How different are the positive and negative numbers?

Question

Is there a fundamental difference between the positive and the negative numbers? Or is the difference like the one with electric charges in physics, where the other type of charge was just decided to be "positive", and therefore the other one was "negative", when actually there is nothing to make the positive one more "positive" than the other one, they've just been defined this way? Are the positive and negative numbers fundamentally different, or are they just opposites of each other; two sides of the same coin?

Answer 1

This question is somewhat unclear but I understand what you're getting at. Intuitively, one would think that numbers $x>0$ and $x<0$ are, in a sense, "symmetric", "the same", etc. That is, they have the same properties/structure, except we just slap on a "$-$" for numbers $<0$.

However, they are not "the same" since the set $\mathbb{R}_{>0}$ is closed under multiplication.

If we consider the sets only under addition, then there isn't really a "difference"; there exists a semigroup isomorphism $x \mapsto -x$.

Answer 2

First some terminology for clarity.

The natural numbers are the counting numbers 1, 2, 3, ... or sometimes 0, 1, 2, 3, ... Makes no difference really. The positive natural numbers (1, 2, 3, ...) are also called the positive integers. The negative numbers are the negative integers; and the positive and negative integers along with zero are called the integers.

The natural numbers are the most intuitive numbers. They have a big drawback, which is that you can't always subtract them from each other. The integers are more sophisticated and much more convenient. They allow subtraction. This makes them a group, which makes them a bridge from everyday arithmetic to higher math.

Viewed this way, the relation between the natural numbers and the integers is one of completion. The natural numbers are prior; the integers are a more sophisticated but vastly improved stage of development.

Humans understood how to count when we lived in caves and made marks on the walls; but negative numbers were not accepted into math till far later. Likewise a human infant first learns to count and only much encounters negative numbers in school. The integers (positive, negative, and zero) are at a much higher level of abstraction than the counting numbers.

Moreover, in the modern formalism of math the natural numbers are logically prior to the integers. In the formalism, everything is built up from sets. We build the natural numbers from the empty set and the integers from the naturals.

So historically, developmentally, and formally, the natural numbers are more fundamental than the integers. The positive and negative numbers are not "equal and opposite." They're not electrons and positrons. They're not Yin and Yang. Rather, they're seed and plant. The counting numbers are first; and the integers are their natural completion.

What is interesting about this point of view is that it turns out to be deep. The generalization of the construction of the integers from the naturals reaches into higher math. It was Grothendieck who realized that this simple little construction is important. https://en.wikipedia.org/wiki/Grothendieck_group (This last bit's beyond my pay grade, please let me know if I'm misconstruing anything).

This all adds up to the conclusion that the natural numbers are prior to the integers. The one grows out of the other. They're not opposites of the type that come into existence at the same time like electrons and positrons. They're seed turning into plant, or caterpillar into butterfly.