Without =
All I want to say here is how interesting I think it is to consider:
For thousands of years people did Mathematics without an equal-sign! The “=” symbol, if memory of a history of Math text serves me correctly, made its first appears due to … Fibonacci? Perhaps I’m getting my mental wires crossed with the fact that he imported Arabic numerals. Maybe it was Recorde? If only there were a way to determine its source without leaving my computer … Well I guess we’ll never know.
But before the 13th century much of mathematics had no symbol for equality, and people merely used the word “equals” or just a blank space where an equality symbol might go. In some ways we might do well to return to that, at least for introductory students, because they often fail to realize that the equal-symbol means the same that the English word “equals” means: That the thing on the left is the same as the thing on the right. (That’s not actually entirely true, equality can be used for the definition of a function, which isn’t precisely, logically, the same thing.)
However, ancient Egyptians had a symbol for it similar to what we now use for proportionality. Certain later Greek speakers had their own symbol although it did not persist through history. Ancient Indians used pha which was a contraction in their language. Today’s equal-sign, “=”, derives from the signification of two parallel lines. This choice is due to the fact that parallelness behaves very similarly to equality. Some of the features of equality that are most useful are:
Reflexivity: Every line is parallel to itself just as every object is equal to itself.
Symmetry: If line m is parallel to line n then line n is also parallel to line m. A similar statement holds for equality.
Transitivity: If m is parallel to n, and n is parallel to p, then m must be parallel to p. Again, the same sort of thing is true for equality.
Any relation satisfying the three properties above is called an equivalence relation. Equality and parallelness are both equivalence relations, and polygon similarity is also an equivalence relation. It is worth noting, though, that equality is stronger than a mere equivalence relation. The distinguishing feature of equality which separates it from all other equivalence relation is that of substitutability. If two quantities are equal, all the same sentences must be true about those numbers. If 11 is prime then, since 11 = 9+2, we must also be justified in saying 9+2 is prime. Substitutability does not hold for parallel lines. Line m may be parallel to n but a distance of 10 units away from it. If substitutability held we would be able to infer that line n is also 10 units away from n. Since that’s ridiculous, we see that parallelness does not permit of substitution.
But the use of a symbol for equality was not practiced even by startlingly modern European mathematicians such as Fermat. In some way I find this inspiring—Math really just is what we could express in many more words in regular English. Math is not categorically different reasoning from other kinds of reasoning, the symbols merely compact a very large amount of information and abstraction.