Your most counterintuitive yet simple math problem

[adriank]

No, not quite, since you're not picking a random function from [f]. I think what Vid was saying was that you should pick a random function from [f]. In any case, it doesn't work; that's what I proved in my post.

 

[M Grandin]

I think few "counterintuitive" math problems remain "counterintuitive" when you at last understand them. Maybe "counterintuitive" before you have grasped it, but not after you have been aquainted to the concept.

But still there is one thing I find counterintitive more than anything I have learnt, although
belonging to physics rather than mathematics: Bernoullis fluid law, where static pressure
gets lower in more narrow passages while the speed of fluid at the same time is higher. I understand the theory, but still I find it very counterintuitive - not the least in practice, where you watch the flowing river surface sinking above for instance a bottom stone.

 

[Hurkyl]

There's a joke that says there exist only two kinds of theorems in mathematics: trivial ones, and deep ones. Trivial theorems are those I undersand, deep theorems are those I don't.

 

[CRGreathouse]

Let f:R->R be any function. No constraints except that its a function. Tell me every value of f except at x_o. Now, we take the set of all functions from R->R and define an equivalence relation where f~g if f and g differ at finitely many points. Take any member g from the equivalence class of f. Since f and g differ on a set of measure zero, f(x_o) = g(x_o) with probability 1.

Of course the fact that we can choose g relies on the axiom of choice.
:D

This reminds me of a paper I read a few years ago, with a title along the lines of 'using the Axiom of Choice to see the future', which discussed similar techniques (as I recall!) to take a glance at a time epsilon in the future.

I found the paper I was thinking of. Citation and quote:
Christopher Hardin and Alan D. Taylor, "A Peculiar Connection Between the Axiom of Choice and Predicting the Future" (2006):

Specifically, given the values of a function on an interval (−∞, t), the strategy produces a guess for the values of the function on [t,∞), and at all but countably many t, there is an ε > 0 such that the prediction is valid on [t, t + ε). Noting that any countable set of reals has measure 0, we can restate this informally: at almost every instant t, the strategy predicts some “ε-glimpse” of the future.

 

[Count Iblis]

http://www-personal.umich.edu/~jpboyd/boydactaapplicreview.pdf"

 

[HallsofIvy]

Of course, the word "converge" is (intentionally) incorrect there.

 

[TXCraig1]

How about the sum of all positive integers s= 1 + 2 + 3 + 4 + ... = -1/12

 

[Char. Limit]

How about the sum of all positive integers s= 1 + 2 + 3 + 4 + ... = -1/12

Technically that's not true, at least not in terms of a normal sum. I believe it is true if you consider the Abel sum of all positive integers.

 

[TXCraig1]

Technically that's not true, at least not in terms of a normal sum. I believe it is true if you consider the Abel sum of all positive integers.

It's not Abel summable, but there are several ways to derive the result such as analytic continuation of the Riemann zeta function or Ramanujan Summation, but the simplest was proposed by Euler. Perhaps more interestingly, it has been experimentally measured, in the Casimir effect for example.