4 ms·

>In mathematics, a monad is defined as a monoid in the category of endofunctors. Sure. But like everything in math what it really is is a thing that the mathem

by juped 7y ago

>In mathematics, a monad is defined as a monoid in the category of endofunctors. Sure. But like everything in math what it really is is a thing that the mathematician has come to know through application of effort. The definition is a portion of a cage built to hold the concept in place so the next mathematician can come along and expend effort to know it. You've probably done this yourself, just maybe not that deep in the abstraction tower (though you'd be surprised how abstract some everyday things can be). For example, you may have internalized the fact about division of integers that every integer has a unique prime factorization. This is an important part of seeing what multiplication is, but it's not part of the tower of abstraction upon which multiplication is built. Mathematicians tend to end up being unconscious or conscious Platonists because mathematicians are trained to see the mathematics itself.

QuesnayJr 7y ago
This is an incredible quote: > But like everything in math what it really is is a thing that the mathematician has come to know through application of effort. The definition is a portion of a cage built to hold the concept in place so the next mathematician can come along and expend effort to know it.
mumblemumble 7y ago
But then there's https://shitpost.plover.com/m/monad.html https://shitpost.plover.com/m/monad.html > The more I think about this, the more it seems to me that a monad is not at all a monoid in the category of endofunctors, but actually a monoidal subcategory. > That's the problem.
- juped 7y ago
  Sometimes the definition is wrong (which should be a big hint that the definition is just pointing at something). Like the definition of prime that I was taught in primary school, "a prime is a positive integer whose only divisors are itself and 1". The right definition is "a prime is a positive integer with exactly two divisors", because otherwise all interesting statements about primes would have to be confusingly rewritten as statements about primes greater than 1. (Sometimes we have theorems about "odd primes", i.e., not 2, but the important set, primes, definitely includes 2.) I don't get the monad joke fwiw; I've done a ton of Haskell and a ton of math, but the Haskell wasn't deep abstraction and the math was differential geometry and the vicinity.
  - gjm11 7y ago
    To make matters worse, it turns out that there are actually two different concepts commonly labelled by the word "prime", and in some highbrow contexts they are different -- so mathematicians actually define "prime" in an entirely different way: p is prime iff p|ab => (p|a or p|b). That is, a prime is something that can't divide the product of two numbers without dividing at least one of the numbers. The thing about having no nontrivial divisors is called being "irreducible". (When are they different? When you're working with some kind of number other than the ordinary integers. For instance, suppose you decide to treat sqrt(-5) as an integer, so your numbers are now everything of the form a + b sqrt(-5) where a,b are ordinary integers. In this system of numbers, 3 is irreducible, but it isn't prime because (1+sqrt(-5)) (1-sqrt(-5)) = 6 which is a multiple of 3 -- but neither of those factors is a multiple of 3 even in this extended system of numbers.)
  - eximius 7y ago
    > primes greater than 1 I mean, many theorems about primes do not apply to 1. 1 is very often not considered a prime. And, as another commenter noted, there is not just one definition of prime.
djedr 7y ago
> The definition is a portion of a cage built to hold the concept in place so the next mathematician can come along and expend effort to know it. Well put. I'd say this extends beyond mathematics. All definitions are like that.
kian 7y ago
words form a net hunt writhing concepts some escape
sriku 7y ago
If I understand what you're saying, you're asking to not confuse mathematical communication with actually doing mathematics ... sort of how Ramanujan was doing amazing math but in a way that wasn't communicable to the convention until the language was introduced by Hardy?
deleted 7y ago
[deleted]
manthideaal 7y ago
The definition of a monoid here is not the usual definition, is a new definition for the special case of a strict category as defined MacLane's Book Categories for the Working Mathematicean. Since you maybe thinking about composition of endofunctors and unit endofunctors you get a confused picture. The monad as a monoid in the category of endofunctors is a way to show that you can confuse people using two different definition of an usual concept and both give different results. I got this from (1), look for "main confusion": monoid in the category of endofunctors is defined in a new way, and is not the expected monoid in the set of all endofunctors. The definition of monoid in monomial categories is on page 166 and Monads in a category are on page 133. As a math person I know what is a monoid (usual term) but I did not know what is a monoid in a monomial category (well I know now because is on page 166 of the book). (1) https://stackoverflow.com/questions/3870088/a-monad-is-just-a-monoid-in-the-category-of-endofunctors-whats-the-problem https://stackoverflow.com/questions/3870088/a-monad-is-just-...
- manthideaal 7y ago
  I wonder what's the point of using such a phrase, it doesn't help you to grok the concept of monad. It can help you to know that someone has given a new definition to sound cute, shame on them. By the way I admire MacLane as a math person, but people seem to use category theory to sell snake oil. Category theory is a tool to give names to some diagrams and properties that are used frequently to avoid repeating the same argument in proofs, is a dry (don't repeat yourself, as in the ruby motto). If you are an expert in category theory you can give short proofs of known facts. Category theory is like pipes in unix, you pipe diagrams to show properties. Grep, sed, awk analoges are functors, categories and natural transformations. The input is a collection of diagrams and the output is a new diagram that has a universal property and it receives a name and a collection of properties that are supposed to be useful to proof new theorems.
  - dddbbb 7y ago
    The phrase appears 6 chapters into a graduate mathematic text on category theory. If one reads the preceding chapters, it is a useful but pithy explanation for what a monad is, using terms which have all already been covered. Its use outside of that context is basically just a joke towards the Haskell community being overly mathematical.

by juped 7y ago

QuesnayJr 7y ago

This is an incredible quote: > But like everything in math what it really is is a thing that the mathematician has come to know through application of effort. The definition is a portion of a cage built to hold the concept in place so the next mathematician can come along and expend effort to know it.

mumblemumble 7y ago

But then there's https://shitpost.plover.com/m/monad.html https://shitpost.plover.com/m/monad.html > The more I think about this, the more it seems to me that a monad is not at all a monoid in the category of endofunctors, but actually a monoidal subcategory. > That's the problem.

juped 7y ago
Sometimes the definition is wrong (which should be a big hint that the definition is just pointing at something). Like the definition of prime that I was taught in primary school, "a prime is a positive integer whose only divisors are itself and 1". The right definition is "a prime is a positive integer with exactly two divisors", because otherwise all interesting statements about primes would have to be confusingly rewritten as statements about primes greater than 1. (Sometimes we have theorems about "odd primes", i.e., not 2, but the important set, primes, definitely includes 2.) I don't get the monad joke fwiw; I've done a ton of Haskell and a ton of math, but the Haskell wasn't deep abstraction and the math was differential geometry and the vicinity.
- gjm11 7y ago
  To make matters worse, it turns out that there are actually two different concepts commonly labelled by the word "prime", and in some highbrow contexts they are different -- so mathematicians actually define "prime" in an entirely different way: p is prime iff p|ab => (p|a or p|b). That is, a prime is something that can't divide the product of two numbers without dividing at least one of the numbers. The thing about having no nontrivial divisors is called being "irreducible". (When are they different? When you're working with some kind of number other than the ordinary integers. For instance, suppose you decide to treat sqrt(-5) as an integer, so your numbers are now everything of the form a + b sqrt(-5) where a,b are ordinary integers. In this system of numbers, 3 is irreducible, but it isn't prime because (1+sqrt(-5)) (1-sqrt(-5)) = 6 which is a multiple of 3 -- but neither of those factors is a multiple of 3 even in this extended system of numbers.)
- eximius 7y ago
  > primes greater than 1 I mean, many theorems about primes do not apply to 1. 1 is very often not considered a prime. And, as another commenter noted, there is not just one definition of prime.

djedr 7y ago

> The definition is a portion of a cage built to hold the concept in place so the next mathematician can come along and expend effort to know it. Well put. I'd say this extends beyond mathematics. All definitions are like that.

kian 7y ago

words form a net hunt writhing concepts some escape

sriku 7y ago

If I understand what you're saying, you're asking to not confuse mathematical communication with actually doing mathematics ... sort of how Ramanujan was doing amazing math but in a way that wasn't communicable to the convention until the language was introduced by Hardy?

deleted 7y ago

[deleted]

manthideaal 7y ago

The definition of a monoid here is not the usual definition, is a new definition for the special case of a strict category as defined MacLane's Book Categories for the Working Mathematicean. Since you maybe thinking about composition of endofunctors and unit endofunctors you get a confused picture. The monad as a monoid in the category of endofunctors is a way to show that you can confuse people using two different definition of an usual concept and both give different results. I got this from (1), look for "main confusion": monoid in the category of endofunctors is defined in a new way, and is not the expected monoid in the set of all endofunctors. The definition of monoid in monomial categories is on page 166 and Monads in a category are on page 133. As a math person I know what is a monoid (usual term) but I did not know what is a monoid in a monomial category (well I know now because is on page 166 of the book). (1) https://stackoverflow.com/questions/3870088/a-monad-is-just-a-monoid-in-the-category-of-endofunctors-whats-the-problem https://stackoverflow.com/questions/3870088/a-monad-is-just-...

manthideaal 7y ago
I wonder what's the point of using such a phrase, it doesn't help you to grok the concept of monad. It can help you to know that someone has given a new definition to sound cute, shame on them. By the way I admire MacLane as a math person, but people seem to use category theory to sell snake oil. Category theory is a tool to give names to some diagrams and properties that are used frequently to avoid repeating the same argument in proofs, is a dry (don't repeat yourself, as in the ruby motto). If you are an expert in category theory you can give short proofs of known facts. Category theory is like pipes in unix, you pipe diagrams to show properties. Grep, sed, awk analoges are functors, categories and natural transformations. The input is a collection of diagrams and the output is a new diagram that has a universal property and it receives a name and a collection of properties that are supposed to be useful to proof new theorems.
- dddbbb 7y ago
  The phrase appears 6 chapters into a graduate mathematic text on category theory. If one reads the preceding chapters, it is a useful but pithy explanation for what a monad is, using terms which have all already been covered. Its use outside of that context is basically just a joke towards the Haskell community being overly mathematical.