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> but I needed a good reason that would justify the time investment It should be understood though that there are cases when math(-like) language is abused, le

> but I needed a good reason that would justify the time investment It should be understood though that there are cases when math(-like) language is abused, leading to overcomplication and obscurantism [1]. In mathematics, there is always the temptation of formalizing for formalization's sake. Indeed, 99% of pure math is non-constructive ("there exists a group such that", "the algorithm converges in O(N) steps"), as opposed to the practical CS and applied math ("here are the runtimes on real world data") which are likely the primary concerns of the HN crowd. None of this can diminish the sheer impractical appeal of pure math and pure CS, not unlike that of poetry, but I would rather not oversell either of the two. [1] A good illustration is a rant by Cosma Shalizi at http://bactra.org/notebooks/nn-attention-and-transformers.html http://bactra.org/notebooks/nn-attention-and-transformers.ht..., recently posted on HN.

bsdpufferfish 3y ago
> leading to overcomplication and obscurantism Mathematicians attempt to express ideas in the most readable and clear way possible. It's actually code that must be obfuscated by the constraints of the language and computer. Mathematicians have no constraints preventing them from presenting something in the way that makes the most sense. The part that can be called "Obscurantism" is when they use a high-level abstraction you are unfamiliar with. This is mostly driven by the audience. > Indeed, 99% of pure math is non-constructive Citation needed? > the algorithm converges in O(N) steps That doesn't sound non-constructive. Even the group example is usually done by constructing such a group. Also the best part about math is you can use it to approach the problems you want with constraints you want. Knuth uses math to solve real CS problems.
- davorak 3y ago
  > Mathematicians attempt to express ideas in the most readable and clear way possible. I buy this from the mathematicians and scientists that I know and have interacted with. I also think mathematicians spend more time trying to discover/play with new math than optimizing the communication of what already exists and is communicable. My speculation is that this naturally leads cruft that needs to be worked through by people entering the field. The cruft can not get too big or people don't enter the field so people are motivated to keep the cruft below a certain level but not the minimum. The cruft makes it harder to enter the field and once you have over come that hurdle you move on to do things in the field not reduce the cruft. Other things that make it hard to reduce cruft 1. not everyone is going to agree what is cruft 2. Person X spend time on reducing cruft in sub field Y may find out that Y is no longer hot topic so while there is less cruft there are not many people taking advantage of the reduced cruft in Y. 3. Mathematicians and scientists are reward more for new and interesting things than better pedagogical practice/techniques. 4. Optimizing for communication/pedagogy is mostly a different skill than science/mathematics so you have to split your focus or not dive as deeply into one or both. 5. I am sure there are others. This seems reasonable to me. It is s system where most everyone is well meaning and want to improve things and where things do improve over time, but where it is still easy to find areas that would benefit from substantial from improvement.
- woolion 3y ago
  No, "there exists a group" is not constructive, in general. You just need to prove that non-existence leads to a contradiction. The whole deal about "constructive mathematics" is to have that if you can prove something exists, you can also construct it. I think the success of the constructive mathematics program is really debatable, but in any case I don't think it leads to more 'natural' mathematics. (The terms used by GP are very confused and I agree with most of your reply)
  - bsdpufferfish 3y ago
    > You just need to prove that non-existence leads to a contradiction. Indeed however this is the exception, not the rule. The general way to do an existence proof is to construct it.
Tainnor 3y ago
To this day, I fail to understand why some people cannot enjoy constructive mathematics or practical engineering without shitting on traditional maths or CS. The "runtime on real data" thing is a trope by now, an algorithm that is exponential is in general not going to miraculously be very fast on "real-world" data, and even if it is, chances are, it won't be anymore once you change your data (with some few exceptions like the Simplex algorithm).
raincole 3y ago
Man, this article is an ad for Math Academy. I'm 100% sure that at the level of what Math Academy teaches, you don't need to worry about "non-constructive" or "pure math".
nyssos 3y ago
> Indeed, 99% of pure math is non-constructive ("there exists a group such that", "the algorithm converges in O(N) steps") Almost all mathematicians work with classical logic, but that doesn't mean that they always use all of its power. On the contrary, most of what you would see in an undergrad math program goes through constructively with at most a few minor modifications.

bsdpufferfish 3y ago

> leading to overcomplication and obscurantism Mathematicians attempt to express ideas in the most readable and clear way possible. It's actually code that must be obfuscated by the constraints of the language and computer. Mathematicians have no constraints preventing them from presenting something in the way that makes the most sense. The part that can be called "Obscurantism" is when they use a high-level abstraction you are unfamiliar with. This is mostly driven by the audience. > Indeed, 99% of pure math is non-constructive Citation needed? > the algorithm converges in O(N) steps That doesn't sound non-constructive. Even the group example is usually done by constructing such a group. Also the best part about math is you can use it to approach the problems you want with constraints you want. Knuth uses math to solve real CS problems.

davorak 3y ago
> Mathematicians attempt to express ideas in the most readable and clear way possible. I buy this from the mathematicians and scientists that I know and have interacted with. I also think mathematicians spend more time trying to discover/play with new math than optimizing the communication of what already exists and is communicable. My speculation is that this naturally leads cruft that needs to be worked through by people entering the field. The cruft can not get too big or people don't enter the field so people are motivated to keep the cruft below a certain level but not the minimum. The cruft makes it harder to enter the field and once you have over come that hurdle you move on to do things in the field not reduce the cruft. Other things that make it hard to reduce cruft 1. not everyone is going to agree what is cruft 2. Person X spend time on reducing cruft in sub field Y may find out that Y is no longer hot topic so while there is less cruft there are not many people taking advantage of the reduced cruft in Y. 3. Mathematicians and scientists are reward more for new and interesting things than better pedagogical practice/techniques. 4. Optimizing for communication/pedagogy is mostly a different skill than science/mathematics so you have to split your focus or not dive as deeply into one or both. 5. I am sure there are others. This seems reasonable to me. It is s system where most everyone is well meaning and want to improve things and where things do improve over time, but where it is still easy to find areas that would benefit from substantial from improvement.
woolion 3y ago
No, "there exists a group" is not constructive, in general. You just need to prove that non-existence leads to a contradiction. The whole deal about "constructive mathematics" is to have that if you can prove something exists, you can also construct it. I think the success of the constructive mathematics program is really debatable, but in any case I don't think it leads to more 'natural' mathematics. (The terms used by GP are very confused and I agree with most of your reply)
- bsdpufferfish 3y ago
  > You just need to prove that non-existence leads to a contradiction. Indeed however this is the exception, not the rule. The general way to do an existence proof is to construct it.

Tainnor 3y ago

To this day, I fail to understand why some people cannot enjoy constructive mathematics or practical engineering without shitting on traditional maths or CS. The "runtime on real data" thing is a trope by now, an algorithm that is exponential is in general not going to miraculously be very fast on "real-world" data, and even if it is, chances are, it won't be anymore once you change your data (with some few exceptions like the Simplex algorithm).

raincole 3y ago

Man, this article is an ad for Math Academy. I'm 100% sure that at the level of what Math Academy teaches, you don't need to worry about "non-constructive" or "pure math".

nyssos 3y ago

> Indeed, 99% of pure math is non-constructive ("there exists a group such that", "the algorithm converges in O(N) steps") Almost all mathematicians work with classical logic, but that doesn't mean that they always use all of its power. On the contrary, most of what you would see in an undergrad math program goes through constructively with at most a few minor modifications.