Infinite, Renaissance Idea of
Reflection upon the infinite (apeiron in Greek, infinitum in Latin) crosses the whole history of thought and involves several fields of knowledge: philosophy itself, theology, cosmology, and logic-mathematics. The Greek term apeiron stands for something that possesses neither physical limitations nor determined qualities, an entity lacking a conceptual determination defining it (see Anaximander). During Antiquity, only few philosophers allowed for the existence of actual infinite reality; rather, Aristotle’s view of a merely potential existence of the infinite was what prevailed. From the sixteenth century onwards, astronomical discoveries challenged the Aristotelian-Ptolemaic cosmological system: the estimated distance between the center of the Earth and the last sphere grew progressively larger. This scientific advancement was joined in theology-philosophy by the thought that God, who is infinite in act, infinitely good, and the origin of the created universe, does not create finite reality. Rather, He manifests his infinite power by creating an infinite cosmos (this assumption is shared by Nicholas of Cusa and Giordano Bruno).
Heritage and Rupture with the Tradition
The Greek concept of the infinite deeply influenced Medieval and Renaissance thought and cosmology in particular, it being founded upon the Aristotelian-Ptolemaic system. Except for a few instances, Antiquity displayed a profound horror infiniti: as a matter of fact, any lack of boundaries caused bewilderment among the Greeks. Ontological perfection was thus believed to be only found in the finite: through its inherent boundaries, it was deemed to encompass the wholeness of being and exhaust any likelihood of becoming.
However, reflections upon the infinite can indeed be found in some ancient naturalist philosophers, namely, Anaximander, the Eleatics, and the Pythagoreans. In a strictly astronomical sense, many narratives depict a man standing at the extreme boundaries of the universe: with a hand or arrow, he crosses the limit of the last sphere – that of fixed stars – thereby opening a gap within the infinity of the cosmos. This image is found in Archytas (see fr. 47 A 24 DK), Lucretius (De rerum natura, I, 968 ss.), and Cicero (De natura deorum, I, 20–54). It then reaches the Renaissance cosmological debate with Giordano Bruno (De l’infinito, universo e mondi, “Proemiale epistola”) and John Locke (An Essay on Human Understanding, II, 17).
However, the Renaissance will mostly be concerned with Aristotelian and Thomistic views. In Physics, III, Aristotle wonders about the very essence of the infinite, claiming that an infinitely large body cannot exist in actuality, since infinity is only allowed in potentiality, that is, in a processual sense. The concept of the infinite is indeed linked to that of continuity; thus the apeiron is not something that is indefinitely spread, rather an entity that is infinitely divisible. The act is thus reiterated without limits, but the object to which the division process is applied is not indefinite. Aristotle allows for a potential infinity, mainly by articulating it in infinite division and indefinite subtraction; thus, the infinite tends to smallness, but it does not reach a minimum limit. Rather, it continues the logical reduction process of a given magnitude. The philosopher also denies increase, thus rejecting the possibility whereby an object can increase its size endlessly; on the other hand, he deems the cosmos’s limits as the maximum possible magnitude a body can increase toward. Surpassing the universe’s magnitude is therefore physically impossible for both an actual entity and a potential one, since its dimension is the measure of the whole (Physics, IV). Therefore, unlike ancient philosophers, who asserted the cosmos was infinite, Aristotle sees the apeiron in a negative manner. Outside it there is no void; rather, something always exists beyond it. Therefore, it can be likened to indefiniteness.
During the Middle Ages, Thomas Aquinas reworded the alternative between actuality and potentiality and their application to the limitless. The theologian reconsidered the potential infinite as syncategorematic, by giving way to the unlimited process of logical-mathematical repetitions of the finite. On the other hand, categorematic infinity is always the relative property of something which surpasses any actual finite measure. Since Medieval interpretation of the potential infinite requires a shift toward actual infinity, this somehow resolves the opposition between potency and act, thus contemplating the idea of the unlimited. Thomas can be likened to Aristotle in the latter’s denial of actual infinity and the idea according to which only finite things are perfect. However, Aquinas narrows these arguments to the field of created reality, the reflection of God, which is perfect overall because it is finite. In God, rather, infinity and perfection coincide: the former also includes the property of perfection, since form finds perfection in its infiniteness while matter within its limits. Therefore, it is only in the Subsistent Being, the pure act, that infinity becomes actual while still being impossible in created reality (Summa theologiae, I, q. 7).
Innovative and Original Aspects
The Renaissance saw an evolution in the concept of infinity which not only prevailed in metaphysics and theology but also in physics and cosmology, to the detriment of the Aristotelian-Ptolemaic system. Thus, the scientific revolution tackled the infinite in both a strictly philosophical manner (namely, by the likes of Nicholas of Cusa and Giordano Bruno) but also from an astronomical point of view. This resulted in the failing of Medieval cosmology: heliocentrism replaced the geocentric system and its celestial spheres (which made up and supported the universe by revolving around the Earth). This new astronomical model did not include the last sphere that of the fixed stars; rather, it was replaced by the idea of an infinite universe and the homogeneity of the matter of which it was made, thus departing from the traditional separation between the lunar and sublunary worlds.
The pioneer of Renaissance theories on the infinite was Nicholas of Cusa, as recognized by Bruno, Kepler, and Descartes. In his De docta ignorantia, Nicholas states that human knowledge is only conjecture; therefore, the mind cannot fully apprehend God, according to a gnoseological thesis that always preserved the transcendence of the creating principle. At the same time, the universe too is impossible to grasp, since it is an explicatio in space and time of the divine complicatio. Though he maintained a clear ontological separation between the One (God) and multiplicity (His sensitive contractio), Nicholas found that both are impossible to apprehend and, most importantly, infinite. The divine cause is, in its nature, infinite; therefore, there cannot be but a plurality of worlds. The space around them, it too a reflection of the One-Whole, must necessarily be infinite: an infinite universe populated by infinite worlds. Therefore, the primary infinite cause – the efficient cause – matches the effect’s infinity. The various means of divine contraction into the world are the sign of coincidentia oppositorum, when the infinite coincides with the finite and the maximum with the minimum. However, there still remained a substantial difference between infinity in the Divine and in creatures, something that profoundly marked Cusa’s and Bruno’s systems, though the former maintained God’s transcendence. For Nicholas of Cusa, God alone is the absence of all limits – thus a negative infinite – since He Himself is the fullness of all possibility. In the case of creatures, on the other hand, their infinity is a privative one connected to the concept of the indeterminate.
According to Giordano Bruno, too, the need for an infinite universe made up of infinite worlds is justified by a specific theological assumption: the overflowing and limitless power of the first principle generates an equally inexhaustible effect. Since God is the Act, the world – which is His reflection – is the manifestation of all possible forms; hence, matter is made up of a plurality of entities which through vicissitude follow the process of cosmic development. Deeming the infinite as actual is what sets Bruno apart from Aristotelian-Scholastic philosophy. Unlike Cusanus, however, Bruno honors a naturalist view that is strongly immanentist by asserting that divine infinity and material infinity actually coincide: God is thus both natura naturans and natura naturata. On a cosmological level, Bruno accepts the Copernican heliocentric model endowing the notion of infinity with both a quantitative and a qualitative value; the Copernican-quantitative dimension of limitless space is joined by the qualitative dimension of reality, which according to hermetic tradition is ruled by a vitalistic principle that eludes geometry and mathematics. On the topic, Bruno’s Italian dialogues on cosmology are an excellent read: La cena de le Ceneri, De la causa principio et uno, and De l’infinito universo e mondi.
On the other hand, Renaissance astronomical discoveries gradually made the Ptolemaic system fail, whereas the possibility of an infinite universe was increasingly welcomed. Bruno himself accepted the Copernican heliocentric system and praised the Polish scientist for having stood out in astronomy much more than Ptolemy, Hipparchus, or Eudoxius (see La cena de le Ceneri, I). Copernicus also recognized the need to move the universe’s last sphere and he maintained that the universe is immensum in its indefinite extension. In the same way, even though he did not accept infinite space, astronomer Christoph Scheiner stated that the distance between Earth and the fixed stars is equal to an enormously large figure compared to what the Scholastics had calculated; nor did Nicolaus Mercator (Nikolaus Kauffmann) mention the size of the last sky in his tables of fixed stars longitudes. Therefore, the shift between an enclosed and limited dimension to an “indeterminate” magnitude marked an important milestone in the affirmation of an infinitely large universe.
As for Galileo, one must distinguish an astronomical and a strictly mathematical level in his writings. In the Dialogo sopra i due massimi sistemi del mondo, Galileo states that the universe has no center, the fixed stars being other suns or other centers. However, he does not affirm the universe’s infinity: in his opinion, this is a metaphysical issue that must not be tackled by astronomers. In mathematics, on the other hand, Galileo stated that a line segment can be divided into an infinite number of parts which are not further divisible; since they are an infinite quantity, they must necessarily lack size, for even a small size would necessarily make up a line segment that includes a finite number of parts.
Therefore, with Galileo, the debate over the existence of actual infinity seemed to reach a crucial point: for the first time and on a metaphysical level at least, cosmological infinity and the possibility of dividing matter into infinite parts with an infinitely small size both became possible (the argument was subsequently reexamined by Newton in an astronomical way).
Impact and Legacy
The Renaissance debate over infinity had a great impact on the seventeenth and eighteenth centuries. According to Descartes, the concept of infinity was functional in proving God’s existence and was thus both metaphysically and theologically crucial. Thanks to the cogito, mankind acquires clear and evident knowledge of the innate idea of God, mainly by underlining His infinity. In proving God’s existence, based on the assumption that the cause of an idea must be at least as real as the idea itself, Descartes claims that the idea of the infinite does belong to humans but is not caused by them alone; mankind is corruptible, therefore finite. Thus, since humans can feel a sense of their own limit and finiteness, there must necessarily exist a primary cause (superior to mankind), which carries the same sense of infinity humans experience with deprivation, whose highest manifestation is methodical doubt. Descartes also believes that the concept of indefinite is reserved for contingency – thus for humans, too – while the concept of the positive infinite is only reserved for God (see Principia Philosophiae, I, 27).
Again, in metaphysics, Spinoza built a markedly immanentistic system by strongly drawing on Giordano Bruno’s theories. Like Bruno, he believed that the modes of the infinite substance proceed in an infinite way through its infinite attributes: an argument that significantly involved the subject of freedom in divine action.
Leibniz, on the other hand, viewed the world as an organic entity whose substance is made up of infinite monads interacting with each other within the body that contains them (the physical world). Substance is thus guided by its very own finalism, which human knowledge can grasp through mathematics; through it, one can investigate the mechanism regulating mutual actions and interactions between the parts of corpuscular matter. Therefore, Leibniz accepted the actuality of the infinitely small in a qualitative sense by denying the potential infinite of the geometrical continuum.
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