13.6.5.3 Measure, circles, spheres and God

Brient asked

In what sense is the infinite the measure for the finite? ¹

Hegel answered

Measure, like the other stages of Being, may serve as a definition of the Absolute: God, it has been said, is the Measure of all things.²

Cusanus used three mathematical metaphors to explain the possibility of this proposition. The first is the number series – that we can recognise the comparative as comparative implies an awareness of the superlative. The second is the division of the continuum and of a finite line in particular. He used this to indicate that the infinite, as its essence, is the measure of the finite.³ The third metaphor is of an n-sided polygon inscribed in a circle. The increase of its sides brings it ever closer to coincidence with the circumference of the circle, thus illustrating that the infinite is the measure of the finite as its goal and perfection.⁴ In all three, mathematical infinity is a metaphor for God’s infinity. For Cusanus, mathematics symbolises the creative power of the divine.

Cusanus further explored the metaphorical potential of an infinite line arguing that it is a triangle and, drawing on two concepts that are Neoplatonic markers, that the maximum triangle is a circle and a sphere.⁵ He likened the infinite circle to oneness.⁶

In De visione Dei he wrote

the angle of Your eye, 0 God, is not of a certain magnitude but is infinite. Moreover, the angle of Your eye is a circle—or better, an infinite sphere-because Your sight is an eye of sphericity and of infinite perfection.⁷

When praising this world as a copy of the intelligible, Plotinus asked

And what globe more minutely perfect than this, or more admirably ordered in its course, could have been conceived in the image of the self-centred circling of the World of Intelligibles? And for a sun figuring the Divine sphere, if it is to be more splendid than the sun visible to us, what a sun it must be!⁸

All-Soul and Intellectual-Principle (the Divine Sphere) form concentric circles around the One – the third hypostasis around the second, both around the first. This Divine Triad, a unity, is the Divinity itself which is also named, as a totality, the Divine Circle.

Prop. 33 In The Elements of Theology states

All that proceeds from any principle and reverts upon it has a cyclic activity.

The explanation is

For if it reverts upon that principle whence it proceeds (prop. 31), it links its end to its beginning, and the movement is one and continuous, originating from the unmoved and to the unmoved again returning. Thus all things proceed in a circuit, from their causes to their causes again. There are greater circuits and lesser, in that some revert upon their immediate priors, others upon the superior causes, even to the beginnings of all things. (my italics) For out of the beginning all things are, and towards it all revert.⁹

Prop. 146 states

In any divine procession the end is assimilated to the beginning, maintaining by its reversion thither a circle without beginning and without end.¹⁰

In On the Theology of Plato Proclus wrote

And intellect is that which converts itself to the principles, conjoins the end with the beginning, and produces one intelligible circle.¹¹

Cusanus wrote in Book I of De docta ignorantia

Others who have attempted to befigure infinite oneness have spoken of God as an infinite circle. But those who considered the most actual existence of God affirmed that He is an infinite sphere, as it were. I will show that all of these [men] have rightly conceived of the Maximum and that the opinion of them all is a single opinion. …the infinite line, which is a triangle, is also a circle. And [this is] what was proposed [for proof].

Moreover, that an infinite line is a sphere becomes very obvious in the following way: The line AB is the circumference of the maximum circle—indeed, it is the [maximum] circle, as was just proved. And, in the triangle ABC, AB was brought from B to C, as was previously stated. But BC is an infinite line, as was also just proved. Hence, AB [which is the maximum circle] reached C by a complete coming around upon itself. (my italics) And since this is the case, it follows of necessity that from such a coming around of a circle upon itself (my italics) a sphere is originated. And given that we previously proved that ABC is a circle, a triangle, and a line, we have now proved that it is also a sphere. And these are [the results] we set out to find.¹²

In his Science of Logic Hegel wrote

The essential requirement for the science of logic is not so much that the beginning be a pure immediacy, but rather that the whole of the science be within itself a circle in which the first is also the last and the last is also the first.

 We see therefore that, on the other hand, it is equally necessary to consider as result that into which the movement returns as into its ground. …the line of the scientific advance becomes a circle.¹³

The image of the progress to infinity is the straight line, at the two limits of which alone the infinite is, and always only is where the line – which is determinate being – is not, and which goes out beyond to this negation of its determinate being, that is, to the indeterminate; the image of true infinity, bent back into itself, becomes the circle, (my italics) the line which has reached itself, which is closed and wholly present, without beginning and end.¹⁴

Finally, on the second-last page of the Science of Logic, Hegel not only echoes Plotinus, Proclus and Cusanus, writing

By virtue of the nature of the method just indicated, the science exhibits itself as a circle returning upon itself, the end being wound back into the beginning, (my italics) the simple ground, by the mediation; this circle is moreover a circle of circles, for each individual member as ensouled by the method is reflected into itself, so that in returning into the beginning it is at the same time the beginning of a new member.¹⁵

in the next sentence, as I argued at 13.6.2.4 of §575 in the Philosophy of Mind/Spirit, he again refers to the ‘links of the chain’ of the Proclean triad and his Encyclopaedia, in developmental order – logic, nature and spirit.¹⁶

Links of this chain are the individual sciences of logic, nature and spirit. each of which has an antecedent and a successor – or, expressed more accurately, has only the antecedent and indicates its successor in its conclusion.¹⁷

Notes

1. Brient, ‘How Can the Infinite be the Measure of the Finite?’, op. cit., 210 ↩
2. Hegel, Hegel’s Logic, op. cit., 157 ↩
3. ‘an infinite line is the essence of a finite line. Similarly, the unqualifiedly Maximum is the Essence of all things.’ Nicholas of Cusa, De docta ignorantia (‘On Learned Ignorance’), op. cit., I, 17, 47 ↩
4. ‘Cusanus uses each of these mathematical metaphors to illustrate general metaphysical principles that pertain to the relationship between the infinite and the finite, namely, between God and creation. He uses the example of the number series to figure the unfolding of creation in all its multiplicity from divine unity. His reflections on the nature of the continuum, in turn, serve to articulate his conception of the immanence of the infinite in the finite and illustrate important aspects of his metaphysics of contraction. Finally, the the maximum polygon, which is resolved into identity with the circle, figures the link between the infinite and the finite – the locus of enfolding and unfolding as a limit-concept, one in fact that posits the coincidence of the privative infinity of the universe with the absolute infinity of God.’, Brient, ‘How Can the Infinite be the Measure of the Finite?’, op. cit., 224-225 ↩
5. De docta ignorantia, Bk I, Chapter 15, ‘The maximum triangle is a circle and a sphere.’ ↩
6. Ibid., Bk I, Chapter 21, ‘The likening of an infinite circle to oneness.’ ↩
7. Nicholas of Cusa, De visione Dei (‘The Vision of God’), op. cit., 8, 32 ↩
8. Plotinus, The Enneads (Abridged), op. cit., II.9.4 ↩
9. Proclus, The Elements of Theology, op. cit., Prop. 33 ↩
10. Ibid., Prop. 146 ↩
11. Proclus, On the Theology of Plato, op. cit., Bk. III, Ch. IX ↩
12. Nicholas of Cusa, De docta ignorantia (‘On Learned Ignorance’), op. cit., I, 12, 34; I, 15, 40-41 ↩
13. Hegel, Hegel’s Science of Logic, op. cit., 71-72 ↩
14. Ibid., 149 ↩
15. Ibid., 842 ↩
16. §575 ‘It is this appearing which originally gives the motive of the further development. The first appearance is formed by the syllogism, which is based on the Logical system as starting-point, with Nature for the middle term which couples the Mind with it. The Logical principle turns to Nature and Nature to Mind.’ Hegel, Hegel’s Philosophy of Mind, op. cit., 314 ↩
17. Hegel, Hegel’s Science of Logic, op. cit., 842 ↩

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