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from itself, and tending toward itself; but by opinion and sensation—for these forms of knowledge are directed toward external things and act upon them—and they contain the causes of things known. Mathematical science begins worthily with recollection original: anamnesis; the Platonic idea that learning is actually the soul remembering truths it knew before birth, and it ends in the formal principles original: logoi; the rational structures or ratios that define reality. It proceeds in a straight path from subsequent things back to prior things, reaching toward the sources of all things. Its activity is motionless, like that of the intellect original: noera; pure, intuitive thought that sees the whole at once, yet it unfolds through motion—not spatial motion, but demonstrative motion This refers to the step-by-step process of a mathematical proof, which "moves" the mind from one truth to the next.. Unlike the senses, it unrolls through a vital motion and surveys the incorporeal order of principles, sometimes proceeding from principles to results, and at other times traveling back in the opposite direction.

It moves from things already known to things sought after, and sometimes from things sought after to things prior in terms of knowledge. Therefore, it is neither established above all inquiry as the self-sufficient intellect is, nor does it end at something external as sensation does; rather, through inquiry it turns toward discovery, and leads from the imperfect to perfection. It possesses twofold powers of inquiry: some that carry the principles out into a multitude This is the process of "synthesis," where one builds complex proofs from simple axioms. and reveal the many-formed outpourings of contemplation, and others that collect them from many paths back into their proper hypotheses This is the process of "analysis," where one works backward from a problem to find the underlying principles that explain it.. For since the starting points—the One and the Many, the Limit original: peras and the Unlimited original: apeiron—preceded it, and their underlying subjects are gathered together into our middle nature...