Comment by CogitoCogito | Hacker News Reader

Comment by CogitoCogito | Hacker News Reader

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> There's so much "pure" and "universal" about math, but the humans who write about it are too lazy to write about it in a rigorous manner.

Are you sure it's laziness? Maybe it's a result of there not actually being any universal notation (not even within subfields) or the exactness you refer to really isn't necessary. This doesn't mean that unclear exposition is a good thing. Mathematical writing (as with all writing) should strive towards clarity. But clarity doesn't require some sort of minutely perfectly consistently notation which would be required by a computer because humans are better than computers at handling exactly those kinds of situations.

> People can't be bothered to be exact about things because being exact is hard and people avoid hard work.

I think you have it wrong. People can't be bothered to be as exact because they don't need to. People can understand things even if they are inexact. So can mathematicians. Honestly this is a feature. If computers would just intuitively understand what I tell them to do like a human assistant would, that would be a step up not a step down in human computer interfaces.

> But clarity doesn't require some sort of minutely perfectly consistently notation which would be required by a computer

I made this point in another comment, but I think it bears repeating and elaboration: Consistency isn't required (at least outside any single paper), but explicitness would be a tremendous boon.

Software incorporates outside context all the time, but it pretty much always does it explicitly (though the explicitness may be transitive, ie. dependencies of dependencies). Math papers often assume context that is not explicitly noted in the citations, nor those papers' citations, etc.

Instead, some of the context might only be found in other papers that cite the same papers you are tracking down. You sometimes need to follow citations both backward and forward from every link in the chain. And unlike following citations backward (ie. the ones each author considered most relevant), the forward links aren't curated and many (perhaps most) will be blind alleys (there also may be cycles in the citation graph, but these are relatively rare). But somehow you have to collect knowledge (or at least passing familiarity) with an encyclopedic corpus in order to at least recognize and place the context left implicit in any one paper in order to understand it.

It's maddening.

I totally agree. I think that many mathematical papers aren't explained as well as they can be. My advisor was pretty adamant that papers should not be written in some proof-chasing style like you describe and that the author should clearly include the arguments they need (citing those authors they might have learned it from) unless those arguments are truly standard. No "using a method similar to [author] in Lemma 5 of [some paper]" and instead just including it in your paper and making sure if fits in well.

That is just an example of bad exposition in my opinion. It's also not technically "unclear" in any notational sense so it's a bit of an aside from this argument. But I agree with you 100% that it is bad bad bad. This is a perfect example of why arguments like "does this proof make coq happy" totally misses the point.

> That is just an example of bad exposition in my opinion [and] a perfect example of why arguments like "does this proof make coq happy" totally misses the point..

In theory, some kind of checker could validate the semantics of a paper to just tell you whether the arguments made are complete. Not whether there is a correct formal proof, just pointing out any obscured leaps of faith[0]. A rough analogue to test suite coverage for code (which is also not any sort of guarantee of correctness, just basic reassurance that all (or most of) the code is tested and isn't broken in any obvious way, especially while making changes).

I'm trying to think of an equivalent for prose, and am coming up with examples like detecting conflicting descriptions of locations or named characters, or whether the author lost track of which character said which lines in the dialog.

> It's also not technically "unclear" in any notational sense

Perhaps not necessarily, but unfamiliar/borrowed/idiosyncratic notation is a perfect (and common) place for insufficient exposition to be hiding.

[0] https://imgur.com/gallery/ApzhVFj

People can also understand each other through combinations of obscure slang, garbled audio, thick accents, and drunken slurring. It's still an unpleasant way to communicate.

Shall we be satisfied with the same low standards in a technical field, because it is how it is?

Hands-on users of math notation are complaining that it sucks. I'm not sure why a dismissive "works for me" is so often the default response.

> Hands-on users of math notation are complaining that it sucks. I'm not sure why a dismissive "works for me" is so often the default response.

It is really easy to complain. People also complain about every popular programming language, but it is really hard to make something that is actually better. It is easy to make something that you yourself think is better, but it is hard to make something that is better in practice.

Personally, I stopped complaining about Java, Perl, and PHP when I didn't have to read or write them anymore for work.

> Hands-on users of math notation are complaining that it sucks. I'm not sure why a dismissive "works for me" is so often the default response.

Are you sure this is because the notation is unclear/imprecise or because you just don't like it? I like certain programming languages and certain programming styles and really don't like others. But in none of the cases (those I like nor those I don't) are they not 100% "clear". The code compiles and executes after all so there really isn't much of an argument that somehow it's underspecified.

The same thing exists in mathematics. There are certain fields of math whose traditional notation/style/approach/etc. are totally incomprehensible to me. There are also many mathematicians who would say the same about my preferences as well.

So my point is that all people are _different_. Some people like certain things and some people like others. How can you hope to please everyone simultaneously? In my experience, there is no field at all that is as precise as mathematics. Sure "code" is precise, but (imo) professional programmers are nowhere near as precise in any general design or conversation than mathematicians. So I find the attack on supposedly bad mathematical notation a bit odd.

Mathematicians constantly try to come up with better methods of explaining things. They put more effort into it than basically any field in my experience. The problems are really that we as humans don't all think the same and that mathematics is just plain hard. We've improved mathematical communication immensely throughout history and we will continue to do so. But we'll never reach some sort of perfect communication style because no single such style could ever exist.

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