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!
> 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.
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.
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.
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.
Bear in mind that computer science is quite young -- clarity, unambiguity and formalism in mathematics were not much better in XVII and XVIII centuries. Everything will improve over time, I believe.
I agree. In pretty much all of my theoretical computer science courses, whenever it came to the math there always seemed to be a lot of hand-waving and noise, but not a lot of content. Some were better than others, but when I consider some of the scores I got on tests (much too high!) for basically just puking up random math formulas, I was beginning to suspect that the TAs (and possibly the professors) didn't understand this stuff much better than I did.
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.
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!