How to Read Mathematics

http://web.stonehill.edu/compsci/History_Math/math-read.htm#

by ColinWright • 15 years ago

115 62 15 years ago

web.stonehill.edu

wccrawford• 15 years ago

I had never really thought of mathematics having its own language, but the parallels to learning a second language (after childhood) are interesting.

kruhft• 15 years ago

I remember back in school that doing the math was not really a problem, but actually reading the textbooks was. I was the same way with programming when I started, where I could write but not read the language, but I'm suprised that I got through 4 years of university math without really being able to read the proofs that explained it.

romaniv• 15 years ago

That's natural, since mathematical notation is heavily slanted towards writing on paper and symbol manipulation. Use of one-letter variable names and highly context-sensitive notation reduces the number of pen strokes, but doesn't help the reader.

Modern programming languages and conventions are quite a bit better in that respect, since people design them to be maintainable. A program using one-letter variable names for everything and a set of similar macros meaning entirely different things in different files would be considered unreadable by pretty much any programmer.

IMO, the world would immensely benefit from another version of mathematical notation designed specifically for explaining things, rather than doing calculations on paper.

nocipher5• 15 years ago

The problem is that the mathematical notation, while not perfect, works really well for those adept with its use. It's an efficient shorthand. For explaining things in depth we have pictures and and natural language. I fail to see how creating a new set of symbols would do anything but hinder communication within the math community. If you have any deeper argument to support your position, I'd be interested to hear it.

ckuehne2• 15 years ago

http://news.ycombinator.com/item?id=1576969

http://news.ycombinator.com/item?id=85743

http://news.ycombinator.com/item?id=583977

http://news.ycombinator.com/item?id=1509758

ColinWright9• 15 years ago

Of those, only the most recent one has any substantial comments, and even that is now 10 months old and can no longer be commented on:

http://news.ycombinator.com/item?id=1576969

levesque• 15 years ago

I enjoyed the Calculus for fools link :)

http://www.gutenberg.org/files/33283/33283-pdf.pdf

Rhapso• 15 years ago

The big ones: Watch out for proof by contrapositive and contradiction. Until you really get used to proving things by assuming bits of them are false it throws you off to see it used in textbooks and papers because most papers do not tell you they are using them.

ckuehne2• 15 years ago

Could you cite an example?

eru3• 15 years ago

Look at this proof of Fermat's Little Theorem [0], can you spot the indirect bit of the proof?

By the way, when you construct a proof you almost always have some leeway in how big your indirect section are. E.g. variant A: Assume X, do bits Y, Z, then contradiction. Variant B: Do bits Y', Z', assume X, then contradiction. In the second variant Y' and Z' could be useful on their own, and might be easier to understand without the inversion. For that reason, I often try to keep the contra-factual parts of the proof as small as possible. Though as with writing any code, clear writing trumps general rules.

[0] http://primes.utm.edu/notes/proofs/FermatsLittleTheorem.html

btilly• 15 years ago

To find the maximum of a function on a closed interval it suffices to look for places where the derivative is 0, undefined, or at the boundary of the interval.

Proof, Suppose that x in the middle of the interval has a derivative > 0. Then from the definition of the derivative there exists h > 0 such that x+h is in the interval and 0 < (f(x+h) - f(x))/h. Multiply by h and rearrange to see that f(x+h) > f(x) and therefore x is not where the maximum is achieved. The argument for f'(x) < 0 is similar except that f(x) is exceeded by f(x-h) instead.

Therefore if f achieves a maximum at x, then x is either a boundary point, or a spot where the derivative is 0 or undefined.

scotty79• 15 years ago

Line by line carefully. Never skip anything in hopes it will clear up later.

You may allow to yourself not to know with what intention author wrote the line but you should never allow yourself not to know why the author could write this line.

eru3• 15 years ago

You can skip on first reading to get an overview of where the author wants to take you, but don't expect it will clear up later on its own. Read and re-read. Fill in all the little gaps, especially when a sentence begins with "Clearly" or "As we can easily see".

amcintyre• 15 years ago

"As we can easily see" == doesn't seem implausible after you read the paper countless times, re-work all the derivations from scratch at least 7 times, read the cited docs, get out your old analysis texts to look up some theorem you'd forgotten existed, and sacrifice a chicken.

I must say, there's nothing like reading a math paper to remind me that there's no shortage of people in the world that are way smarter than me.

nocipher5• 15 years ago

The best way I've found to read math texts is to take it concept by concept. Line by line is too mechanical. Approach each concept and get a "gist" for what is trying to be accomplished. Once you understand where they are starting from for a given concept, where they are going with it, and have a rough idea of how they'll get there, then you read the details. The details are essentially meaningless if you don't know the bigger picture.

dmlorenzetti3• 15 years ago

I just had a reviewer criticize a report I wrote for containing "sentence fragments". The paper says things like, "Then x = y/2", which my reviewer interpreted as a one-word sentence ("Then"), followed by an equation. It made me realize how unnatural it is for some people to actively read mathematics-- for example, turning "=" into the verb "equals".

hvs• 15 years ago

What kind of reviewer, for what kind of paper? Any paper in which I wrote "Then x = y/2" would almost certainly be a math (or computer science) paper, so the reviewer would know what I was saying. If it was for a more general audience, I would probably find a better way to state my point.

dmlorenzetti3• 15 years ago

It's about modeling specific physical processes in buildings. It's computer science in the sense that the paper focuses on efficient implementation. However, the audience is the indoor environment community.

The reviewer is a skilled experimenter, whom I enlisted to check my understanding of the physical processes. (By the way, this person knew what I was saying, just didn't like the way I said it. For our internal reviews, we tend to get pretty picky.)

It's definitely not for a general audience-- there'll probably be five people in the world who can use the paper directly (once the results are embedded in software, there'll probably be several hundred who will use it).

grannyg00se• 15 years ago

That's a very simple example. Then x equals y divided by 2

How can anybody with a grade school level education not understand that? Are you sure the reviewer couldn't turn it into a sentence? Or perhaps the reviewer didn't feel that was the proper formatting for a sentence? I'm not sure how else you would include equations in a sentence, but if my reviewer didn't understant grade school level mathematics I'd be horrified that this person was in a position to be reviewing anything at all.

dmlorenzetti3• 15 years ago

You're right, this was a format complaint only.

When a sentence had an identifiable "other" verb-- for example "Then substituting gives x = y/2"-- everything was fine.

The reviewer actually made some incredible suggestions for restructuring the paper to bring out the important points. I'd hate for people to come away from my original comment thinking the reviewer is anything other than a very bright person who was pushing me to make the paper as clear as possible.

pixcavator• 15 years ago

I agree “Then x = y/2” is fine. However, your reasoning why this is OK (“=” is “equal”) is incomplete. Consider “Then the answer = y/2.”, which is grammatically correct but improper.

vbtemp• 15 years ago

While it may be a bit pretentious to quote oneself, I'll go ahead and do just that from another post similar to this topic:

"When I come across some academic mathematics paper, even if I'm somewhat familiar with the field, I generally find other things far more interesting, like the coffee stain on the floor. Effectively reading a mathematics paper requires a print out (not just of the paper itself but also many of the papers cited) and a pencil to work through some of the definitions and take notes for yourself, and maintaining laser-like focus for a sustained period of time."

I think that capture the gist of that article. Basically, it can be an exhausting and draining affair, especially for someone with my cognitive capacity.

lkozma• 15 years ago

" Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? "

-Paul Halmos, inventor of "iff" and the ∎ symbol ( http://en.wikipedia.org/wiki/Paul_Halmos )

nandemo• 15 years ago

Interesting:

> Halmos argued that mathematics is a creative art, and that mathematicians should be seen as artists, not number crunchers. He discussed the division of the field into mathology and mathophysics, further arguing that mathematicians and painters think and work in related ways.

alderssc2• 15 years ago

The great G.H. Hardy felt similar:

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

as did Bertrand Russell (one of my favorite quotes):

Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

I suspect many mathematicians view themselves, at least to some extent, as creative types similar to poets.

dfan• 15 years ago

My secret for reading difficult math and physics texts: write out, by hand, every equation as you encounter it. Firstly, it slows me down so I'm not tempted to skim it like a novel. More importantly, my standards for what I write are much higher than my standards for what I read; if I see something I'm not completely convinced of, I may shrug and move on, but I'm not willing to write something down unless I really understand it.

dlib• 15 years ago

This is what I also do, any step in a derivation I would glance over when read must be fully understood and proved whenever I write it down. Every non-trivial result in my math books is written down so I can fully comprehend what's going on.

tintin2• 15 years ago

I understand math, but I can't read mathematical symbols. Does anyone know a good website/book for applying and understanding mathematical symbols?

Deutscher• 15 years ago

EDIT: I hope these help:

https://secure.wikimedia.org/wikipedia/en/wiki/Table_of_math...

https://secure.wikimedia.org/wikipedia/en/wiki/List_of_logic...

roundsquare2• 15 years ago

Sorry, what exactly does that mean? Can you give some examples?

tintin2• 15 years ago

Well I learned to read math the 'computer' way. So here two examples of 'Math' vs 'Computer'

  ≠ -> !=
  √ -> sqrt

pak3• 15 years ago

The most irritating trope of math and CS literature has to be whenever something interesting or useful is "left as an exercise to the reader." I always felt that phrase has no place in the internet age, where the concept of a "page limit" is laughable and a simple hyperlink can point me to chapters upon chapters of appendices. Calling something "trivial" is swallowable, and perhaps this article's example of "it easily follows that..." is the mathematician's secret handshake that the steps in between are boring, but pointing out an interesting conclusion and then saying the reader deserves an exercise to actually make sense of it, I mean... isn't the point of academic writing to communicate research and results already performed? Wouldn't it be better to show the work for your assertion so that others could critique it or offer even better solutions?

I know somebody is about to point out to me that I'm being "lazy" and should enjoy doing more work to learn so and so, but being asked regularly in the literature to reinvent somebody else's wheel seems to run counter to everything we do in CS (and academia in general, I would think). This article aside, I do sometimes feel that academic writing in the math/CS realm occasionally reeks a little bit of snobbery, where communication is held secondary to keeping up appearances.

ColinWright9• 15 years ago

The most irritating thing to me is when someone says they want to understand something, but then they are unwilling to put in any real effort. It is a misapprehension, misunderstanding and misrepresentation to say:

    ... reinventing the wheel runs counter
    to everything we do in CS.

You can read all you like about how to juggle - if you don't put in the hours, you won't be able to do it.

You can read all you like about how to unicycle - if you don't put in the hours, you won't be able to do it.

You can read all you like about how to program - if you don't put in the hours, you won't be able to do it.

Skills require practice. These instances of "left as an exercise for the interested reader" are to help you really learn and properly understand the material. And if you don't care, don't bother. If you are unwilling to put in the time then the chances are that you would end up thinking you understand an explanation, whereas in fact you don't.

The above is only for good writing, of course. There are plenty of instances of bad writing, but that's not then a complaint about the subject, it's a complaint about the writer.

Someone once said of Feynman that his lectures were beautifully clear, and that those who listened gained real insight and understanding. Until they had to use it. Then they realised that they didn't really understand it at all.

If you want understanding, do the exercises.

So there you are, I've done as you predicted and pointed out that you're just being lazy, and that if you really want to learn then you have to put in the work. Just because you preempted it, doesn't mean it's wrong. It's right, you already know it. Complaining won't help you.

Put in the time.

pak3• 15 years ago

You must have only seen this phrase in places where it was used in the bright, sparkling, professorial manner assumed by your comment. I take no offense when it appears in a textbook, where the intent is to have me practice a skill. I do object to its appearance in a journal article, however; that is a context where you are supposed to communicate findings as clearly and concisely as possible. In particular, I've come across its use more and more often where one of the following likely applies:

1. it is doubtful that the author has actually done the work to prove his own assertion, and uses the phrase to cast his own burden on the reader,

2. the author has possibly done the work, but can't be bothered to condense it to the quality required for publication,

3. showing the work would clearly be useful for the target audience, but the author is more concerned with making the material appear difficult, or

4. the author is being ironic, because the assertion is either superfluous, outright incorrect, or known to be unprovable.

In fact, all these bad use-cases are common sources of humor among math nerds (http://www.jargon.net/jargonfile/e/exerciseleftasan.html), (http://abstrusegoose.com/12), (http://fasterdonuts.tumblr.com/post/4516834904/the-proof-is-...) ... so I think I'm being fair in calling this phrase a trope. As in, its overuse is well-known, tolerated, and occasionally ridiculed.

tldr: I am indeed complaining about bad writing, not the subject--specifically that "left as an exercise" has become a common idiom behind which bad writers in math and CS hide, and its inappropriate use is now all too common.

contravert• 15 years ago

I completely agree. This phrase appears in even some of the most hallowed tomes of computer science, such as Introduction to Algorithms. The textbook puts important theorems and proofs in the exercises section, and does not even provide solutions online (at least for the later sections). Keep in mind that this book is used as a reference manual for many implementations of algorithms and data structures.

pyninja2• 15 years ago

> I mean... isn't the point of academic writing to communicate research and results already performed?

Obviously mathematicians don't leave "exercises for the reader" in research papers, but it is common in textbooks or expository writing.

> I always felt that phrase has no place in the internet age, where the concept of a "page limit" is laughable and a simple hyperlink can point me to chapters upon chapters of appendices.

The constraint is not page length, of course, but time. It is a much better use of a mathematician's time to leave easily reproducible proofs to the reader, and focus on the interesting ideas.

Anyway, doing these kinds of exercises is actually very useful in understanding new ideas.

pak3• 15 years ago

>Obviously mathematicians don't leave "exercises for the reader" in research papers

ORLY? I'll leave this right here for you:

http://scholar.google.com/scholar?hl=en&q=left+as+an+exe...

That's >100k hits from what Google Scholar can index alone--a few are from books, but the vast majority are journal articles.

nothis• 15 years ago

That's a very elegant way of saying: "Communication in math is just as fucked up as you'd think."

As an outsider who only ever needs to use maths as a "tool" for very specific things, maths books and general attitude by professors and teachers is just nightmarish. Somewhere behind that curtain of inaccessibility, it has to be their fault. Especially, since once you actually do understand certain mathematical concepts you suddenly realize they're easily explainable with a few words of plain text or --gasp-- a "childish" drawing to illustrate.

nocipher5• 15 years ago

Most math concepts -are- fairly easy to explain and, from the sounds of it, you don't actually want a math book. You want a book that gives you the executive summary or a run-down of specific tools for specific situations. The flaw is not in the books or the teaching, it is what you are expecting from mathematicians.

Math is, in a very general sense, about understanding and reasoning about highly structured objects. Mathematicians have developed methods and notation to achieve that end, not to make it easily digestible by the uninitiated. Rigor is an important part of this. It is what allows mathematicians to be so sure their work is correct and it is often what makes math seem so arcane. It is not always necessary for teaching, but professional mathematicians are often the teachers and the ideas are intimately tied to their rigorous formulations in their head.

For example, continuity is a fairly intuitive concept in calculus, but the rigorous epsilon-delta definition is necessary in proofs. The basic concept, while easy to explain and understand is useless, while the precise formulation, though much more opaque, is ubiquitous in analysis simply because it is an incredible tool... for mathematicians.

An engineer likely just needs the machinery built on top of the analysis: the derivative and integral. If that's all you need, then pick a book that is focused on applications and a development of general intuition, not a book designed for mathematicians-in-the-making.

ColinWright9• 15 years ago

Do you really think that continuity is an obvious concept?

Are you aware, for example, that there are functions that are continuous at all irrational points and discontinuous at all rational points?

ColinWright9• 15 years ago

I'm working on something at the moment to try to explain exactly why what you've said here is, to a large extent, simply wrong. It's too long to include here, and it's not yet ready to "publish". You're about two weeks too early.

But let me ask you this. It's easy to cut a square into identical pieces so that all the pieces touch the center point.

In slightly more detail, the pieces are disjoint sets such that their union is the whole square. The pieces are identical except perhaps for details as to the boundaries. To say that they all "touch" the center point means that every non-zero radius disk centered at the center contains some points from each piece.

So now, how many ways can this be done? No, it's not five. And no, it's not six either.

When you start trying to work it out you find that the details matter, and they can't just be covered by a "childish" drawing to illustrate.

Details matter, and some of them are hard.

Yes, most math teaching is atrocious. We all know that. But it's not always just the teacher's fault. Sometimes it's at least partly the fault of the readers expecting everything to be made simple and immediately accessible with neither work nor effort.

kenjackson3• 15 years ago

So now, how many ways can this be done? No, it's not five. And no, it's not six either.

An infinite number of ways this can be done.

Someone2• 15 years ago

I have my doubts about that claim. In most cases, the 'you' that thinks something 'is easily explainable with a few words of plain text' often is a different 'you' from the one that struggled grasping the concept. That struggle is what made you think the concept is 'easy'.

Next time you encounter such a 'aha erlebnis', try saying the 'few words of plain text' to someone who still has to master the subject, and _immediately_ test him or her on the subject. I predict that, oftentimes, (s)he understood nothing of what you said.

Case in point: at university, I had a prof who said that it was perfectly OK if you could not do any practice integral. Trust me, he said, if, a month from now, you do these exercises, you will not understand what is hard about them.

invalidOrTaken• 15 years ago

I was just looking at this the other day. It reminded me of this:

"What readability-per-line does mean, to the user encountering the language for the first time, is that source code will look unthreatening. So readability-per-line could be a good marketing decision, even if it is a bad design decision...The math paper is hard to read because the ideas are hard."

How to Read Mathematics

Related Stories