Mathematics

Are positive numbers in some way more basic than negative numbers?

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In more than one way, the answer is yes. It's clear that psychologically, as it were, positive numbers are more basic; we learn to count before we learn to subtract, for instances, and even when we learn to subtract, the idea of a negative number takes longer to catch onto. Also, the non-negative numbers were part of mathematics long before the full set of integers were. (In fact, treating zero as a number came later than treating 1, 2, 3... as numbers.

Also, we can start with the positive numbers and define the set of all integers. The positive numbers are usually called the natural numbers in mathematics, and N is the usual symbol for the natural numbers. The integers Z are sets of ordered pairs of natural numbers on the usual definition. The integer that "goes with" the natural number 1 is the set of pairs

{(1,2), (2,3), (3,4), 4,5)...}

(By "goes with" I mean it's the integer that, when we're through with the construction, we can in effect, treat as the same thing as the natural number 1.) The integer that goes with the natural number 2 is the set of pairs

{1,3), (2,4), (3,5), (4,6)...}

So far, all these integers are positive; notice that the second natural number in the pair is bigger than the first. The integer 0 is the set of pairs

{1,1), (2,2), (3,3), (4,4)...}

What about negative numbers? They're the pairs in which the first natural number is bigger. The integer -1 is the set

{(2,1}, 3,2), (4,3)...}

The integer -2 is

{(3,1), (4,2), (5,3)...}

and so on. There's more to the story than I've presented; to present the full story we'd need to talk about addition how the definition of addition leads naturally to treating the set

{1,1), (2,2), (3,3), (4,4)...}

as 0, and how this is intimately related to treating {(2,1}, 3,2), (4,3)...} as -1, etc. The Wikipedia article at http://en.wikipedia.org/wiki/Negative_number does a good job covering the basics.

And so the positive numbers are the "backbone: of the construction of the integers.

We could go on to define rational numbers (1/2, 3,4, -5/17, etc.) as sets of pairs of integers, with in turn are sets of pairs of natural numbers. We could then define real numbers as infinite sequences of rational numbers. So in an important mathematical sense, the positive numbers (the natural numbers) are more basic.

Are the natural numbers "really" more basic in some deep metaphysical sense? I'll confess that I don't know for sure what this question means or what would count as a good answer. But other panelists who are better-versed in philosophy of mathematics may be able to say something worthwhile on that question.

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