Two Kinds of Infinity: Nicholas of Cusa and Giordano Bruno
Paul Richard Blum (Loyola University Maryland, Baltimore)
It is a commonplace that Giordano Bruno learned from Nicholas of Cusa. He quotes extensively the De beryllo and other texts of Cusanus in order to make his philosophy of the infinity of the world philosophically acceptable. Already in the 19th century, when the Bruno fad among the would-be atheists morphed into the discovery of the ‘forgotten’ initiator of modern thought, it has been discovered, that Cusanus even had entertained the centrality of the sun (instead of the earth) antedating Copernicus by a century. Philosophically speaking, those are matters of dogmatics. The serious problem in the familiarity and difference between the early and the late Renaissance thinker is the notion of the infinite, its constructive function in the philosophical system, its epistemological and metaphysical status. Whereas Cusanus postulates the infinite as the unavoidable condition for the correlation between reality, origin, and understanding, Bruno harvested the fruits of such daring approach to what used to be the domain of Aristotelian ontology and turns the transcendence of the infinite into the immanence of truth in reality.
To propose to speak about two kinds of infinity must appear to be a joke. As we all know, infinity cannot be added to, or subtracted from, infinity; and the reason is that infinity has no comparison. Hence there cannot be two different kinds of infinity, unless both were species of an infinite genus, in which case they were at the same time impossible for their being limited and for there being subordinate kinds of what by definition is devoid of subdivision. However, this calculation raises the question: why should one speculate about infinity at all? In Cusanus and in Bruno we can observe how and why infinity enters the philosophical discourse. We can also see what “comes out” of infinity. And for this pursuit we may paradoxically speak about two kinds of infinity in the sense of different ways to direct philosophical considerations to the infinite and the argumentative result of such considerations. Do we, in the end, have to judge and confer the palm to one of the two ways? Probably not, provided both philosophers philosophize seriously and, as a result, leave us equally puzzled about the quandaries of the infinite. The major thesis I want to convey is this: both Cusanus and Bruno speculate about the infinite for converse reasons or purposes. Cusanus aims at understanding God, Bruno aims at understanding the world. Out of the vast body of available texts I dare to choose just one instance, but I am confident, I could make the case with any number of parallel evidence.
As is well known, in the fifth dialogue of De la Causa Bruno stated: “chi vuol sapere massimi secreti di natura, riguardi e contemple circa gli minimi e massimi de gli contrarii e opposti. Profonda magia è saper trar il contrario dopo aver trovato il punto de l’unione.” (… There is a profound magic in knowing how to extract the contrary …, after having discovered their point of union.)  Friedrich W. J. Schelling’s interpretation of this passage is helpful:
But concerning the shape of philosophical science, and the challenge to cultivate the sturdy seed of this principle of indifference to its fullest flower, the ultimate goal is to achieve a perfect harmony with the very framework of the universe. To this end, both for ourselves and for others, we can prescribe no maxim more excellent to constantly keep before our eyes than that contained in these words, handed down to us by an earlier philosopher: To penetrate into the deepest secrets of nature ...
The somewhat wordy paraphrase by Schelling contains a number of fundamental insights into the philosophy of the infinite.
First of all we need to understand that the One, the punto de l’unione, is indeed the infinite, as both Cusanus and Bruno are keen on arguing. This is also the presupposition for the fact that there cannot be two kinds of infinity. Then, Schelling suggests that indifference is the endpoint of universal harmony. Bruno’s observation about the magic of finding diversity after unity amounts to outlining harmony, but harmony is always of the diverse; and such harmony is not the end but the “maxim” of proper philosophy and its guiding rule. From the idealist point of view, Bruno discovered the principle of true philosophy by way of setting the goal. However, the surprising fact for Schelling is that not the One is the goal but the multitude. Plurality, diversity, and difference is the object of philosophical research based on the understanding that every distinction needs to be grounded in indifference, that is, union, that is ideally the One. In other words, philosophy is not about unity but diversity, not about the infinite but the finite. The finite, however, is grounded in the infinite. Therefore, Teofilo explains a few lines later:
Quella unità è tutto la quale non è esplicata, non è sotto distribuzione e distinzione di numero, e tal singularità che tu intendereste forse; ma che è complicante e comprendente. (That unity is all, which is not unfolded, not distributed and distinguished by way of number, and such singularity as you might intend; rather, such that is enclosing and encompassing.)
Unity is all, but it is not a thing; not a one thing that is singular but a one that contains the diverse. Unity is not a thing but the singularity of things. Teofilo gives the examples of decade and centenary numbers that contain and enclose the lower numbers. The decade is that singularity that comprehends the then digits.
Now in a first approach we may be content with recognizing the well known pattern of unity and diversity and their functions in ordering our comprehension of the finite world; at the same time we are happy to see that everything is capped in some way under a canopy of higher order. But since we do not expect Bruno (or any philosopher) to state the obvious, we need to see why does it matter to him. Therefore we should look at where he was coming from, namely, Nicolaus Cusanus.
Bruno adopted some of Cusanus’ geometrical examples; so the question is, what did these examples stand for? He introduces them with the suggestion that “all things are one in the same way as every number, even or odd, finite or infinite, goes back to unity, which posits number if reiterated with the finite and with the infinite negates number.” All is one – that appears to be plausible. (Of course it is not at all! Otherwise, philosophers from Heraclitus to Hegel had not struggled with it.) All numbers go back to unity – fine. There are four classes of numbers: pair and odd, finite and infinite. So infinite numbers are numbers; and they equally go back to unity? The qualification that follows immediately suspends that understanding: it is unity that confirms or even produces number when repeated with the finite; but it negates number if unity is associated with the infinite. Let us assume that actually does not make sense.
The fact that both odd and even numbers can be reduced to unity appears to be plausible, but only if we consider that either class of numbers is a repetition of the counting of one. Whatever the arithmetic properties of even and uneven digits, they are nothing but units. But Teofilo’s additional remark states that such digits arise from the iteration of one “with the finite.” The only plausible explanation appears to be that ‘finite’ is not a property but a principle such that unity can join the finite or not join it. If it does, then we have number. Number is unity in the realm of the finite. The second alternative (“con l’infinito”) annihilates number. There is no infinite number. Which is implied in the statement that number is by definition finite. Consequently, the infinite considered as a realm, which negates the finite, negates number. But infinity does not negate unity. Unity and infinity go well together.
Bruno illustrates this claim with geometrical examples taken from Cusanus’ De mathematica perfectione. Straight line and curve converge on the level of the maximum and minimum.
Ecco dumque come non solamente il massimo et il minimo convegnono in uno essere …, ma ancora nel massimo e nel minimo vegnono ad essere uno et indifferente gli contrari. (Here, then, is how not only the maximum and the minimum come together into one being … , but also how, in the maximum and the minimum, contraries come to be one and indistinct.
The extremes share identical being and, consequently, opposites are one and indifferent on that level. That may be read as saying that they are not annihilated ontologically but rather they have their foundation in the infinite indifference. In the same way as numbers are grounded in the infinite, insofar as they are finite in terms of numbers, so is any finite geometrical figure as being distinct from other geometrical figures grounded in the extreme that does not know the distinctions but founds them. Apologies for a lame pun: the Infinite is indistinct from, but not indifferent to, the finite.
The second geometrical simile, borrowed from Cusanus, is that of triangles of varied sizes. The principle stated from the outset is: “Just as in all genera, the analogous predicates draw their degree and order from the first and loftiest of the genus.” Obviously, the author is invoking the principle primum in aliquo genere, which contemplates the prime instantiation of a genus to be also the foundation of that genus and thus establishing the ontological status and the gnoseological validity of any member of that class. His way to apply it here is to remind us that this prime is the foundation of analogy and order. Analogy, here, does not mean a weak epistemological approximation to something beyond truth and falsehood. (For instance: the predicate ‘good’ can be said about God only analogically because God remains unknown.) In this context, analogy is the relationship of belonging to the same genus for things that on the finite level of reality are distinct and yet of the same kind – triangles, for instance. Therefore, the ‘first and greatest’ of the class of analogues establishes that class and all of its members. The triangle is convenient as an example, Bruno says, because among the plane figures with angles it is the most elementary that cannot be dissolved into other figures. However, it can be of different sizes. But in terms of triangle there is no distinction between them. This insight is measured against the infinite:
Però se poni un triangulo infinito (…) , quello non arà angolo maggiore che il triangolo minimo finito, non solo che mezzani et altro massimo. (However, if you posit an infinite triangle (…) , it will not have an angle greater than that of the smallest finite triangle, and likewise for that of any intermediate triangle and of another, maximum triangle.)
Inserted in this osservation is the qualifier:
(non dico realmente et assolutamente, perché l'infinito non ha figura: ma infinito dico per supposizione, e per quanto angolo dà luogo a quello che vogliamo dimostrare) [(I do not mean really and absolutely, since the infinite has no figure; I mean infinite hypothetically, insofar as its angle is useful for our demonstration.)]
Bruno avers us that a real infinite triangle does not exist, as infinite numbers don’t. A triangle by definition is finite. However we may conceive of the infinite hypothetically in order to show on the example of the angle the fact that angles remain angles; and that can mean that the infinite triangle, if it were real, is the first of any angle and makes it gnoseologically and geometrically possible. But the overarching argumentative goal is this:
Quindi per similitudine molto espressa si vede come la una infinita sustanza può essere in tutte le cose tutta, benché in altri finita, in altri infinitamente; in questi con minore, in quelli con maggior misura. (Through this very elaborate simile one sees in which way one infinite substance can be whole in all things, although in some in a finite way in others in an infinite way and in some to a minor and in others to a greater degree.)
God, who else could be meant by this “infinite substance”? But is pantheism or panentheism Bruno’s philosophical intention? I don't’ think so. On a first level he is arguing that oneness is the foundation of the many. It is not said that God is actually and essentially present in all things. What he says is that the mode of ‘being in’ of the one is finite and infinite in various and specific degrees. God’s presence in things is analogical. And, as we saw, analogy is not identity but the condition of being the same and yet individual. Therefore, if we search again for the philosophical import of Bruno’s maxim, we see clearly that he is seeking for the foundation of the reality that is by definition finite. There is no multitude without oneness; there is no singularity without the singularity of the One.
Now let us have a look at how Cusanus explained how he employed geometrical examples. In
Amplius, non satiabilis noster intellectus cum maxima suavitate vigilanter per praemissa incitatus inquirit, quomodo hanc participationem unius maximi possit clarius intueri. (Furthermore, our insatiable intellect, stimulated by the aforesaid, carefully and with very great delight inquires into how it can behold more clearly this participation in the one Maximum.)
Initially, his is an epistemological enterprise: how can the human intellect understand that things participate in the maximum One? Bruno will transform the notion of participation into that of analogy and harmony. Also, he is not skeptical about the human understanding but more concerned with the ontological status of finite beings. About the curved and the straight line Cusanus explains:
Et sicut finita recta in hoc quod recta – in quod quidem rectum curvitas minima resolvitur – secundum simpliciorem participationem participat infinitam, et curvum non ita simplicem et immediatam sed potius mediatam et distantem, quoniam per medium rectitudinis quam participat: ita aliqua sunt entia immediatius entitatem maximam in seipso subsistentem participantia, ut sunt simplices finitae substantiae, et sunt alia entia non per se, sed per medium substantiarum entitatem participantia, ut accidentia. (A finite straight line, insofar as it is straight (minimal curvature is a reduction to that which is straight) participates in the infinite line according to a more simple participation, and a curve [participates in the infinite line] not [according to] a simple and immediate participation but rather [according to] a mediate and remote participation; for [it participates] through the medium of the straightness in which it participates. Similarly, some beings—viz., simple finite substances—participate more immediately in Maximum Being, which exists in itself. And other beings—viz., accidents—participate in [Maximum] Being not through themselves but through the medium of substances.)
It is critical that the curved and the straight line participate in the maximum in various degrees and that the curved, in terms of participation, depends on the straight. The thrust of the argument goes towards establishing such maximum on which the particulars depend. The maximum is not ‘the largest thing around’, it is the only true thing; it is the measure of what there is and it is the reality of what appears to be there. A particular thing is not more thing than any other; but in terms of derivative from the one and only true reality (the maximum) it is a closer or weaker approximation to that.
Et in hoc aperitur intellectus illius, quod dicitur substantiam non capere magis nec minus. Nam hoc est ita verum, sicut linea recta finita in eo, quod recta, non suscipit magis et minus; sed quia finita, tunc per diversam participationem infinitae una respectu alterius maior aut minor est, nec umquam duae reperiuntur aequales. Curvum vero, quoad participationem rectitudinis, recipit magis et minus; et consequenter per ipsam participatam rectitudinem sicut rectum recipit magis et minus. (In this is disclosed an understanding of the statement that substance does not admit of more or less. This statement is as true as a finite straight line, insofar as it is straight, does not admit of more and less; but because it is finite, — through a difference of participation in the infinite line— one line is longer or shorter in relation to another; and no two lines are ever found to be equal. But a curve admits of more and less as it participates in straightness; and consequently, due to this participated straightness it admits of more or less as the straight line does.)
The maximum does not entail comparison but is the canon of comparability.
Substantiae igitur et accidentis una est adaequatissima mensura, quae est ipsum maximum simplicissimum; quod licet neque sit substantia neque accidens, tamen ex praemissis manifeste patet ipsum potius sortiri nomen immediate ipsum participantium, scilicet substantiarum, quam accidentium. (There is, then, one most congruent measure of substance and of accident—viz., the most simple Maximum. Although the Maximum is neither substance nor accident, nevertheless from the foregoing we see clearly that it receives the name of those things which participate in it immediately, viz., substances, rather than [the name] of accidents.
The maximum, that is, the infinite that is God, is the ontological measure of things while not being one of the things. In this way the human mind may fathom epistemologically that the finite reality refers to an ontological foundation. Therefore we may say that Cusanus, as any medieval theologian, takes the existence of the Creator-God as a given and labors to make this plausible to the human mind. On the other hand, Bruno accepts Cusanus’ lead, but he now focuses on how this God can possibly help understand the creation. In Cusanus, the epistemological challenge is to understand God through things; in Bruno the challenge is to understand things with the invocation of God.
The geometrical example of the triangle clearly deviates from Cusanus. The author of Learned Ignorance had argued that “patet lineam esse infinitam triangulum maximum.” If we extend a triangle to the maximum by stretching the angles, we arrive at a straight line, and an infinite one.
Quare si per positionem angulus valeret duos rectos, resolveretur in lineam simplicem totus triangulus. Unde cum hac positione, quae in quantis impossibilis est, iuvare te potes ad non-quanta ascendendo; in quibus, quod in quantis est impossibile, vides per omnia necessarium. (Therefore, if, by hypothesis, an angle could be two right angles, the whole triangle would be resolved into a simple line. Hence, by means of this hypothesis, which is impossible in quantitative things, you can be helped in ascending to non-quantitative things; that which is impossible for quantitative things, you see for non-quantitative things to be altogether necessary .)
The entire exercise is intended to show that by extending the empirical and geometrical knowledge to the infinite, we may come to an understanding of what it is like to be infinite.
So, do we have two kinds or two versions of the infinite? Cusanus’ infinite was
- epistemologically challenging but accessible by way of similes
- the notion of God that may be gained from His creation.
Bruno’s infinite was
- the ontological foundation of finite things
- epistemological tool of understanding reality
- a useful hypothesis.
Bruno accepts Cusanus’ speculation that yields an intellectually acceptable notion of the infinite and he turns this infinite into a scientifically inevitable hypothesis that establishes harmony in the finite world.
 This study is a result of research funded by the Czech Science Foundation as the project GA ČR 14-37038G “Between Renaissance and Baroque: Philosophy and Knowledge in the Czech Lands within the Wider European Context”.
 Giordano Bruno, Dialoghi italiani, ed. Giovanni Gentile and Giovanni Aquilecchia (Firenze, Sansoni, 1958), 340. I quote this edition as the most accessible through http://bibliotecaideale.filosofia.sns.it. Giordano Bruno, Cause, Principle, and Unity and Essays on Magic, trans. Robert De Lucca and Richard J. Blackwell (Cambridge, U.K.: Cambridge University Press, 1998), 100: “There is a profound magic in knowing how to extract the contrary from the contrary, after having discovered their point of union.” The words “from the contrary” are not in the text and inconsistent, for the idea is to derive opposites from the very One. The translator of Schelling's quotation of this passage has it right: "To discover their point of union is not the greatest task, but to do this and then develop its opposite elements out of their point of union, this is the genuine and deepest secret of art." Friedrich Wilhelm Joseph von Schelling, Bruno, or On the Natural and Divine Principle of Things , trans. Michael G. Vater (Albany: SUNY Press, 1984), 222.
 Schelling, Bruno, 222.
 Bruno, Dialoghi italiani, 341. My translation.
 Bruno, 335: “che le cose tutte sono uno: come ogni numero tanto pare quanto ímpare, tanto finito quanto infinito, se riduce all'unità, la quale iterata con il finito pone il numero, e con l'infinito nega il numero.” My translation.
 Nicholas of Cusa, Opera (Basel: Petri, 1565), vol. 3, p. 1120; Nicolaus de Cusa, Opera omnia. Vol. 11.1: De beryllo, ed. Carolus Bormann and Iohannes Gerhard Senger (Hamburgi: Meiner, 1988) n. 41, p. 47. Cusanus is quoted from http://cusanus-portal.de.
 Bruno, Dialoghi italiani, 336; Bruno, Cause, 97 (modified).
 Bruno, Cause, 97; Bruno, Dialoghi italiani, 337: “come in tutti geni li predicati analogi tutti prendeno il grado et ordine dal primo e massimo di quel geno.”
 Plotinus, Opera omnia: Cum latina Marsilii Ficini interpretatione et commentatione, ed. Stéphane Toussaint, [Basel: Perna, 1580] (Villiers-sur-Marne: Phénix, 2005), 3.5, chapt. 5, p. 289: “Quo vero sublimius simpliciusque corpus est, eo tardius videtur ab anima deferendum, caeleste vero nunquam. In omni siquidem genere quod primo fit particeps, semper est particeps.” The argument is this: the first participation founds the lower ranking participations.
 Bruno, Dialoghi italiani, 336f.; Bruno, Cause, 98 (modified).
 Bruno, Dialoghi italiani, 337; Bruno, Cause, 98 (modified).
 Nicolaus de Cusa, Opera omnia. Vol. 1. De docta ignorantia, ed. Ernestus Hoffmann and Raymundus Klibanksy (Lipsiae: Meiner, 1932), book 1, chapt. 18, n. 52, p. 35; Nicholas of Cusa, On Learned Ignorance: A Translation and an Appraisal of De Docta Ignorantia, trans. Jasper Hopkins (Minneapolis: Benning Press, 1981), 29, http://jasper-hopkins.info/DI-I-12-2000.pdf.
 Nicolaus de Cusa, Opera 1, book 1, chapt. 18, n. 52, p. 36; Nicholas of Cusa, On Learned Ignorance, 29.
 Nicolaus de Cusa, Opera 1, book 1, chapt. 18, n. 53, p. 36; Nicholas of Cusa, On Learned Ignorance, 29f. (modified).
 Nicolaus de Cusa, Opera 1, book 1, chapt. 18, n. 54, p. 36f. Nicholas of Cusa, On Learned Ignorance, 30.
 Nicolaus de Cusa, Opera 1, book 1, chapt. 14, n. 39, p. 28f. Nicholas of Cusa, On Learned Ignorance, 23 (modified).