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Ask HN: How did you learn math notation?

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Re: Ask HN: How did you learn math notation?

#171

Post by CogitoCogito »

webmaven wrote: post #160
CogitoCogito wrote: post #67

Earlier quoted context omitted.

> 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 writin…

> 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 (thoug…

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.

Re: Ask HN: How did you learn math notation?

#172

Post by [deleted] »

ColinWright wrote: post #154
klibertp wrote: post #134

Earlier quoted context omitted.

Yes. What's wrong with changing math notation? Why wouldn't you do it if you know that it would make it easier for others to approach? What's the rationale behind doing exactly nothing to make the notation more approachable for the masses?

Math notation has evolved to be what it is because it is useful for the actual doing of math, and the communication of math to those who have sufficient background. It's not deliberately designed to keep people out, and there are literally hundreds of thousands of books that introduce people to the notations used, to help on-board them. Haskell is unreadable to one who has not trained in it or similar languages ... w…

[deleted]

Re: Ask HN: How did you learn math notation?

#173

Post by erichocean »

ColinWright wrote: post #27
NamTaf wrote: post #24

Earlier quoted context omitted.

I feel you're coming at this without appreciating your body of prior knowledge. Intended or not, your statment "But that can't be it, because just two lines above it we have..." assumes a whole lot of knowledge. You and I both know that it reads as one term, but for someone unfamiliar with calculus but exposed to algebra they are drilled to understand separate graphemes as separate items, because the algebraic 'multi…

I agree completely that I'm coming at this with a lot of background knowledge, but if I'm reading in an unfamiliar field and I see a symbol I don't recognise, I look in the surrounding text to see if the symbol appears nearby. As I say, "Δt" appears immediately above ... that's a clue. As you say, it's drilled in at school that everything is represented by a single glyph, and if these are juxtaposed then it means mul…

> a symbol

That's literally the whole problem: he sees two symbols.

Re: Ask HN: How did you learn math notation?

#174

Post by CogitoCogito »

My advisor's advise was basically "find a notation that you yourself like and understand well" and stick consistently to it. He said this in a context of having seen many standard notations before (so he's not saying to re-invent the wheel), but his point was just that notations and ways of thinking are personal. Try to be clear and precise (for yourself and others), but realize that you are crafting something that reflects you and your way of thinking.

It's kind of a cop-out, but to be fair it's basically what I would say for programming as well. Try to simultaneously write code that clear to yourself and clear to others. There's no perfect method. Just constantly self-critique and try to improve.

Re: Ask HN: How did you learn math notation?

#175

Post by marcosdumay »

f38zf5vdt wrote: post #59
cabalamat wrote: post #50

Earlier quoted context omitted.

> I think a real problem in this area is the belief that there is "one true notation" and that everything is unambiguous and clearly defined. One main cause for this belief is that in a programming there is one true noation (or rather, a separate one for each language) that is unambiguous and clearly defined. I dislike maths notation as I find it lacks rigour.

I'm glad I'm not the only person like this. I've never liked tradition math notation and found it about as useful as traditional musical notation, that is, hard to read for the layman and for no other reason than "this is how people have been doing it for a long time". Maybe I'm the minority, but when I read a CS paper I mostly ignore the maths and then go to the source code or pseudocode to see how the algorithm was…

Hum... Apart from math, music notation maps linearly from the symbols to the instrument positions and time.

It's absolutely not used only because "this is how people have been doing it for a long time". It's a very efficient notation to decode.

Re: Ask HN: How did you learn math notation?

#176

Post by erichocean »

Math papers can be pretty sloppy, and you don't realize this until you start working with formal mathematics—then it's obvious.

Almost all hand "proofs" in math papers have minor bugs, even if they're mostly correct in the big picture sense.

Even math designed to support programming (e.g. in computer graphics) is almost always incomplete/outright wrong in some meaningful way.*

But with a struggle, it's still largely usable/useful.

I've used advanced mathematics most of my career to do work (i.e. read a paper, implement it), but the ability to actually use math to do new things in computer science that mattered only to me only happened after I learned TLA+, which took a few weeks of solid study to click. Since then, it's been a pleasure. My specs have never been this good!

Lamport's video course on TLA+ is pretty good, but honestly I've read everything I can find on the topic so it's difficult to know what helped me the most.

*I think this is because, short of doing formal mathematics, there's no way to "test" your math. It's the equivalent of expecting programmers to write correct code the first time with no tests, and without even running the code.

Re: Ask HN: How did you learn math notation?

#177

Post by ColinWright »

Grustaf wrote: post #164
ColinWright wrote: post #156

Earlier quoted context omitted.

I don't know your background, but I wonder how broad it is in terms of mathematical topics. The notations used in Algebraic Topology vs Category Theory vs Algebraic Number Theory vs Analytical Combinatorics vs Complex Analysis. This isn't a criticism, it's just that notations vary wildly in those areas, and there's lots of cross-over of notations, not all of which agree with each other. I'm not an expert, but I've ha…

I studied diff geo at phd level and met stat at undergrad level, plus a sprinkling of category theory, some discrete mathematics and some physics, so I’ve been exposed to most of these. I presumed we were talking about basic mathematics here since new notation is the least of your worries when your thinking about fibre bundles and cohomologies, but I can’t really think of any significant overlap in notation that woul…

I'm trying to be more general than specific questions at the mid-undergrad level, because looking in from the outside, people seem to thing that if only the notation weren't so mysterious then they could understand everything. But this comment -- https://news.ycombinator.com/item?id=29344238 -- gives a flavour, talking about coming across "π" in different contexts and having to give different interpretations.

But I remember sketching an algorithm to someone and just inventing notations on the fly as I did so, knowing that they would simply be ways to remember the underlying ideas.

Even so, at 1st year undergrad the notations used in Mathematical Physics vary from those used in Introductory Graph Theory, and again from Real Analysis. But once the reader knows the underlying semantics, the actual notation is mostly a non-issue (as you know).

Re: Ask HN: How did you learn math notation?

#178

Post by klibertp »

Jensson wrote: post #168
klibertp wrote: post #163

Earlier quoted context omitted.

> it isn't worth investing in a team to go through and update all of those papers to properly add definitions for everything they missed. Thank you. There's nothing else left to discuss.

Thanks for understanding, math is a small field without money for things like this, there is no way anyone should expect those niche papers to be as well documented as big programming projects used by millions. If you still think that is a problem then start some open source organization to fix that. Nobody has done that yet though since so few people care about math papers, but since you feel so strongly about this…

No, I mean, well, it's very understandable when you describe it that way. Actually, I think your post here changed my perception of the problem the most out of all discussions I had on the subject. It made me think about people who are behind the papers. I somehow missed it. Thank you.

(And, sorry for being a jerk in this thread. I said too much in a few places, exactly because I didn't think of innocent mathematicians who might read it. I'm still convinced that there is a lot that math can borrow from CS and SE, but I'm definitely going to argue this differently.)

Re: Ask HN: How did you learn math notation?

#179

Post by contravariant »

ReleaseCandidat wrote: post #48

Well, the real fun is deciphering a lower case xi - ξ - when written on the blackboard (or whiteboard), specially compared to a lower case zeta - ζ (fortunately way less commonly used). As all the others already told you. you don't learn by reading alone.

Ah, yes. I remember the time when I saw someone write something vaguely like the following

[0,ξ[={x|0Which was fun trying to figure out when written in handwriting where ξ,{,} all look the same.

If you can't figure out what it's supposed to be, this equation starts with a half-open interval denoted: [ξ,0[. This notation has some advantages but can be make things hard to read.

Re: Ask HN: How did you learn math notation?

#180

Post by Jensson »

klibertp wrote: post #170
ColinWright wrote: post #154

Earlier quoted context omitted.

Math notation has evolved to be what it is because it is useful for the actual doing of math, and the communication of math to those who have sufficient background. It's not deliberately designed to keep people out, and there are literally hundreds of thousands of books that introduce people to the notations used, to help on-board them. Haskell is unreadable to one who has not trained in it or similar languages ... w…

> It's not deliberately designed to keep people out, It looks that way, to many people, even in this thread. > why don't they change the syntax to make it more readable? They do, actually. Quite often at that. It's called releasing new version. > Look! Computer programs are easy and intuitive and everyone can understand them, even without training! Make math like that! No. Computer code is as far from intuitive as it…

> Why, do you believe it's not possible for it to be that way? See: https://en.wikipedia.org/wiki/Pythagoras#Prohibitions_and_re...

Did you just use an example from 2600 years ago to make a point? You don't think much changed since then?

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