Comment author:rkr1410
03 January 2011 04:55:36PM
0 points
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There are some points of view that sometimes do require mathematical statements to be dependent on reality (i.e. constructivism, actual versus potential infinity debate, etc).

Sometimes it is intuitive to require mathematics to behave this way, i.e. 'natural' numbers are called that for a reason, and they better behave like the apples or I'm postulating a change in nomenclature.

P.S. Ii seems to me the OP's wording wasn't precise enough. I can very well imagine a situation in which some basic addition would yield non obvious results (like addition inside modulo N number space).

Comment author:Will_Sawin
03 January 2011 08:44:15PM
3 points
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When I reason inside a fully axiomatized formal system, the axioms don't depend on reality, but the rules for manipulating symbols depend on ... something. You could define it as "if I perform these manipulations in reality, I will get this result" but what if performing the manipulations in different places gets different results?

What if, when you applied the rule "(x+Sy) => S(x+y)" twice and the rule "(x+0)=>x" once, to "(SS0+SS0)", you got "SSS0" instead of "SSSS0"?

Comment author:rkr1410
05 January 2011 11:14:32PM
0 points
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I guess when one reasons inside a fully axiomatized formal system, this something the rules for symbol manipulation depend on is the set of axioms.

Now I'm putting on my uneducated hat, so excuse me if this is heresy:
Starting with the axioms you apply logic to formulate more specific rules (in this case the abstract is empirically falsifiable, since we're working on natural numbers).

So, to arrive at SS0+SS0=SSS0, you'd have to venture outside the realm of reason I'm afraid.Tthat would maybe manifest itself as magic - getting 4 apples on the table during night, but 3 during day when you put 2 and 2 apples side by side. And could mean ability to produce something from nothing by clever arrangement of apples. and waste disposal would become easy :)

In other words my opinion is it's not possible even as thought experiment unless you introduce some random factor from beyond the scope of axioms.

Comment author:Will_Sawin
06 January 2011 02:10:29AM
*
3 points
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well there's the special other thing, the reason you can't explain Peano Arithmetic to a rock, which is that axioms are static sequences of signals, but in addition you have these dynamics.

These dynamics are contained within the structure of our thoughts, which is why they're preserved in a thought experiment. But we still have to actually check our thoughts, which are part of reality.

P.S. Ii seems to me the OP's wording wasn't precise enough. I can very well imagine a situation in which some basic addition would yield non obvious results (like addition inside modulo N number space).

Hm... not precise enough for what? I think we all know what was meant... unless Eliezer did a ninja edit after you posted ;) this seems to cover it:

There are redefinitions, but those are not "situations", and then you're no longer talking about 2, 4, =, or +.

What you suggested, that's not a "basic addition" any more, is it?

## Comments (390)

OldThere are some points of view that sometimes do require mathematical statements to be dependent on reality (i.e. constructivism, actual versus potential infinity debate, etc).

Sometimes it is intuitive to require mathematics to behave this way, i.e. 'natural' numbers are called that for a reason, and they better behave like the apples or I'm postulating a change in nomenclature.

P.S. Ii seems to me the OP's wording wasn't precise enough. I can very well imagine a

situationin which some basic addition would yield non obvious results (like addition inside modulo N number space).When I reason inside a fully axiomatized formal system, the axioms don't depend on reality, but the rules for manipulating symbols depend on ... something. You could define it as "if I perform these manipulations in reality, I will get this result" but what if performing the manipulations in different places gets different results?

What if, when you applied the rule "(x+Sy) => S(x+y)" twice and the rule "(x+0)=>x" once, to "(SS0+SS0)", you got "SSS0" instead of "SSSS0"?

I guess when one reasons inside a fully axiomatized formal system, this something the rules for symbol manipulation depend on is the set of axioms.

Now I'm putting on my uneducated hat, so excuse me if this is heresy: Starting with the axioms you apply logic to formulate more specific rules (in this case the abstract is empirically falsifiable, since we're working on natural numbers).

So, to arrive at SS0+SS0=SSS0, you'd have to venture outside the realm of reason I'm afraid.Tthat would maybe manifest itself as magic - getting 4 apples on the table during night, but 3 during day when you put 2 and 2 apples side by side. And could mean ability to produce something from nothing by clever arrangement of apples. and waste disposal would become easy :)

In other words my opinion is it's not possible even as thought experiment unless you introduce some random factor from beyond the scope of axioms.

*3 points [-]well there's the special other thing, the reason you can't explain Peano Arithmetic to a rock, which is that axioms are static sequences of signals, but in addition you have these dynamics.

Best source on this is Lewis Carroll http://www.ditext.com/carroll/tortoise.html

These dynamics are contained within the structure of our thoughts, which is why they're preserved in a thought experiment. But we still have to actually check our thoughts, which are part of reality.

Sorry if this wasn't very coherent.

Hm... not precise enough for what? I think we all know what was meant... unless Eliezer did a ninja edit after you posted ;) this seems to cover it:

What you suggested, that's not a "basic addition" any more, is it?