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A paradoxical math ratio

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"This itself is a rather surprising result: Euler’s constant e comes out of nowhere!"

Is it? There's an integral with a 1/x term[1]. How exactly is it surprising that logarithms start showing up? I realise integrating by hand is a pain, but surely it's a good idea to at least think about the integral expression before typing it into Wolfram Alpha. (EDIT: to clarify - the author takes integration for granted, yet is surprised when the integral evaluates to a logarithm; this makes little sense to me, integrating polynomials is the first thing they teach you)

[1] 1/(1-x) is transformed into 1/y in a pretty straightforward substitution. When you do that with our actual integrals, you find yourself with 2 identical integrals, as you well should, given the symmetry. My solution was actually to look at the distribution of breaks already sorted into short & long, which means doing only one of the integrals (short sticks are 0..1/2) and divide by the range, since we're taking an average (i.e. multiply by 2).

I like cheese. I also find your tone disturbing.

To get really blown away, consider (again if necessary) that the average spacing between neighboring primes (primes!) less than N roughly approaches ln N [1]... Stunning.

[1] http://en.wikipedia.org/wiki/Prime_number_theorem

I think the bonus question is more counter-intuitive, not only is the inverse ratio not the reciprocal, it doesn't exist.

Exactly. There's nothing paradoxical about this at all, just a bit counter-intuitive.

Well, he does say 'seemingly paradoxical', and look what happens when you try and express this (naively) in English:

"The shorter stick is typically (about) 2/5 the length of the longer stick, but the longer stick is typically much more than 3 times the length of the shorter stick."

To be honest, I can't figure out how to summarize the two results in English correctly (short of saying 'the mean of x/y~=2/5, but the mean of y/x does not exist, where x is the shorter and y the longer length')

There've been some psychological studies on averageness, and I believe one common case (though I'd have to look it up) is that people think of "typical" as a representative example in the middle of a distribution, something like a peak of a distribution and/or a median, and then imagine the distribution as a cloud of "stuff on either side of that".

So the intuitive answer to the question would require there to exist a "typical" stick. Once you pick one specific stick, of course things work intuitively: the stick's dimensions are x=A, y=B, its shorter/longer ratio is min(A,B)/max(A,B), and its longer/shorter ratio is max(A,B)/min(A,B).

An interesting question might be whether there's a definition of "typical stick" that in any useful way provides the answer that's intuitively wanted here.

Our brains aren't exactly wired to handle uniform distributions too, I guess. Nature is full of normal distributions anyway.

I think the notion of the average ratio makes little sense. Maybe a geometric mean? I dunno.

Your problem here is that “typically” and “the mean” are two very different concepts. Taking the mean of a set of ratios is frankly not very meaningful, and so saying that the shorter stick is typically about 2/5 the length of the longer is highly misleading.

Strong counter-intuitiveness is a widely accepted use of the word “paradox”, even by logicians. (See, e.g., W. V. Quine’s The Ways of Paradox.)

Hmm. Well, Quine and I will just have to agree to disagree, then.

But in this case, I would hardly call the counter-intuitiveness "strong."

Now I wonder whether you are brushing off Quine out of the conviction that it is a matter of opinion anyway and that his actual argument therefore doesn't matter? In which case I would like to point out the advantages of challenging your own opinions. Sometimes opinions turn out to have been, well, let's say: much harder to defend than originally thought and even though you can defend them as 'mere opinion', you just don't want to anymore. Decent philosophical treatises have at least changed my 'opinions' about some things.

On the contrary: I think Quine is being sloppy with language, as many philosophers reserve "paradox" to stand for something far stronger than mere counterintuitiveness.

At the moment, I'm in the middle of writing a dissertation on Nagarjuna-- now, there was a master of paradox-- and I agree completely with you that decent philosophical can change a reader's opinions profoundly. I just don't put Quine into that categorization.

Naturally, YMMV.

Why would you expect the mean of a set of numbers to be the reciprocal of the mean of reciprocals of those numbers? It seems pretty obvious that those two operations (taking the reciprocal of the numbers, taking the mean) can’t be reordered.

To take a trivial example, the mean of 1 and 2 is 3/2, but the mean of 1 and 1/2 is 3/4. It’s easy to see that 3/2 ≠ 4/3. Now let’s try something that pumps the difference up. The mean of 1/1000 and 1 is 1001/2000, about half. On the other hand, the mean of 1000 and 1 is 1001/2, about 500. Clearly 500 is not the reciprocal of 1/2.

In the case where we have the two sticks, the ratio of small to big is going to have to be somewhere between 0 and 1, because we’re taking the mean of a set of numbers between 0 and 1. On the other hand, in the ratio of big to small, we’re taking the mean of a set of numbers between 1 and ∞.

It's a funny trick, but it touches on something that I can't help but feel is completely nonsensical: repeatedly picking random numbers from an infinite set.

It's basically a longer version of the following trick:

1) Pick a random real number in the range [0, 1]

2) Pick another

3) What is the probability that they are identical?

For another example, see: http://en.wikipedia.org/wiki/Bertrand_paradox_%28probability...

It seems to me the probability that they are identical is zero (or at least zero at the limit).

My hunch is that it's meaningless at the limit.

I like this alternative problem statement (not for the ratio part): If you break a stick in two places, what is the probability that the pieces can make a triangle?

There is a nice geometrical solution which is described here: http://www.cut-the-knot.org/Curriculum/Probability/TriProbab...

This is pretty basic math. A fun puzzle for high school students maybe, but anyone remembering any of their comp sci or engineering courses should be able to solve this very quickly.

Though it's a nice bit of interesting Maths for a Tuesday afternoon!

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