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I think you don’t realise how much math you know. I’ve heard of set theory, but never studied it.

by sjducb 3y ago

mjburgess 3y ago
sure, but you dont need to understand actual set theory -- it's just notation for mostly obvious programming stuff, R means float, Z means int R^2 is actually notation from linear algebra, but here it means a point is 2 floats with a measure of distance between points etc. A lot of this could just be given in a "crib sheet" for tech people, and you'd get 80% of it straightaway. It's years of work to understand this notation as used by the professional domains it was invented by, but it's a couple weeks for most "good, technical, software engineers" -- since they arent really using notation in much more than superficial ways. Of course there are hard mathematical papers, etc. but they're rare in ML/AI mainstream papers; and not something most would read. If you need to be able to read, eg., some sort of adv. statistical time series research in economic modelling, you'd already have the background to do that. If all you want is to be able to read 90% of the popular papers, the notation in them is just syntax sugar for things you could state easily in english or python
- 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 - ...
  - 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
- dimal 3y ago
  What they’re saying is that you may be falling for the curse of knowledge. From Wikipedia: > The curse of knowledge is a cognitive bias that occurs when an individual, who is communicating with others, assumes that others have information that is only available to themselves, assuming they all share a background and understanding. This bias is also called by some authors the curse of expertise. You know which parts of the AI math notation is shallow and which not. Someone else might not have that same knowledge, so they might not know exactly what to put on the crib sheet and what they need to go deeper on. By the way, if you wanted to make such a crib sheet and publish it online, I think a lot of people would be very grateful!

I think you don’t realise how much math you know. I’ve heard of set theory, but never studied it.

by sjducb 3y ago

I think you don’t realise how much math you know. I’ve heard of set theory, but never studied it.

sure, but you dont need to understand actual set theory -- it's just notation for mostly obvious programming stuff, R means float, Z means int R^2 is actually notation from linear algebra, but here it means a point is 2 floats with a measure of distance between points etc. A lot of this could just be given in a "crib sheet" for tech people, and you'd get 80% of it straightaway. It's years of work to understand this notation as used by the professional domains it was invented by, but it's a couple weeks for most "good, technical, software engineers" -- since they arent really using notation in much more than superficial ways. Of course there are hard mathematical papers, etc. but they're rare in ML/AI mainstream papers; and not something most would read. If you need to be able to read, eg., some sort of adv. statistical time series research in economic modelling, you'd already have the background to do that. If all you want is to be able to read 90% of the popular papers, the notation in them is just syntax sugar for things you could state easily in english or python

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 - ...
- 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
dimal 3y ago
What they’re saying is that you may be falling for the curse of knowledge. From Wikipedia: > The curse of knowledge is a cognitive bias that occurs when an individual, who is communicating with others, assumes that others have information that is only available to themselves, assuming they all share a background and understanding. This bias is also called by some authors the curse of expertise. You know which parts of the AI math notation is shallow and which not. Someone else might not have that same knowledge, so they might not know exactly what to put on the crib sheet and what they need to go deeper on. By the way, if you wanted to make such a crib sheet and publish it online, I think a lot of people would be very grateful!