3 ms·

> sure, but you dont need to understand actual set theory How would you know? You sound like you understand set theory. Ironically, a body needs a rather exten

by roenxi 3y ago

> sure, but you dont need to understand actual set theory How would you know? You sound like you understand set theory. Ironically, a body needs a rather extensive grounding in the theory and practice of set theory to look at something and think "I don't need to use the tools of set theory here" with deserved confidence. A lot of maths is about having a huge repertoire of tricks that you know won't help solve the problem at hand. Saves weeks of fruitless attempts. But I'll bet you didn't notice the things you know not to do. That does matter for interpreting these papers.

aleph_minus_one 3y ago
> > sure, but you dont need to understand actual set theory > How would you know? You sound like you understand set theory. Easy: most mathematicians don't really care about set theory (beyond the fact that it exists and is the standard foundation). Understanding set theory goes deep. For example: why does the Banach-Tarski paradox exist in ZFC (C: Axiom of Choice (https://en.wikipedia.org/wiki/Axiom_of_choice https://en.wikipedia.org/wiki/Axiom_of_choice)), but not in ZFD (D: Axiom of Determinacy (https://en.wikipedia.org/wiki/Axiom_of_determinacy https://en.wikipedia.org/wiki/Axiom_of_determinacy))? Or another one: why does the statement "every field has an algebraic closure" not hold in ZF (but in ZFC)? Never heard of this stuff despite having studied mathematics? Just like I said: most mathematicians don't really care about set theory.
- Tainnor 3y ago
  Not every mathematician needs to understand axiomatic set theory, but probably every working mathematician needs to understand at least basic set operations, simple identites like De Morgan's laws (I don't care if you know the name or not), as well as what cartesian products, relations and functions are.
  - aleph_minus_one 3y ago
    > every working mathematician needs to understand at least basic set operations, simple identites like De Morgan's laws (I don't care if you know the name or not), as well as what cartesian products, relations and functions are. This is stuff that you learn in the 5th or 6th grade in school, and is about as far removed from what mathematicians call "set theory" as basic arithmetic operations are from college math courses for soon-to-be mathematicians.
    - gmadsen 3y ago
      that is a little bit of hyperbole. No school in 5th or 6th grade is giving formal definitions of relations and functions. There is a reason nearly all undergrad math books in analysis, topology, algebra, all devote an entire first chapter to it.
      - mjburgess 3y ago
        It would be dangerous to teach anyone in ML what a function was, since they might realise, the premise of the entire activity is false. (ie., that in the vast majority of cases there are no empirical functions to model, no f: Pixel -> Animal) So better continue the current practice where few could distinguish a relation and a function; and fewer still are aware that they arent approximating empirical functions.
      - aleph_minus_one 3y ago
        > No school in 5th or 6th grade is giving formal definitions of relations and functions. I remember that my math teacher pretty surely did. > There is a reason nearly all undergrad math books in analysis, topology, algebra, all devote an entire first chapter to it. Indeed there exist multiple good reasons: - recapitulation - setting up the notation - clarifying how the textbook defines the relevant mathematical objects, because the definitions of some concepts might differ depending on the textbook - making clear what existing standard the textbook expects from the learner - ...
        gmadsen 3y ago
        how do you motivate formal functions and relations to 5th graders, and more importantly why? Relations are important because of the natural partitions of a set they create and the development of group theory. Functions are useful in calculus, but not really the algebraic properties, those are glossed over, i.e kids learning calculus are usually not learning the formalization of functions. That isn't important until analysis or abstract algebra, hence why its included in the textbooks
        3y ago
JohnKemeny 3y ago
He's saying you don't have to understand set theory to use and understand set notation. In the same way that you don't have to understand number theory and abstract algebra to use and understand numbers and their basic operations (such as +, -, ·, /).
- Jtsummers 3y ago
  It would help if instead of saying "set theory" we emphasize the algebra of sets. Not by that name, but this level of teaching sets was common in middle school math late last century, not sure about this century, in the US. And this would be enough for most people trying to read technical documents using set notation. https://en.wikipedia.org/wiki/Algebra_of_sets https://en.wikipedia.org/wiki/Algebra_of_sets

by roenxi 3y ago

aleph_minus_one 3y ago

> > sure, but you dont need to understand actual set theory > How would you know? You sound like you understand set theory. Easy: most mathematicians don't really care about set theory (beyond the fact that it exists and is the standard foundation). Understanding set theory goes deep. For example: why does the Banach-Tarski paradox exist in ZFC (C: Axiom of Choice (https://en.wikipedia.org/wiki/Axiom_of_choice https://en.wikipedia.org/wiki/Axiom_of_choice)), but not in ZFD (D: Axiom of Determinacy (https://en.wikipedia.org/wiki/Axiom_of_determinacy https://en.wikipedia.org/wiki/Axiom_of_determinacy))? Or another one: why does the statement "every field has an algebraic closure" not hold in ZF (but in ZFC)? Never heard of this stuff despite having studied mathematics? Just like I said: most mathematicians don't really care about set theory.

Tainnor 3y ago
Not every mathematician needs to understand axiomatic set theory, but probably every working mathematician needs to understand at least basic set operations, simple identites like De Morgan's laws (I don't care if you know the name or not), as well as what cartesian products, relations and functions are.
- aleph_minus_one 3y ago
  > every working mathematician needs to understand at least basic set operations, simple identites like De Morgan's laws (I don't care if you know the name or not), as well as what cartesian products, relations and functions are. This is stuff that you learn in the 5th or 6th grade in school, and is about as far removed from what mathematicians call "set theory" as basic arithmetic operations are from college math courses for soon-to-be mathematicians.
  - gmadsen 3y ago
    that is a little bit of hyperbole. No school in 5th or 6th grade is giving formal definitions of relations and functions. There is a reason nearly all undergrad math books in analysis, topology, algebra, all devote an entire first chapter to it.
    - mjburgess 3y ago
      It would be dangerous to teach anyone in ML what a function was, since they might realise, the premise of the entire activity is false. (ie., that in the vast majority of cases there are no empirical functions to model, no f: Pixel -> Animal) So better continue the current practice where few could distinguish a relation and a function; and fewer still are aware that they arent approximating empirical functions.
    - aleph_minus_one 3y ago
      > No school in 5th or 6th grade is giving formal definitions of relations and functions. I remember that my math teacher pretty surely did. > There is a reason nearly all undergrad math books in analysis, topology, algebra, all devote an entire first chapter to it. Indeed there exist multiple good reasons: - recapitulation - setting up the notation - clarifying how the textbook defines the relevant mathematical objects, because the definitions of some concepts might differ depending on the textbook - making clear what existing standard the textbook expects from the learner - ...
      - gmadsen 3y ago
        how do you motivate formal functions and relations to 5th graders, and more importantly why? Relations are important because of the natural partitions of a set they create and the development of group theory. Functions are useful in calculus, but not really the algebraic properties, those are glossed over, i.e kids learning calculus are usually not learning the formalization of functions. That isn't important until analysis or abstract algebra, hence why its included in the textbooks
        3y ago

JohnKemeny 3y ago

He's saying you don't have to understand set theory to use and understand set notation. In the same way that you don't have to understand number theory and abstract algebra to use and understand numbers and their basic operations (such as +, -, ·, /).

Jtsummers 3y ago
It would help if instead of saying "set theory" we emphasize the algebra of sets. Not by that name, but this level of teaching sets was common in middle school math late last century, not sure about this century, in the US. And this would be enough for most people trying to read technical documents using set notation. https://en.wikipedia.org/wiki/Algebra_of_sets https://en.wikipedia.org/wiki/Algebra_of_sets