Ask HN: How did you learn math notation? • Hacker News Board

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

#181

Post by Jensson » Thu, Nov 25, 2021, 8:12 PM UTC

klibertp wrote: post #178

Jensson wrote: post #168

Earlier quoted context omitted.

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 mathematicia…

I wrote one math paper before I went into programming. It is a lot of work, like code reviewing but much much longer. It isn't fun. A big reason I got into programming is because that process is so much work. Of course I, the professor who reviewed it and the professors who looked at it afterwards understood it, but I can't guarantee that someone who hasn't read a lot about research level topology or combinatorics will easily understand much at all. However I doubt that anyone who didn't do those things will ever read it since it is an uninteresting niche topic. I'd be surprised if even 10 people read it fully.

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

#182

Post by klibertp » Thu, Nov 25, 2021, 8:13 PM UTC

Jensson wrote: post #180

klibertp wrote: post #170

Earlier quoted context omitted.

> 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?

> Did you just use an example from 2600 years ago to make a point?

Yes? What's wrong with that?

I'm pointing out the most widely known example, to make a point, which is: "it is possible to design notation specifically for keeping outsiders out". I'm not saying that modern math notation is like that. I think, as a layman, that it probably evolved over a long time and so is full of idiosyncrasies that made perfect sense back when they were introduced (my GP seems to describe it in similar terms, so I hope I'm not that far removed from reality).

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

#183

Post by rackjack » Thu, Nov 25, 2021, 8:18 PM UTC

All math notation was created by mathematicians who wanted to quickly represent something, either to:

- better see the structure of the problem; or

- reduce the amount of ink they need to write the problem

Very similar to how programmers use functions, in fact.

To this end, mathematicians in different fields have different notation, and often this notation overlaps with different meaning. Think how Chinese and Japanese have overlapping characters with different meanings.

As others have stated, there is no "one true notation" -- all notation is basically a DSL for that math field.

Instead, choose a topic you are interested in, find an introductory text, and start reading. They will almost certainly explain the notation. Unfortunately, even within a field, notation can vary, but once you have a grasp of one you will probably grasp the rest quick enough.

I will mention, though, that some notation is "mostly" universal. Integrals, partial derivatives, and more that I can't recall right now all use basically the same notation everywhere, since they underlie a lot of other math fields.

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

#184

Post by srcreigh » Thu, Nov 25, 2021, 8:18 PM UTC

First math course at university of Waterloo: https://cs.uwaterloo.ca/~cbruni/Math135Resources/courseNotes...

Learning everything about math is nearly impossible like knowing everything about all code that exists.

That course should teach some basics for proof strategies. Ex here on page 2, there are definitions with examples: https://cs.uwaterloo.ca/~cbruni/pdfs/Math135SeptDec2015/Lect...

Specialized math tends to have specialized notation. For ex Linear Algebra, Calculus, Combinatorics. Any decent textbook will have an appendix or table with what the notation means.

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

#185

Post by hansvm » Thu, Nov 25, 2021, 8:19 PM UTC

kazinator wrote: post #159

ColinWright wrote: post #2

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. Yes, conventions have emerged, people tend to use the same sort of notation in a given context, but in the main, the notation should be regarded as an aide memoire , something to guide you. You say that you're struggling because of "the math notations and zero explanation of it i…

> 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. No, that belief isn't the problem; that actual status quo itself is obviously the problem. There are numerous notations and authors don't explain what they are using, assuming everyone has recursively read all of their references depth-first before reading their paper.

Why does RTFM not apply to mathematics dependencies?

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

#186

Post by merlinran » Thu, Nov 25, 2021, 8:25 PM UTC

Had been in the same situation for years. Read a paper, encounter the first equation, scratch my head and search around trying to understand it, give up. That changed half a month ago, after watching the Linear Algebra and Calculus course at https://www.youtube.com/c/3blue1brown/playlists?view=50&sort....

Let me explain a little bit. Just like a foreign language you stopped learning and using after high school, what prevents you from using it fluently is not just the vocabulary and grammar, but also the intuition and the understanding of the language as a whole. Luckily, math is a human designed language, with linear algebra and calculus being the fundamentals. And again, learning them is about building intuition on why and how they are used, so whenever you encounter transformation, you think in terms of vectors and matrices, and derivative for anything relevant to rate of change. By using carefully designed examples and visual representation, Grant Sanderson greatly smoothed the learning curve in the video courses. Try it out and you'll see.

Beyond that, different fields do have slightly different notation. When you first encounter them, just grab some introduction books or online courses and skim over the very first chapters.

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

#187

Post by Grustaf » Thu, Nov 25, 2021, 8:32 PM UTC

ColinWright wrote: post #177

Grustaf wrote: post #164

Earlier quoted context omitted.

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 r…

Alright, but are there really any overlapping concepts between graph theory and analysis? There can’t be many!

The comment you linked to is pretty strange, given the limited number of symbols in the Greek and Latin alphabets, there’s obviously going to be a lot of reuse, but I can’t see how that could really cause any confusion though, unless you’re just grabbing books from the shelf and opening them at random. And even then, it should almost always be clear from context if pi is a number or a plane, and if it’s a function that will be visually distinguished.

I’ve seen non-mathematicians use words as names of variables and functions, it always makes me shudder. I unsuccessfully tried to introduce Hebrew letters as an alternative,when I discovered how to use the in Latex, but it never caught on…

I actually find math notation incredibly intuitive and effective, I think it’s close to optimal. In fact it’s only after getting into programming that it even occurred to me how elegant and magical it is. I understand what things mean and can write things myself, without being able to exactly explain how, or to translate it into a fully specified system that a computer would understand.

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

#188

Post by forgetfulness » Thu, Nov 25, 2021, 8:37 PM UTC

kens wrote: post #66

rullopat wrote: post #11

Earlier quoted context omitted.

I was trying to grasp some of the papers linked in the Valhalla DSP block, for example this one: http://www2.ece.rochester.edu/courses/ECE472/resources/Paper... There is a formula with a triangle and I don't get what's that about, for example.

Everyone is talking about the Δ symbol, but the real problem that you'll encounter will be later in the paper where they start talking about H(ω), which is the Fourier transform of the impulse function (equation 4 and following). You'll need to know a fair bit about Fourier transforms and impulse responses and filter design to get through this section. The notation is the least of the problems. One place to start is…

Wikipedia is truly atrocious for learning math, the articles are like man pages in that they precisely describe the concepts in terms that will only make sense if you already know the thing. They just aren't written for pedagogy.

Like in 300 BC today, there's no royal road to geometry.

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

#189

Post by klibertp » Thu, Nov 25, 2021, 8:39 PM UTC

Jensson wrote: post #181

klibertp wrote: post #178

Earlier quoted context omitted.

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 mathematicia…

I wrote one math paper before I went into programming. It is a lot of work, like code reviewing but much much longer. It isn't fun. A big reason I got into programming is because that process is so much work. Of course I, the professor who reviewed it and the professors who looked at it afterwards understood it, but I can't guarantee that someone who hasn't read a lot about research level topology or combinatorics wi…

Yeah, I didn't think about it at all - I didn't realize that what I'm saying is basically demanding people to work for free (and on things that won't be useful to anyone in 99% of cases), and that's on top of already huge effort that is writing the paper in the first place. Honestly, I was behaving like people who open tickets in an open source project just to demand that someone implements a particular feature, just for them, and right now. I dislike such behavior, and realizing that I'm doing the same hit me hard :)

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

#190

Post by j7ake » Thu, Nov 25, 2021, 8:57 PM UTC

Syzygies wrote: post #127

I'm a math professor, and my students find it revelatory to understand math as I talk and draw. Math notation is not math, any more than music notation is music. Notably, the Beatles couldn't read sheet music, and it didn't hold them back. The best comparison would be is reading someone else's computer code. At its best computer code is poetry, and the most gifted programmers learn quickly by reading code. Still, let…

Great insight! I’ve definitely encountered mathematically inclined people but who cannot read or write math. Now it makes sense to me.

Also I’ve found the converse true. There are people who can manipulate mathematical symbols very well but actually don’t understand the big picture or general direction. The analogy would be that there are people who can write and read music notes (even transpose to different keys) without hearing it in their head (I was one of them).