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Normally we consider simple arithmetic to be related to the world of objects. So the sum $3+2=5$ means $3$ three apples and $2$ apples gives $5$ apples. But is there an alternative interpretation which does not have anything to do with discrete objects?

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    I don't think this sum has to have a 'physical' interpretation. Do you know about the Peano axioms?2011-11-12
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    @Stijn But the problem is that the Peano axioms describe a methodology but do not necessarily restrict themselves to a good notion of what a number actually is.2011-11-12
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    @analysisj then how about the set-theoretic definition of natural numbers? You could interpret every number to be a set.... never mind, I see that you just gave that as an answer :)2011-11-12
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    I guess the sentence "Normally we consider simple..." is your assumption (we) is not quite accurate here! - Just kidding :)2011-11-12
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    Looking at the answers, perhaps some clarification would be helpful. Are you looking for an interpretation that doesn't involve physical objects (in which case, look to the set-theoretic answer), or one that doesn't involve objects that come only in whole number values (in which case, the rope answer may serve)?2011-11-12
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    The title of this post is misleading.2011-11-13
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    I voted to close this question. I do not think the intention is clear and am unclear on the intended interpretation. My favorite answer is Hardy's, which contradicts and answers the OP in equal portions.2011-11-13
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    The answer is yes. one can give such an interpretation (in terms of symbolic definition, with no actual *meaning* or underlying substratum). Consider this: i define **bourda + mourda = noima**, what does it mean? Apart fom the definition of the behaviour of symbols does not have any other connection. Of course it is useless as is2014-05-27

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