Thanks for the comment! A few thoughts:

One thing I worry about is whether our application of terms like ‘applied mathematics’ and ‘pure (or proper) mathematics’ is anachronistic. Why think that such a distinction is made or even conceived of by the Ancients?

If the distinction is between the sort of mathematics that investigates particular problems in calculation, geometry, astronomy, harmonics vs. the sort of mathematics that investigates the general rules and principles that underlie all of these subjects, then I’m inclined to say that this latter sort of investigation is more akin to philosophy, in Plato’s mind, than mathematics. The process by which one tries to discover and defend the basic rules of mathematics looks to be what dialectic is supposed to do.

Consider the account of mathematics that we find in the Line. There, mathematicians use the hypothetical method and use the hypotheses to work down to conclusions. They don’t try to justify or account for the hypothesis they posit by putting it into an explanatory framework or by positing some higher hypothesis to explain the lower one. They’re interested in solving a particular problem, not in working out the basic underlying principles that tie together all of mathematics. (This is reiterated in Plato’s discussion of dialectic at 533bc.) By the by, insofar as they use the hypothetical method, I think they probably also work with theorems and proofs of some sort.

I tend to think of Plato’s mathematicians as being concerned with problems and issues confined to a particular discipline (or even more restricted to particular problems within some discipline). And it’s important for the philosophers in training to learn those disciplines, because doing so gives them important insight into the nature of these intelligible objects and it teaches them important methodological skills (employing the hypothetical method; using proofs and theorems). But equally important for the philosophers in training is to realize that there is an underlying unity and connection between the various problems and disciplines; an underlying unity that contemporary (to Plato) mathematicians are either not aware of or unconcerned with. I think these are probably the basic laws and principles that you mention. But in thinking about and discovering these, I don’t really think they’re doing mathematics any more. They’re working toward an unhypothetical first principle and drawing together the disparate subjects into one, unified, whole. And this seems, to me at least, very much like dialectic and philosophy.

Another way I suppose to think about it is to say that one can do mathematics in a dialectical and a non-dialectical (I’ve heard it called ‘dianoetic’) way. Initially the philosophers in training are asked to learn and think about mathematics in the non-dialectical/dianoetic manner. They’re asked to do mathematics in a way very similar to the way that it is done at that time. Then, at the age of 20, they’re asked to start reflecting on math in a dialectical way. I’m happy to say this…indeed I think something like this must be right. But I then want to put emphasis on the use of dialectic (and by implication philosophy) to argue that the youths are being asked to do something pretty fundamentally different than what they were asked to do prior to then.

And one implication of all of this – an implication that is pretty important to me – is that the youths of Kallipolis are very sophisticated thinkers. By the time they’re twenty they’re already going to be very accomplished mathematicians. They’re just not yet accomplished philosophers, which is what they’re going to start working toward from the age of twenty on.

–> I think you’re erring in your terminology. Harmonics, geometry, calculation etc are, roughly speaking, applied mathematics. Plato could just be saying that at that point in their education, students should be directed to *mathematics proper*–i.e. the proofs, theorems, and laws that make calculations valid.