The city residents certainly do math, although astronomy is the only subject specifically mentioned. Interestingly, they have a Keplerian orrery showing elliptical orbits, presumably in a heliocentric model. This seems the sort of thing that would fit in well, and doesn't require any fundamental innovations beyond the conic sections Hellenistic mathematics was well acquainted with. (You can use hyperbolas to trisect angles, for example.) But the practice of astronomy is tricky.

Would the Just City use Greek numerals? They are somewhat cumbersome for high-precision arithmetic, though of course the Greeks did pretty well (and had abaci.) But compared to place-value notation they are far from ideal, and so shouldn't a city that strives toward excellence adopt the ahistoric Indo-Arabic numerals?

Similarly, astronomy requires a lot of precomputed tables of trigonometric functions to do well. The ancients had such tables--- but we know that many of them were buggy. Excellence ought to require using a modern table, but would the masters be willing to accept one? (Particularly since it would require allowing modern numerals.)

But if you're going to allow a modern place table, are you going to forbid the modern analysis which constructed it? Ancient techniques were very clever but again, cumbersome in practice. Even leaving aside the workers and the sciences involved in electricity, many of the masters would be conversant with Euclid's parallel postulate and the philosophical debate surrounding it. Would they really prefer that the library of the city

*not*have the answer?

Jo is clear that philosophy in the City has a cut-off date. No moderns (and certainly no Christians.) History ends with the Battle of Lepanto in 1571. It's less clear to me where you draw that line in a principled way for mathematics. Of course, the Just City does not shy from unprincipled ways of making decisions, and practical concerns mean that much of mathematics that the masters might allow is in practice inaccessible, just because nobody among the masters understands and can teach it. But Kepler, who was born in 1571, used mathematics in a way that is so different from the Hellenistic tradition, and it would be difficult to include his work without including his foundation.

In fact, Simmea learns the calculus, which dates to the late 17th century. From a philosophical standpoint, this is particularly troubling as the logical problems of infinitesimals were well-understood at the time (and certainly could be apparent to anyone trained in the Socratic tradition.) They were only resolved in the 19th century with modern real analysis. I don't think this material could be included without bringing in aspect of modern philosophy as well, even though most practicing mathematicians default to a rough-and-ready Platonism.

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