Is there a common theme between modulus, moduli and modulo?

Question

The moduli space is some sort of space of parameterizing space, modulo as in modular arithmetic and modulus as in the modulus of a complex number.

Is there a reason all these words are the so similar?

Answer 1

They all have the same root. "Modulus" means "measure"; a modulus is a measure of something. The modulus of a complex number measures its length as a vector. The term "modular arithmetic" comes from Gauss who used "modulus" as follows:

Si numerus a numerorum b, c differentiam metitur, b et c secundum a congrui dicuntur, sin minus, incongrui; ipsum a modulum appelamus. Uterque numerorum b, c priori in casu alterius residuum, in posteriori vero nonresiduum vocatur. [If a number $a$ measure the difference [emphasis mine] between two numbers $b$ and $c$, $b$ and $c$ are said to be congruent with respect to $a$, if not, incongruent; $a$ is called the modulus [emphasis mine], and each of the numbers $b$ and $c$ the residue of the other in the first case, the non-residue in the latter case.]

So in modular arithmetic $n$ is the modulus and working $\bmod n$ means measuring the difference between two integers in units of $n$. Remember that the Western mathematical tradition is deeply steeped in Euclidean geometry and for over a thousand years a number was a length. "Measuring the difference" means that $n$ divides the difference but the term "measuring" is meant (I believe, anyway) to evoke the geometric picture of a stick of length $n$ being used to literally measure the length of the difference.

"Moduli" is the plural of modulus so "moduli space" should be interpreted as "space of moduli," in the sense that a moduli space describes all possible ways an object of some type can vary (and we can measure these variations).