Looking up dictionary definitions of algebra geometry is pretty unsatisfying as they are usually along the lines of
"the branch of mathematics in which letters and other general symbols
are used to represent numbers and quantities in formulae and
equations.""the branch of mathematics concerned with the properties and relations
of points, lines, surfaces, solids, and higher dimensional analogues"
which don't answer the question much further than 'geometry is what geometers do' and an even more cringeworthy 'algebra is black scribbles on a page for people doing algebra'. The only real difference I can find is the reference to quantities (but what is a quantity?). There are also often remarks about geometry referring to 'real world' objects such as shapes and solids and relations between them. I'm hesitant to accept these because they fail to recognise the distinction between observations and mathematics.
So to ask the question more precisely, how do you take a set of axioms and know they describe a 'geometry' or an 'algebra' or any other subject for that matter? It seems unlikely that mathematicians would label things differently without having a clear distinction between them. To the best of my understanding, algebra defines operations on 'things' and geometry is to do with how 'things' stay the same. Now that I say it it doesn't make much sense. Any help?
Best Answer
You are right, that the attempted distinction makes sense only from a rather naive viewpoint, popular though it may be. Among professional mathematicians, these inherited "legacy" labels have a surprising endurance, which I think is mostly/only due to their widespread recognition among amateurs and professionals alike, despite their inaccuracy. Traditions die hard, even in the face of supposedly rational considerations.
200+ years ago, anyone who could "really prove" things, as opposed to giving a physical/physics-y quasi-heuristic, was a "geometer"... but this didn't mean you were "doing geometry", it only meant something about conforming to the alleged standards of proof of the ancient Greek geometers. "Arithmetization" of analysis in the mid-to-late 19th century really meant "finding a way to make a rigorous foundation", as opposed to invocation of "physical intuition" about continuity and such. (This in contrast to both Newton's and Leibniz' explanations of how to do calculus, in terms that were difficult to "ground" until A. Robinson.)
So, no, these labels do not accurately refer to much, but they are popular, and people speak in terms of them. :)