Do positive integers necessitate negative integers?

[JanEnClaesen]

If counting/positive numbers exist, do they imply the existence of negative numbers?

I'd say yes, because there's always a bijection that maps the lowest counting number of the set to the highest, then the second lowest to the second highest, etc. This reversal of order/mirroring is possible for any set with a strict order. The negative numbers are then some sort of dual space of the positive numbers. This bijection is a mirror permutation, can the idea of permutation groups be applied more generally to integers/sets?

I apologize for the borderline vague statements.

 

[micromass]

If counting/positive numbers exist, do they imply the existence of negative numbers?

What do you mean with "exists"? Do you somehow want to construct the negative numbers from the positive counting numbers?

 

[HallsofIvy]

They certainly exist in any mathematical sense and, yes, they can be constructed from the positive counting numbers- so the negative integers exist in then same sense that the positive integers do.

 

[JanEnClaesen]

"exists" as in constructing: mapping the highest to the lowest, second highest to the second lowest, etc.
This construction is a permutation, do permutations have a more general use for ordered sets?

 

[micromass]

"exists" as in constructing: mapping the highest to the lowest, second highest to the second lowest, etc.
This construction is a permutation, do permutations have a more general use for ordered sets?

That's not the usual construction of the integers. The usual construction is called the Grothendieck construction. It's not that your proposal of constructing the negatives is wrong, but there are annoying details such as defining addition and multiplication and checking the properties.

And yes, permutations are useful outside constructing the integers too. For example, in group theory.