TITLE: Is every integer a unary operation?
QUESTION [13 upvotes]: I have been thinking about mathematical operations, and I was trying to ponder what an operation simpler than addition would be. I started thinking about what each operation did (exponents: repeated multiplication, multiplication: repeated addition). I figured that addition was repeated something, and I came up with a unary operation, 4 = 1 + 1 + 1 + 1, where each number is the input, and its value is the output. That would mean that our number system's numerals are just shorthand for repeated addition of one. (To extend this to negative numbers, we could define this unary function's opposite: -4 = -1 - 1 - 1 - 1.) I am an eighth grader taking high school geometry, and this sort of shocked me. Am I totally wrong, or is every integer just the value of a unary operation: itself?

REPLY [2 votes]: While is is true that to every integer $k$ one can associate a unary operation $n\mapsto n+k$ of adding$~k$ (which you can see as a function $\Bbb Z\to\Bbb Z$, or in case $k\in\Bbb N$ as a function $\Bbb N\to\Bbb N$) and this correspondence is injective (different integers give different unary operations), it would be wrong to define integers as such operations, and fairly confusing even to identify integers with such operations.
The problem is, these operations have to act on something, namely on integers, but if you want to define the integers as those operations, the integers that they must act upon have not been defined yet. So you end up with a circular definition.
You might want to define integers first, and then the operations on the integers, and then identify $k$ with the operation $n\mapsto n+k$. This would be similar to how one defines the integers, then the rational numbers in therms of them (a somewhat subtle affair), after which (for the purposes of arithmetic) the integer $k$ is identified with the rational number$~\frac k1$. This is not logically inconsistent, and would allow you to write $1(1)=2$ or even $1\circ 1=2$ instead of $1+1=2$, but there is not too much to be gained by this, and it would be less clear than what is usually done. Also it is fairly arbitrary: why identify $k$ with $n\mapsto n+k$ rather than with $n\mapsto nk$ o something else yet?