Commentary on Euclid's Elements

Indeed, we must first follow this path of inquiry. For if we were to suppose, as some say, that "true opinion" is implanted in mathematical concepts through a kind of technical skill original: technē; here referring to the mental craft of abstracting rules from experience., despite the soul not truly possessing or encompassing their underlying principles original: logoi; the rational structures or "blueprints" of reality., then how does mathematics appear to bring such great order to the soul and generate such profound contemplation?

How could the soul distinguish between things that have grown together—those which are fertile offspring—and those which are unchanging and self-governing? By what laws does the soul use mathematics to tame the sensations within us? Furthermore, if the soul does not possess the very essence original: ousia; the fundamental being or "what-ness" of a thing. of these concepts, how could it generate such a vast variety of reasons and ratios? If we followed that logic, we would be making the very existence of mathematical objects a matter of chance, referring to no established boundary or limit.

Rather, are mathematical forms the offspring of the soul? Does the soul possess these principles and this knowledge at the same time as it perceives sensory things? It is from those internal principles that these mathematical objects are projected In Neoplatonism, "projection" (probolē) is the process by which the soul unfolds its innate, simple ideas into the complex, extended shapes we see in our "mental screen" or imagination.. They arise from the soul’s own essence. It does not seem impossible that what is eternal should come from within the soul’s own home.

Secondly, if we were to gather the principles of mathematics from "below"—that is, from sensory objects—how could we claim that the proofs within us are of unequal value? For example, how could we say that proofs derived from the "Good" are superior, or those that follow more universal and higher paths? We maintain that in every case, the "causes" used in a proof must be appropriate to the nature of the thing being investigated.

If, then, particular things were the causes of universals, and sensory things were the causes of intelligible things original: nooumena; things understood by the mind rather than the senses., what possible mechanism would allow the limit of a proof to refer only to the universal rather than the specific? And how could the intelligible, which excels in its very essence, be considered the source of truth?