antiform

Why is this completely necessary? What would be an acceptable proof to you of this statement?

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).

I believe the paper the author is referring to is

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.

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.

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.

From the real number axioms:

Let x \in R. Then we have that

  x*0 + x*0 = 0*x + 0*x = (0+0)*x = 0*x = x*0

by commutativity of multiplication, distributivity, the existence of a neutral element 0 for addition, and the commutativity of multiplication once again. Thus, we have that

  x*0 + x*0 = x*0.

For clarity, define

  A = x*0.

Then we have that

  A + A = A
  A + A + (-A)  = A + (-A), by the existence of additive inverses.
  A + (A + (-A)) = (A + (-A)), by associativity of addition
  A + 0 = 0, by the definition of additive inverses
  A = 0, by the definiton of the neutral element 0 of addition

which gives us our desired result.

Because the concentrated effort needed to "not do something" seems to increase the likeliness that you will do exactly that, especially under stress. For some reason, it seems that it is not the will or intent of the thinker that is important, but what the thinker is actually thinking that controls reflexive actions.

It also tends to work better than not telling anybody about it at all.

If you tell people that you are quitting something, there is a social pressure on you to follow through on your promise. Your friends and family might also be willing to help you achieve your goal, which can be encouraging and push you that much further.

Well worth it, if you ask me. Best of luck to the author.

Some schools are fighting back against this. For example, I know that the UCLA math department has a policy that professors are required to search for a replacement textbook once a textbook goes beyond the third edition, unless there is a significant addition of material (e.g. entirely new chapters needed as a result of recent developments). The idea being: "If the author hasn't got it by the third edition, they'll never get it."

I believe this eliminated the majority of calculus/linear algebra/diff-eq texts from introductory courses, so it has become easier to find used copies.

I'm glad I'm not the only person who has #3. For some reason, I seem to get miraculous insights into problems when I'm exercising, so I've taken to keeping a tiny pen and a couple index cards within reach at all times. While running, this has sometimes resulted in me crouching on the side of a street, scribbling madly while constantly looking for oncoming cars, but when I get back home and finally finish that tricky proof or implement that evasive idea into code, I find it is well worth the trouble.

"It's not a shortcoming, it's a feature!"

While I am now an Apple fan, this reminds me of the ridiculous excuses Apple fanboys gave me when they compared their early-generation iPods to my MP3 player at the time (a Rio Karma).

  "It doesn't support gapless playback because in modern times, you should focus songs not albums!"
  "It only holds 10 gigs, because you should only listen to your best songs on the go!"
  "You don't need a button for skipping tracks, because you should spend your time listening to music, not switching between songs!"

Does anybody here know somebody that has used this program to aid their learning of computer science concepts? It's certainly an admirable cause, but is this a better introduction to programming than something like Logo, which has been used to teach programming to kids for many years?

Alice is impressive, and you can see the obvious programming influence, but it seems like "Moviemaker with control flow and objects" more than anything else. What problem is it trying to solve?

For me, the hardest part of learning elementary computer science was not things like OOP or if-statements, it was data structures, algorithms, and the difficulty of keeping many different interactions in your headspace. It seems like in many CS programs, the data structures or algorithms class is the "weeder" that separates the CS majors from the wannabe CS majors. Personally, I believe the drop-off at this level is more important to address than how many students enroll in CS 101.

These are the times when I wish that this technology was open-source (correct me I'm wrong), so if it was lacking, it could be added. I would love to have some similar way of turning commutative diagram sketches into TeX-ready form. It would save me hours of times fiddling with the current LaTeX packages.

No way. There are tons of people in the sciences with very interesting lives, they just don't have people knocking down their door to write popular biographies like Feynman, Erdös, or Tesla.

I say that living mathematicians like John Nash Jr., Shing-tung Yau, and Alexander Grothendieck have much more interesting lives than the examples you mentioned.

I believe this link was flagged and killed when posted before.

  http://news.ycombinator.com/item?id=523065
  http://news.ycombinator.com/item?id=544076

Does HN not remember flagged/killed posts?

The reason that short, frequent publishing is so pervasive on the web is that you have to compete with everything else on the internet that is shiny, flashing, and begging for your attention. I think that we, as humans, have a predilection to focus on real-time updates, on what's happening NOW. It is the only reason that I can possibly understand the success of things like Twitter or Facebook status updates.

Furthermore, there are so many benefits to publishing frequently that it's hard to argue against it. For instance, if you publish frequently, you will have a better Google ranking, be seen as an "active" member of the community, get more links on social news sites that link submissions to URLs, constantly show up in peoples' RSS readers, etc. Also, if you write a long, substantive article, many people will not take the time to read it. Most will just skim to see if there's anything interesting, and if they don't find anything, they will leave.

Also, if you want to make money on the Internet, it's something that you need to do daily, or at least very regularly. Dollars follows peoples' eyeballs, and peoples' eyeballs follow constantly updated content. I can't think of an example off the top of my head of a blogger, videoblogger, etc. making a living by posting infrequent posts but I can list a couple dozen people that make very good money with either gawker-like blogs on certain topics, a regularly updated promotional (video)blog, or through advertising dollars on popular YouTube channels.

I know in my rational mind that it is probably better to read content-rich articles on the web, but my reptilian mind is constantly drawn to the new, the hot, the now. Until somebody finds a solution, there will always be a short battle with myself every time I fire up Google reader or think of checking HN.

A good way for me to maintain motivation when slogging through information-dense nonfiction books (like textbooks, programming language manuals) is to read through actively (writing notes/questions in margins or post-its) and then to quickly "generate" something from what I just learned. For instance, if I am reading a math textbook, I will play with the theorems, adding and subtracting conditions and generating counterexamples, and then do all the problems and TeX them up. If I am reading a philosophy book, I will summarize the accounts of the section and then argue my own perspective, essentially writing a brief philosophy paper. If I am reading something that I tend to mostly disagree with, like GEB, I will even try and write a "devil's advocate" paper, where I try suspend by disbelief and figure out why the particular perspective has appeal by arguing against myself.

In the short run, this may mean that you take a lot more time to finish books, but it has drastically improved my ability to recall what I have learned and my ability to not abandon the process after I hit a wall. This not only because you are immediately reinforcing what you have just learned, but I've found that if I see the fruit of my labors, even if it's just a steadily growing PDF on my desktop or a short, elegant program in my home directory, it provides just enough to motivation to finish the next chapter, and the next chapter, until you're finally done.

In my opinion, the original quote by Aldous Huxley is more accurate:

"An intellectual is a person who has discovered something more interesting than sex." [http://en.wikiquote.org/wiki/Aldous_Huxley]

It's a review of the book by the guy who wrote two articles from the Atlantic article that were popular on HN recently: [http://www.theatlantic.com/doc/200501/kirn] [http://www.theatlantic.com/doc/200711/multitasking]

First of all, I am no expert in psychology, and most of my exposure to it was through readings in AI courses.

For a good example of a milestone in modern psychology, consider the Rescorla-Wagner Model--an attempt to model classical (Pavlovian) conditioning in a quantitative manner--which essentially implies that organisms only learn from unexpected events. It not only predicts the degree to which an organism would react to stimuli, but also compensates for something called the "blocking effect," which occurs when something stimulates more than one association and one of those associations overpowers a weaker association, prevent conditioning of the weak association. It is similar to the Widrow-Hoff Least Mean Squares (LMS) learning algorithm that you might learn in a neural networks class. This model is something that has been tweaked and tested under many different conditions and experiments, and the results have been distilled through thorough statistical analysis.

The R-W Model is one part of a dramatic change in our understanding of learning that has occurred in the last couple decades. For a summary of some of these results, Rescorla has an article published about 20 years ago called "Pavlovian Conditioning: It's Not What You Think It Is," [http://jsackur.free.fr/classiques/Rescorla1988.pdf] which should be pretty accessible.

I really dislike it when people (especially people of a technical mindset like math/physics/CS majors) summarize the above article in the way, and use it to justify their dislike of the social sciences. It's not about the problems of methods in psychology in general, but about blindly applying methods (of any field) without understanding the motivation, underlying assumptions, or unintended consequences behind them.

Most modern psychological research is surprisingly rigorous and depend on sophisticated statistical methods. The results may not be as strong or conclusive as those of, say, chemistry, but for a discipline that has to study something as mercurial as human behavior, it is certainly a useful and worthwhile.

Also consider that if you're a musician, people are less likely to say that they dislike music to your face, out of politeness.

I don't think it's so much cultural pressure as it is unusual. It's like meeting somebody who doesn't like art. It is so common and there are so many kinds out there that categorically disliking it entirely is strange. Most people probably do not want to be seen as strange.

It's also possible that in Western culture, not liking music is traditionally associated with something sinister. For instance, in Julius Caesar, Cassius is described as follows

  "...He reads much,
  He is a great observer, and he looks
  Quite through the deeds of men. He loves no plays,
  As thou dost, Antony; he hears no music.
  Seldom he smiles, and smiles in such a sort
  As if he mock'd himself, and scorn'd his spirit
  That could be mov'd to smile at anything.
  Such men as he be never at heart's ease
  While they behold a greater than themselves, 
  And therefore are they very dangerous."

which contrasts with the more sympathetic character, Brutus, and his love of music.

If I could choose one book on sales, it would be

Non-Manipulative Selling by Tony Alessandra.

This is a diamond in the rough among sales books. Many other books on sales, at least in the few that I've read and others I've skimmed, seem to be full of subtle and not-so-subtle psychological manipulation that always gave me an uneasy feeling in my gut once the conversation ended. This book is different. However, even if you believe in the "hard sell" philosophy, you should read this, since at the very least, you learn many tips to prevent negotiations from getting hostile.

This was discussed in this thread, but this tends to be more of a trait of immigrant parents rather than Asian parents. I grew up in places like New York and California which was full of first-generation immigrants (mostly Asian and Middle Eastern) who had come to the U.S. to kick ass and take names academically and they were the hardest on their kids. However, if you meet second-, third-, or fourth-generation Asian-Americans in places like Hawaii or the Japanese or Cantonese-speaking parts of the Bay Area, they tend to fit the profile you describe.

This is certainly an interesting approach, but does anybody have any firsthand experience with learning a technical discipline from primary historical sources? While I do enjoy reading the original papers for many mathematical ideas, it seems like if it were done in a course, it would be very slow-going and leave much of the material—especially in a huge subject like Discrete Math—uncovered.