I was watching a lecture where the professor was describing the mathematics of the ancient Greeks and said they had division because "that's just fancy subtraction." That line got me thinking because it doesn't seem to be quite true. Multiplication is certainly "fancy addition" because you can turn the equation 3 x 5 = 15 into an addition problem where you have "=15" at the other side. But division can't be turned into a subtraction problem. You can't take 15/3=5 and make a subtraction problem with "=5" at one side.
I have read in many places that division is subtraction in the sense of 15/3=5 turned into 15-3-3-3-3-3=0 and then you count the number of times you subtracted 3. But I take a lot of issue with that explanation of division because a) it does not parallel multiplication b) it does not have "=5" on one side of the new equation and c) it's simply a "count" of something. It's like viewing the equation from the outside and seeing how many times you did something, like a counter in programming. But it's not fundamental to the equation.
Essentially, I'm hoping you can give me some guidance on the essential theory of division in terms of how it relates to subtraction and how it parallels multiplication. I have been turning this question over in my head for many weeks now and I've come up with a theory (I'm not close to being even an amateur mathematician but I love solving problems) but it has to do with redefining multiplication and I think it would obfuscate the point.