Comment by time_management | Hacker News Reader

Comment by time_management | Hacker News Reader

time_management

It's true that you can't put a uniform probability measure on Z, the integers, in the way that you can on a finite set {1..n} or, for that matter, [0, 1). However, to say that one can't put any probability measure on countable finite sets, the minimum that would be required to have a definition of "most", is clearly false.

I might say "most words have vowels", and few people would disagree with me that this is a valid claim. The denominator is infinity, in the sense that I'm not constricting "words" to those existing now, and language is an open system. The set of strings that could conceivably become words is infinite. (Who would have predicted "blog" and "pwn"?) On the other hand, if we place a probability measure on words according to frequency of use (allowing that m("zwrskwzpn") might be some very, very small positive number instead of 0) we can now make statements such as "most words contain vowels", with the tacit understanding that our claim of "most" involves a weighting according to frequency of use.

Sure you can: 50% of positive integers (Z+) are divisible 2.

It just doesn't do much and can be easily proved to be true.

Most of the time when people use qualifiers like that it's to express some type of instinct - which seems to be what the author is against.

"Most of the time when people use qualifiers like that it's to express some type of instinct - which seems to be what the author is against."

Hah! Wonderful sentence. Love the irony. Also concur. Sorry if this is a bit off topic, but I couldn't let such quality wordsmithing go unheralded.

Saying that "50% of positive integers (Z+) are divisible 2" is probably true no matter what definition you use. But it's also fair to say that whatever you do mean by that does not generalize in the natural ways you'd expect it to from dealing with finite sets.

(Edited, 6:06 pm)

Good point, and it's probably reasonable to make statements like "most integers are not prime" or "perfect numbers are very rare among the integers". However, I'd still disagree with the use of percentages. For example, "0% of integers" are prime, in the sense that primes become arbitrarily sparse as N -> infinity, but primes clearly exist, which doesn't conform with our concept of "0%".

The set of even numbers intuitively seems to be "50%" the size of the integers, but they're actually sets of the same "size", because there is a bijection between them. So I'm not fully comfortable with saying "50% of integers are even" even though it's intuitively true. A lot of intuitively true things are false with infinite sets. For example, we generally assume commutativity of addition to the point that we might believe:

1/1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... = 1/1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + ...,

because both sums contain the same terms, only in different order, and in finite sums we can rearrange terms in any way we wish. However, the sums converge to different values, and terms of such a sum (conditionally but not absolutely convergent) can be rearranged so as to converge to any real value.

However, we can both agree that the probability of choosing an even integer from {1..n} (or {-n..n}) -> 0.5 as n -> infinity.

True. Dealing with probability and infinite sets can cause some problems.

It's a bit weird how the size of positive integers is equal to the size of the positive even integers and yet the probability of choosing one from the other is not 100%.

Another more extreme example is that the size of the Z+ is equal to the size of the positive rational numbers (a/b where a,b in Z+) but what is the probability that a rational number is an integer?

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