Comment author:Wei_Dai
30 December 2012 09:46:58AM
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8 points
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How come we never see anything physical that behaves like any of of the non-standard models of first order PA? Given that's the case, it seems like we can communicate the idea of numbers to other humans or even aliens by saying "the only model of first order PA that ever shows up in reality", so we don't need second order logic (or the other logical ideas mentioned in the comments) just to talk about the natural numbers?

Comment author:Qiaochu_Yuan
30 December 2012 11:56:01AM
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7 points
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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.

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?

Comment author:[deleted]
13 July 2013 05:54:05PM
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1 point
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(0
children)

Comment author:[deleted]
13 July 2013 05:54:05PM
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1 point
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Still, they could count with beads or rocks, à la the magic sheep-counting bucket.

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.)

Comment author:Wei_Dai
30 December 2012 08:08:19PM
3 points
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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.

Is it just coincidence that these nonstandard models don't show up anywhere in the empirical sciences, but real numbers and complex numbers do? I'm wondering if there is some sort of deeper reason... Maybe you were hinting at something by "delicate"?

If I wanted to communicate the idea of numbers to aliens, I don't think I would even talk about logic.

Good point. I guess I was trying to make the point that Eliezer seems a bit obsessed with logical pinpointing (aka categoricity) in this post. ("You need axioms to pin down a mathematical universe before you can talk about it in the first place.") Before we achieved categoricity, we already knew what mathematical structure we wanted to talk about, and afterwards, it's still useful to add more axioms if we want to prove more theorems.

Comment author:Qiaochu_Yuan
30 December 2012 09:52:41PM
5 points
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Is it just coincidence that these nonstandard models don't show up anywhere in the empirical sciences, but real numbers and complex numbers do?

The process by which the concepts "natural / real / complex numbers" vs. "nonstandard models of PA" were generated is very different. In the first case, mathematicians were trying to model various aspects of the world around them (e.g. counting and physics). In the second case, mathematicians were trying to pinpoint something else they already understood and ended up not quite getting it because of logical subtleties.

I'm not sure how to explain what I mean by "delicate." It roughly means "unlikely to have been independently invented by alien mathematicians." In order for alien mathematicians to independently invent the notion of a nonstandard model of PA, they would have to have independently decided that writing down the first-order Peano axioms is a good idea, and I just don't find this all that likely. On the other hand, there are various routes alien mathematicians might take towards independently inventing the complex numbers, such as figuring out quantum mechanics.

Before we achieved categoricity, we already knew what mathematical structure we wanted to talk about, and afterwards, it's still useful to add more axioms if we want to prove more theorems.

I guess Eliezer's intended response here is something like "but when you want to explain to an AI what you mean by the natural numbers, you can't just say The Things You Use To Count With, You Know, Those."

Comment author:shminux
31 December 2012 12:03:30AM
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2 points
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How come we never see anything physical that behaves like any of of the non-standard models of first order PA?

Umm... wouldn't they be considered "standard" in this case? I.e. matching some real-world experience?

Let's imagine a counterfactual world in which some of our "standard" models appear non-standard. For example, in a purely discrete world (like the one consisting solely of causal chains, as EY once suggested), continuity would be a non-standard object invented by mathematicians. What makes continuity "standard" in our world is, disappointingly, our limited visual acuity.

Another example: in a world simulated on a 32-bit integer machine, natural numbers would be considered non-standard, given how all actual numbers wrap around after 2^32-1.

Exercise for the reader: imagine a world where a certain non-standard model of first order PA would be viewed as standard.

Comment author:cousin_it
23 January 2018 10:33:15AM
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0 points
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How come we never see anything physical that behaves like any of of the non-standard models of first order PA?

Qiaochu's answer: because PA isn't unique. There are other (stronger/weaker) axiomatizations of natural numbers that would lead to other nonstandard models. I don't think that answer works, because we don't see nonstandard models of these other theories either.

wedrifid's answer: because PA was designed to talk about natural numbers, not other things in reality that humans can tell apart from natural numbers.

My answer: because PA was designed to talk about natural numbers, and we provably did a good job. PA has many models, but only one computable model. Since reality seems to be computable, we don't expect to see nonstandard models of PA in reality. (Though that leaves the mystery of whether/why reality is computable.)

## Comments (339)

Best*8 points [-]How come we never see anything physical that behaves like any of of the non-standard models of first order PA? Given that's the case, it seems like we can communicate the idea of numbers to other humans or even aliens by saying "the only model of first order PA that ever shows up in reality", so we don't

needsecond order logic (or the other logical ideas mentioned in the comments) just to talk about the natural numbers?*7 points [-]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.

*0 points [-]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?

*1 point [-]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,

contrapop-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.)Is it just coincidence that these nonstandard models don't show up anywhere in the empirical sciences, but real numbers and complex numbers do? I'm wondering if there is some sort of deeper reason... Maybe you were hinting at something by "delicate"?

Good point. I guess I was trying to make the point that Eliezer seems a bit obsessed with logical pinpointing (aka categoricity) in this post. ("You need axioms to pin down a mathematical universe before you can talk about it in the first place.") Before we achieved categoricity, we already knew what mathematical structure we wanted to talk about, and afterwards, it's still useful to add more axioms if we want to prove more theorems.

The process by which the concepts "natural / real / complex numbers" vs. "nonstandard models of PA" were generated is very different. In the first case, mathematicians were trying to model various aspects of the world around them (e.g. counting and physics). In the second case, mathematicians were trying to pinpoint something else they already understood and ended up not quite getting it because of logical subtleties.

I'm not sure how to explain what I mean by "delicate." It roughly means "unlikely to have been independently invented by alien mathematicians." In order for alien mathematicians to independently invent the notion of a nonstandard model of PA, they would have to have independently decided that writing down the first-order Peano axioms is a good idea, and I just don't find this all that likely. On the other hand, there are various routes alien mathematicians might take towards independently inventing the complex numbers, such as figuring out quantum mechanics.

I guess Eliezer's intended response here is something like "but when you want to explain to an AI what you mean by the natural numbers, you can't just say The Things You Use To Count With, You Know, Those."

*2 points [-]Umm... wouldn't they be considered "standard" in this case? I.e. matching some real-world experience?

Let's imagine a counterfactual world in which some of our "standard" models appear non-standard. For example, in a purely discrete world (like the one consisting solely of causal chains, as EY once suggested), continuity would be a non-standard object invented by mathematicians. What makes continuity "standard" in our world is, disappointingly, our limited visual acuity.

Another example: in a world simulated on a 32-bit integer machine, natural numbers would be considered non-standard, given how all actual numbers wrap around after 2^32-1.

Exercise for the reader: imagine a world where a certain non-standard model of first order PA would be viewed as standard.

This is basically the theme of the next post in the sequence. :)

*0 points [-]Qiaochu's answer: because PA isn't unique. There are other (stronger/weaker) axiomatizations of natural numbers that would lead to other nonstandard models. I don't think that answer works, because we don't see nonstandard models of these other theories either.

wedrifid's answer: because PA was designed to talk about natural numbers, not other things in reality that humans can tell apart from natural numbers.

My answer: because PA was designed to talk about natural numbers, and we provably did a good job. PA has many models, but only one

computablemodel. Since reality seems to be computable, we don't expect to see nonstandard models of PA in reality. (Though that leaves the mystery of whether/why reality is computable.)