a lot of insight in the connection between various algebraic structures. But I think it would be completely asinine to use category theoretic formalisms in a first course covering algebraic structures like groups, rings and fields. That level of abstraction doesn't make the distinctions between each structure more intuitive; if anything it obfuscates them and requires setting up far more machinery to explain what they even are. It would be highly inappropriate for an undergrad's first exposure. That is my impression of the top level comment's point, and I largely agree with it. If you understand what a group is, I can explain what a ring is to you in a very straightforward way that still shows you the connections between the two things. But setting up category theoretic language would take up a nontrivial portion of the course itself, and at the end you'd have a much less specific understanding of the relevant theorems on rings and groups. You'd be missing all the trees for the forest.
    - zozbot123 8y ago
      Look, it's 2019. "Setting up category-theoretic language" is something that should begin in the first lesson of the first course in formal, non-K12 math, starting with the use of ETCS or a comparable "structural" set theory as the 'naïve' foundation of choice. It provides countless advantages not just in abstract algebra, but in a wide variety of other subfields of math as well.
      - throwawaymath 8y ago
        To be clear, how much category theory are you envisioning setting up in a first course on abstract algebra? Are you basically just talking about a modern treatment of morphisms in the abstract sense, or do you also mean discussing things like functors? My perspective on this is basically informed by the following: I like category theory a lot. I've chatted with Baez about this on a few occasions and insofar as I need to use algebra, I like the category theoretic formalisms that come along with it. I think if you had students with a lot of mathematical maturity but no prior exposure to abstract algebra (except maybe a rigorous treatment of linear algebra), you could accomplish what you're proposing. To your point, one of my favorite math textbooks is Aluffi's Chapter 0. You're probably familiar with it, but if not: it builds up abstract algebra in a rigorous and modern fashion alongside category theory. However, there are a few caveats here: 1. Aluffi explicitly states it's a textbook for upper level undergrads and preferably graduates. In my experience, when authors say this they actually mean it's appropriate as a year one graduate course. That's not the time to learn abstract algebra for the first time! 2. Aluffi does a great job of covering things like morphisms and categories, but the size of the book is massive. He doesn't have any nontrivial coverage of deeper category theoretic concepts like functors until much later in the book; it's probably far enough in that you couldn't reasonably cover it in a single semester course. 3. I really don't think most math majors would benefit from it. Those who are applied math majors will get questionable benefit from a slower, more comprehensive approach to algebra than a faster approach that lets them get to applications. Moreover, not all pure math majors have the mathematical maturity to approach category theory before grad school. Proofs in traditional abstract algebra are a lot less abstract than category theory, and it's easier to motivate them even if they aren't ultimately as elegant.
- Koshkin 8y ago
  > knowing that a ring is “like a group...” is useful Turns out, not so much, though. The ring theory bears surprisingly little resemblance to the group theory, to the point that these concepts seem to have nothing to do with each other...
  - nerdponx 8y ago
    Except that a ring is literally a group under addition? Do you seriously mean to suggest there's no pedagogical value in using groups to teach rings?
    - ginnungagap 8y ago
      There's actually a big debate between math educators on whether abstract algebra courses should follow a groups first or a rings first approach so it is totally possible to teach rings without appealing to groups. (I'd still argue that both approaches have their advantages and disadvantages and that there can be value in using groups to teach rings though, I just wanted to say it's not as obvious as you might think)
    - throwawaymath 8y ago
      I could be wrong here, but my read of that comment is that they're talking about the structural differences between rings and groups at the research level. Which I guess makes sense, since you're in very different territories if you're talking about something like representation theory of finite groups... Otherwise I agree with you, there's a lot of pedagogical value in combining coverage of algebraic structures.
    - dnautics 8y ago
      Yes obviously you need to know what a group is to even begin ring theory but ring theory looks and feels nothing like group theory.
- SatvikBeri 8y ago
  Yup, and I think it's actually a completely rigorous definition – all rings are commutative groups under one operation (often referred to as "addition"), plus another operation ("multiplication"), plus rules for how they interact.
  - zozbot123 8y ago
    The point of the category-theoretical definition is to explain how these "rules for how they interact" might arise, and in what sense these might be understood as 'minimal' rules. These are not "equivalent" definitions in any real sense, and to think of them as such is arguably missing the whole point of how category theory can be useful here.
- joppy 8y ago
  Of course I'm not recommending learning ring theory without learning the (usual) definition of a ring. This is absolutely rigorous and useful, and immediately after you learn it you can start playing around with rings, and start proving some interesting theorems. The analogue of this approach in Haskell is probably something like sections 1 and 2 of "Tackling the awkward squad: monadic input/output, concurrency, exceptions, and foreign-language calls in Haskell" [1], which is quite an old paper (from around 2000 or something?). It does not even contain the word "monoid", but instead shows what an IO action is, and how functions like >> and >>= can be used to sequence and thread them. It also provides a lot of context as to why monadic IO exists. After reading this, a new programmer could start playing around with making new IO actions using these operators. After reading about monoid objects, a new programmer is no closer to being able to write a Haskell program. [1]: https://www.microsoft.com/en-us/research/wp-content/uploads/2016/07/mark.pdf https://www.microsoft.com/en-us/research/wp-content/uploads/...
EGreg 8y ago
All mathematics is abstraction. Numbers are abstractions Variables are higher level abstractions You can have properties that you abstract and then see what you have. Banach spaces. Metric spaces. Three different ways to define compactness. I liked to teach calculus by teaching the limit of a sequence first and then defining limits of functions by considering all sequences x_n that converge to x, and then looking if y_n converge. Building up definitions from simpler definitions, instead of the STUPID handwavy definition they use in Calculus 1 that can’t even handle limits of functions like cos 1/x and answer about limits of certain subsequences of x_n that approach 0. I started a course about mathematics this way, building up from the very basics: https://m.youtube.com/channel/UCuge8p-oYsKSU0rDMy7jJlA https://m.youtube.com/channel/UCuge8p-oYsKSU0rDMy7jJlA
throwawaymath 8y ago
Very well said. The practical reality is that the “category theory” used in functional programming paradigms (like Haskell’s) has only a passing resemblence to formal category theory in mathematics. It’s a nice idea to formalize some concepts in functional programming algebraically, but learning formal category theory won’t realistically improve your programming ability. There is some conceptual overlap, but they’re just very different things.
- PaulAJ 8y ago
  On the one hand I know what you mean: I know practically no category theory (basically I know enough to expect "co" as a prefix to mean "everything the other way around"), but I can still produce good Haskell code. But on the other hand, I look at some of the stuff Edward Kmett has produced, particularly his lens library which I use all the time, and its very obvious that the design has been informed by a deep knowledge of category theory.
klyrs 8y ago
Agreed. More often than not, it's just a layer of jargon that obscures the topic at hand rather than clarify it. Worse, I see a lot of folks using it to belittle and intimidate -- recently I met an instructor at a startup-focused programming school who put his (very novice) students through a lecture which began, "Is a burrito a monoid" and went down hill from there -- he was actually bragging about how lost his students were. And he told me about all this under the presumption that I didn't know category theory (easy mistake, I guess, I'm a woman after all) and my biggest regret is that I didn't ask him to explain Poincare duality to me. If only he knew that my paralysis was a result of biting my tongue at his pedagogical trashfire...
- dominotw 8y ago
  > startup-focused programming school Curious what are these? They place into a startup after you are done? surprised that they are teaching category theory if that is the case.
  - klyrs 8y ago
    That's just the thing. Category theory isn't really a part of their standard curriculum -- it's something that a lone instructor decided to put into his syllabus. My one-line description of the business doesn't really do it justice -- it's programming school with a focus on web & app development, and they have fairly good (dev-)community engagement. They also teach some business skills along the way, and students often end up with startups, or founding their own. Also many of their instructors currently work at startups.
- scarejunba 8y ago
  That guy is an imbecile and your life is better for not having engaged with him.
- agumonkey 8y ago
  > Poincare duality to me next time don't pass the opportunity
- apta 8y ago
bitwize 8y ago
The fact that "A monad is a monoid in a category of endofunctors" explains everything while clarifying nothing is exactly why Haskells repeat that phrase -- as a joke.
danharaj 8y ago
As a contrasting point of view, the excellent graduate level introduction to algebra, Algebra Chapter 0 by Paolo Aluffi introduces category theory right from the start and it's a very good book. It just uses the concepts of a diagram and a universal construction, which have a high power-to-weight ratio. I agree that the danker category theory is more useful for research than for teaching. In particular when you want to try to import definitions and theorems from one context to another, or you want to decompose a proof into its abstract skeleton and concrete particularities. As an example of how "A monad is a monoid... etc." can be useful for the abstraction (i.e. library) builder, if you import this definition of a monad into the bicategory of endoprofunctors, you get arrows [1]. [1] http://www.riec.tohoku.ac.jp/~asada/papers/arrStrMnd.pdf http://www.riec.tohoku.ac.jp/~asada/papers/arrStrMnd.pdf
- throwawaymath 8y ago
  I love Aluffi's Chapter 0 and mentioned it in another comment. But I don't think it's appropriate as a first pass on abstract algebra, even for pure math majors. Maybe for an Honors Intro to Abstract Algebra course...otherwise, I think Aluffi is pretty clear in that while he thinks category theory should be covered early on, it's really a text for graduates and not undergrads. Abstract algebra should be covered before you get to grad school.
  - autocorrect23 8y ago
    What is a good textbook in abstract algebra for the beginner?
    - umanwizard 8y ago
      It depends what "beginner" means -- what's your level of sophistication? Are you a budding mathematician or just an interested programmer?
      - deleted 8y ago
        [deleted]
nubslayer 8y ago
>and means you have to define things like the tensor product of abelian groups properly before you can even define a ring Hmm, someone managed to do it at Wikipedia: "In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions."[0] [0] https://en.wikipedia.org/wiki/Ring_%28mathematics%29 https://en.wikipedia.org/wiki/Ring_%28mathematics%29
- TheDong 8y ago
  You missed the point of the parent comment. They were saying that the definition you found, on wikipedia, is better than the definition of "a monoid object in the category of abelian groups" because the definition on wikipedia doesn't require knowing about tensor products etc. You're proving their point. Wikipedia doesn't define it with that language for exactly the reasons the comment pointed out.
  - nubslayer 8y ago
    Dunno if they had a point or not. The statement is not true.
- throwawaymath 8y ago
  Yes, exactly. That's not category theoretic language, which is why that definition is so approachable. There are far fewer prerequisites to understanding what's going on in that definition. That's precisely what the parent commenter is getting at.
  - nubslayer 8y ago
    Seems like all language would be "category theoretic".
    - throwawaymath 8y ago
alexandercrohde 8y ago
I'd go a step further. If you're trying to do functional programming in scala/js/ruby/php you don't need to care about any of those words (functor/monoid/monad/category). You can write high reusable, generalized, clean functional code without knowing a darn thing about the theory behind it. Moreover, if you do learn these words you probably shouldn't use them around your peers because they'll serve to socially alienate you and confuse them.
- bunderbunder 8y ago
  What the hey, I'll pile on more -- Category theory type stuff is a very useful thing to know when designing programs, because it is useful to know things like, "Well, if I give this data structure these properties, then it'll be well-behaved when I map over it." But you do not need to learn all the technical jargon for the concepts in order to develop a command of them, any more than you need to memorize a natural language's formal grammar rules in order to speak it competently. (Since the grammar needs to be explained using natural language, one could even argue that it isn't really possible to do the one before the other, in a big picture sense.) Also similar to learning a natural language, learning all the technical, formal stuff isn't necessarily all that useful for helping you to achieve practical competence. To stretch that analogy even further, getting too enamored of the category theory stuff is likely to earn you a certain measure of benevolent disdain from experienced functional programmers in the same way that getting too enamored of grammar rules is likely to earn you a certain reputation of disdain from people who write for a living. It's not that they don't value these things, it's that talking about them too much sends signals that you're more interested in reading the rulebook than playing the game.
- SatvikBeri 8y ago
  In my view, "Functor" and "Monad" are the wrong emphasis, while "map" and "flatMap" are the parts that really matter, and are much easier to teach in comparison. Most programmers are familiar with "map" today, and "flatMap" is not much of a stretch. But once you understand these two things, you're 90% of the way to understanding monads and why they're useful.
strictfp 8y ago
Having studied abstract algebra before using FP, I must say that I instantly made this connection when I came into contact with monads, and that it helped me a lot to understand them. So I wouldn't be so quick to dismiss the idea of associating these concepts. After all, category theory is all about making that kind of associations.
- nine_k 8y ago
  This is a good observation. Though, most programmers don't study, and don't have to study, enough abstract algebra before being exposed to promises, optionals, and even lists (which all exhibit some form of monadic behavior). What worked best for me was both playing with code, reading the definitions in code, and then look up some CT to get to the mathematical roots and where they fit in the intuitive picture. (Then you can continue with arrows, monad transformers, etc.) Since my primary purpose for learning monads (and other algebraic constructs) was and is their use in programming, I concentrate on "practical" intuitions, and periodically check if I can go down to the algebraic base of them in a particular case. If I still can, I suppose I know enough math to get by in the particular area. If not, I open a book and clarify my understanding, and maybe glean something new.
- AnimalMuppet 8y ago
  If you already know category theory, sure, that's fine. (Or maybe if you already know abstract algebra really well.) I'm going to go out on a limb, though, and speculate that the average person starting to use FP does not already know category theory.
mathgenius 8y ago
For me category theory is more concrete. The basic reason is simple: categories provide type information. If all you want to do is grind out some calculations to pass an exam, then yeah, memorize the ring axioms and have at it. Anything more than that, and you need direction, or context, Ie. types, categories. Once you de-categorify that's when things become mysterious and confusing, and the impressive calculations abound. But it's just because you forgot the concrete thing that you are calculating. It's all a bit paradoxical, and I'm not explaining it very well. (And I have no clue about the Haskell stuff.)
- joppy 8y ago
  I fail to see how a category theory perspective is going to help with "grinding out" any interesting theorems about commutative rings, such as: - Every commutative ring has a maximal ideal. - The quotient of a commutative ring by a maximal ideal is a field. - The quotient of a commutative ring by a prime ideal is an integral domain. - The classification of finitely generated modules over a principal ideal domain. There are some universal properties which define what it means to be a quotient, which can help you work with them. But for example, the only "categorical" definition of prime ideals that I know is based on that third theorem. I think it is easy to forget just how much underlying theory you need to know before you can check that the categorical analogues of definitions are even the same as the original ones.

by joppy 8y ago

nerdponx 8y ago

On the flipside, knowing that a ring is "like a group (which we already covered), with another operation that follows these axioms" is useful, and still mostly rigorous. It's better than "eh just use it a lot and you'll figure it out."

throwawaymath 8y ago
True, but that likeness is normally taught in abstract algebra without requiring category theory. For example this: https://i.redd.it/yqnbigi032i11.png https://i.redd.it/yqnbigi032i11.png doesn’t require any category to understand.
- mlevental 8y ago
  that's definitely a neat diagram but the examples (the blue) almost entail already know the definitively.
- nerdponx 8y ago
  Not my point. For education, there is a happy medium between providing theoretical context and hand-waving over technicalities and learning by example. Very nice diagram though.
  - throwawaymath 8y ago
    To give credit where it's due, that diagram is from a /r/math thread: https://www.reddit.com/r/math/comments/99y547/my_digram_of_abstract_algebra_and_other_stuff/ https://www.reddit.com/r/math/comments/99y547/my_digram_of_a...
zozbot123 8y ago
And "these axioms" are indeed the axioms of a monoid operation (for multiplication), plus some sensible notion of "consistency" with the underlying abelian-group structure. Which is all that the "monoid object" characterization is saying, actually. So, I disagree that the perspective is inherently unintuitive.
- throwawaymath 8y ago
  I think you may have a different definition of intuitive than the top level comment, or else you two are speaking to different things. Category theory is certainly "elegant" and provides a lot of insight in the connection between various algebraic structures. But I think it would be completely asinine to use category theoretic formalisms in a first course covering algebraic structures like groups, rings and fields. That level of abstraction doesn't make the distinctions between each structure more intuitive; if anything it obfuscates them and requires setting up far more machinery to explain what they even are. It would be highly inappropriate for an undergrad's first exposure. That is my impression of the top level comment's point, and I largely agree with it. If you understand what a group is, I can explain what a ring is to you in a very straightforward way that still shows you the connections between the two things. But setting up category theoretic language would take up a nontrivial portion of the course itself, and at the end you'd have a much less specific understanding of the relevant theorems on rings and groups. You'd be missing all the trees for the forest.
  - zozbot123 8y ago
    Look, it's 2019. "Setting up category-theoretic language" is something that should begin in the first lesson of the first course in formal, non-K12 math, starting with the use of ETCS or a comparable "structural" set theory as the 'naïve' foundation of choice. It provides countless advantages not just in abstract algebra, but in a wide variety of other subfields of math as well.
    - throwawaymath 8y ago
      To be clear, how much category theory are you envisioning setting up in a first course on abstract algebra? Are you basically just talking about a modern treatment of morphisms in the abstract sense, or do you also mean discussing things like functors? My perspective on this is basically informed by the following: I like category theory a lot. I've chatted with Baez about this on a few occasions and insofar as I need to use algebra, I like the category theoretic formalisms that come along with it. I think if you had students with a lot of mathematical maturity but no prior exposure to abstract algebra (except maybe a rigorous treatment of linear algebra), you could accomplish what you're proposing. To your point, one of my favorite math textbooks is Aluffi's Chapter 0. You're probably familiar with it, but if not: it builds up abstract algebra in a rigorous and modern fashion alongside category theory. However, there are a few caveats here: 1. Aluffi explicitly states it's a textbook for upper level undergrads and preferably graduates. In my experience, when authors say this they actually mean it's appropriate as a year one graduate course. That's not the time to learn abstract algebra for the first time! 2. Aluffi does a great job of covering things like morphisms and categories, but the size of the book is massive. He doesn't have any nontrivial coverage of deeper category theoretic concepts like functors until much later in the book; it's probably far enough in that you couldn't reasonably cover it in a single semester course. 3. I really don't think most math majors would benefit from it. Those who are applied math majors will get questionable benefit from a slower, more comprehensive approach to algebra than a faster approach that lets them get to applications. Moreover, not all pure math majors have the mathematical maturity to approach category theory before grad school. Proofs in traditional abstract algebra are a lot less abstract than category theory, and it's easier to motivate them even if they aren't ultimately as elegant.
Koshkin 8y ago
> knowing that a ring is “like a group...” is useful Turns out, not so much, though. The ring theory bears surprisingly little resemblance to the group theory, to the point that these concepts seem to have nothing to do with each other...
- nerdponx 8y ago
  Except that a ring is literally a group under addition? Do you seriously mean to suggest there's no pedagogical value in using groups to teach rings?
  - ginnungagap 8y ago
    There's actually a big debate between math educators on whether abstract algebra courses should follow a groups first or a rings first approach so it is totally possible to teach rings without appealing to groups. (I'd still argue that both approaches have their advantages and disadvantages and that there can be value in using groups to teach rings though, I just wanted to say it's not as obvious as you might think)
  - throwawaymath 8y ago
    I could be wrong here, but my read of that comment is that they're talking about the structural differences between rings and groups at the research level. Which I guess makes sense, since you're in very different territories if you're talking about something like representation theory of finite groups... Otherwise I agree with you, there's a lot of pedagogical value in combining coverage of algebraic structures.
  - dnautics 8y ago
    Yes obviously you need to know what a group is to even begin ring theory but ring theory looks and feels nothing like group theory.
SatvikBeri 8y ago
Yup, and I think it's actually a completely rigorous definition – all rings are commutative groups under one operation (often referred to as "addition"), plus another operation ("multiplication"), plus rules for how they interact.
- zozbot123 8y ago
  The point of the category-theoretical definition is to explain how these "rules for how they interact" might arise, and in what sense these might be understood as 'minimal' rules. These are not "equivalent" definitions in any real sense, and to think of them as such is arguably missing the whole point of how category theory can be useful here.
joppy 8y ago
Of course I'm not recommending learning ring theory without learning the (usual) definition of a ring. This is absolutely rigorous and useful, and immediately after you learn it you can start playing around with rings, and start proving some interesting theorems. The analogue of this approach in Haskell is probably something like sections 1 and 2 of "Tackling the awkward squad: monadic input/output, concurrency, exceptions, and foreign-language calls in Haskell" [1], which is quite an old paper (from around 2000 or something?). It does not even contain the word "monoid", but instead shows what an IO action is, and how functions like >> and >>= can be used to sequence and thread them. It also provides a lot of context as to why monadic IO exists. After reading this, a new programmer could start playing around with making new IO actions using these operators. After reading about monoid objects, a new programmer is no closer to being able to write a Haskell program. [1]: https://www.microsoft.com/en-us/research/wp-content/uploads/2016/07/mark.pdf https://www.microsoft.com/en-us/research/wp-content/uploads/...

EGreg 8y ago

All mathematics is abstraction. Numbers are abstractions Variables are higher level abstractions You can have properties that you abstract and then see what you have. Banach spaces. Metric spaces. Three different ways to define compactness. I liked to teach calculus by teaching the limit of a sequence first and then defining limits of functions by considering all sequences x_n that converge to x, and then looking if y_n converge. Building up definitions from simpler definitions, instead of the STUPID handwavy definition they use in Calculus 1 that can’t even handle limits of functions like cos 1/x and answer about limits of certain subsequences of x_n that approach 0. I started a course about mathematics this way, building up from the very basics: https://m.youtube.com/channel/UCuge8p-oYsKSU0rDMy7jJlA https://m.youtube.com/channel/UCuge8p-oYsKSU0rDMy7jJlA

throwawaymath 8y ago

Very well said. The practical reality is that the “category theory” used in functional programming paradigms (like Haskell’s) has only a passing resemblence to formal category theory in mathematics. It’s a nice idea to formalize some concepts in functional programming algebraically, but learning formal category theory won’t realistically improve your programming ability. There is some conceptual overlap, but they’re just very different things.

PaulAJ 8y ago
On the one hand I know what you mean: I know practically no category theory (basically I know enough to expect "co" as a prefix to mean "everything the other way around"), but I can still produce good Haskell code. But on the other hand, I look at some of the stuff Edward Kmett has produced, particularly his lens library which I use all the time, and its very obvious that the design has been informed by a deep knowledge of category theory.

klyrs 8y ago

Agreed. More often than not, it's just a layer of jargon that obscures the topic at hand rather than clarify it. Worse, I see a lot of folks using it to belittle and intimidate -- recently I met an instructor at a startup-focused programming school who put his (very novice) students through a lecture which began, "Is a burrito a monoid" and went down hill from there -- he was actually bragging about how lost his students were. And he told me about all this under the presumption that I didn't know category theory (easy mistake, I guess, I'm a woman after all) and my biggest regret is that I didn't ask him to explain Poincare duality to me. If only he knew that my paralysis was a result of biting my tongue at his pedagogical trashfire...

dominotw 8y ago
> startup-focused programming school Curious what are these? They place into a startup after you are done? surprised that they are teaching category theory if that is the case.
- klyrs 8y ago
  That's just the thing. Category theory isn't really a part of their standard curriculum -- it's something that a lone instructor decided to put into his syllabus. My one-line description of the business doesn't really do it justice -- it's programming school with a focus on web & app development, and they have fairly good (dev-)community engagement. They also teach some business skills along the way, and students often end up with startups, or founding their own. Also many of their instructors currently work at startups.
scarejunba 8y ago
That guy is an imbecile and your life is better for not having engaged with him.
agumonkey 8y ago
> Poincare duality to me next time don't pass the opportunity
apta 8y ago

bitwize 8y ago

The fact that "A monad is a monoid in a category of endofunctors" explains everything while clarifying nothing is exactly why Haskells repeat that phrase -- as a joke.

danharaj 8y ago

As a contrasting point of view, the excellent graduate level introduction to algebra, Algebra Chapter 0 by Paolo Aluffi introduces category theory right from the start and it's a very good book. It just uses the concepts of a diagram and a universal construction, which have a high power-to-weight ratio. I agree that the danker category theory is more useful for research than for teaching. In particular when you want to try to import definitions and theorems from one context to another, or you want to decompose a proof into its abstract skeleton and concrete particularities. As an example of how "A monad is a monoid... etc." can be useful for the abstraction (i.e. library) builder, if you import this definition of a monad into the bicategory of endoprofunctors, you get arrows [1]. [1] http://www.riec.tohoku.ac.jp/~asada/papers/arrStrMnd.pdf http://www.riec.tohoku.ac.jp/~asada/papers/arrStrMnd.pdf

throwawaymath 8y ago
I love Aluffi's Chapter 0 and mentioned it in another comment. But I don't think it's appropriate as a first pass on abstract algebra, even for pure math majors. Maybe for an Honors Intro to Abstract Algebra course...otherwise, I think Aluffi is pretty clear in that while he thinks category theory should be covered early on, it's really a text for graduates and not undergrads. Abstract algebra should be covered before you get to grad school.
- autocorrect23 8y ago
  What is a good textbook in abstract algebra for the beginner?
  - umanwizard 8y ago
    It depends what "beginner" means -- what's your level of sophistication? Are you a budding mathematician or just an interested programmer?
    - deleted 8y ago
      [deleted]

nubslayer 8y ago

>and means you have to define things like the tensor product of abelian groups properly before you can even define a ring Hmm, someone managed to do it at Wikipedia: "In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions."[0] [0] https://en.wikipedia.org/wiki/Ring_%28mathematics%29 https://en.wikipedia.org/wiki/Ring_%28mathematics%29

TheDong 8y ago
You missed the point of the parent comment. They were saying that the definition you found, on wikipedia, is better than the definition of "a monoid object in the category of abelian groups" because the definition on wikipedia doesn't require knowing about tensor products etc. You're proving their point. Wikipedia doesn't define it with that language for exactly the reasons the comment pointed out.
- nubslayer 8y ago
  Dunno if they had a point or not. The statement is not true.
throwawaymath 8y ago
Yes, exactly. That's not category theoretic language, which is why that definition is so approachable. There are far fewer prerequisites to understanding what's going on in that definition. That's precisely what the parent commenter is getting at.
- nubslayer 8y ago
  Seems like all language would be "category theoretic".
  - throwawaymath 8y ago

alexandercrohde 8y ago

I'd go a step further. If you're trying to do functional programming in scala/js/ruby/php you don't need to care about any of those words (functor/monoid/monad/category). You can write high reusable, generalized, clean functional code without knowing a darn thing about the theory behind it. Moreover, if you do learn these words you probably shouldn't use them around your peers because they'll serve to socially alienate you and confuse them.

bunderbunder 8y ago
What the hey, I'll pile on more -- Category theory type stuff is a very useful thing to know when designing programs, because it is useful to know things like, "Well, if I give this data structure these properties, then it'll be well-behaved when I map over it." But you do not need to learn all the technical jargon for the concepts in order to develop a command of them, any more than you need to memorize a natural language's formal grammar rules in order to speak it competently. (Since the grammar needs to be explained using natural language, one could even argue that it isn't really possible to do the one before the other, in a big picture sense.) Also similar to learning a natural language, learning all the technical, formal stuff isn't necessarily all that useful for helping you to achieve practical competence. To stretch that analogy even further, getting too enamored of the category theory stuff is likely to earn you a certain measure of benevolent disdain from experienced functional programmers in the same way that getting too enamored of grammar rules is likely to earn you a certain reputation of disdain from people who write for a living. It's not that they don't value these things, it's that talking about them too much sends signals that you're more interested in reading the rulebook than playing the game.
SatvikBeri 8y ago
In my view, "Functor" and "Monad" are the wrong emphasis, while "map" and "flatMap" are the parts that really matter, and are much easier to teach in comparison. Most programmers are familiar with "map" today, and "flatMap" is not much of a stretch. But once you understand these two things, you're 90% of the way to understanding monads and why they're useful.

strictfp 8y ago

Having studied abstract algebra before using FP, I must say that I instantly made this connection when I came into contact with monads, and that it helped me a lot to understand them. So I wouldn't be so quick to dismiss the idea of associating these concepts. After all, category theory is all about making that kind of associations.

nine_k 8y ago
This is a good observation. Though, most programmers don't study, and don't have to study, enough abstract algebra before being exposed to promises, optionals, and even lists (which all exhibit some form of monadic behavior). What worked best for me was both playing with code, reading the definitions in code, and then look up some CT to get to the mathematical roots and where they fit in the intuitive picture. (Then you can continue with arrows, monad transformers, etc.) Since my primary purpose for learning monads (and other algebraic constructs) was and is their use in programming, I concentrate on "practical" intuitions, and periodically check if I can go down to the algebraic base of them in a particular case. If I still can, I suppose I know enough math to get by in the particular area. If not, I open a book and clarify my understanding, and maybe glean something new.
AnimalMuppet 8y ago
If you already know category theory, sure, that's fine. (Or maybe if you already know abstract algebra really well.) I'm going to go out on a limb, though, and speculate that the average person starting to use FP does not already know category theory.

mathgenius 8y ago

For me category theory is more concrete. The basic reason is simple: categories provide type information. If all you want to do is grind out some calculations to pass an exam, then yeah, memorize the ring axioms and have at it. Anything more than that, and you need direction, or context, Ie. types, categories. Once you de-categorify that's when things become mysterious and confusing, and the impressive calculations abound. But it's just because you forgot the concrete thing that you are calculating. It's all a bit paradoxical, and I'm not explaining it very well. (And I have no clue about the Haskell stuff.)

joppy 8y ago
I fail to see how a category theory perspective is going to help with "grinding out" any interesting theorems about commutative rings, such as: - Every commutative ring has a maximal ideal. - The quotient of a commutative ring by a maximal ideal is a field. - The quotient of a commutative ring by a prime ideal is an integral domain. - The classification of finitely generated modules over a principal ideal domain. There are some universal properties which define what it means to be a quotient, which can help you work with them. But for example, the only "categorical" definition of prime ideals that I know is based on that third theorem. I think it is easy to forget just how much underlying theory you need to know before you can check that the categorical analogues of definitions are even the same as the original ones.