Wittgenstein, according to this book I'm reading [1], is quoted as saying "Suppose you had all done arithmetic within this room only and suppose you go into the next room. Mightn't this make 2+2=5 legitimate? If you came back from the next room with 20x20=600, and I said that was wrong, couldn't you say 'But it wasn't wrong in the other room.'"

Yes, you could say that. You wouldn't be correct, though.

2, 5, 20, 600 are all natural numbers. These numbers arise from the human brain's natural ability to atomicize objects into countable entities. Let's say you have a bunch of stones in your hand. When you look at your group of stones, you would unconciously atomicize that group into either the number of stones in your hand, or into groupings based on their shape, size, color, etc.

This isn't merely because you were taught to count as a child but because counting is innate. The fact that the theory of natural numbers and basic arithmetic developed in multiple cultures entirely independant of each other is evidence of the human instinct to atomicize. I have a hard time imagining any human (of average intelligence) who couldn't logically deduce that 2 stones added to 2 stones isn't the same as a group of 5 stones but is the same as a grouping of 4 stones.

Since counting is such an innate ability, I wonder if Wittgenstein's theory is an example of the P-and-not-P paradox.

Wittgenstein might also be playing games with the names of numbers (i.e. xxxxx isn't 5 x's but some other "number"), although that seems far beneath his abilities. Although he was into linguistic puzzles I'm not sure I would classify that as such.

Like usual, email on the subject is welcome. Or even an argument in your own blog! Blog Battles are fun.

[1]: Wittgenstein's Poker ISBN:0-06-6211244-8, pg 14. Whether Wittgenstein actually said this could possibly be in dispute but it's poor philosophy regardless.