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.