Gaal Yahas ([info]gaal) wrote,
@ 2009-04-26 21:19:00
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overmarking
Wikipedia is sometimes entertainingly ludicrous. In an entry about Conway's Chained Arrow notation, which is a system for writing freakishly huge numbers, we have:

2→4→3

= 2→(2→(2→(2)→2)→2)→2 (by 1) The four copies of X (which is 2 here) are in bold to distinguish them from the three copies of q (which is also 2)
= 2→(2→(2→2→2)→2)→2 (rrp)
[omitting several steps -GY]
= 2→(2→(2→(...16...)))) (3)
= 22...2 (a tower with 216 = 65536 stories)

which is unimaginably large.[citation needed]


I can't imagine that somewhere, there's someone who can imagine how large this is. But hey, I guess now they can cite this post and everything's all right?

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WP:RS
[info]Daniel [oeconomist.com]
2009-04-27 01:22 am UTC (link)
Eh, if they try citing your 'blog, then sanctions may ensue.
http://en.wikipedia.org/wiki/Wikipedia:Reliable sources

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Re: WP:RS
[info]gaal
2009-04-27 05:48 am UTC (link)
So now humanity is in pursuit of a cognitive scientist who would tell it the limits of imagination?

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Re: WP:RS
[info]Daniel [oeconomist.com]
2009-04-27 05:58 am UTC (link)
Wikipedia can be a pretty inhuman place, actually.

(Reply to this) (Parent)

[info]Daniel [oeconomist.com]
2009-04-28 03:22 am UTC (link)
Seriously, though, I'm not sure in what sense we want to claim that the tower is unimaginably large.

In some sense, the number 6 is unimaginably large for most people. It is only distinguished from 7 by reconceptualizing one as something such as 5 with 1 and the other as something such as 5 with 2.

Ask a typical person if he-or-she can imagine $655.36 worth of pennies, and he-or-she will probably say yes. And I believe that there is a sense in which he or she will be right.

The thought of a 65536-storied building perturbs me primarily as a structure that I would expect to be from 87 to 125 miles tall. (Being an American, I of course think in terms of G_d's units, rather than kilometers.) The thought of a one-storied building in that range would cause about the same difficulties for me (though I admit that it would likely involve less builing material).

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[info]gaal
2009-04-28 05:43 am UTC (link)
Um. The particular value under discussion is rather larger than 65536. (It's not the tower that's too big, it's the number it represents.)

Here's one sense in which it's too large. It would be very difficult for you to collate two numbers two numbers in this area. Here's another: it is even difficult for me to give you two numbers in this area -- the Conway notation skips over a lot of the integers, meaning to supply you with two numbers in that area I'll end up having to do something like to let x = 2→4→3; and then to let y = f(x) for some f. On "imaginable" numbers, there's symbolic manipulation I can do that would yield y directly; with x's neighborhood, there's simply no route except by this express elevator. (Writing in decimal is a log10n problem, which we manage for numbers in the (say) millions.)

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[info]Daniel [oeconomist.com]
2009-04-28 08:50 am UTC (link)
The principal point that I was trying to make could perhaps be captured by saying that we classify a number as imaginable or unimaginable relative to how we imagine using the number.

I believe that your collation example illustrates that point, as it notes a use effectively removed by virtue of the size of the number. Still some uses would remain to us, and for those uses the number is (as I contend) imaginable.

Except insofar as the imagined use were the translation itself from Conway's notation to some more familiar notation, I'm doubtful about a proposition that the numbers are unimaginably large based upon the difficulty in translating. (We could devise another notation such translation costs rose dramatically as numbers fell within a range that we would otherwise regard neither as large nor as unimaginably small.)

If the sense of unimaginably meant in the article is exactly that — that the translation were unbearably costly — then the claim could be replaced by some measure of the computational requirement.

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[info]gaal
2009-04-28 09:20 am UTC (link)
You could go with the use, but you can also interpret my collation example as a demand for distinction.

If you're going for complexity, you can do things like Kolmogorov complexity. The number originally discussed in my post actually has a reasonably enough small complexity in that regard, *but* it exists in an area of numbers where (I presume) the density of small-Kolmogorov-complexity-numbers is astoundingly small. This is perhaps a way to formalize the "distinction" criterion.

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[info]gaal
2009-04-28 09:22 am UTC (link)
(I was abbreviating; "n has small K" means "a program producing n has small K"...)

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[info]Daniel [oeconomist.com]
2009-04-28 10:25 am UTC (link)
Okay, I think that distinction is a special case of use, but I could also see that it could be a very special case.

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[info]gaal
2009-04-28 10:40 am UTC (link)
Wittgenstein would agree!

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[info]gaal
2009-04-28 10:53 am UTC (link)
Hmmm, there's a problem with my Kolmogorov complexity approach; it doesn't have much to do with largeness of the value at all. If we're dealing with integers, that's no big deal. But even in the reals, there are lots of small values, say in (0, 1), that require significant effort to "get to".

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[info]themoniker
2009-04-28 09:55 am UTC (link)
You don't need to dig much to find an abused "citation needed" in Wikipedia. But I'm just here to put the needed citation for "wikipedia's sourcing policy is ripe for parody":

http://wondermark.com/291/

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[info]gaal
2009-04-28 10:36 am UTC (link)
Hee!

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