What comes after 1?

To answer this question, we have to first define what kind of mathematical "space" we're dealing with. If we're dealing with the "natural numbers", ie 1, 2, 3, ... , then what comes next is 2. Same with the integers ( ... -2, -1, 0, 1, 2...). This is where it gets tricky. If we are dealing with the rational numbers, i.e. fractions made from elements of the integers, then there is no "next" - between every two distinct rational numbers, another rational number can always be constructed. There is no "next" number in this space (as a side note, this particular "space", the rational numbers, is an example of what mathematicians call a "field".)

For the "real" numbers (containing the rational numbers and irrational numbers such as pi), which calculus and most practical applications of mathematics deal in, , there is not only an infinite number of elements between any two distinct numbers, this number is UNCOUNTABLY infinite - they can't even be described by an infinitely long list. Phew!

But, to return to the natural numbers, what is the justification, really, for 2 coming after one, and not before? The answer lies in set theory. The foundational construction of the natural numbers consists of a series of "nested" sets, each containg the last. A "set" is just some collection of elements. The number 0 is defined as the empty set, the set that includes no element. 1 is defined as the set CONTAINING the empty set (not the same thing as the empty set!) 2 contains both 1 and 0 - so it is a set containing both the empty set, and the set containing the empty set. 3 contains 2, 1, and 0 as elements - and so on and so forth. How does this relate to your question? Because these "nested" sets, through inclusion, define what mathematicians call a "well-order" - where every element is strictly less or greater than every distinct other element. 0 is contained in 1, so 0 is less than 1. There is no number that both contains 0 and is less than 1 - so 1 is the "next" number. The same logic works for 1 going to 2. This is a lot to digest, but I hope it made sense!