
How to Read Mathematics - ColinWright
http://web.stonehill.edu/compsci/History_Math/math-read.htm#
======
lkozma
" 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
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.

~~~
alderssc
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.

------
vbtemp
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.

------
dmlorenzetti
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
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.

~~~
dmlorenzetti
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).

------
dfan
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
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.

------
ckuehne
<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>

~~~
ColinWright
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
I enjoyed the Calculus for fools link :)

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

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

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

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

    
    
      ≠ -> !=
      √ -> sqrt

~~~
roundsquare
Ah. Well, its not like your missing or unable to follow a fundamental concept,
you just use different symbols. A bit of practice with reading things the
"math way" and you'll be fine.

------
pak
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.

~~~
ColinWright
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.

~~~
pak
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-...](http://fasterdonuts.tumblr.com/post/4516834904/the-proof-is-left-as-
an-exercise-for-the)) ... 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.

~~~
ColinWright
Yes, it's used too often. Yes, it's used sometimes when it's inappropriate.
Yes, sometimes I suspect that the author doesn't really know the answer. I'm
less sure that the author uses it it all seriousness when the know the result
is unproven or unprovable.

But it is a phrase that has its place, there are times when it's exactly right
to use it. Don't reject it outright. I've used it in publication where the
editor actually asked me to remove details and put the phrase in its place.

It's not always the author.

------
kruhft
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
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.

~~~
nocipher
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.

~~~
slowpoke
>I fail to see how creating a new set of symbols would do anything but hinder
communication within the math community.

I don't think this is just about the math community. The problem - at least as
I see it, as a student myself - is that Math is a required part of many fields
of studies. Fields which not necessarily have much to do with Math. And that's
a problem for many of the students. I won't say that Math is entirely unneeded
and should be completely abolished, but what I am saying is that many of these
students (including myself) don't want to be part of said "math community".

We don't want to have to learn the arcane notations and "math language" just
to be able to use some of the tools we might indeed need. Yes, I might sound
like a whiny student who's butthurt over having to actually study for
something. I may be. However, seeing as /so many/ other students (who, for the
most part, are competent in their own chosen field) are in danger of failing
their studies or struggle heavily with their math lectures simply because they
are /not/ competent at getting into the "math community" is ridiculous to say
the least.

For the record, I study Informatics at a university in Germany, so the
situation might be different elsewhere, but I've read about similar issues at
American colleges and universities, and every now and then, related articles
pop up here at HN.

~~~
alderssc
As a math major, my view is probably biased, but I feel like one of the
reasons mathematics is so widely used and applied in other fields such as
physics, CS, and even biology these days is that they express or model, to a
high degree of accuracy, patterns that recur in all these fields. The
mathematical notation is the most general and abstract form of expressing
these patterns. My feeling is that to not learn the language of math dooms one
to reinvent the wheel because they don't realize that the problem she is
working on has been solved before in a more general case by some one else. In
those cases where the problem hasn't been solved before, the solution leads to
new areas or notation which can show up in unexpected ways in different
branches of science.

Without this common language, we would have a specialized language for every
field, allowing little "cross-pollination". As an example, in evolutionary
biology, take the NK model of epistatic interactions which models gene
interactions. It just so happens that this model is extremely similar to a
thing in statistical physics called a spin-glass. Without mathematics the
biologist would have to sit and work out all the details of these epistatic
interactions before getting to actually _do_ biology. So in short, mathematics
saves you time by expressing common patterns or ideas.

~~~
ColinWright
There was an example not so long ago of a paper published in, I think,
biology, where the author had invented a wonderful new technique for more
accurate estimates of the area under a curve.

It was Simpson's rule.

CORRECTION:

It was worse than that. It was the trapezoidal rule:

<http://care.diabetesjournals.org/content/17/2/152.abstract>

The author named it after themselves:

    
    
        ... The Tai model allows flexibility
        in experimental conditions ...
    

and got 75 citations ...

[http://fliptomato.wordpress.com/2007/03/19/medical-
researche...](http://fliptomato.wordpress.com/2007/03/19/medical-researcher-
discovers-integration-gets-75-citations/)

Unbelievable.

------
nothis
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.

~~~
ColinWright
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.

~~~
kenjackson
_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.

~~~
ColinWright
Yes, but there's more that you can say. In particular, there is more than one
infinite family. How many are there? And do all solutions belong to an
infinite family? Or are there isolates/sporadics?

Can you characterise the solutions? How many pieces do they contain? Some
solutions have two pieces. Some have four. Are there other possibilities?

~~~
kenjackson
I don't know what an infinite family, isolate, or sporadic is. A quick Google
search didn't make it obvious.

I think 8 pieces also works, but I think that may be it.

~~~
nocipher
An infinite family seems to refer to an infinite collection of solutions that
are all related in some sense. An isolate or sporadic solution is a solution
that is unique and unrelated to other solutions. They are both terms used to
categorize solutions.

~~~
kenjackson
Thanks nocipher for the definitions. From this I see no sporadics. And two
infinite families, at least how I'd classify a family.

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

------
invalidOrTaken
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."

------
Rhapso
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.

~~~
ckuehne
Could you cite an example?

~~~
eru
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>

~~~
ColinWright
I expect you mean this bit:

    
    
        Suppose that ra and sa are the same modulo p,
        then we have r = s (mod p), so the p-1 multiples
        of a above are distinct and nonzero ...
    

More completely, I expect you want them to say:

    
    
        Consider the (p-1) multiples of a given by:
    
            a, 2a, 3a, ... (p-1)a.  (mod p)
    
        These are all distinct.  To see this, consider
        otherwise, and suppose ra=sa (mod p)
    

... and so on.

Is that what you meant?

The point is that all writing is aimed at an audience. I wouldn't expect
someone with no experience of Science Fiction to be able to read "Quantum
Thief," and I wouldn't expect anyone with a reading age of 6 to be able to
read "Lord of the Rings." Similarly, that proof requires some degree of
familiarity with the structure of proofs. This is not an especially difficult.

I've always found that indirect proofs, or proofs by contradiction, or proofs
by the contrapositive, are _mostly_ obvious as to what they are doing,
although not always.

In short, I agree with what you say, don't think the problem is as bad as you
are portraying, think the example you have given is not especially good, but
I'd be hard pressed to find a better one.

And finally, it's possible to come up with bad writing in every context. Some
proofs are badly written, badly expressed, and badly explained. I know - I've
not only read many of them, I've written some as well.

No surprise there.

Addendum: Sometimes the proofs that gave me the greatest understanding were
the ones that were the most badly written, forcing me to work through the
material on my own terms and understand it in my own way. Perhaps well-
written, well-expressed and well-explained proofs are actually a bad thing.

~~~
eru
Yes, I just went for something really basic to give an example of a semi-
implicit indirect proof.

I spent years of my life reading mathematics, so I do not trust myself to
judge how hard a piece of mathematics is for outsiders. I find the proof cited
is easy to read.

About your addendum: You could have a look at Alexander Schrijver's
"Combinatorial Optimization: Polyhedra and Efficiency". The interesting thing
about its style is, that the author manages to make all lines require constant
thought, while in most books there are really hard and really easy parts.

~~~
Someone
Maybe I am too well-trained in this, but I think using 'Suppose' (or "assume")
is a dead giveaway for a proof by contradiction.

The only semi-implicit ways to start a proof by contradiction I can think of
are the phrases "if x is..." or (less implicit) "if x were...".

------
scotty79
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.

~~~
eru
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
"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.

