I've always been a "yea proofs" kind of person, and I was right on board with the WTM/LCC recommendation of proof-based geometry. And then I ran across this homeschooling couple that does proofs in algebra, too. They (the male half, at least) strongly advocate teaching "pure math" - i.e. starting with axioms and definitions and then using them to prove theorems. And they are doing so with their children. How awesome is that?
I was completely inspired - this is what (neo) classical math is supposed to be. I searched online used bookstores and bought all their rec'd, oop textbooks. I started (slowly) learning to do proofs myself. I was invigorated by all the possibilities.
But what about the elementary years? Were we condemned to just mark time doing fake math 'till we're ready for the real stuff in algebra?
And then I found CLAA, and their Classical Arithmetic program. No marking time here! It deductively builds up, from the very beginning, the whole of arithmetic. From the beginning, the focus is on proofs. For 9 year olds! I was in love - *real* math, from the beginning. Now that I know it was possible, I started thinking of exactly how to do it.
But should I do classical axiomatic math or modern axiomatic math? For one, I'm not wanting to enroll my dc in CLAA, so it would be *me* taking the courses, and then synthesizing the info into something I could teach my kids. And I know nothing about how the ancients did math. But I suspect it is quite different than modern axiomatic math - and that those differences have important philosophical ramifications.
One of the more obvious differences is that ancient math seems to rely on words and thus on traditional logic, while modern axiomatic math uses symbolic logic. I've read Martin Cothran's essay, "Logic is not Math," as well as the section in Kreeft's "Socratic Logic" that contrasts traditional and symbolic logic (and certainly they are not the same thing), which has given me an idea of some of the philosophical issues that led to the development of symbolic logic, but I've no idea how that plays out in the use of symbolic logic in math. I feel that it has a deeper significance than just that one uses symbols and the other words, but as to what that significance actually *is*... only vague impressions.
A less obvious, but possibly more fundamental, difference is how they treat axioms. (Certainly there seems to be a big difference in axiom choice b/w ancient arithmetic and the modern properties of the real number field.) From the CLAA article, the ancients picked their axioms based on what was self-evidently true. I'm guessing they just have one master set of axioms for each major branch of the Quadrivium (or just one set of axioms, period). However, moderns have different sets of axioms associated with all different kinds of sets, and they select their axioms based not on self-evident truths, but on trying to generate the smallest set of axioms possible (they hate the whole idea of having to *assume* anything - clear philosophical implications here! - and so want to have the fewest assumptions possible). This can result in axioms which are very much *not* self-evidently true, as well as multiple valid set of axioms to describe the same structure or branch of mathematics.
As well, moderns are interested in what happens when certain axioms don't apply, with little to no concern if it the results seem to correspond with an external reality or not, so long as they are internally consistent (e.g. non-Euclidean geometry came about from imagining what would happen if parallel lines *did* intersect, even though it seemed self-evident that they don't). I'm not sure that this concept would make sense to the ancients - would they care about the results that came from assuming that a self-evident truth was, in fact, not true? (Interestingly enough, though, several branches of mathematics that seemed to have no real world connection when they were first discovered were later found to be exactly what was needed to mathematically describe a new physics theory.)
What do the above differences in approach to axiomatic math reveal about the underlying philosophical differences between the ancients and modern mathematicians?
I don't know, yet. But I'm trying to sort it all out.
(Of course, its not like moderns even have a consensus on the underlying philosophy of math in the first place. Are we discovering things that have always existed, or is math just a human construction? Why is math so strangely effective at describing the real world? For all I know, the ancients were just as divided.)