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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?

#141

Post by Jtsummers »

klibertp wrote: post #138
Jtsummers wrote: post #136

Earlier quoted context omitted.

Mathematicians do document and comment, that's what papers and textbooks are: commentary on the math. They don't throw out formulae and equations and call it a day. Attaching a full tutorial for every level of reader is tantamount to attaching Stroustrup's C++ books to every C++ program, or K&R to every C program. You wouldn't do that, you'd expect the reader to ask you for references or to seek them out themselves.

> or K&R to every C program. That's actually doable... ;) K&R is rather terse, what, 1/5 of Stroustrup or something like that. But I digress. More on topic: there's also a class of programs that DO come with a book attached - or rather, multiple books, for every level; if not included outright in the distribution then at least linked to in the "learn" tab on a homepage. They're called programming languages. So, it ca…

> 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 also appendices which cover (some of) the foundational material. And most include sections (often in the front and back covers) that show the symbols and their names, if not their definitions.

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

#142

Post by jonnybgood »

Jtsummers wrote: post #136
klibertp wrote: post #133

Earlier quoted context omitted.

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…

Mathematicians do document and comment, that's what papers and textbooks are: commentary on the math. They don't throw out formulae and equations and call it a day. Attaching a full tutorial for every level of reader is tantamount to attaching Stroustrup's C++ books to every C++ program, or K&R to every C program. You wouldn't do that, you'd expect the reader to ask you for references or to seek them out themselves.

I think the GP post is criticizing the lack of documenting syntax. Math papers tend to document semantics, whereas the understanding of the syntax by the reader is presumed.

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

#143

Post by yongjik »

Try reading a good undergraduate calculus textbook. It would be hefty and a bit wordy, and it may take a few months to go through, but calculus requires surprisingly little amount of prior knowledge - even the concept of limit should be defined in the textbook (the famous epsilon-delta).

Also remember that math notations are meant for people. If you learn the sigma summation notation, and if you wonder "So I understand what is \Sigma_{i=0}^{10}, but what is \Sigma_{i=0}^{-1}?" then you're wondering irrelevant stuff. If a math notation is confusing to use, good mathematicians will simply not use it and devise an alternative way to express it (or re-define it more clearly for their purpose).

Also, don't skip exercises. Try to solve at least 1/3 of them after each chapter. Exercises are the "actually riding a bike" part of learning how to ride a bike.

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

#144

Post by bradlys »

https://www.amazon.com/Introduction-Mathematical-Reasoning-N...

I’d highly recommend this book. It’s what I had for my intro to proofs class in college and it was the best book I found for understanding. I found many other books on this topic to be kinda garbage but this one was amazing.

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

#145

Post by Jensson »

Hermitian909 wrote: post #85
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.

As someone who's studied math and CS extensively, it's not that mathematicians don't need that rigor it's only certain sub-fields have a culture of this kind of notational rigor. You absolutely see little bubbles of research, 2-4 professors, get sealed off from the rest of the research community because their notational practices are so sloppy that no one wants to bother whereas others make it easy to understand thei…

> CS are just better about this, and I can only explain it as a part of the culture and tradition.

CS professors writes just as incomprehensible math as everyone else, as you can see many here brings up examples of CS professors writing incomprehensible math in their papers.

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

#146

Post by klibertp »

Jtsummers wrote: post #141
klibertp wrote: post #138

Earlier quoted context omitted.

> or K&R to every C program. That's actually doable... ;) K&R is rather terse, what, 1/5 of Stroustrup or something like that. But I digress. More on topic: there's also a class of programs that DO come with a book attached - or rather, multiple books, for every level; if not included outright in the distribution then at least linked to in the "learn" tab on a homepage. They're called programming languages. So, it ca…

> 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 libraries you depend on. It has literally nothing to do with explaining the syntax close to where it's used.

> appendices which cover (some of) the foundational material.

Ha, ha, ha. No. If it's not front and center, then it doesn't count. I'm sorry, but I'm really tired of this subject. I would be willing to compromise more if that wasn't the case, believe me.

> show the symbols and their names, if not their definitions.

Ok. Putting that on the cover is a bit strange, but ok. That's a nice, but very small, step in the right direction. Please iterate and improve upon it!

EDIT: again, because I missed it at first:

> Programs don't do this, why do you expect every math paper to do it?

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; you can also trace execution, stop the execution, rewind the execution (if you have good debugger), experiment with various expressions evaluated at different points.

The most important difference between math and programming (or CS)is that programmers can (and do) build automated tools that help the next generation of newbies get into programming, while mathematicians can't. It's just that they don't want to admit this is a weakness, and only fortify more in their ivory towers.

TLDR: I just can't see how you can even put math papers and programs on the same scale in terms of accessibility!

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

#147

Post by Tronno »

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…

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.

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

#148

Post by eointierney »

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…

Super answer! I wish you were one of my professors, and I had excellent professors.

If I may humbly add, try making your own notation and playing around with it. Very rapidly one realizes just how hard a problem good notation is.

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

#149

Post by macrowhat »

My pre-calc teacher would regularly use smiley faces and christmas trees in place of common symbols. After a while you start to start to see past the funny symbols and look at it as pure symbol manipulation. Very interesting approach, and I went on to get (literally) a 100 in calc 3--with the curve ;)

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

#150

Post by Jensson »

webmaven wrote: post #135
Jensson wrote: post #94

Earlier quoted context omitted.

> All behavior in an implementation can be teased-out if given sufficient time. Can it? Given what? You would need to understand how the CPU is supposed to execute the compiled code to do that. In order to understand the CPU you would need to read the manual for its instruction set, which is written in human language and hence not any better defined than math. At best you get the same level of strictness as math. If…

> Every math paper is its own DSL in this context with its own small explanations for how it does things. That's really the point though: not every piece of software defines it's own DSL, nor does it necessarily incorporate a DSL from some library or framework (which in turn may or may not borrow from other DSLs, etc.). It is also impossible to incorporate something from other software without actually referencing it…

> It is also impossible to incorporate something from other software without actually referencing it explicitly.

No, some programming languages just injects symbols based on context. You'd have to compile it with the right dependencies for it to work, so it is impossible to know what it is supposed to be.

And even if they reference some other file, that file might not even be present in the codebase, instead some framework says "fetch this file from some remote repository at this URL on the internet" and then it fetches some file from the node repository, which can be another file tomorrow for all we know. This sort of time variance is non-existent in math, so to me math is way more readable than most code.

And you have probably seen a programming tutorial or similar which uses library functions that no longer exists in modern versions, tells you to call a function but the function was found in a library the tutorial forgot to tell you about, or many of the other things that can go wrong.

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