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"We will assume no background knowledge on behalf of the student, starting from scratch on both the programming and mathematics." This is a fantastic "side eff

"We will assume no background knowledge on behalf of the student, starting from scratch on both the programming and mathematics." This is a fantastic "side effect" of the fact that category theory isn't built on any other mathematical knowledge. You don't even need even any arithmetics for that.

qppo 6y ago
to get super meta, one could say that the opposite is true: arithmetic requires categories
- dan-robertson 6y ago
  Except this isn’t really true in the common sense meaning (schoolchildren do arithmetic fine without knowing about category theory) or in the formal sense you’re trying to get at (there are formal axiomatic foundations which arithmetic can be based on which do not need category theory. A simple proof is by causality: arithmetic was successfully formalised before category theory was invented)
- housecarpenter 6y ago
  You should read the famous book by Linderholm, Mathematics Made Difficult
spirographer 6y ago
At a meta level, Category Theory requires some comfort with abstraction, which really only comes with a mathematical education. So while it may stand apart from much math, it relies on your strong mathematical foundations.
- dkarl 6y ago
  As so many undergraduate math textbooks say, "No background is assumed beyond sufficient mathematical maturity."
  - mcguire 6y ago
    "Sufficient mathematical maturity": https://c8.alamy.com/comp/HY1PD9/elderly-math-teacher-HY1PD9.jpg https://c8.alamy.com/comp/HY1PD9/elderly-math-teacher-HY1PD9...
    - deleted 6y ago
      [deleted]
  - pizza 6y ago
    Which is kind of ironic, because students take classes exactly because they feel 'immature' with respect to that subject.. Honestly, most of my smoothest educational experiences with hard topics assumed some immaturity on my part, and that that was OK
    - dan-robertson 6y ago
      People don’t generally take category theory because they don’t really understand how proofs work or how to read definitions. The maturity required is about being able to cope with proving things and following proofs based on definitions which will probably seem somewhat bizarre at first and unmotivated at first. The immaturity you seem to talk about is people taking category theory because they don’t know category theory but that’s different and not what is meant by mathematical maturity. That said, a background in mathematics helps with category theory. Things like group theory, topology (particularly algebraic topology), Galois theory and set theory can be useful in motivating a lot of category theory. I’m yet to see much of a strong motivation from programming (where is there a functor that isn’t an endofunctor?)
      - exdsq 6y ago
        Isn’t an endofunctor a morphism from/to the same category? So a functor would be any morphine from/to different categories, and that wouldn’t be an endofunctor? Endofunctor: A -> A, for category A Functor: A -> B, for categories A and B
- gnufx 6y ago
  I think I got more appreciation of abstraction from physics and programming than what mathematical education I had; I'm weak at maths -- experimental physics doesn't need much :-/ -- but I do understand that weakness, and am quite comfortable with abstraction.
picklenerd 6y ago
This is almost every upper division undergrad math class. It was always fun watching people squirm when they pulled out some useful fact from their past 14 years of math education and then got told they had to prove it before they could use it.
- Someone 6y ago
  If only it were limited to facts learned from math education. For example, there’s the Jordan curve theorem (https://en.wikipedia.org/wiki/Jordan_curve_theorem https://en.wikipedia.org/wiki/Jordan_curve_theorem), which I guess most four-year olds ‘know to be true’ from their experience with coloring books.
  - Tainnor 6y ago
    Yeah, but from that same Wikipedia article: > It is easy to establish this result for polygons, but the problem came in generalizing it to all kinds of badly behaved curves, which include nowhere differentiable curves, such as the Koch snowflake and other fractal curves, or even a Jordan curve of positive area constructed by Osgood (1903). So to some extent, the reason why such an "obvious" statement requires a complicated proof is because our everyday notions of what a "closed curve" is are much more restricted than what we consider in mathematics. This is kind of common in maths, especially in fields with a lot of visual intuition.

qppo 6y ago

to get super meta, one could say that the opposite is true: arithmetic requires categories

dan-robertson 6y ago
Except this isn’t really true in the common sense meaning (schoolchildren do arithmetic fine without knowing about category theory) or in the formal sense you’re trying to get at (there are formal axiomatic foundations which arithmetic can be based on which do not need category theory. A simple proof is by causality: arithmetic was successfully formalised before category theory was invented)
housecarpenter 6y ago
You should read the famous book by Linderholm, Mathematics Made Difficult

spirographer 6y ago

At a meta level, Category Theory requires some comfort with abstraction, which really only comes with a mathematical education. So while it may stand apart from much math, it relies on your strong mathematical foundations.

dkarl 6y ago
As so many undergraduate math textbooks say, "No background is assumed beyond sufficient mathematical maturity."
- mcguire 6y ago
  "Sufficient mathematical maturity": https://c8.alamy.com/comp/HY1PD9/elderly-math-teacher-HY1PD9.jpg https://c8.alamy.com/comp/HY1PD9/elderly-math-teacher-HY1PD9...
  - deleted 6y ago
    [deleted]
- pizza 6y ago
  Which is kind of ironic, because students take classes exactly because they feel 'immature' with respect to that subject.. Honestly, most of my smoothest educational experiences with hard topics assumed some immaturity on my part, and that that was OK
  - dan-robertson 6y ago
    People don’t generally take category theory because they don’t really understand how proofs work or how to read definitions. The maturity required is about being able to cope with proving things and following proofs based on definitions which will probably seem somewhat bizarre at first and unmotivated at first. The immaturity you seem to talk about is people taking category theory because they don’t know category theory but that’s different and not what is meant by mathematical maturity. That said, a background in mathematics helps with category theory. Things like group theory, topology (particularly algebraic topology), Galois theory and set theory can be useful in motivating a lot of category theory. I’m yet to see much of a strong motivation from programming (where is there a functor that isn’t an endofunctor?)
    - exdsq 6y ago
      Isn’t an endofunctor a morphism from/to the same category? So a functor would be any morphine from/to different categories, and that wouldn’t be an endofunctor? Endofunctor: A -> A, for category A Functor: A -> B, for categories A and B
gnufx 6y ago
I think I got more appreciation of abstraction from physics and programming than what mathematical education I had; I'm weak at maths -- experimental physics doesn't need much :-/ -- but I do understand that weakness, and am quite comfortable with abstraction.

picklenerd 6y ago

This is almost every upper division undergrad math class. It was always fun watching people squirm when they pulled out some useful fact from their past 14 years of math education and then got told they had to prove it before they could use it.

Someone 6y ago
If only it were limited to facts learned from math education. For example, there’s the Jordan curve theorem (https://en.wikipedia.org/wiki/Jordan_curve_theorem https://en.wikipedia.org/wiki/Jordan_curve_theorem), which I guess most four-year olds ‘know to be true’ from their experience with coloring books.
- Tainnor 6y ago
  Yeah, but from that same Wikipedia article: > It is easy to establish this result for polygons, but the problem came in generalizing it to all kinds of badly behaved curves, which include nowhere differentiable curves, such as the Koch snowflake and other fractal curves, or even a Jordan curve of positive area constructed by Osgood (1903). So to some extent, the reason why such an "obvious" statement requires a complicated proof is because our everyday notions of what a "closed curve" is are much more restricted than what we consider in mathematics. This is kind of common in maths, especially in fields with a lot of visual intuition.