Unit II — Incorruptible Certitude & the Three-Step Method of Mathematical Transsumption — Structural Synopsis

Argumentative Architecture & Interactive Diagram Guide · DI I.11–12 (1440)

De Docta Ignorantia I.12 — Three-Step Method of Mathematical Transsumption ( O1 D ): (x - (0))^2 + (y - (-1))^2 - (1) = 0 ( O2 D ): (x - (0))^2 + (y - (-2))^2 - (4) = 0 [ O2 D ]: (1)*x + (0)*y + (0) = 0 [ E G ]: (0)*x + (1)*y + (0) = 0 [ B C ]: (0)*x + (1)*y + (1/2) = 0 [ P Q ]: (0)*x + (1)*y + (1) = 0 E = (-2, 0) G = (2, 0) O1 = (0, -1) O2 = (0, -2) D = (0, 0) B = (-sqrt(3)/2, -1/2) C = (sqrt(3)/2, -1/2) H = (0, -1/2) P = (-sqrt(3), -1) Q = (sqrt(3), -1) E G O1 O2 D B C H P Q
Book I, Chapters 11–12 — Three-Tier Epistemological Ascent of Transsumptio: (I) Finite Figures in Ratio -> (II) Coincident Infinite Limit in Intellectus -> (III) Transsumption to the Figure-Free Simple Infinite.

I. Argumentative Synopsis

  1. Ontological Status of Mathematical Entities (`DI I.11`): Visible things are symbolic mirrors of invisible realities (1 Cor. 13:12), yet sensible bodies fluctuate continually owing to material possibility. Mathematical entities, abstracted from sensible flux while retaining imaginable figure, possess incorruptible stability and certainty. Cusa marshals an unbroken canon—Pythagoras, Plato, Aristotle (“as the triangle is in the quadrangle, so the lower form is in the higher,” De Anima II.3, 414b29), Augustine, and Boethius—culminating in the mathematical refutation of Epicurean indivisible atoms via geometric divisibility.

  2. The Three-Step Method of Transsumption (`DI I.12`): Since every imaginable figure is finite whereas the Absolute Maximum is figure-free, simple analogy fails; one must “leap beyond simple likeness” (simplicem similitudinem transilire) through a rigorous three-tier transsumption:

    • Step 1 (Finite Ratio): Examine finite geometric figures together with their exact metric properties and ratios (passiones et rationes).

    • Step 2 (Infinite Limit in Intellectus): Transfer those ratios correspondingly to infinite figures of the same kind, where finite oppositions collapse into identity.

    • Step 3 (Transsumptive Leap): Transsume the intelligible ratios of those infinite figures still higher to the Simple Infinite wholly absolved from every figure whatsoever.

  3. Unifying the Patristic & Medieval Geometric Symbols (`DI I.12 §2`): Cusa announces that Anselm's infinite rectitude, the infinite right-angled triangle, the infinite circle, and the infinite sphere are not competing metaphors but mathematically identical in the infinite limit.

II. Interactive Diagram Guide (cusa_transsumptio_ascent.svg)

  • Step 1 (Finite Ratio): Finite chord B–H–C and circles centered at O1 and O2 through contact vertex D remain mutually distinct under finite ratio.

  • Step 2 (Infinite Limit): Tangent horizon E–D–G absorbs expanding circles into coincident rectitude under intellectus.

  • Step 3 (Transsumptive Axis): Upper transcendent axis P–Q marks the leap beyond every spatial figure.