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Comment author: rkr1410 21 January 2011 05:25:01PM *  1 point [-]
Comment author: Will_Sawin 03 January 2011 08:44:15PM 3 points [-]

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 [-]

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: Gray_Area 28 September 2007 12:21:09AM 3 points [-]

The core issue is whether statements in number theory, and more generally, mathematical statements are independent of physical reality or entailed by our physical laws. (This question isn't as obvious as it might seem, I remember reading a paper claiming to construct a consistent set of physical laws where 2 + 2 has no definite answer). At any rate, if the former is true, 2+2=4 is outside the province of empirical science, and applying empirical reasoning to evaluate its 'truth' is wrong.

Comment author: rkr1410 03 January 2011 04:55:36PM 0 points [-]

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).