  Versions of these problems also affect Mill's empiricism (1843). For Mill, mathematics and logic are natural sciences, and their principles are laws of nature. Arithmetic, for instance, is the theory of aggregates, i.e., the theory of collections of concrete entities. Geometry is the theory of idealised limits of concrete entities—lines, points, planes and so on—whose principles are “real facts with some of their circumstances exaggerated or omitted” (Mill 1843, vol. 1, bk. II, ch.v). Millian philosophy of mathematics is susceptible to the cardinality problem just given. (Of course this is an anachronistic criticism, since infinitary set theory had yet to be developed in Mill's time.) As for the contingency of mathematics, Mill bit the bullet and accepted it.