I've been thinking, for whatever reason, what ET math would be like. I bet a lot of people would think that it and our math will not only be comparable, they will likely be essentially identical. For example, the induction principle in arithmetic would likely also be a theorem in their math. The Pythagorean theorem is probably among their theorems, as another example though their name for it (if it has a name) might be just "Theorem #691,865,863,532,987".

To that end, I was wondering how various ET mathematicians might define the word equation. A lot of us on earth are taught first off that an equation is just a formula or expression that contains the following symbol: =. But the ETs won't have that symbol in their language. So when fabricating the Rosetta Stone which will provide a means to translate their language to ours and vice versa, to give the proper meaning of our word equation, we have to capture the essence of what equality is.

Equality is a "principle" of sorts that exists only in certain formal systems. Many formal systems have nothing like equality while other formal systems have an equality. If xEy (which intuitively means x=y) is a grammatically-correct utterance in the formal system and if E is a symbol in the formal system possessing the following properties, then I would say E represents equality:

1. What E points to is an equivalence relation, meaning

1a. for all x, xEx (reflexivity)

1b. for all x and y, if xEy then yEx (symmetry)

1c. for all x, y and z, if xEy and yEz then xEz (transivity)

2. every equivalence class defined by E has cardinality exactly 1.

An equivalence class defined by E means that, given x, the equivalence class generated by x is the set of all things E-equivalent to x. Everything in one equivalence class is E-equivalent to everything else in that equivalence class. Saying it has cardinality one means that it has one element.

If we did not have criteria #2 in the definition of "equation," then it might be the case that aEb although a and b are different. In that case, the equivalence class generated by a has at least one other element, b; so the cardinality of that equivalence class would not be 1, it would be at least 2.

Looking at the three criteria under #1, it is clear that equality behaves so, and equivalence relations just generalize equality.

In a class where the word equation is defined, soon to follow is the definition of the word solution, as in "a solution to an equation." I haven't thought much about what "solution to an equation" might mean to an ET mathematician.