[Book] One

Pure intellectual intuition looks toward the unhypothesized principle itself original: noēsis; the highest form of knowledge which grasps truth directly and instantly, without needing steps or proofs.. Mathematical things, then, have neither been allotted an indivisible subsistence original: hypostasis; the underlying reality or "being" of a thing. that is separate from all division and complexity, nor a subsistence that is known through sense perception, which is ever-changing and many-parted. It is entirely evident that they are granted their own place according to their essence.

Understanding original: dianoia; the discursive faculty of the mind that reasons step-by-step, specifically associated by Plato with mathematical thinking. stands over them as a criterion, just as sense perception stands over sensible things, and as all opinion original: doxa stands over external things.

Hence Socrates Socrates is the primary speaker in Plato's Republic, the work Proclus is commenting upon here. defines opinion as a knowledge less honorable than the primary science referring to noēsis or Dialectic., but better than the reach of mere opinionated impression.

very reasonable

For the one [mathematics] is divisible and proceeds through a course of theory, having an advantage over intuition in its multiplicity, while the other [intellectual intuition] is stable and immutable, surpassing opinion. And while the one [mathematics] starts from a hypothesis original: ex hypotheseōs; mathematics assumes certain definitions and axioms (like "a line is this") without proving them, unlike the highest philosophy. in its pursuit, it has been allotted a share of the primary science. However, these objects were not brought forth in the forms of matter, though they appear according to the finality of sensory things.

to them

Therefore, we define the criterion of all mathematics, according to the mind of Plato, as the Understanding (dianoia), which is established above opinion but falls short of pure intellectual intuition.

What is the essence of mathematical things, from where do they come to us, and how do they subsist?

It follows, surely, that we must observe what essence we should say all mathematical things and their genera possess. Should we concede that we pursue this essence from sensory objects—either by abstraction original: aphairesis; the Aristotelian view that we "extract" mathematical concepts from physical objects, such as seeing a circle in a wooden plate., as people are accustomed to saying, or by the collection of divided parts into one common reason? Or is their subsistence granted prior to these sensory things, as Plato deems worthy, and as the progression of the whole universe suggests? First, then...

Proclus, Mathematics, Dialectic, Plato, Essence, essence, Understanding, discursive thought, Intelligible, intuition, Socrates