Liber IV — Imagination, Phantasma & Mathematical Abstraction — Structural Synopsis

Argumentative Architecture & Bekker Conspectus (Bekker 427a17–432a14)

Architectural Thesis

In De Anima III.3–8 (427a17–432a14), Aristotle unveils the cognitive psychology that makes geometry possible for an embodied intellect: distinguishing imagination from both sensation and judgment, distinguishing receptive and active intellect, explaining how the mind thinks undivided continuous lengths and indivisible boundaries (430b6–20), and establishing the canonical law that the soul never thinks without a phantasm (431a16, 432a8) and thinks mathematicals (ta ex aphaireseōs, 431b12–17) within imaginative forms.

Logical & Noetic Progression

De Anima III.3 (427a17–429a9) — Isolation of Phantasia Between Sensation & Belief

Separates perceiving, imagining, and judging: perception of proper sensibles is true, belief (doxa) entails conviction (pistis), whereas phantasia is a movement arising from actual perception (κίνησις ὑπὸ τῆς αἰσθήσεως τῆς κατ’ ἐνέργειαν γιγνομένη, 429a1–2) capable of Voluntary figurative construction before the mind’s eye (πρὸ ὀμμάτων γὰρ ἔστι ποιήσασθαι, 427b18).

De Anima III.4–5 (429a10–430a25) — Receptive & Active Intellect; Continuum vs. Essence of Straight

Shows that intellect must be unmixed to receive all intelligible forms (429a18). Crucially distinguishes (429b10–22) three tiers: (1) flesh and ‘the snub’ (sensible matter + form), (2) ‘the straight’ (to euthy, analogous to the snub because it involves continuous extension, μετὰ συνεχοῦς γάρ, 429b19) vs. ‘to be straight’ (to euthei einai), and (3) wholly immaterial essences. Introduces productive intellect (nous poiētikos) acting like light (ὡς ἕξις τις, οἷον τὸ φῶς, 430a15).

De Anima III.6 (430a26–431a1) — Thinking of Indivisibles, Undivided Length & Point

Explains how the mind thinks an actually undivided line segment (μῆκος ἀδιαίρετον ἐνεργείᾳ) in an undivided time, whereas a point and every division are made known like privation (ὥσπερ ἡ στέρησις, 430b20–21).

De Anima III.7–8 (431a1–432a14) — Noesis in Phantasmata & Objects of Abstraction (431b12–17)

Declares that to the thinking soul phantasmata serve as percepts (431a14–17), that the faculty of knowledge thinks the forms in the images (τὰ μὲν οὖν εἴδη τὸ νοητικὸν ἐν τοῖς φαντάσμασι νοεῖ, 431b2), and that the mind thinks mathematical objects of abstraction (τὰ ἐν ἀφαιρέσει λεγόμενα, 431b12) just as one might think concavity in isolation from the nose-flesh in which it occurs (431b12–17). Concludes in III.8 that intellect is the ‘form of forms’ (εἶδος εἰδῶν, 432a2) and that without perception and phantasmata nobody could ever learn or understand anything (432a7–9).

Chapter & Bekker Span Conspectus

Treatise & Chapter	Bekker Span	Thematic Scope	Tracks
De Anima III.3	427a17–429a9	Distinction of Imagination (Phantasia) from Perception & Judgement	EN · EL
De Anima III.4	429a10–430a9	Receptive Intellect, Flesh vs. Snub, & Continuum vs. Essence of Straight	EN · EL
De Anima III.5	430a10–430a25	Active & Productive Intellect (Nous Poietikos) as Light	EN · EL
De Anima III.6	430a26–431a1	Thinking of Indivisibles: Undivided Length & Point/Division	EN · EL
De Anima III.7	431a1–431b19	Thinking in Phantasmata & Grasping Mathematicals (Ta ex Aphaireseos, 431b12–17)	EN · EL
De Anima III.8	431b20–432a14	The Soul as Place of Forms & Intrinsic Dependence of Noesis upon Phantasmata	EN · EL

Reception Snapshot in the GEOMETOR Canon

This unit encompasses 4 distinct Bekker loci cited across geometor/euclid, geometor/hoenen, and geometor/pacioli. See the companion Critical Notes & Dual Commentary and the master Bekker Concordance for full folio and page back-references.