
Thirteen words that lose their meaning when the denominator approaches infinity - dbreunig
http://www.longtail.com/the_long_tail/2008/08/thirteen-words.html
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Chocobean
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
"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 :)

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hugh
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.

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Flemlord
Words that result in buffer overflows:

* more

* double

* triple

* ...

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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.

~~~
dangoldin
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.

~~~
time_management
(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.

~~~
dangoldin
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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trezor
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
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.

