Two Kinds of Infinity: Nicholas of Cusa and Giordano Bruno
Paul Richard Blum (Loyola
University Maryland, Baltimore)
Presented at
Giordano Bruno: Will, Power, and Being
(Festival Bruniano, 3rd Edition: Tours, 2627 April 2018 – Wittenberg, 1718 May 2018)
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 19^{th}
century, when the Bruno fad among the wouldbe 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.[1]
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.) [2]
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 ...[3]
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.)[4]
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.”[5]
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.[6]
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.[7]
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.”[8]
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.[9]
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.)][10]
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.)[11]
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.)[12]
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.)[13]
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.)[14]
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.[15]
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 CreatorGod 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 nonquanta 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 nonquantitative
things; that which is impossible for quantitative things, you see for nonquantitative things to be
altogether necessary .)[16]
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

exemplary

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.
[1]
This study is a result of research funded by the Czech Science Foundation as
the project GA ČR 1437038G “Between Renaissance and Baroque: Philosophy and
Knowledge in the Czech Lands within the Wider European Context”.
[2]
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 [1802], trans. Michael G. Vater (Albany: SUNY Press, 1984), 222.
[3]
Schelling, Bruno,
222.
[4]
Bruno, Dialoghi
italiani, 341.
My translation.
[5]
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.
[6]
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://cusanusportal.de.
[7]
Bruno, Dialoghi
italiani, 336; Bruno, Cause, 97 (modified).
[8]
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.”
[9]
Plotinus, Opera
omnia: Cum latina Marsilii Ficini interpretatione et commentatione, ed.
Stéphane Toussaint, [Basel: Perna, 1580] (VillierssurMarne: 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.
[10]
Bruno, Dialoghi
italiani, 336f.; Bruno, Cause, 98 (modified).
[11]
Bruno, Dialoghi
italiani, 337; Bruno, Cause, 98 (modified).
[12]
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://jasperhopkins.info/DII122000.pdf.
[13]
Nicolaus de Cusa, Opera
1, book 1, chapt. 18, n. 52, p. 36; Nicholas of Cusa, On Learned
Ignorance, 29.
[14]
Nicolaus de Cusa, Opera
1, book 1, chapt. 18, n. 53, p. 36; Nicholas of Cusa, On Learned
Ignorance, 29f. (modified).
[15]
Nicolaus de Cusa, Opera
1, book 1, chapt. 18, n. 54, p. 36f. Nicholas of Cusa, On Learned
Ignorance, 30.
[16]
Nicolaus de Cusa, Opera
1, book 1, chapt. 14, n. 39, p. 28f. Nicholas of Cusa, On Learned
Ignorance, 23 (modified).
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