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

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

#151

Post by TheTrotters » Thu, Nov 25, 2021, 7:07 PM UTC

klibertp wrote: post #133

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…

Why are you telling OP what his problem is? Shouldn't you address his pain points, not your rationalization of them? I wrote it many times already and am bit tired of it, so just a quick summary: - programmers[1] also use cryptic notation and tend to think in concepts rather than syntax - nevertheless, programmers spend a lot of time commenting the code, documenting it, specifying it, and so on. - why can't mathemati…

Note that the OP is asking about college-level math, not cutting-edge papers.

Textbooks routinely have a list of symbols and their definitions.

But, from my experience, notation is rarely the problem. I’d bet that the root cause of OP’s frustration is lack of understanding of concepts, not notation. (But, of course, it’s hard to say more without specific examples).

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

#152

Post by Grustaf » Thu, Nov 25, 2021, 7:10 PM UTC

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…

There pretty much is one true notation. There could be some slight variations, like bolding vectors, putting an arrow over them or not distinguishing them at all from scalars. But 95% of the time everyone uses the same notation.

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

#153

Post by whatsakandr » Thu, Nov 25, 2021, 7:12 PM UTC

I have a masters in engineering, but there was a lot of pure math things that I never understood until recently. I found the same approach to learning software concepts and APIs. Just start at the one you don't know and recursively explore the concepts until you find stuff you do know.

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

#154

Post by ColinWright » Thu, Nov 25, 2021, 7:13 PM UTC

klibertp wrote: post #134

ColinWright wrote: post #58

Earlier quoted context omitted.

> I dislike maths notation as I find it lacks rigour. I see this a lot from programmers, but in essence, you seem to be complaining that maths notation isn't what you want it to be, but is instead something else that mathematicians (and physicists and engineers) find useful.

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 ... why don't they make the syntax more readable? Or C++ with its modern templating ... why don't they change the syntax to make it more readable?

You might be tired of wandering into someone else's area of expertise and telling them:

You must change! You must make it more accessible!

Believe me, mathematicians are tired of non-mathematicians wandering up and saying:

Look! Computer programs are easy and intuitive and everyone can understand them, even without training! Make math like that!

Do you really believe that math notation is deliberately designed to make it hard for people untrained in math to learn how to use it? Do you really believe that no one has tried to make it more accessible?

Do you really believe you know more about why math notation is what it is than mathematicians and trained mathematics educators do?

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

#155

Post by Jensson » Thu, Nov 25, 2021, 7:16 PM UTC

klibertp wrote: post #146

Jtsummers wrote: post #141

Earlier quoted context omitted.

> What is so wrong about attaching additional few pages to every paper that nobody wants to do it? Pages with explanation of the syntax used, even the common bits. Programs don't do this, why do you expect every math paper to do it? > Link to external resources with explanations! This is called a bibliography, every book that isn't so old that it is the definition and paper includes one. In many textbooks there are a…

> Programs don't do this, why do you expect every math paper to do it? Well, I don't. It was you moving the goalpost. I talked about "a few pages", and you made "a book" out of it. I simply don't agree with you here and so I have very little to add at this point, sorry. > This is called a bibliography, every book that isn't so old that it is the definition and paper includes one. No. Bibliography is like a list of li…

> Programs do come with man pages! And tutorials, interactive tours, contextual help, and more. Emacs comes with 3 books, and a tutorial. (GNU) libc has a book to it. Firefox has a whole portal (MDN) as its documentation. Visual Studio comes with MSDN and a huge amount of explanatory material. And when it comes down to code, you have auto-completion, go to definition, search for callers; you can hover over a symbol and you get a popup with documentation and types. I just can't see how you can even put math papers and programs on the same scale in terms of accessibility!

You are comparing big teams and products to a single guy writing a paper intended for a niche audience and to be read maybe a few hundred times if he is lucky. People makes mistakes and sometimes forget to document everything, they try to document everything though as can be seen in their papers where most things are documented well, but sometimes they miss things and unlike code you don't have compiler warnings telling you about it. And given how few people read those papers 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.

The equivalent to those programs in math would be high school textbooks, and they are extremely well documented and easy to read in most cases.

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

#156

Post by ColinWright » Thu, Nov 25, 2021, 7:18 PM UTC

Grustaf wrote: post #152

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…

There pretty much is one true notation. There could be some slight variations, like bolding vectors, putting an arrow over them or not distinguishing them at all from scalars. But 95% of the time everyone uses the same notation.

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 had some exposure to the problem(s).

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

#157

Post by kmill » Thu, Nov 25, 2021, 7:20 PM UTC

throwaway31338 wrote: post #54

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.

Came here to say the same thing harshly and laced with profanity. I guess I can back off a bit from that now. I was filled with crushing disappointment when I learned mathematical notation is "shorthand" and there isn't a formal grammar. Same goes for learning writers take "shortcuts" with the expectation the reader will "fill in the gaps". Ostensibly this is so the writer can do "less writing" and the reader can do…

Here's a take from a mathematician-in-training, and it's biased toward research-level math, or at least math from the last hundred years:

Math is difficult, and a lot of what we have is the result of the sharpest minds doing their best to eke out whatever better understanding of something they can manage. Getting any sort of explanation for something is hard enough, but to get a clear theory with good notation takes an order of magnitude more effort and insight. This can take decades more of collective work.

Imagine complaining about cartographers from a thousand years ago having sketchy maps in "unexplored" regions. Maps are supposed to be precise, you say, there's actual earth there that the map represents! But it takes an extraordinary amount of effort to actually send people to these places to map it out -- it's hardly laziness. Mathematics can be the same way, where areas that are seemingly unrigorous are the sketches of what some explorers have seen (and they check that their accounts line up), then others hopefully come along and map it all in detail.

When reading papers, there's a fine balance of how much detail I want to see. For unfamiliar arguments and notation, it's great to have it explained right there, but I've found having too much detail frustrating sometimes, since after slogging through a page of it you realize "oh, this is the standard argument for such-and-such, I wish they had just said so." You tend to figure that something is being explained because there is some difference that's being pointed out.

I've been doing some formalization in Lean/mathlib, and it is truly an enormous amount of work to make things fully rigorous, even making it so that all notation has a formal grammar. It relies on Lean to fill in unstated details, and figuring out ways to get it to do that properly and efficiently, since otherwise the notation gets completely unworkable.

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

#158

Post by Jensson » Thu, Nov 25, 2021, 7:23 PM UTC

Tronno wrote: post #147

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…

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.

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

#159

Post by kazinator » Thu, Nov 25, 2021, 7:27 PM UTC

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.

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

#160

Post by webmaven » Thu, Nov 25, 2021, 7:28 PM UTC

CogitoCogito wrote: post #67

throwaway31338 wrote: post #54

Earlier quoted context omitted.

Came here to say the same thing harshly and laced with profanity. I guess I can back off a bit from that now. I was filled with crushing disappointment when I learned mathematical notation is "shorthand" and there isn't a formal grammar. Same goes for learning writers take "shortcuts" with the expectation the reader will "fill in the gaps". Ostensibly this is so the writer can do "less writing" and the reader can do…

> 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 (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.