Non-functional proof of operations on both sides

Question

Practically in every algebraic problem that I am asked to deal with there is an issue of applying the basic properties to equations in order to simplify them, however the only proof that would justify operations on both sides, that is the only proof I found here, relies on the introduction of functions which seems rather odd, because the concept of a function had not emerged until 17 century, and surely the algebra was there before that. So, is there a non-functional proof for the operations on both sides; what kind of justification was used before introduction of functions?

Answer 1

Two of the "Common Notions" in Euclid's Elements (see here) are ...

• If equals are added to equals, then the wholes are equal.
• If equals are subtracted from equals, then the remainders are equal.

These are, of course, special cases of the modern notion of "doing the same thing to both sides of an equation", which Euclid might have expressed as ...

• If equals are manipulated equally, then the results are equal.

As the discussion linked above indicates, the "equals added to equals yield equals" Common Notion is nowadays formalized as a "Substitution Axiom" we assign to the addition operation.

• If $x = y$, then $x + z = y + z$. (i.e., "adding $z$ to both sides of an equation" works.)

Of course, we generalize this to subtraction, multiplication, etc, and the application of arbitrary functions. In any case, "equals manipulated equally yield equals" isn't something we prove; it's something we assume via appropriate axioms.

Answer 2

There is a lot of literature on logic as developed by Leibniz, who was worried about rigor as much as we are. Symbolic notation originates with Viete a century earlier. Set theory wasn't developed until the 1870s but the opinion that rigor originates with set theory is a hypothesis that many historians reject.