Nicholas of Cusa Writes that the Universe is Privately Infinite

In my previous post, An Infinite Sphere Whose Centre is Everywhere and Circumference is Nowhere, I wrote that Nicholas of Cusa’s (1401-1464) sentence “the world-machine will have its center everywhere and its circumference nowhere” gave the impression that Cusa believed the universe to be infinite.

The question remains, did Cusa actually believe the universe to be infinite? I will focus on this question in this post.

One well-known supporter of the idea that Cusa supported the infinite universe was René Descartes (1596-1650), who wrote the following in 1647 in a letter to Pierre Chanut (1601-1662), ambassador to Sweden and negotiator of the Peace of Westphalia¹.

En premier lieu, ie me souuiens que le Cardinal de Cusa & plusieurs autres Docteurs ont supposé le monde infiny, sans qu'ils ayent iamais esté repris de l'Eglise pour ce suiet; au contraire, on croit que c'est honorer Dieu, que de faire conceuoir ses œuures fort grands. Et mon opinion est moins difficile à receuoir que la leur; pource que ie ne dis pas que le monde soit infiny, mais indefiny seulement.

For those who read modern French, and are having trouble reading the above text, remember that at that time, today’s ‘j’ and ‘v’ were mostly written as ‘i’ and ‘u’, as for Latin. Here is my own English-language translation.

First of all, I remember that Cardinal Cusa and other Doctors supposed the world to be infinite, without ever being taken to task by the Church about this subject; on the contrary, it is supposed that it is to honour God to conceive of his works as great. And my opinion is not as difficult to accept as theirs; for I say that the world is not infinite, but only indefinite.

It should be remembered that Descartes was quite wary of the Holy Inquisition of the Roman Church. For example, his Le Monde [The World] which he wrote during the period 1629-1633, was not published until 1664, well after his death. Descartes had been very perturbed by the condemnation of Galileo Galileo (1564-1642) in 1633. By writing that the world was indefinite, rather than infinite, Descartes was being cautious.

However, not all readers of Cusa have agreed with Descartes. For example, Jasper Hopkins (1936-), who translated into English many of Cusa’s texts, claims in his notes to On Learned Ignorance² that Cusa did not believe in an actual infinity. We consider two notes.

Hopkins’s first note is for Book I:

Nicholas does not believe that there is an actually existing infinite and eternal circle. Nor does he believe that the infinite circle is a supra-Platoniclike Idea or even an Idea in the mind of God. Rather, his point here is purely conceptual and illustrative: the “logic” of infinity is such that an infinite circle would have to be eternal, just as its circumference would have to be its center. [Book 1, pp.54-55, note 111]

This is a note from the following paragraph in Chapter Twenty-One, which is entitled, “The likening of an infinite circle to oneness”.

All these [points] are exhibited by the infinite circle, which is eternal, (111) without beginning and end, indivisibly the most one and the most encompassing. Because this circle is maximum, its diameter is also maximum. And since there cannot be more than one maximum, this circle is most one to such an extent that the diameter is the circumference. Now, an infinite diameter has an infinite middle. But the middle is the center. Therefore, it is evident that the center, the diameter, and the circumference are the same thing. [Book 1, Chapter Twenty-One, p.35, paragraph 64, my emphasis]

In Book I, Cusa conflates the infinite line, the infinite triangle and the infinite circle. The infinite line is related to oneness, hence to God, while the infinite triangle is related to triness, hence to the Holy Trinity. And since Cusa clearly believes in God and in the Holy Trinity, I cannot see on what basis Hopkins can claim that “Nicholas does not believe that there is an actually existing infinite and eternal circle”.

Hopkins’s second note is for Book II:

According to Nicholas there are an indefinite number of stars—from the point of view of the human mind. He does not, however, believe that there is an actual infinity, of stars (or of anything else). [Book II, p.110, note 138]

This is a note from the following paragraphs in Chapter Twelve, which is entitled “The conditions of the earth”.

Hence, since that entire region is unknown to us, those inhabitants remain altogether unknown. By comparison, here on earth it happens that animals of one species—[animals] which constitute one specific region, so to speak—are united together; and because of the common specific region, they mutually share those things which belong to their region; they neither concern themselves about other [regions] nor apprehend truly anything regarding them. For example, an animal of one species cannot grasp the thought which [an animal] of another [species] expresses through vocal signs—except for a superficial grasping in the case of a very few signs, and even then [only] after long experience and only conjecturally. But we are able to know disproportionally less about the inhabitants of another region. We surmise that in the solar region there are inhabitants which are more solar, brilliant, illustrious, and intellectual—being even more spiritlike than [those] on the moon, where [the inhabitants] are more moonlike, and than [those] on the earth, [where they are] more material and more solidified. Thus, [we surmise], these intellectual solar natures are mostly in a state of actuality and scarcely in a state of potentiality; but the terrestrial [natures] are mostly in potentiality and scarcely in actuality; lunar [natures] fluctuate between [solar and terrestrial natures]. We believe this on the basis of the fiery influence of the sun and on the basis of the watery and aerial influence of the moon and the weighty material influence of the earth. In like manner, we surmise that none of the other regions of the stars are empty of inhabitants— as if there were as many particular mondial parts of the one universe as there are stars, of which there is no number. (138) Resultantly, the one universal world is contracted—in a threefold way and in terms of its own fourfold descending progression—in so many particular [parts] that they are without number except to Him who created all things in a [definite] number. [Book II, Chapter Twelve, paragraphs 171-172, pp.96-97, my emphasis]

Once again, Hopkins, as far as I can see, provides no substantiation whatsoever to support his claim that Cusa “does not, however, believe that there is an actual infinity, of stars (or of anything else)”.

Of course, what is important is what Cusa actually wrote in his On Learned Ignorance. Here is a paragraph from Chapter One of Book II:

Therefore, only the absolutely Maximum is negatively infinite. Hence, it alone is whatever there can at all possibly be. But since the universe encompasses all the things which are not God, it cannot be negatively infinite, although it is unbounded and thus privatively infinite. And in this respect it is neither finite nor infinite. For it cannot be greater than it is. This results from a defect. For its possibility, or matter, does not extend itself farther. For to say “The universe can always be actually greater” is not other than saying “Possible being passes over into actually infinite being.” But this latter [statement] cannot hold true, since infinite actuality—which is absolute eternity, which is actually all possibility of being—cannot arise from possibility. Therefore, although with respect to God's infinite power, which is unlimitable, the universe could have been greater: nevertheless, since the possibility-of-being, or matter, which is not actually extendible unto infinity, opposes, the universe cannot be greater. And so, [the universe is] unbounded; for it is not the case that anything actually greater than it, in relation to which it would be bounded, is positable. And so, [it is] privatively infinite. Now, the universe exists actually only in a contracted manner, so that it exists in the best way in which the condition of its nature allows. For it is the creation, which, necessarily, derives from Absolute and unqualifiedly Divine Being—as subsequently and by means of learned ignorance I will very briefly show, as clearly and simply as possible. [Book II, Chapter One, paragraph 97, p.61, my emphasis]

Just to be sure, I checked the Latin original³:

Solum igitur absolute maximum, est negatiue infinitum, quare solum illud, est id quod esse potest, omni potentia. Vniuersum uerò cum omnia complectatur quae Deus non sunt, non potest esse negatiuè infinitum, licet sistine termino, & ita priuatiue infinitum, & hac consideratione, nec finitum nec infinitum est. Non enim potest este maius quam est, hoc quidem ex defectu euenit, possibilitas enim siue materia, ultrà se non extendit. [Liber II, Capitulum 1, p.24, my emphasis]

So according to Cusa, the universe is “privately infinite”, hence “neither finite nor infinite”. It is not “negatively infinite”, because the universe is not God, but, rather, His creation. The universe is not physically bounded, but bounded by the fact that it is lesser than God; as such it is only “privately infinite”. To me, this reasoning sounds remarkably similar to that of Descartes’s, who does not dare call the universe “infinite”—even though he recognises that it is not physically bounded—but retains the word “indefinite”.

It would only be with Giordano Bruno that anyone would claim openly, with no reservations, that the universe is infinite. In so doing, Bruno paid the ultimate price.

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1. Œuvres de Descartes. Volume 5, Correspondance, Mai 1647 - Février 1650. Lettre CDLXXXVIII. Descartes à Chanut.  La Haye, 6 juin 1647, pp.51-52.

2. Jasper Hopkins. Nicholas of Cusa on Learned Ignorance. A Translation and an Appraisal of De Docta Ignorantia. Minneapolis: The Arthur J. Banning Press. 2nd ed., online. 1985.

3. Nicolai de Cusa. Opera. Basileæ: Ex officina Henricpetrina, 1565.