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This is fantastically timed and very well explained. Our company is working deeply with Category Theory right now as we believe programming in the future will find itself focused primarily on the task of orchestrating AI to code.
Earlier quoted context omitted.
The word function has a sort of precis mening in math as a rule mapping each NUMBER in some domain to some NUMBER in a codomaon. The word arrow however can refer to any directed relation including less than or divides or what not, as long as they are composable and associative
Further math is sort of a stone age technology
This is fantastically timed and very well explained. Our company is working deeply with Category Theory right now as we believe programming in the future will find itself focused primarily on the task of orchestrating AI to code.
What company is that?
Earlier quoted context omitted.
Functions are one kind of arrow. There are other arrows besides functions. Renaming "arrow" to "function" is like replacing "number" with "1".
What other types of arrows are there?
You can call them morphisms, too.
Why don't things like Category Theory and Set theory update their terminology to align with more modern words of describing these? Why is a "function" and "arrow"? A "function" implies an operation that is part of a larger system of organized operations. An "arrow" implies stone age technology. One is infinity more descriptive in my opinion. Is there a reason for this?
W.r.t the "modernity" of the terms: I think category theorists are glad if they find any word that fits the concept in their minds. "Mathspeak" is not supposed to be intuitive to outsiders (and everyone is such an outsider at one point), but once you get more into it the names do start to make sense.
Generally mathematics is a field that breaks everyday-intuition constantly. Having a technolect, a "foreign language in a language" if you will, helps coping with that.
Earlier quoted context omitted.
Functions are one kind of arrow. There are other arrows besides functions. Renaming "arrow" to "function" is like replacing "number" with "1".
What other types of arrows are there?
(0) Any monoid can be viewed as a category with a single object. The arrows from the object to itself are the monoid's elements.
(1) Any preorder can be viewed as a category such that, between any two objects, there is at most one arrow, precisely when the source is less or equal than the target.
(2) Given a directed graph, or more generally a quiver[0], there is a small category[1] whose objects are the quiver's nodes, and whose arrows are the paths (finite sequences of edges) from a source node to a target node.
Earlier quoted context omitted.
The word function has a sort of precis mening in math as a rule mapping each NUMBER in some domain to some NUMBER in a codomaon. The word arrow however can refer to any directed relation including less than or divides or what not, as long as they are composable and associative
Further math is sort of a stone age technology
Why don't things like Category Theory and Set theory update their terminology to align with more modern words of describing these? Why is a "function" and "arrow"? A "function" implies an operation that is part of a larger system of organized operations. An "arrow" implies stone age technology. One is infinity more descriptive in my opinion. Is there a reason for this?
Category theory uses weird terminology all over the place partly because it needs to distance itself from concepts that are "sort of like in set theory but not quite" and there are a lot of concepts that need naming thusly. Browse https://ncatlab.org/nlab/show/category+theory and you'll see a ton of examples of this. W.r.t the "modernity" of the terms: I think category theorists are glad if they find any word that fi…
Isn't that antithetical to the entire concept of an organized science? As I understand we're meant to make commutative models of the world around us that have predicative properties. If it's not possible to easily communicate your model, then there isn't much of a point in using the terminology.
Think back to Newton's notations of calculus. They are improper and poor ways to demonstrate the information that is being spoken about [0]. I don't know anyone who doesn't actually use Leibniz's notations.
I don't think anyone can convince me that "Mathspeak" should be unintuitive.
[0] - ttps://en.wikipedia.org/wiki/Notation_for_differentiation#Newton.27s_notation