Thirteen words that lose their meaning when the denominator approaches infinity

http://www.longtail.com/the_long_tail/2008/08/thirteen-words.html

by dbreunig • 18 years ago

6 12 18 years ago

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.

dangoldin2• 18 years ago

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.

OneSeventeen• 18 years ago

"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.

bdr• 18 years ago

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.

time_management2• 18 years ago

(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.

Chocobean• 18 years ago

We've always had sets that approach infinity, and we've never had any trouble using these words to describe them. Grains of sand/rice, stars, cells in our biosphere, drops of water...etc

And "All" or "no/none"? That's completely ridiculous. All water droplets on earth contain Hydrogen and Oxygen atoms. All Bloogles are Bloogles. No Floozles are non-Floozles.

The only thing new about "blogs" and "videographers" isn't the number of them, but that the definition of these words and their set boundaries are still being refined, and as such the number varies depending on one's criteria. A much better thesis would be "It's difficult to make statements about things difficult to define." which could be shortened to "Duh".

sysop073• 18 years ago

"A much better thesis would be 'It's difficult to make statements about things difficult to define.' which could be shortened to 'Duh'."

Hence the need for obfuscation :)

Flemlord• 18 years ago

Words that result in buffer overflows:

* more

* double

* triple

* ...

hugh• 18 years ago

This is a silly post. As others have said, there's no problem making "most elements of the set P have property Q", even when the set P is very large or infinite. "Most real numbers are not integers. In fact, almost all real numbers are not integers."

The problem with making pronouncements about "most bloggers" or "most youtube videos" is that we just don't know enough about those sets. They're large and new and rapidly-changing, and although many of us have experience with a large number of blogs or youtube videos, our experience is strongly biased towards the ones we find interesting, so we don't know much about the averages.

trezor• 18 years ago

While I get the basic idea of the article, the author obviously has no idea what he is talking about.

(absolute numbers are still meaningful in marketplaces where the number of products grows by orders of magnitude overnight, but percentages are not)

Anyone care to explain to me how this is even mathematically possible?

Kate• 18 years ago

He's simply making the point that even when your business can't capture a fraction of the total market that's worth discussing in numerical terms the numerator can still be quite noticeable. Your average small web company wouldn't mind having 100K paying users right now and they probably won't when 1M is less than 0.05% of net users. Reasoning about percentages and limits is relevant if your target user base is of the same scale as the total market or you're counting on a VC pitch based on winner takes all. It's not mandatory if all you care about is building a sustainable business: My local coffeehouse probably has a few hundred regular customers and their share of the worldwide coffee business is effectively 0%, but they're doing just fine financially.

Thirteen words that lose their meaning when the denominator approaches infinity

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