How Natural is Zero?

[Eric Wawerczyk]

The concept of "1", the idea of quantity is only relevant when compared to the absence of quantity. In a universe completely void of objects, empty would be the norm. However, there would be no concept of quantity in this universe, since there is nothing to compare it to. In a universe with only one object, the concept of "1" (and quantity for that matter) is still unintelligible because there is no concept of the absence of this object. There is only knowledge of the object and nothing else. No other objects exist to compare the quality "quantity".

We are able to identify "red" by comparing it to the other colors and similarly the characteristics "big" and "small" are relative to the things with which they describe. Quantity is a characteristic that an aggregate (collection of objects) may possess, so to have something we must first have the concept of nothing.

From an axiomatic standpoint, the set {} can be constructed with only Existence and Comprehension axioms (with Extensionality for uniqueness). The set { {} } requires the axiom of pairing as well! So Zero does, indeed, come before 1!

It is from these justifications that I say yes, Zero is a Natural Number.

[MY]

Why was the zero, being such a natural number according to some people, introduced and used so late in the development of Mathematics? Unlike 1, say.

[Eric Wawerczyk]

It is conceded by many that the beginning of mathematics on Earth was counting. It's origin lies in it's immediate application to survival and development of civilization. The notion of an aggregate, a grouping/collection of individual objects, was a recognizable pattern. Uses of counting was first limited to the hand or to some other religious object of by the use of bijections. For physical objects in reality that we experience it is impossible to "point" at zero cattle or ears of corn, or necklaces. It may be that until the occurrence of a written language of the concept of quantity did the necessity of the concept of zero appear necessary for the authors of the time. Mathematics began as a tool of civilization for accurate measurement, so the "need" of zero arising in the language of math can be seen probable to appear in cultures needing it.

It is my assertion that the concept of 0 is natural in a more fundamental and necessary way than 1 in the view of logic rather than empirical necessities of measurement. I will agree that the "Counting Numbers" need not contain zero, but of the many Natural Numbers, zero is the most natural of all.

[camilo]

I agree that zero is a very special number, and in terms of logic it precedes 1, in a way similar to existentialism where existence precedes essence, but I am not sure if 0 should be given membership in N.

Besides my main point that it changes nothing mathematically, I will argue that zero does not act like the other natural numbers. Where 0=0+0+0+0+..., this does not work for any other n in the Natural numbers. That alone distinguishes 0 from 1, 2, 3... It seems to me that zero looks more like a concept (like infinity) than a number. (Where these two concepts are like reciprocals/inverses of each other, zero and infinity)

At the end of the day, I'm completely indifferent to the question of zero's membership in the Naturals, until something is actually affected by it, it's just a matter of taste; which do you prefer, N=positive integers={1,2,3...} or N=non-negative integers={0,1,2,3,..._}?

But philosophically, I would include zero for aesthetic reasons, not because it acts how I would expect a natural number to act. (intuitively, it doesnt!)