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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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closed as not constructive by lhf, mixedmath, Asaf Karagila, VelvetThunder, J. M. Nov 13 '11 at 12:34

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I don't think this sum has to have a 'physical' interpretation. Do you know about the Peano axioms? – Stijn Nov 12 '11 at 20:55
@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 :) – Stijn Nov 12 '11 at 21:01
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)? – Alex Basson Nov 12 '11 at 21:29
The title of this post is misleading. – AnonymousCoward Nov 13 '11 at 2:00
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. – mixedmath Nov 13 '11 at 3:14

There are varying interpretations. One interpretation in set theory is that each number is the set of all numbers prior. For instance, one would be defined as the set containing zero, or the null set so {0} where 0 is the null set. 2 would be {0,1}, or {0,{0}}, etc. This is another interpretation of the numbers.

An interesting note: You should look at the Platonic Theory of Forms. This idea essentially states that there exists a world of mathematical entities separate from the physical reality. Does the number "7" actually exist? That's an interesting question.

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But is there an alternative interpretation which does not have anything to do with discrete objects?

If you have a rope 2 meters long, and join it with 3 meter long rope, you get 5 meter long rope. "meter" is not a discrete quality here, the same interpretation gives 0.2 m + 0.5 m = 0.7 m. (At least assuming ropes are infinitely divisible.)

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"Discrete" objects? Or do you mean physical objects? If the latter, I could say that 43, 21, and 50 are three numbers, and 20 and 100 are two numbers distinct from those, and three plus two is five, so 43, 21, 50, 20, and 100 are five numbers. This time they're not physical objects like apples, but they're numbers themselves.

But three gallons of orange juice plus two gallons of orange juice makes five gallons of orange juice, and orange juice is just as physical as apples, but it's not "discrete".

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I dont think there is any physical unit that you can consistently do all algebra with, $5=2^1+3^1$, so if every number is an apple, what is say (2 apples)^(2 apples) ?

This is the real problem: Everything a human can do in math a computer can do given enough time, and computers are deterministic, so in this sense all maths allow for a physical interpretation as reasoning about the behaviour of computers. And computers are very discrete, 0,10,1,11,0,00101,111. So even if you tried to abstract far away and say that math is just a game with symbols and rules for manipulating them, no matter how far you go, if your rules are deterministic you can map it onto a computer program.

Anyways there are models of the integers which do not satisfy your apple analogogy, but satisfy 2+3=5, this is because there are different non-isomorphic models of arithemtic and the physical apple thing can only correspond to one model.

Also, if x is an object which you call not-discrete. Then just put that x as a unit behind every number to get a non-discrete interpretation

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$(2 apples)^{2 apples}$ belongs to apples (ordinal) function space $apples^{apples} = \{x^y | x,y:apples\}$! For numbers this space is isomorphic with $apples$. :) – Kaveh Nov 19 '11 at 18:11

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