Diophantus of Alexandria, a 2nd Century Greek mathematician, had a lot of the concepts needed to develop an Algebra. However, he was unable to fully generalize his methods of problem solving, even if he invented some interesting methods.
Ancient math was written in paragraphs, using words for the most part, thus making reading it very, very painful compared to the compact elegance of modern mathematical notation. However, I was surprised to see Diophantus (or his very early editors at least) develop some interesting and helpful notation in his algebra.
Final sigma ‘ς’ represented the unknown variable, but there were different symbols for variables of every power so for x^2… x^6 each had a unique variable. In fact, this situation persisted into the 17th century, even Fermat used N for unknown and S for the unknown-squared and C for the unknown cubed!
The problem with this is that it meant Diophantus couldn’t devise general methods to solve algebraic problems which had multiple unknowns, and it wasn’t obvious to him that one CAN frequently relate x^2 to x.
The cool thing about this notation from the past though, is how it makes obvious something that Algebra I – Algebra II students mess up frequently. You can’t just combine x^2 + x^3, these are different variables whose relation concerns the base. And almost everyone has made this mistake in their early math career. Some never recover.
Although the editor of my copy, Sir Thomas L. Heath, claims that Diophantus experienced limited options as a mathematician because all the letters of the Greek alphabet were in use as letters except for the final sigma, which Diophantus used to represent the unknown variable, I think D could have invented more variables quite easily. We see this in his invention of the subtraction sign as an inverted psi, and his use of a different variable with superscript for an unknown to the nth power up to the sixth. There was also the extinct digamma and all the Egyptian symbols which at least could have cribbed off of. Surely, the problem was not a lack of imagination, but merely satisfaction with the method then in use. Besides, one person can only invent so much, unless that person is Leonard Euler or Von Neumann, neither of whom had any limits. D. merely didn't see the limits of his notation.
Although D’s problems are surprisingly challenging even using modern notation, the logic D. used to solve the problems is obscure. He does not explain his step by step process. Since they are not proofs, and they are merely problems, it’s hard to divine exactly what D. thought the import of his methods were or exactly which steps he took to come to the answer. He seems to have used trial and error to solve some problems frequently, just plugging in numbers until the right answer popped out. He only wanted positive integers in his answers, so the problems are designed to reflect that. However, some problems don’t have an answer as whole number. For those he would estimate the answer. “X is < 11 and > 10.” Sometimes he is wrong on these estimations! I don’t know quite what to make of that. In problems whose answer is a negative number, Diophantus says, “Pthht, absurd!”
This is unfortunate, because if D. had credits and debts in mind when he was putting together these problems, he might have seen the utility of negative numbers and started an accounting revolution 1500 years early.
If Diophantus can teach us one thing about discovery, I believe it is that iterating over different methods of notation might lead us to make conceptual breakthroughs.