Sort of. The way I was taught, a progression is called arithmetic because each term is the arithmetic mean of its predecessor and successor, i.e. if A comes before a term X and B comes after a term X in an arithmetic sequence, then X = (A+B)/2. Similarly, a geometric progression uses the geometric mean, i.e. X = (AB)^(1/2).
Franks, John. "Flow equivalences of subshifts of finite type." Ergodic Theory and Dynamical Systems (1984). 4:53-66. Cambridge University Press.
Also, minor errors in mathematics papers are not uncommon. Usually, they are fixed and included in a later edition of the journal as "Errata to [name of paper]." The main purpose of the refereeing process is to detect game-breaking errors such as a missed condition on a theorem used or a misapplication of a tricky technique (usually from another field of mathematics). Interestingly enough, from what I've seen, budding mathematicians actually make a lot of errors that are similar to that of novice programmers (e.g. off-by-one errors in induction, forgetting about boundary conditions).
There are two biographies that I think should be required reading for wannabe Silicon Valley entrepreneurs: (1) High St@kes, No Prisoners by Charles H. Ferguson and (2) The New New Thing by Michael Lewis, about Jim Clark
And two that, while light on content (i.e. math), never fail to stimulate the childlike nature of my inner aspiring mathematician: (1) The Man Who Knew Infinity by Robert Kanigel, on Ramanujan and (2) The Man Who Loved Only Numbers by Paul Hoffman, on Paul Erdos.
In linear algebra, for a given linear transformation (a certain kind of matrix which generally represents some operation) M, an eigenvalue of M is a scalar c such that given a non-zero vector v (called the eigenvector), Mv = cv, where multiplication on the left is matrix multiplication and multiplication on the right is multiplication of v by the scalar c.
The definition is significant because it says that for a certain vector v, the transformation M, no matter how complicated it may be, just scales v by a factor of c. This is useful, for instance, if you want to determine an axis by which to evaluate the range of the transformation, because by choosing an eigenvector, you are choosing a "simple" or "natural" perspective from which to evaluate the range.
This is the problem I had with Linear Algebra. The first half of the class felt like I was just being drilled definitions. But once all the definitions click, it's rather simple.
The best thing about my linear algebra class was that the professor warned us about this up front. Yet, I still was not prepared for the onslaught of new words, and the bad part of the class is they were all defined in terms of more mathematical words, not concepts like this rubber band example.
That was the point of the "What are eigen values?" article. It was to give the intuition behind the scary mathematical definition.
I find this helpful, as it answers the all important questions 1) Why should I care? and 2) What is the basic problem that the scary math is trying to solve?
With answers to those questions in hand, you can return to the scary math and work out how it maps to those answers.
Write the idea down somewhere. However, ideas are a dime a dozen and are essentially useless unless you have a viable plan of attack. For instance, I would love to work on something world-changing like time-travel, teleportation, or anti-gravity, but there are currently no fruitful ways of even approaching those problems.
I had an advisor tell me that I should always be keeping several "big" problems on the backburners at all times, so that whenever I find out about a new technique or technology, I will be able to immediately think of ways to apply the new tools to the big problems. He said it will fail 99 percent of the time, but on the 1 percent of the time that it does work, everything will somehow fit together and with any luck, you will have found a viable approach.
To me, the article isn't saying to abandon the road to mastery, but merely to take some detours once in a while. Dedicating a lot of time and energy to something is definitely fulfilling and has a compounding effect, but as many astute programmers have noticed, there is an echo-chamber effect if all the information you get is from programmers and the programming world. Worse, you could continue to hold incredibly inaccurate beliefs about other people like the professional photographer in the article, and make judgment calls based on that false information.
Of course you need something to bring to table in modern times, to be a productive member of society. However, that doesn't mean to specialize to the extent that you lose all context. Dig deep, sure, but come up for air sometime and take a look around. You'll be surprised at what you may discover.
To the extent that it is possible to strike that balance, you'll get no argument from me. :-)
I think the reason every reasonable person doesn't just do this without needing to read an article to remind him of the value of doing so is that it's really hard to do in these modern times.
Before I graduated from high school and started working in tandem with college (which rapidly escalated into a full-time career development path and compelled me to drop out, eventually), I used to hold a lot of people in deep contempt for being so ignorant and narrowminded at a time of historically unprecedented access to a never before seen breadth of information; between the Internet, book sellers, public libraries, universities, etc. there's just nothing you can't learn about if you want to these days.
What I had no concept of is that after you come home from a 10-hour day at an intense job, even a technical white-collar job that seems physically undemanding from a superficial perspective, all you want to do is just veg out; even then, there are too many other bullshit errands to do. It's not just strictly a matter of time being available logically; it takes energy, mental and spiritual, to come home at 7 PM and then compose symphonies until bed time.
Some people are more insistent than others at doing what they want to in spite of what they have to, but it's not reasonable to expect most people to do that even if the availability is technically there. We may have it easy compared to our ancestors from many economic and physical points of view, but that doesn't mean we aren't plagued with some of the problems existentially perennial to the economic man.
EDIT: Also, I am not sure that the "echo chamber" that characterises Valley web startup culture is any more sealed or myopic than the echo chambers of other comparably specialised combinations of professional endeavour. Ever seen what financial instruments traders breathe, eat, snort, etc. 24/7? People who participate in insurance industry MLM schemes? People "tracked" for "blue-collar" skilled trades? People who love and excel at working on cars or motorcycles? Realtors, mortgage brokers? I'm talking about the folks that are on top of their game in those respective sectors - the "hackers" - not the most bromidically average, uninspired 9-to-5ers. It's very similar.
Yes but you are working with the assumption that you must be "on top of [your] game". i.e. that you must "win". Alan Kay said that perspective is worth 80 IQ points. Let me give you an example: Suppose you decided to drop out for a year at the risk of sliding back from the top of your game. Suppose you went to a third world country and worked for a charity. Suppose you came back with a different perspective namely: Most of the stuff we do in the west is completely superflous. Its about making life even more convenient and sumptuous for those who already have far too much. That in fact every minute of your amatuer and inefficient time was worth hours of what you were doing professionally because the work you did as an amateur was so much more important. In summary: The persepctive you get is that the game you were trying to be at the top of is a game not worth playing, that really it was doing you and the world no good and that on your deathbed being able to say: "I was the best at X for a while, knew everything there was to know about it" - doesn't amount to much. Note: I haven't done this. I am just saying that I think it is possible based on some experiences I had while travelling (and neglecting my game). Bill Gates seems to have had a similar insight but true to form he went and did something about it.
The persepctive you get is that the game you were trying to be at the top of is a game not worth playing, that really it was doing you and the world no good and that on your deathbed being able to say: "I was the best at X for a while, knew everything there was to know about it" - doesn't amount to much.
Certainly can't argue with that. I'm just trying to provide a useful account of why people who play the game play it with the total commitment that they do.
In my experience, reading papers with math gets a lot faster as you get used to the conventions. It used to take me almost a whole day to digest a single math or CS paper in a field I was familiar with, but now I can get through a couple in an afternoon, provided that I am not interrupted.
The "chunking" that you develop is like that in chess, or programming for that matter. This seems to be one of the primary reasons why journals will reject papers with unconventional notation. Unfortunately, the notation doesn't seem to stay uniform across disciplines (math <-> physics) or even languages (english <-> french) even when using the same mathematical structures.
That certainly seems true, but how much of that is because by doing so you get to go to a school like Caltech, Harvard, or MIT? A more interesting question to me would be whether American IMO contestants have a better chance of doing good mathematical research than other math majors at their school. But it's hard to come to any definite conclusion, especially given the small sample size and the difficulty of classifying what is "good research."
From what my high school math teacher told me, it's because the modern U.S. curriculum was last drastically revamped in wartime to pump out future engineers and physicists, which is why there is so much emphasis on trigonometry and calculus rather than sets, logic, statistics, or discrete math.
