I am learning about Mathematical proofs, and am shocked at how early some of these proofs are seen. For example, the expansion of (a+b)^2 by Euclid(ca. 300 B.C.) or the Binomial Theorem in Baghdad(1007) and China(1100). Was there a large amount of Mathematicians coming up with mistake-ridden proofs and Mathematical fallacies?
One of the things mathematicians have often discussed up through history is how rigorous mathematical proofs need to be, how much you need to spell out and how much the reader can themself "see". Because if everything, and I mean everything, is spelled out then even simple mathematical proofs gets HUGE. Today we have a very high level of expectation to how rigorous the mathematical proofs should be, so from that point alone many older proofs will look mistake ridden to us. As a good example for this look at what the Oxford Dictionary of Scientists say about Euclid:
"However, despite being revered as an almost perfect example of rigorous thinking for almost 2000 years there are considerable defects in Euclid's reasoning. A number of his proofs were found to contain mistakes, the status of the initial axioms themselves was increasingly considered to be problematic, and the definitions of such basic terms as ‘line’ and ‘point’ were found to be unsatisfactory."
Euclid is simply not rigorously enough for modern mathematicians!
The reason we currently have so high demands is the collaps of the "Italian school of algebraic geometry". It is a mathematical movement flourishing from 1885 to 1935 where the proofs slowly became less rigorous, more informal arguments where used and it depended a lot on mathematical intuition. Post world war II people started to find mathematical fallacies in their works and by 1950 there where so many mistakes and fallacies in it that it simply collapsed, because no one dared to use other peoples work for the fear of mistakes.
Keep in mind the selection bias--we only see the proofs that have withstood the test of time and endless scrutiny by fellow mathematicians. Even today there is plenty of bogus math around. Someone compiled this list of 100 published "proofs" of the P=NP problem, which is one of the famous Millennium problems and carries a $1 million reward. So far, no one has received it.
I'd appreciate you asking this question in /r/mathematics and seeing what kind of response you get. It is a good question.
Mistake ridden proofs are not exactly countable, anyone can write a bad proof. There are famous examples of false proofs, like the ones for Fermats last theorem. Fermat himself probably had a false proof, as he wrote in the margins of his writings "truly marvelous proof" and yet it wasn't until 1995, 358 years later was it finally proven.
Classical Italian Algebraic Geometry was mentioned as a place where there were incorrect results (there really were). There are others, and you mentioned examples.
I don't think the main problem of doing "non-rigorous" math (what that means is a little ambiguous) is that you get bad results, though sometimes you do. Rather, when you push to make math "rigorous" (as people did in a big way the 19th and 20th century) you often end up exploring a lot of very abstract, flexible ideas that end up being useful elsewhere/everywhere. Look at the definition of a topological space. It doesn't look like much, but it's the product of a long process of distilling and abstracting the notion of "closeness" that began as an attempt to generalize and make rigorous the work of Leibniz and Newton (which really was not rigorous--there wasn't a good definition of a "limit," for example). Now it's used every day in literally every branch of mathematics. There are a lot of examples like this one -- I'm just selecting a big and not-especially-controversial one.
Actually algebraic geometry is kind of a similar story.
Nerd