Point, position and mathematical ascent

Use in the book

Ficino makes the mind’s everyday ascent explicit. Natural forms require determinate matter, mathematical forms only indeterminate matter, and metaphysical forms neither. Dimensions need both parts and position; a point lacks parts but remains in some way related to position; numbers need parts but no position; unity needs neither. He then rises beyond indivisible accidental unity to substantial unity, the stable ground of changing accidents. The sequence prevents a reader from treating every nonextended object as already an independently subsisting soul. A second list compares the reductions of motion to an instant, natural form to a least degree, dimensions to a point and numbers to unity. Those examples support an argument for indivisible substance rather than establish that all these limiting items are one object.

Places to read

1. I.3.12–132Position, parts and substantiality are successively distinguished
2. I.3.14The point and centre join an analogical argument about each genus’s principle
3. I.5.24Mental extension of numerical and geometrical possibilities supports a further claim about intelligible abundance

Translation in this edition

“Point” translates both punctum and geometrical signum. “Free” for absolutas in I.3.12 concerns release from particular constraints. It does not yet imply that every universal imagined by a human is a separately existing substance.

certus / incertus

certus / incertus. Printed forms: certa; certis; incerta.

Determinate or fixed, and indeterminate. Natural forms require determinate matter while mathematical forms need only indeterminate matter; this is a distinction of dependence, not confidence versus doubt. References: I.3.12. See Point, position and mathematical ascent.

mathematicus / metaphysicus

mathematicus / metaphysicus. Printed forms: mathematicas; metaphysicas.

Mathematical and metaphysical forms. The distinction is drawn through dependence on indeterminate matter or on no matter, within a mental ascent beyond determinate natural forms. References: I.3.12. See Point, position and mathematical ascent.

momentum / signum

momentum / signum. Printed forms: momentum; signum.

An instant and, in geometry, a point. They are limiting items for motion/time and dimensions respectively in an analogy of kinds reduced to indivisible principles. References: I.3.14. See Point, position and mathematical ascent.

punctum / situs

punctum / situs. Printed forms: punctis; punctum; situ; situm.

A point and position. A point lacks parts yet remains related to position; numbers lack position while requiring parts, distinguishing two kinds of non-bodily abstraction. References: I.3.12. See Point, position and mathematical ascent.

unitas accidentalis / substantialis

unitas accidentalis / substantialis. Printed forms: accidentalem; substantialem; unitatem.

Accidental unity and substantial unity. Ficino rises beyond the indivisible unit toward a subsisting essence that grounds accidents; indivisibility alone does not finish the ascent. References: I.3.13. See Point, position and mathematical ascent.

Use in the book

The geometric exercise begins with finite body as long, broad and deep. Its limit of depth is a surface; the surface’s limit of breadth is a line; the line’s limit of length is a point. These successively remove three dimensions, but Ficino does not yet call the point a soul. As the line’s terminus it remains fixed at a determinate site in body and neither fills nor vivifies the whole. Dimensionlessness alone therefore fails to supply the independence being sought.

A fourth step releases the point at least in thought. Ficino supports this by a qualified priority: what bounds the line exists “in some manner” before its function of bounding, and what is prior may be considered apart from what follows. The restriction cogitatione matters. The reader is not cutting a small physical object loose but considering a limit apart from its located relation. Ficino then invites the reader to give this released point an equal regard toward all bodily parts, diffusion throughout them and power to move them. Fundatur and moveat are jussives: these active capacities are introduced through guided supposition, not deduced as a geometrical theorem from absence of length. The irrational soul is assigned its seat at this fourth stage.

The fifth step claims a reality corresponding to the rational ascent. Because rational thought does not deceive in ascending, Ficino argues, it can find not merely a mentally released point but an indivisible something actually freed from body. It fills and moves body, is itself freely active, and can be raised toward higher things. This is called substantial unity and is the level assigned to rational soul. The transition from conceptual consideration to actual reality therefore depends on the stated reliability of rational ascent. It must not be represented as an empirical measurement proving that a material point changes into a soul.

The ensuing comparison ranks the stages by need and passivity. Body requires more dimensional conditions and admits more divisions; surface needs fewer, line fewer still, and point still depends on position. Substantial unity needs neither dimensions nor site. Ficino infers that the less dependent principle is more perfect, powerful and true. The point of the closing question is precisely whether something can be great in effective power without being great in measurable quantity. Removing size need not mean removing reality. When VI.6.5 later says soul is “more indivisible” than a point, the superiority concerns release from geometrical dependence, not possession of a still smaller spatial fraction.

These numbered degrees belong to a particular pedagogical ascent away from bodily bulk. They should not be mechanically equated degree by degree with either the five metaphysical grades of body, quality, soul, mind and God, or Cavalcanti’s five rival accounts. The succeeding concentrated-fire and light exercises pursue the same pedagogical purpose through different images. VI.2.27 finally recalls the liberated point as a substantial unity and expressly calls the chapter an initial excursio and praeludium. Its vivid construction prepares the reader for formal arguments while retaining distinctions between thought, reality, irrational animation and rational elevation.

Places to read

1. VI.2.19, VI.2.20Four conceptual releases are distinguished from the fifth step to rational substantial unity.
2. VI.2.27The exercise is expressly labelled preliminary.
3. VI.6.4, VI.6.5Indivisibility exceeds geometrical dependence and positional containment.

Translation in this edition

The jussives fundatur and moveat in VI.2.19 mean “let it be diffused” and “let it move,” preserving the imaginative construction. Cogitatione, quodammodo and re ipsa distinguish thought-experiment from the claimed real conclusion. No bodily conversion is asserted.

cogitatione separari

cogitatione separari. Printed forms: cogitatione; cogitatione saltem; re ipsa.

At least in thought; contrasted with separation in reality. Ficino explicitly marks the transition from conceptual release of a point to claimed actual substantial unity. References: VI.2.19, VI.2.20. See Point, dimension and substantial unity.

egere; pati

egere; pati. Printed forms: eget patiturque; indiget.

To need and to undergo, a double measure of dependence. Body’s dimensions and divisibility make it more dependent than surface, line, point and ultimately substantial unity. References: VI.2.20. See Point, dimension and substantial unity.

excursio; praeludium

excursio; praeludium. Printed forms: excursio; praeludium.

An exploratory excursion and a preliminary exercise. Ficino expressly limits the chapter’s geometric and luminous persuasion before the ordered proofs begin. References: VI.2.27. See Point, dimension and substantial unity.

finis; limes; terminus

finis; limes; terminus. Printed forms: fine; finem; fines; limite; terminos; terminum; terminus.

End, boundary or limit, spatial, causal or purposive as context requires. A point bounds a line, body has finite dimensions, growth has a governing limit, and God is the soul’s beginning and end; these are not one spatial relation. References: VI.2.3, VI.2.6, VI.2.18, VI.2.19, VI.12.16, VI.12.19, VI.13.8. See Point, dimension and substantial unity.

fundere; movere

fundere; movere. Printed forms: Fundatur; moveat.

Let it be diffused and let it move the parts. These jussives belong to the guided construction of an unlocated animating point, not an observed physical event. References: VI.2.19. See Point, dimension and substantial unity.

liber; solutus; separatus

liber; solutus; separatus. Printed forms: libere; liberius; liberum; seorsum; separata; separatas; soluta; solutum.

Free, released, separated and apart, whose respect must be specified. Conceptual release from position differs from actual freedom from material origin; separated limbs in a reductio do not imply quantitatively severed souls. References: VI.1.1, VI.1.5, VI.2.10, VI.2.13, VI.2.19, VI.2.20, VI.13.1. See Point, dimension and substantial unity.

locus; situs

locus; situs. Printed forms: loci; locis; loco; locum; locus; situ; situm.

Place and determinate position. Dimensionlessness does not remove a point’s positional attachment; soul’s presence is not occupation of another bodily place. References: VI.1.5, VI.2.3, VI.2.19, VI.2.25, VI.9.1, VI.10.1. See Point, dimension and substantial unity.

longitudo; latitudo; profunditas

longitudo; latitudo; profunditas. Printed forms: latitudo; longitudine; longitudo; profunditatis.

Length, breadth and depth. Their ordered removal organizes the first three approaches to incorporeality; bodily magnitude is not reduced to a single length. References: VI.2.19. See Point, dimension and substantial unity.

quantum; quatenus; respectus

quantum; quatenus; respectus. Printed forms: prout; quanto; quantum; Quantum; tanto; tantum.

Insofar as, as much as, and according as: measures of degree or qualifying respect. The spirit/body analogies and soul’s different functions require the specified respect; a comparison of power is not necessarily a comparison of spatial size. References: VI.1.1, VI.2.8, VI.2.20, VI.12.17, VI.12.19. See Point, dimension and substantial unity.

reperire; invenire

reperire; invenire. Printed forms: inveniemus; inveniet; invenimus; invenire; inveniri; invenistis; invenit; reperietis; reperitur; reperiuntur.

To find or discover in inquiry. The preliminary imaginative exercises announce a discovery pedagogically, while the formal self-motion proof seeks a sufficient first inward source. References: VI.2.3, VI.2.8, VI.2.18, VI.2.20, VI.2.22, VI.12.7, VI.12.17. See Point, dimension and substantial unity.

substantialis unitas

substantialis unitas. Printed forms: substantialis quaedam unitas; substantialis unitas.

Unity existing as a substantial principle, not as a merely positioned geometrical limit. The liberated point is renamed through a reality free of dimensions and position and capable of higher activity. References: VI.2.20, VI.2.27. See Point, dimension and substantial unity.

superficies; linea; punctum

superficies; linea; punctum. Printed forms: lineam; punctum; superficiem.

Surface, line and point as successively less dimensional limits. The geometric ascent subtracts depth, breadth and length without yet removing a point’s determinate position. References: VI.2.19. See Point, dimension and substantial unity.

terminus; situs

terminus; situs. Printed forms: certum habet situm; lineae terminus.

A line’s limit and a determinate position. The point is still tied to bodily location even though it lacks dimensions, so geometrical indivisibility is insufficient. References: VI.2.19. See Point, dimension and substantial unity.

virtute; quantitate

virtute; quantitate. Printed forms: quantitate; virtute magnum.

Greatness in power contrasted with measurable size. Soul’s efficacy need not imply extension; the closing question answers the fear that a nonbody would be nothing. References: VI.2.20. See Point, dimension and substantial unity.