Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Why Mathematics is Boring (2007) [pdf]

math.ucr.edu

Post reply on HN
Print view View original

21–30 of 129 posts

Re: Why Mathematics is Boring (2007) [pdf]

#21

Post by gattilorenz »

datastoat wrote: post #13
abdullahkhalids wrote: post #3

If you read any science papers, they start with a clear introduction with the aims, claims, importance and novelty of the work. A lot of math papers (but not all) just start off with a dry statement of what the theorems being proved are, and jump right into the proofs. I always wonder why editors don't understand the importance of these things and don't enforce them.

Relatedly, a lot of recipe books just have a dry statement of what the ingredients are and how to combine them. They should be more like CS and science papers, and explain the narrative of why the recipe is exciting, and where it comes from, and what it means for the cook. (Posted from a parallel universe.)

> why the recipe is exciting, and where it comes from, and what it means for the cook

I know you mean this as a joke, but many cooking books are like that, and I for one value knowing the history (and science, as in "The Food Lab", highly recommended) of what I'm cooking or eating

Re: Why Mathematics is Boring (2007) [pdf]

#22

Post by taneq »

elliekelly wrote: post #17
bluedays wrote: post #2

I believe this is actually a really great point. For instance, I didn't know that it was originally Brahmagupta who started using symbols in math. His initial use of symbols to represent numbers involved using color names like "blue" and "green". More interestingly before using symbols as variables in math Egyptians were capable of doing Quadratic equations without these variables. If I were teaching math today I wou…

I think they’re doing this more in schools. I was helping a first grader with homework recently and I was surprised the math worksheet was essentially simple algebra but with a shape or a little picture to represent the variable instead of a letter: > 3 + circle = 7 > 10 - cloud = 4 It seems like a perfectly reasonable exercise for a first grader. Clouds and puppy faces and circles. But I have to admit if I’d seen “x…

The introductory arithmetic I've seen recently even kind of flips it around and defines subtraction in terms of algebra. "7 - 5 = ?" is presented as "5 plus what equals 7?"

Re: Why Mathematics is Boring (2007) [pdf]

#23

Post by robomartin »

GDC7 wrote: post #16
robomartin wrote: post #15

Earlier quoted context omitted.

> Maybe the field would benefit to go more towards philosophy and logic, explaining it with words. Interesting perspective. I studied Philosophy and Logic in university: https://en.wikipedia.org/wiki/List_of_logic_symbols Much of it was familiar to me because, earlier, I took a class my Physics professor insisted we should take, as he put it, "if you want to get out of the dark ages": Programming in APL. AS it turns…

> https://en.wikipedia.org/wiki/List_of_logic_symbols Among all those symbols only ">" and " Even "=" is derivative of " " because by reasoning you can understand that you get to it by rotating the 2 lines about 30 degrees after realizing that you are dealing with 2 numbers which are in fact the same and not one being bigger than the other

Yes, of course. BTW, I don't think my comment covered the fact that I agree with you 100% on the impenetrability of mathematical notation.

That said, all notations --including the written alphabets of many spoken languages-- are impenetrable until you learn them. As a personal example, for me, learning French and German was a million times easier than learning Chinese and Japanese. In the first two cases I could read and write the languages right away. In the case of the latter two the notation imposed both a significant time drain and a cognitive load that got in the way of learning. I did a lot better with Japanese than Chinese. And BTW, I would not dare say I know these two languages. I can rattle off a bunch of phrases in Japanese and understand them if spoken slowly. My brain has yet to synchronize to Chinese.

My point is that specialized notations have been a part of the human experience forever. From cuneiform to modern written languages. Our brains are pretty good at learning notation. I would not fault mathematics for anything other than, perhaps, practitioners assuming everyone reading a math-heavy text understands the notation as they do.

Personal example: One of my kids is going though an MIT CS class on edX. He got scared when he was presented a formula with a huge sigma "Σ" sign in front of it and numbers below and above it.

It took less than a minute to explain that this just means a sequence of sums, maybe ten seconds. I just wrote down something like: "(a0 * b0) + (a1 * b1) + ... + (an * bn)" and said: "This is what it means. Summation". Done.

The point is, notation doesn't have to be hard.

Re: Why Mathematics is Boring (2007) [pdf]

#24

Post by bluedays »

cabalamat wrote: post #4
bluedays wrote: post #2

I believe this is actually a really great point. For instance, I didn't know that it was originally Brahmagupta who started using symbols in math. His initial use of symbols to represent numbers involved using color names like "blue" and "green". More interestingly before using symbols as variables in math Egyptians were capable of doing Quadratic equations without these variables. If I were teaching math today I wou…

> I think that this would probably stop a lot of the questions like "why are there letters in math!?" Wouldn't people just instead ask "Why are there colours in maths?"? I don't think that this: blue + blue = 2*blue Is any more meaningful or edifying than: x + x = 2*x

Not if you show a really long example with algebra where you are not using any variables, and show how it can be simplified with variables.

Re: Why Mathematics is Boring (2007) [pdf]

#25

Post by enriquto »

taneq wrote: post #20
enriquto wrote: post #8

Earlier quoted context omitted.

I for one welcome the "dryness" of mathematical writing. It feels clean, like reading a story without distracting ads. A beautiful advice that I received as a student was to write mathematics as a series of definitions, propositions and proofs. No text is allowed to exist outside of these three. In practice it is difficult to enforce, but it is helpful to keep this as an aim.

As a non-mathematician this seems to be lacking a little something. Would you not want your paper to provide some indication of what you're attempting to communicate and why? Or is that information (as the joke goes) possible, and therefore trivial, to infer?

A great deal of maths is that you get to use theorems for purposes that they were not intended for. Presenting the theorem in a "pure" form thus allows to approach it without pre-conceptions.

I love the analogy with cooking recipes in another comment. Math papers are like recipe books, deliberately devoid of their social context. The same recipe may mean different things to different cooks, even contradictory! Having a clean, neutral description of the recipe allows both cooks to safely refer to the exact same recipe, without endorsing contexts that they may find odious.

I agree that knowing the context in which a recipe was created, and the contexts where it has been used, is very useful. But it would be extremely annoying to have this explanation interleaved with the recipe description itself. This information is best kept separate. Then, at the beginning of the recipe you can have a list of links to cooks who have written different things about it.

Re: Why Mathematics is Boring (2007) [pdf]

#26

Post by cormacrelf »

erichahn wrote: post #6

It's boring but usually everything is well-defined and hence well-understandable. Not the case for most CS papers.

So many of the difficulties in reading many papers (well, disciplines) boil down to adding links to things the first time you mention them. Good academic writers do this, bad ones don’t. The gist of this paper is that assuming all knowledge prior to your development is very limiting, not only because far fewer people can read it, but because every time you don’t introduce knowledge you also skip over part of a story. Very well, but I think you are correct that CS’ problem is pretty much isolated to the first point, because nearly all papers get to talk about real world applications of the research and that’s the story covered.

A big problem for CS papers, particularly in PL (programming language research), seems to be heavy reliance on assumed knowledge of Greek letters and notation in the very field-specific way you like to use them. People would understand your paper if only they had read your previous three, which alluded to what you might have meant by these Greek letters, but only by figuring out the two citations they each have in common. If you are bad at pronouncing Greek letters and it’s a PDF so you can’t copy them, you can’t even google what you see. Even if you could, it wouldn’t help. Notation is ten times harder to search for.

(I have never, ever had this problem reading a law paper, not even slightly, not even once.)

There’s an interesting demo here from Will Crichton about how to prepare better documents for conveying understanding in PL. He has a thing to show you the “read as” on hover. https://twitter.com/wcrichton/status/1442891297333800966 https://willcrichton.net/nota

Re: Why Mathematics is Boring (2007) [pdf]

#27

Post by GDC7 »

robomartin wrote: post #23
GDC7 wrote: post #16

Earlier quoted context omitted.

> https://en.wikipedia.org/wiki/List_of_logic_symbols Among all those symbols only ">" and " Even "=" is derivative of " " because by reasoning you can understand that you get to it by rotating the 2 lines about 30 degrees after realizing that you are dealing with 2 numbers which are in fact the same and not one being bigger than the other

Yes, of course. BTW, I don't think my comment covered the fact that I agree with you 100% on the impenetrability of mathematical notation. That said, all notations --including the written alphabets of many spoken languages-- are impenetrable until you learn them. As a personal example, for me, learning French and German was a million times easier than learning Chinese and Japanese. In the first two cases I could read…

> It took less than a minute to explain that this just means a sequence of sums, maybe ten seconds. I just wrote down something like: "(a0 * b0) + (a1 * b1) + ... + (an * bn)" and said: "This is what it means. Summation". Done.

I think the real world feedback is quite different, given that math can be explained textually with words , why should we not do it?

The burden of the proof is always on the institution trying to do something. In this case the US government trying to make the US population better at math.

The population is quite okay with the present day situation, it's the government's job to make stuff happen and change things around to obtain the desired result, that is an improvement compared to what we have today.

Math proficiency is in line with new notation foreign languages proficiency from your examples (Chinese, Japanese, Austrian and German to a certain extent), that's because as you said both math and those languages have a different notation.

Given that (unlike foreing languages) math can be explained WITHOUT having to teach a new notation, then why don't we do it?

New notations are necessary for Chinese, but not for math, so why don't we remove this barrier to entry?

New notation is part of the human civilization but it has to be acquired early on to become like a second skin, which is what Latin letters are for us.

One has to be realistic . Mathematical notation will always take the backseat vis-a-vis literal notation. Kids just don't learn (and aren't taught) mathematical notation the same way they learn (and are taught) latin letters.

Instead of fighting against windmills we should take that as a given and try to influence what can be influenced.

As I said the institution trying to make a change in end results, must consider changes in the process...otherwise nothing happens.

Re: Why Mathematics is Boring (2007) [pdf]

#28

Post by ectopod »

robomartin wrote: post #18
amcoastal wrote: post #12

Earlier quoted context omitted.

I'll disagree. I read many papers with mathematics in them, and I get a lot of the concepts but the symbology used doesn't make sense to me so its hard for me to understand what is exactly going on. The sentence after the equation that explains each symbol is necessary for me, and many others as well. Not everyone has taken 8 math classes to know each kroniger delta by heart.

Well, musical notation looks like gibberish to someone who did not learn it. That said, I do agree with you 100% on scientific papers. Without an explanation of the formulas to cater to a wider audience a lot of papers fall into the "and then a miracle occurs" fallacy. Not because that's what they actually do. Not at all. I say this because to a large set of readers the impenetrable math has to be taken as a divine a…

You can start explaining fractional powers with roots, e.g. x^0.5

Re: Why Mathematics is Boring (2007) [pdf]

#29

Post by joeberon »

GDC7 wrote: post #14
_Microft wrote: post #9

Earlier quoted context omitted.

The idea that unfamiliar symbols and alphabets are a huge problem for the accessibility of math is common. As physicist I do not agree. Math is hard. It's damned difficult. Symbols and alphabets are the least of your concerns when dealing with a math paper. I know a lot of these symbols by name, I sometimes understand the notation or could familiarize myself with it but the math itself? Nope, no chance, usually. If o…

Math will only emerge as a field when people will stop treating it as something that one only has to do at the frontier. In other competitive fields such as banking, basketball and football you have the higher ups caring about the pyramid below them, if only as a place to recruit new talents. Among the math higher ups, only Jim Simons cares about the "math pyramid" so to speak. One has to be pragmatic, the goal of ge…

What are you talking about? Mathematics has well and truly "emerged as a field" lmao
Post reply on HN