Mathematics: The Most Misunderstood Subject

http://www.fordham.edu/academics/programs_at_fordham_/mathematics_departme/what_math/index.asp

by spacemanaki • 16 years ago

144 43 16 years ago

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?

snikolov• 16 years ago

The author does imply that great teachers have a lot to do with it. At a later age, when one is out of school or almost out of school, one obviously needs to show more personal initiative. And a lot of people here and elsewhere have asked or written about how to best self-study mathematics. However, the importance of having a teacher is often not mentioned. Even if you are very self-motivated, a teacher can make an enormous difference by showing you the cool stuff that you might miss on your own. One can be very motivated about building a house by themselves, but might have only ever seen mud bricks, and hence will try to diligently build a house out of mud bricks. A proper teacher can illuminate and guide a student through the vast world of other things out there, just like someone trained in construction can show the self-motivated builder how to build a proper house.

Now, how does one find a teacher, you ask (especially when one has left school)? A few options I can think of:

* Go back to school.

* Participate in an appropriate online community.

* Reach out to professors or even students in nearby schools.

spacemanaki• 16 years ago

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...

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.

alexwestholm2• 16 years ago

Thanks! This looks like an excellent way to ground the concepts in more concrete terms.

jpren• 16 years ago

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• 16 years ago

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.

Dilpil• 16 years ago

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• 16 years ago

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• 16 years ago

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.

tokenadult• 16 years ago

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

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

hypersoar• 16 years ago

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.

shou45772• 16 years ago

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.

geebee• 16 years ago

"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."

I was math major, and I did an MS in a very math-related Industrial Engineering program (lots of proofs about convexity and stochastic processes). And honestly, I don't really understand this either.

And I actually know what "vector space", "combinatorics", and "square matrices" are, and I'm aware that algebra is more than a second year high school subject. I wonder if I'd get it if you spent more time explaining.

Unfortunately, your field is extremely abstract and difficult to understand, though it is kind of possible (that PBS special on fermat's last theorem did explain some pretty incredible things to people who don't have a math background).

lacker• 16 years ago

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.

gaurav_v• 16 years ago

I too am super pumped about Lie algebras, and have come up against the issue of how to explain the topic to my less mathematically inclined friends.

You have to start with groups. Group structure is something that's both beautiful and something that can be truly appreciated in a 'cocktail party' setting. And you can't convey anything of substance and true about Lie algebras to someone who doesn't know what a group is.

Then, once your listener is happy and feeling smarter about his new group theory knowledge, you can try to motivate the idea of a Lie algebra however best fits with your research. This way, you are able to tell your listener something comprehensible, and also hint at what else is out there.

At least, I've found that this explanation keeps people pretty happy.

endtime2• 16 years ago

I get the same thing as a CS person. Every time someone refers to me as an "IT major" or expects me to be able to fix their printer drivers I die a little inside.

Natsu• 16 years ago

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.

Jach• 16 years ago

I'm not sure the article itself is likely to be properly understood (or even read) by a lot of the USA. I like the idea of "evil education" though.

protomyth2• 16 years ago

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.

endtime2• 16 years ago

Well, the phrase "liberal arts education" is more common in the US, I think, but I also expect most people would get the meaning.

RiderOfGiraffes2• 16 years ago

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".

xyzzyz2• 16 years ago

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• 16 years ago

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.

shasta• 16 years ago

> 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.

You're missing part of the argument, which has to do with the nature of the kinds of structures that mathematicians find interesting. Otherwise you can use this reasoning to justify anything at all.

HilbertSpace4• 16 years ago

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!

HilbertSpace4• 16 years ago

Part II

Money for Academic Math Applications

Still in academics, if want to make a big splash outside math departments, then math can be one of your best tools and advantages. One approach: Learn some measure theory and functional analysis, standard early math grad school topics. Then, less standard, learn probability, stochastic processes, and statistics based on measure theory. Learn some differential equations -- big part of math. Then, also less standard, learn some optimization and control theory, both deterministic and stochastic. Yes, I'm not nearly the first to suggest such math topics; in recent years they have been proposed as 'the mathematical sciences' (that didn't catch on nearly as well as hoped). Then use this knowledge to build best possible, 'optimal', 'models' that attack the ubiquitous 'uncertainty' in other fields, write papers, teach seminars and then courses, write text books, do consulting, get grants and grad students, etc. Be a prof, maybe, in finance or production in a B-school or in EE in an engineering school.

Math Jobs Outside Academics

Likely the biggest opportunity for 'jobs' in math outside of academics is the US Federal Government, especially with DoD funding, related to national security.

Otherwise, for a 'job' with any very significant role for math, f'get about it. Why? To have such a job, except in very small companies or the DoD path just above, someone needs to understand the work of the job, write a job description, get the job funded in their budget, and put some of their career on the line that the money will be seen by the more senior managers as money well spent. That is, in essentially all larger organizations, the ideas of the factory floor 100 years ago are still in place: The supervisor knows more than the subordinate, and the subordinate is there mostly just to apply more blood and sweat to the work of the supervisor. So, since nearly no supervisors know much math, f'get about such jobs.

Or, suppose there is a mathematician in a large organization. At the top there is the CEO who forgot any calculus they might have learned. Between the two is middle management. So, by a standard math argument, somewhere in that management chain must be a mathematician reporting to a middle manager non-mathematician, and that won't work. So, yes, maybe a mathematician can be 'on staff' to the CEO. Don't hold your breath.

But how do other technical fields such as law and medicine work? From licensing, malpractice threats, professional codes of conduct, professional practice peer review, they have a LOT of professional status -- math doesn't. Also they are applied fields with their graduate education aimed almost entirely at practice -- that is, is 'professional' training -- instead of research, etc.; math isn't like that. In particular, law has a standard that a lawyer can report only to another lawyer.

There's a LOT of advanced math in high end academic EE, but it remains that an electrician's license can be a much better foundation for a career.

The Main Opportunity

Outside of academics and government, a relatively stable career nearly always needs a relatively stable collection of happy, paying customers.

To skip to the bottom line, math can be an advantage if the mathematician owns the business that is, except for the math, much like other businesses from Main Street to Silicon Valley to Wall Street.

So, the mathematician uses the math to construct the crucial, core, powerful, defensible (difficult to duplicate or equal) 'secret sauce' and implements it in software that delivers valuable results. It is the results, essentially only the results, that the happy customers pay for.

Back to my claim of more important than Moore's Law: We already know what the world wants in the famous one word answer, "More!". The main way for more is automation. For that, so far we've been just coding what we already knew how to do by hand or just intuitive or heuristic ideas. The main way to get more powerful software (that is, able to generate more valuable results) is to have it implement more powerful manipulations, and the main way to that is math, yes, complete with theorems and proofs (so that we can have confidence in the work), possibly original based on advanced material. My view is that for the rest of this century, (1) this math direction is (thanks heavily to DoD projects of the past 70 years) well proven and rock solid, (2) progress better than via math is not promising, and (3) progress without the math is not promising. Of course, just now, one advantage is that nearly no one understands the math or accepts this claim!

For the academic math departments, their 'teaching' pyramid is at risk. To get their field going again, they need to 'connect with reality' and deliver value that plenty of other people are willing to pay for, hopefully quite directly, otherwise at least indirectly. "The analytic-algebraic topology of the locally Euclidean metrization of infinitely differentiable Riemannian manifolds" or some pursuit of abstract beauty no one else can appreciate are NOT good directions.

JosefK• 16 years ago

> 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.

HilbertSpace4• 16 years ago

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.

p_nathan• 16 years ago

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.

Tycho• 16 years ago

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• 16 years ago

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...

guscost• 16 years ago

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.

Mathematics: The Most Misunderstood Subject

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