snippetModerate
What is meant by Category theory doesn't yet know how to deal with higher-order functions?
Viewed 0 times
meantorderknowwhattheorywithyetdoesnhigherhow
Problem
In reading Uday Reddy's answer to What is the relation between functors in SML and Category theory? Uday states
Category theory doesn't yet know how to deal with higher-order
functions. Some day, it will.
As I thought Category theory was able to serve as a foundation for math, then it should be possible to derive all of math and higher-order functions.
So, what is meant by Category theory doesn't yet know how to deal with higher-order functions? Is it valid to consider Category theory as a foundation for math?
Category theory doesn't yet know how to deal with higher-order
functions. Some day, it will.
As I thought Category theory was able to serve as a foundation for math, then it should be possible to derive all of math and higher-order functions.
So, what is meant by Category theory doesn't yet know how to deal with higher-order functions? Is it valid to consider Category theory as a foundation for math?
Solution
The issue with higher-order functions is simple enough to state.
-
A type-constructor like $T(X) = [X \to X]$ is not a functor. It should have been.
-
A polymorphic function like ${\it twice}_X : T(X) \to T(X) = \lambda f.\, f \circ f$ is not a natural transformation. It should have been.
If you read Eilenberg and MacLane's original category theory paper, (PDF) the intuitions they present cover those cases. But their theory doesn't. Theirs was a great paper for 1945! But, today, we need more.
The reaction of category theorists to these issues is a bit perplexing. They act as if higher-order operations form a Computer Science idea; they are of no consequence to mathematics. If that is so, then a foundation of mathematics would not be good enough for a foundation of computer science.
But I don't seriously believe that. I believe that higher-order functions would be quite important for mathematics as well. But they have not been seriously explored. I am hopeful that, some day, they will be explored and the limitations of category theory will be realized.
-
A type-constructor like $T(X) = [X \to X]$ is not a functor. It should have been.
-
A polymorphic function like ${\it twice}_X : T(X) \to T(X) = \lambda f.\, f \circ f$ is not a natural transformation. It should have been.
If you read Eilenberg and MacLane's original category theory paper, (PDF) the intuitions they present cover those cases. But their theory doesn't. Theirs was a great paper for 1945! But, today, we need more.
The reaction of category theorists to these issues is a bit perplexing. They act as if higher-order operations form a Computer Science idea; they are of no consequence to mathematics. If that is so, then a foundation of mathematics would not be good enough for a foundation of computer science.
But I don't seriously believe that. I believe that higher-order functions would be quite important for mathematics as well. But they have not been seriously explored. I am hopeful that, some day, they will be explored and the limitations of category theory will be realized.
Context
StackExchange Computer Science Q#9818, answer score: 15
Revisions (0)
No revisions yet.