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army1987 comments on Logical Pinpointing - Less Wrong
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The natural numbers are supposed to be what you get if you start counting from 0. If you start counting from 0 in a nonstandard model of PA you can't get to any of the nonstandard bits, but first-order logic just isn't expressive enough to allow you to talk about "the set of all things that I get if I start counting from 0." This is what allows nonstandard models to exist, but they exist only in a somewhat delicate mathematical sense and there's no reason that you should expect any physical phenomenon corresponding to them.
If I wanted to communicate the idea of numbers to aliens, I don't think I would even talk about logic. I would just start counting with whatever was available, e.g. if I had two rocks to smash together I'd smash the rocks together once, then twice, etc. If the aliens don't get it by the time I've smashed the rocks together, say, ten times, then they're either so bad at induction or so unfamiliar with counting that we probably can't meaningfully communicate with them anyway.
The Pirahã are unfamiliar with counting and we still can kind-of meaningfully communicate with them. I agree with the rest of the comment, though.
I was ready to reply "bullshit", but I guess if their language doesn't have any cardinal or ordinal number terms ...
Still, they could count with beads or rocks, à la the magic sheep-counting bucket.
It's understandable why they wouldn't really need counting given their lifestyle. But I wonder what they do (or did) when a neighboring tribe attacks or encroaches on their territory? Their language apparently does have words for 'small amount' and 'large amount', but how would they decide how many warriors to send to meet an opposing band?
Here's a decent argument that they probably don't have words for numbers because they don't count, rather than the other way round, contra pop-Whorfianism. (Otherwise I guess they'd just borrow the words for numbers from Portuguese or something, as they probably did with personal pronouns from Tupi.)