Aristotle's Theory Unity of Science

29 Genus, Abstraction, and CommensurabilityOnly certain theorems could be proved at the general level. Others still hadto be treated specifically. Still others, as we just observed, could be treatedboth ways. The question 'In what genus is the theorem of alternatingproportion proved 7' does not have a single unambiguous answer. Ratherit is proved in two genera in two different ways. As a result, numericalproportion, for example, cannot be treated within a single genus, but insteadsome theorems must be proved in a more general genus, others in amore specific. This is a very frequent occurrence in Aristotelian science aswell.AbstractionConfusion in determining the correct qua-level for a demonstration arisesmost frequently among qua-levels that are related to one another by abstraction(acpatpftTls). We can move from one qua-level to a more generalqua-level when we abstract certain features of the object under consideration. For example, to move from isosceles triangle to triangle we abstractthe general triangular nature and ignore the fact that it has two sides equal.Similarly, from the straight line we may eliminate all of its geometricalnature, including its genus, line, and leave only a necessary accident, itsbeauty. As this last example makes clear, the process of abstraction occursnot just among the qua-levels of mathematics, but among objects moredistantly related as wel l. In fact, Aristotle's most famous application of theprocess abstracts mathematical entities from their physical substrates:Just as the universal part of mathematics deals not with objects which existseparately, apart from magnitudes and from numbers, but with magnitudes andnumbers, not however qua such as to have magnitude or to be divisible, clearlyit is possible that there should also be both formulae and demonstrations aboutsensible magnitudes, not however qua sensible but qua possessed of certain definitequalities. For as there are many formulae about things merely considered as inmotion, apart from the essence of each such thing and from their accidents, andas it is not therefore necessary that there should be either something in motionseparate· from sensibles, or a separate substance in the sensibles, so too in the caseof moving things there will be formulae and sciences which treat them not quamoving but only qua bodies, or again only qua planes, or only qua lines, or quadivisibles, or qua indivisibles having position, or only qua indivisibles. (Met. M.3l077b17-30jAccording to Aristotle mathematical objects are ontologically dependent onnatural substances. We abstract the quantitative nature of substance and