4 ms·

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 "arr

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?

tobbe2064 10y ago
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
- tobbe2064 10y ago
  Further math is sort of a stone age technology
  - jonsterling 10y ago
    wtf are you on about? That is not how math works.
    - deleted 10y ago
      [deleted]
    - tobbe2064 10y ago
      I apologise for my drunken and incorrect silliness. Aside from that, would you please enlighten as to what you meant by "that's not how math works"?
      - jonsterling 10y ago
        Sorry for my harsh comment; here's what I'm thinking of... In math, you are dealing with many different kinds of object, not just numbers. In fact, one of the big realizations that led to modern mathematics is that not all mathematical objects can even be coded as numbers! Your remark about "arrow" being more general than function is correct; but functions do not map only between sets of numbers, but between arbitrary sets, some of which contain elements that are not numbers, or even encodable as such.
  - GFK_of_xmaspast 10y ago
    Bronze age, sure, but stone age might be pushing it.
- startling 10y ago
  That's not even true. Functions can map between lots of things, not necessarily numbers.
startling 10y ago
Functions are one kind of arrow. There are other arrows besides functions. Renaming "arrow" to "function" is like replacing "number" with "1".
- gravypod 10y ago
  What other types of arrows are there?
  - systoll 10y ago
    State machines & object dependencies [and function composition, and... categories] are often drawn as directed graphs. When drawn like that, the circles/boxes represent objects, and the arrows represent arrows. The name comes from that kind of representation. You can call them morphisms, too.
    - gravypod 10y ago
      Then from what I'm understanding it's not an "arrow", it's something closer to a "step" or a "process". Some method to get from one location to another. An arrow means nothing without background information. If I say it takes "10 arrows to get from A to x(A)" as a way to explain there are 10 things happening in the function X, then wouldn't it be better to say it takes "10 steps/processes to get from A to x(A)"? > The name comes from that kind of representation. The representation, the way I conceive it, is a mode for displaying computational or data in a spacial manor. As such relating a computational or mathematical process to a spacial terminology would be a "step" not a stick with a rock at the end of it.
      - kian 10y ago
        Does step help when thinking of the objects as individual data elements, the arrows as cons-links, map being a functor that respects cons-links [i.e. does not change the structure of the list]? The problem here is that implying process, ordering, or anything else is too narrow of a definition for what an arrow/morphism can be used to represent. That's why arrow (a thing that points to another thing), or morphism (a thing that helps define the shape) are actually better terms. Because Category Theory isn't about the things, it's about the relationships between the things, how these can be composed, and how higher-order operators that respect these relationships [and the new ones they represent for even higher-order operators] can form new relationships which can also be composed.
RGamma 10y ago
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 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 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.
- gravypod 10y ago
  > "Mathspeak" is not supposed to be intuitive to outsiders 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

tobbe2064 10y ago

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

tobbe2064 10y ago
Further math is sort of a stone age technology
- jonsterling 10y ago
  wtf are you on about? That is not how math works.
  - deleted 10y ago
    [deleted]
  - tobbe2064 10y ago
    I apologise for my drunken and incorrect silliness. Aside from that, would you please enlighten as to what you meant by "that's not how math works"?
    - jonsterling 10y ago
      Sorry for my harsh comment; here's what I'm thinking of... In math, you are dealing with many different kinds of object, not just numbers. In fact, one of the big realizations that led to modern mathematics is that not all mathematical objects can even be coded as numbers! Your remark about "arrow" being more general than function is correct; but functions do not map only between sets of numbers, but between arbitrary sets, some of which contain elements that are not numbers, or even encodable as such.
- GFK_of_xmaspast 10y ago
  Bronze age, sure, but stone age might be pushing it.
startling 10y ago
That's not even true. Functions can map between lots of things, not necessarily numbers.

startling 10y ago

Functions are one kind of arrow. There are other arrows besides functions. Renaming "arrow" to "function" is like replacing "number" with "1".

gravypod 10y ago
What other types of arrows are there?
- systoll 10y ago
  State machines & object dependencies [and function composition, and... categories] are often drawn as directed graphs. When drawn like that, the circles/boxes represent objects, and the arrows represent arrows. The name comes from that kind of representation. You can call them morphisms, too.
  - gravypod 10y ago
    Then from what I'm understanding it's not an "arrow", it's something closer to a "step" or a "process". Some method to get from one location to another. An arrow means nothing without background information. If I say it takes "10 arrows to get from A to x(A)" as a way to explain there are 10 things happening in the function X, then wouldn't it be better to say it takes "10 steps/processes to get from A to x(A)"? > The name comes from that kind of representation. The representation, the way I conceive it, is a mode for displaying computational or data in a spacial manor. As such relating a computational or mathematical process to a spacial terminology would be a "step" not a stick with a rock at the end of it.
    - kian 10y ago
      Does step help when thinking of the objects as individual data elements, the arrows as cons-links, map being a functor that respects cons-links [i.e. does not change the structure of the list]? The problem here is that implying process, ordering, or anything else is too narrow of a definition for what an arrow/morphism can be used to represent. That's why arrow (a thing that points to another thing), or morphism (a thing that helps define the shape) are actually better terms. Because Category Theory isn't about the things, it's about the relationships between the things, how these can be composed, and how higher-order operators that respect these relationships [and the new ones they represent for even higher-order operators] can form new relationships which can also be composed.

RGamma 10y ago

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 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 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.

gravypod 10y ago
> "Mathspeak" is not supposed to be intuitive to outsiders 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