
Mathematics:  The Most Misunderstood Subject - spacemanaki
http://www.fordham.edu/academics/programs_at_fordham_/mathematics_departme/what_math/index.asp
======
hypersoar
I'm pursuing a career as a math professor (currently an undergrad). I'm not
shy about my passion for math, and this has lead to countless conversations
like the ones below:

"What do you want to do with your math degree?"

"I want to go to graduate school and eventually become a math professor."

"Oh, so you want to teach!"

"No, I want to do research."

(Here they give some expression of confusion. I've had this particular
conversation dozens of times)

Also

"So, what do you do in math research? Do you just sit around and solve
equations all day?

I do my best to explain to these people what math is like and why I do it, but
I usually don't feel like I'm getting through. I have been told by
mathematicians many times, and have experienced myself, that doing mathematics
requires lots and lots of frustration (I'm sure many people here, including
those who don't do pure math, know what I'm talking about).

But for me the most frustrating and disheartening part of math is the fact
that most people don't know what it is. It's not just that they don't
understand the details, or what happens at high levels. It's certainly not
just that they look at an end product (analogous to, say, a product from a
startup) but don't get where it came from. It's that most people fundamentally
don't understand what I do. They think of math as the capricious monotony they
were put through in grade school and can't fathom why anyone would consider
dedicating a life to it. Most aren't even willing to try. My love of math is a
very big part of me, and it's a part that very few people understand.

~~~
shou4577
Agreed. I'm currently in your same position, only a few years ahead (in my
second year of grad school, currently). When I try to explain math research to
people, I too am left with strange looks.

I think most people honestly think that mathematicians sit around and multiply
larger and larger numbers together.

Part of the problem of explaining the true nature of mathematical research is
one of language. For example, I study Lie algebras and representations. People
ask me "What are Lie algebras?" and pretty much the only definition I can give
that is understandable is "Lie Algebras are a kind of algebraic structure that
is useful in many fields of science, including quantum physics." This is an
answer many people understand (on the surface), but it really doesn't say
anything.

What's worse, most of the time, when we justify or explain our research, it is
by connecting it to fields that the person may be more used to (astronomy,
biology, physics, economics, etc.). But for those of us in pure math, we
really do not think about these fields in our day-to-day work. That is, I
study Lie algebras because of their beautiful structure and the interesting
combinatorics behind them, not because they are useful in some other field.

So a better answer to the question "What are Lie algebras?" would be something
like "Lie algebras are vector spaces with additional algebraic structure that
gives rise to beautiful and deep combinatorics. They occur naturally as
certain sets of square matrices, and are a kind of generalization of the ideas
of symmetry." However, this is mostly unintelligible to most people who
haven't taken some mathematics beyond calculus, and I find that it sounds
condescending to a lot of people, which turns them off from listening to any
more explanation.

What I've taken to saying lately in response to a question along the lines of
"What do you do in math research?" is something like "Mathematicians create
new knowledge from existing knowledge. They take things that the human race
already knows, and using only logic, they deduce new things. This allows them
to find fascinating relationships all throughout the world."

I find this response to be pretty good. It's mostly accurate, it's mildly
interesting, and best of all, it's short.

~~~
lacker
It is hard to explain the value of higher math, but it's worth it.

Perhaps a better answer to "What are Lie algebras?" is to respond in terms
that mean something to your audience. Avoid words like "vector" and
"combinatorics".

Instead use metaphors. Like Rubix cubes. Tell them Lie algebra is a way to
solve Rubix cubes faster. And also other similar puzzles that are way harder
than Rubix cubes. It's true enough for casual conversation and probably more
interesting than a vaguer answer.

~~~
shou4577
I realize that using words like "vector" and "combinatorics" are poor choices.
This is part of the problem. It is difficult to come up with a good metaphor
that is simultaneously interesting and meaningful. I think the Rubik's cube is
a great example. Thanks, I'll be using that in the future.

~~~
sliverstorm
Telling them what it can do or what it is used for is almost always the best
way to talk about something you do with the totally uninitiated.

Make sure it's something they've heard of before, e.g. the Rubik's cube
example.

~~~
pbhjpbhj
>" _Telling them what it can do or what it is used for is almost always the
best way to talk about something you do with the totally uninitiated._ "

Although I'd guess for most researchers in Maths what it can be used for is
many years away (if it actually has a "practical use") and what it can do is
far too esoteric for the layman.

------
RiderOfGiraffes
When I'm asked about math there are a couple of things.

Firstly, I ask them about Pythagoras. Most people know of it, and I phrase it
in terms of cardboard cutout squares. Take three squares cut from a heavy
material, and make them so that A and B together weigh the same as C. Arrange
them so their sides lie on a triangle. Not only is it always possible, but the
triangle you get always has a right-angle.

Why? How do we know? As it happens, the reasoning as to why it's true is
wonderfully elegant, and totally accessible.

Secondly, I ask - do you think mathematicians know about numbers? Here's
something. Take any positive number. If it's even, halve it. Otherwise, triple
and add one. Keep doing this, and what happens. So far every number anyone has
every tried ends up in a ...->1->4->2->1->... cycle. Does it always happen?

No one knows.

Possibly it's useless, but there's a bunch of stuff people thought would be
useless, and they've given us micro-processors, SatNav, cryptography, error-
correcting codes, and a million other things.

Who knows what will be useful? After all, if we knew what we were doing, it
wouldn't be called "Research".

~~~
xyzzyz
Oh, Collatz problem. I usually use Goldbach or twin prime conjectures to
explain the difficulty of seemingly simple problems, but come to think of it,
Collatz is even better, because it does not involve any complicated concept at
all -- Goldbach and twin prime conjectures are about primes, and sometimes
people do not even know what primes are.

~~~
rfurmani
How about the problem of, given a polynomial equation in at least two
variables and integer coefficients, figuring out whether it has any integer
solutions. This has directed a lot of modern mathematics and, on the face of
it, doesn't seem like it should be so hard. Plus you can build off of this.
They may remember that for two variable quadratics the real solutions form an
oval or hyperbola or quadratic or two lines. In general you'll get some higher
dimensional surface, and the "shape" of it (and how many "holes" it has) is
closely tied with how many integer solutions there can be.

~~~
RiderOfGiraffes
If you try that you get:

* What's a polynomial?

* What are variables?

* What's an integer?

* ... a completely blank stare.

You're absolutely right about the usefulness, but most people really won't get
past the word "polynomial".

------
Dilpil
A very similar, much longer exploration of the problems with high school (and,
honestly, most undergraduate) math education:
<http://www.maa.org/devlin/LockhartsLament.pdf>

~~~
aik
Very good book/essay. Every math professor/teacher should read this in my
opinion. I found this book was incredibly enlightening and refreshing after
years of school math.

------
treo
This reminds me of "A Mathematican's Lament" by Paul Lockhart (
<http://www.maa.org/devlin/LockhartsLament.pdf> ). It is a great read in any
case.

------
alexwestholm
The author has rightfully directed his words at those considering becoming
math majors. Consequently, he makes it clear that most basic math education is
inadequate to really understand math and its practicality, but leaves as the
only solution enrolling as a math student. This is a great call to action for
students, but for the broader audience, it leaves me thinking the following:

My math background is clearly inadequate, and his description of mathematical
reasoning sounds like something I'd like to be familiar with, so how do I get
there? What's the route to the prize other than enrolling in Fordham's math
program?

While the professor in question has no reason to answer this, at least on the
Fordham website, I wonder what answers the HN crowd might have... thoughts?

~~~
spacemanaki
One series of articles which might serve as a start is Steven Strogatz's
series which ran in the NY Times earlier this year:

[http://topics.nytimes.com/top/opinion/series/steven_strogatz...](http://topics.nytimes.com/top/opinion/series/steven_strogatz_on_the_elements_of_math/index.html)

It's written for a layperson (he starts with counting on Sesame Street) but
nonetheless touches on some very advanced topics, and describes them in simple
ways. It won't give you a firm foundation in mathematics (the articles are
pretty short, and there are only 15 or so), but it might serve as a guide for
which subtopics in particular pique your interest, and there are numerous
references and suggestions for further reading. I believe he's collecting the
articles in a longer book, hoping to publish it in 2012.

~~~
jpren
Very apt that you brought up Steven Strogatz. I took Prof. Strogatz's calculus
for engineers class back in college (Spring 2004). If you like the way he
discusses math curiousities in his NYTimes column, you'll surely enjoy his
lectures as well!

In my life at least, it's been rare to come across a math professor/educator
who pushes harder for understanding (connecting the dots) rather than
knowledge (disjointed lists of equations and theorems). Mathematicians'
laments aside, inspired teachers are already changing the way that some
students -- albeit a small minority -- view the field of mathematics.

------
CallMeV
Now this article is brilliant. I'm reposting the link to this article to a
maths blog.

My old maths tutors in my old alma mater would also love to read the article,
so I'll email them the link. If nothing else, it'll remind them that
mathematics teaching does make a _great_ difference.

------
Tycho
I remember at school there was a bunch of kidsd who were good at maths
seemingly due go natural aptitude, and some who were keen on the academic
credibility it gave them... but genuinely I can only think if one person who
actually displayed a 'passion' or at least a deep interest in the subject.

I was very enthusiastic about physics and lots of people were about English
and we'd have creative writing groups and stuff, but maths was just... Some
folk quietly excelled at it and that was that.

------
dannyb
My only beef with his interesting post is that he uses graphics of completely
understandable stuff like diagonalizing a matrix or the map of the phase space
map of the logistic equation and then talks about current developments in
math. Anything new is almost completely incomprehensible to me - a couple of
years ago, I got a book about fractional calculus. I read about 5 pages away
and gave it to a mathematician friend of mine...

------
p_nathan
I have a mathematics minor as part of my bachelor's in computer science. I
have never once regretted my mathematics training. I sometimes wish I had
more.

------
guscost
One pet peeve about math education: calling "imaginary" numbers that makes it
seem like there is such a thing as a number that is "real" in a literal sense.
All numbers are actually imaginary, they exist as isomorphisms to things in
the real world that have numbers or can be counted.

------
tokenadult
Another good article about mathematics education, by William P. Thurston, a
Fields medalist:

<http://www.math.sunysb.edu/~mustopa/thurston_edu.pdf>

------
Natsu
The article's phrase "liberal education" is unlikely to be properly understood
in a lot of the USA. The political meaning has so eclipsed the ordinary
meaning that the latter seems all but unknown.

~~~
protomyth
I don't know about that. I come from what's described as a "red state"(1) and
the term "liberal education" is pretty well understood.

1) The funny part about the "red state" thing is that for years the 2 Senators
and 1 House Rep from ND have been Democrat. This changed this year.

------
HilbertSpace
Part I

Yet again we are flagellated, excoriated, eviscerated, etc. about
'mathematics'.

Still, some crucial points are missing. Been there; done that; learned the
lessons; and below are some crucial ones.

Yes, candidate understatement of the millennium is that people don't
understand math! Yup, they don't! That is, except mathematicians, and they are
a tiny fraction of the population.

I review some of the main directions and then give my view of the crucial
points and direction.

Best Undergraduate Major

Yes, in many ways math is a terrific subject. I recommend it as in many ways
(not all) as the best undergraduate major.

Why? First, because in all the rest of the academic subjects of physical
science, economics, social science, engineering, computer science, and now
even parts of biology and medical science, 'mathematization' of the field is
widely regarded as the best academic 'research progress'. E.g., mathematical
(theoretical) physics is the most prestigious part of physics, and the
situation is similar in the other fields. Second, because in all those other
fields, nearly all the people feel that they very much need to know more
mathematics. And, any mathematician who reads their work will readily agree!

In particular, the level of math in academic computer science research made
some progress with Knuth and since then has, in a word, sucked.

Outside of academics, the level of knowledge of math is so poor that at the
right time and place knowing some relevant math, that might not be very
advanced, can be one heck of an advantage.

For such an advantage, there is a general principal, a double edged sword: For
some knowledge to be a big advantage, it is nearly necessary that very few
other people understand it. So, if you really do have an idea that can put $1B
in the bank, before the money is coming in at a rate that makes the $1B look
likely, explaining the knowledge to anyone else will give only contempt,
laughter, anger, or silence. Generally people will give respect for something
they admire, say, making $1B, but some knowledge they don't understand
(without something like money clearly attached) will mostly just make them
angry. In particular, for such math knowledge, people in business won't
understand the math, and people in math won't understand the business. It can
be lonely at the top, or as a pioneer, etc. Generally, having a big advantage
later can be valuable but at first can be lonely.

Getting Paid

Since for nearly everyone, most of their career has to be directed to getting
paid, we need to say how math can contribute.

My guess is that for at least the rest of this century, math will be more
important for computing than Moore's law is, will be, or, really, so far has
been. So generally I'm optimistic. On this point, I expect that so far nearly
no one will agree with me. Still, such importance can be a long way from
getting paid.

Money for Academic Math

For the more technical academic fields, there has been one main source of
money -- the US Federal Government. Why? Before 1940, f'get about it! After
1945, D. Eisenhower, J. Conant, V. Bush and others were so impressed by the
role of math in WWII that Eisenhower supposedly said "Never again will US
science be permitted to operate independent of the US military." Conant, et
al., deliberately set up several sources of funding -- NSF, ONR, etc. -- so
that there would be no one place to cut off the flow of money. The Cold War
and the Space Race added more funding. By 1960, there was so much money for
research, including math, that a joke went "While you are up, get me a
grant.". Now commonly the top US research universities get about 60% of their
budget from NSF, NIH, DoE, etc.

Scenario: You are a university dean of the School of Science with the math
department, and they want to hire some profs. As the dean you look mostly at
(1) prestige for the university, (2) demand for courses, and (3) opportunity
for research grants. There (1) is okay for, maybe, 50 mathematicians today.
For (2), mostly f'get about it: The other departments and the math profs agree
that the math department shouldn't teach 'service' courses. So, the other
departments want to teach the math themselves or just f'get about it. Besides
now there is a history of math department service courses taught by people who
didn't speak English, and bitterness remains. For (3), some years ago there
was an Exxon executive David who lead the writing of a report that basically
claimed that the research and teaching in the math departments was next to
useless 'abstract nonsense'. One result was that the NSF, etc. felt more
justified in cutting back grants for math. Math had too little support in
Congress, and there were plenty of other fields that wanted the grants
instead. Net, in the research universities, the math departments went on
meager rations. They still are.

So in academic math, where is the 'action'? Well, there is plenty of screaming
that K-12 needs math teachers. Okay, so there are colleges with math
departments that specialize in such 'math teacher training'. Those colleges
need some profs who got math Ph.D. degrees from, say, a state university.
There the math profs got their Ph.D. degrees from research universities. And
there the math profs do research on generalized abstract nonsense that may not
go useless forever. So there is a pyramid with several levels, the lowest of
which is K-12 math teaching and the top of which are the math departments at
the usual suspects Stanford, Berkeley, Princeton, Harvard, etc. Of course what
a Princeton math prof does is essentially irrelevant to anything in K-12 math.
Being irrelevant is economically risky!

This pyramid is at risk: E.g., college departments of education might just do
their own teaching of math to students headed for K-12 math.

So, here's the good news about academic math: The stuff on the library shelves
isn't going anywhere!

~~~
JosefK
> In particular, the level of math in academic computer science research made
> some progress with Knuth and since then has, in a word, sucked.

Could you elaborate on this? As someone currently pursuing a PhD in
theoretical computer science, I often simply tell (nontechnical/nonacademic)
people that I study mathematics rather than computer science, so this offhand
remark is somewhat at odds with my personal experience.

~~~
HilbertSpace
Uh, when I wrote that, I didn't have you in mind!

I agree: The key, maybe nearly all the content, of theoretical computer
science is math.

As we design more complex systems in the future, we will need math to know at
least about correctness, performance, and economy.

If you are learning the math, then terrific, and you have one heck of an
advantage.

While independent study is often crucial, I advise you to have essentially all
of an undergraduate major in pure math and a carefully selected Masters in
math with also a lot of pure math. Getting all that on your own or while being
a 'computer science' student will be tough.

For "sucks", I've just read far too much material by CS profs where they try
to use or do math and make a mess. The first symptom is that they don't know
how to write math, say, as in Rudin, Birkhoff, Feller, Doob, Coddington,
Dieudonne, Bourbaki, etc. The second symptom is that they didn't absorb the
standard but rarely explained 'rules' for notation. E.g., there is the
disaster NP which by the usual notation just CANNOT be a name and, instead,
just MUST be a product of some kind. NP is borrowing from common programming
language notation based on, say, limitations of punched cards! Next, there is
the problem of failing to understand that, in English speaking communities,
math is written in complete English sentences. Then the common practice of
using mnemonic variable names as a substitute for English is totally
unacceptable and totally missing in good math. Next, beyond writing and
notation, when the CS profs start to get into the actual math, they blow it
again. E.g., there is a love for saying 'map' and then just stopping,
apparently believing something meaningful has been said. It has NOT! Instead,
just saying 'map' omits the DEFINITION of the 'map'. Saying 'map' without a
definition is meaningless. Such writing and notation is snake oil instead of
medicine, cardboard instead of carpentry.

One of the most recent disasters I saw was just screaming out for the 101
level of statistical hypothesis tests but totally missed it. Statistical
hypothesis testing was understood in at least some detail by K. Pearson over
100 years ago; the social scientists have had this material cold for over 60
years.

Next, I wrote a paper in computer science. Looking for a journal, I sent
copies to several computer science journals, including some of the best ones.
From two of the editors in chief, I got back essentially the same: "Neither I
nor anyone on my board of editors has the prerequisites to review your paper."
For one editor in chief, of one of the best journals, I wrote him tutorials
for two weeks before he gave up.

I thought about submitting to a theoretical computer science journal, and the
editor wrote me that my paper looked good for his journal. But I submitted to
Elsevier's 'Information Sciences' instead to get wider readership for the part
of my paper that was for practice. Then came the review process: It was grim.
I suspect that in the end the editor in chief walked the paper around his
campus; some mathematicians told him my math was okay but they didn't know
about the importance for computing, and some CS profs said it was nice for
computing but they didn't know about the math.

The way Knuth did and wrote math in 'The Art of Computer Programming' was
mostly not very advanced but fine. Since then my impression is suckage.

There's no royal road to math, and it's not a spectator sport. The
prerequisites I listed take about six years, and more experience is helpful.
Nearly no CS profs have those prerequisites, and it shows.

Net, the dichotomy is clear: Essentially every math prof I ever had or ever
read at least knows how to write and do math, especially with definitions,
theorems, and proofs. Other than Knuth, essentially every CS prof I ever had
or read at best floundered.

~~~
winxordie
You should read more of Chazelle or Tarjan; they both write their maths very
clearly.

Thank you for an amazing series of faux-blog posts on this thread. As an
undergraduate hoping to enter math research, these have been invaluable to me.
I can't upvote you enough times.

~~~
HilbertSpace
Sure, Tarjan is a good mathematician. Maybe CS wants to claim him, but math
should keep him for themselves! And the math of operations research -- Cinlar,
Nemhauser, Kuhn, etc. -- is good math.

