Enormous Integers

> "These higher levels of largeness blur, where one is unable to sense one level of largeness from another."

IMO that's the really neat thing about enormous integers. They simply pass beyond our ability to reason about them in the same ways that we reason about numbers in our everyday life. We can conceive of two and three digit numbers simply by counting to them. We can conceive of millions and billions by imagining piles of material (a 2 foot cube of sand might be 1-2 billion grains). We can conceive of larger numbers by counting the number of digits, or equivalently, using logarithms -- often comparing them to the number of particles in the solar system or the known universe, or thinking in terms of how many bits are needed to store the number.

At some point, though, we start dealing with numbers so big that Knuth's up-arrows or Conway Chained Arrow Notation become necessary. We reach a point where the numbers no longer correspond to objects, distances, or physical quantities. These sorts of large numbers are truly abstract -- the only way we have to measure them is with other ridiculously large numbers, which themselves are only understood as notation.

See, for example, http://en.wikipedia.org/wiki/Large_numbers#Examples_of_numbe...

>> It’s like seeing a seasoned tiger hunter running through the jungle, yelling “Help! It’s a giant ant!”

I don't think anyone can top this analogy in the near future :)

On a tangentially related note, “Dark Integers”[1] is now perhaps the best short story I have ever read.

[1] http://www.asimovs.com/_issue_0805/DarkINtegers.shtml

What about busy beaver numbers? http://en.wikipedia.org/wiki/Busy_beaver

Large integers are sometimes easier to think about in terms of infinities. Coming to mind is the extremely counterintuitive Goodstein's theorem, which becomes comprehensible when one uses (infinite) ordinal numbers:

http://en.wikipedia.org/wiki/Goodsteins_theorem#Goodstein_se...

Anyone else wondering about the problem with 26 letters?