Pretty much. According to someone on Stack Exchange, "The most serious difficulties with Euclid from the modern point of view is that he did not realize that an axiom was needed for congruence of triangles, Euclids proof by superposition is not considered as a valid proof." But making mistakes in a formal treatment of a subject does not negate the fact that it is a formal treatment of the subject IMHO, and AFAICR none of the theorems in Elements is wrong; i.e., Elements is formal enough to have avoided a mistake even though the axioms listed weren't all the axiom that are actually needed to support the theorems.
18th Century European math was much more potent than ancient Greek math, and although parts of it like algebra and geometry were, for a long time, most of it was not understood at a formal or rigorous level for a long time even if we accept the level of rigor found in Elements.
Isn't how formal or rigorous something is just a social convention? Grammer Nazi's used to make online speech be formal with perfect rigor. Isn't it all relative to what your society defines?
No. That's the colloquial definition of formal. In mathematics, the word formal refers to something more specific: one or more statements written using a set of symbols which have fully-defined rules for mechanically transforming them into another form.
A formal proof is then one which proceeds by a series of these mechanical steps beginning with one or more premises and ending with a conclusion (or goal).
If you're a formalist in philosophy of math, then math is neither true nor false, it's merely a bunch of meaningless symbols you transform via mechanical rules.
To an extent. A truly completely formal proof, as in symbol manipulation according to the rules of some formal system, no. It's valid or it isn't.
But no one actually works like this. There are varying degrees of "semiformality" and what is and isn't acceptable is ultimately a convention, and varies between subfields - but even the laxest mathematicians are still about as careful as the most rigorous physicists.
18th Century European math was much more potent than ancient Greek math, and although parts of it like algebra and geometry were, for a long time, most of it was not understood at a formal or rigorous level for a long time even if we accept the level of rigor found in Elements.