**The sixteenth-century astronomical debate** began with Nicolaus Copernicus’s discovery of the planetary heliocentric (or rather "heliostatic") system.1 After the first reception of this work by German mathematicians, who were primarily interested in the calculation of planetary positions,2 the debate focused on general hypotheses and geometrical models, on cosmology and natural philosophy, and also involved the epistemological status of mathematics. This comparative survey of three influential Copernicans—the mathematician Johannes Kepler, the pre-Galilean physicist Giovanni Battista Benedetti, and the philosopher Giordano Bruno—is aimed at reflecting on the role of philosophical and epistemological assumptions for the development of a new mathematical science. Moreover, it will shed new light on the so-called Copernican or astronomical revolution, questioning the historical commonplace that Copernicanism was a standard world view, rather than the eclectic composition of diverging cosmologies. The work of Benedetti, relatively unknown to scholars, will be particularly helpful in illuminating just what it was to learn and know in the context of post-Copernican cosmological debate.

## Empirical Astronomy

**In his Mysterium cosmographicum** (Tubingen, 1596)—"a work … of tiny bulk, of modest effort, of contents in every way remarkable"—Kepler revealed the archetypal reasons for heliocentrism, which he asserted to have been part of the secret doctrines of the Pythagoreans.3 This attempt at a metaphysical foundation of the Copemican system opposed the empirical approach of Kepler’s immediate forerunners, who, particularly in the 1570s and 1580s, concentrated on the observation of the heavens and the recording of new data. Copernicus had already taken a certain number of accurate observations for the improvement of celestial parameters. In the De revolutionibus, Copernicus also described his instruments (similar to those of Ptolemy): the solar quadrant, the armillary or spherical astrolabe, and a "parallactic instrument" called the triquetrum.4 The empirical ground of De revolutionibus was very much appreciated in the early-modern period. Sometimes it was even overemphasized, as in the case of Giovanni Antonio Magini, professor of mathematics at the University of Bologna, who assured that his ephemerides agreed with "Copernicus’s observations."5 Actually, they relied merely on Erasmus Reinhold’s Copernican tables (Tabulae prutenicae, Tubingen, 1551).6

**Other members of the generation** before Kepler also prosecuted the project to develop astronomy through records of new empirical data. In Kassel, the Landgrave Wilhelm, patron of astronomy, charged his skilful craftsman Jost Burgi with building observational instruments and planetary models. Moreover, he worked together with some of the major astronomers of the time (Rothmann, Brahe, Flaml0se, Wittich, Ursus), who visited him or resided at his court. He exchanged data and opinions with Brahe through an intense correspondence published in 1596.7 Brahe himself was a keen celestial observer: his observatory of Uraniborg is one of the technical marvels of sixteenth-century science.8 He published the images and the description of his instruments in a work titled Mechanica "for the restoration of astronomy" (Wandesburg, 1598). Later, Brahe’s data enabled Kepler to perfect his celestial geometries, to discover the laws of planetary motion, and to compute the Rudolphine Tables (Ulm, 1627).

**Mastlin,** the master of Kepler in Tubingen, shared the conviction that astronomy should rely on the precise recording of data. In Ephemerides novae (Tubingen, 1577), he urged astronomers to begin with celestial observations instead of intellectual or abstract speculations.9 In order to measure the elevation of sun and planets, and the distances of the celestial bodies, he indicated two instruments: a quadrans magnus and a radius. Like Wilhelm of Hesse and Brahe, Mastlin planned to perfect astronomy, in particular its predictive part, which he called astronomia practica. He adhered to Copernicus’s hypotheses, though he preferred not to write openly about them.10

**Nonetheless,** in the astronomical disputation De astronomiae hypothesibus sive de circulis sphaericis et orbibus theoricis (Heidelberg, 1582), Mastlin dealt with an epistemological issue. He gave a solution to the problem of the real (material) existence of astronomical circles and orbits, placing them between conventionalism and realism.11 Celestial orbs, he wrote, are deduced a posteriori, and not a priori, because no one has privileged access to the ethereal region. Therefore, the names of the celestial zones are partly conventional. Yet they correspond to something real, as for instance "orient" refers to the place where the sun rises, and "ecliptic" refers to its path.

## The Cosmological Turn

**Reflection on the natural and physical** consequences of the new astronomy matured in the 1590s. This "conceptual revolution," regarded by Koyre and Kuhn as the "astronomical" or "Copernican" revolution tout court,12 received a decisive impulse from discussion about geo-heliocentric world systems. Such models were invented in order to combine Copernicus’s geometry (as Kepler saw it, the "unity" of the celestial phenomena based on a new conception of the relationship earth-sun-other planets),13 with Aristotle’s physics (which implied the centrality and immobility of the earth). Granada has convincingly dated the "cosmological turn" back to 1588,14 when two printed topics presented for the first time the geo-heliocentric model: Ursus’s Fundamentum astronomicum, and Brahe’s De mundi aethereis recentioribus phaenimenis.15 Also in 1588, Bruno printed a Latin treatise directed at German scholars, Acrotismus comoeracensis, in which he described and argued for an infinite universe containing an infinite number of solar systems. He and other supporters of Copernicus needed to elaborate answers, both physical and philosophical, to the geo-heliocentric challenge. This confrontation forced them to deepen the epistemological problem of the status of mathematics in natural science. A reflection on the relationships of hypotheses to nature, and mathematics to physics, was necessary in order to demonstrate the plausibility of the Copernican model: that is, to establish whether it was just a skilful "invention," or rather a "discovery."

## Kepler’s a priori

**Kepler,** however, did not clearly distinguish between invention and discovery. As is well known, Kepler believed that there were binding geometrical reasons, founded in God’s mind, for heliocentrism. He found a surprising correspondence between the five regular solids (the so-called "Platonic solids"), and the six planets. In fact, he succeeded in inscribing and circumscribing the celestial spheres in these geometrical figures, in full observance of the astronomical distances. He regarded this remarkable coincidence as the intelligible proof of cosmic harmony and Divine Providence.16

**Mastlin was** first informed of this ratio a priori in a letter dated October 3, 1595. He appreciated Kepler’s inventio so much that he took upon himself the responsibility of publishing the Mysterium cosmographicum, together with a new edition of Rheticus’s Narratioprima and his own calculations of planetary distances from the center of the world (a kind of verification of Kepler’s model). Though Mastlin had earlier considered aprioristic astronomy impossible, in the preface to the Mysterium he approved his pupil’s attempt to deduce the planetary theory from archetypal principles ("a fronte"), instead of from the effects ("a terga"), in contradistinction to his predecessors.17 Mastlin thought that Kepler’s metaphysical discovery/invention would lead to universal acceptance of Copernicus’s teaching.

On exactly this basis, his adherence to the heliocentric hypotheses became fervent.18

**To be sure,** Kepler also investigated the physical causes of planetary motions. In the twentieth topic of the Mysterium, he considered the sun as the "universal motive soul," cause of all planetary motions through distant action. He regarded it as the luminous image of the first person of the Holy Trinity, and called it also "world hearth," "king," "emperor," and "visible God." Kepler considered remoteness from the centre as the reason for the different periods of the planets, in agreement with a well-known Aristotelian dictum. In later work, Kepler kept the main assumptions of this cosmology. In the Astronomia nova (1609) he tried to develop a new physics, bringing together explanations from material causes and rigorous mathematical demonstrations.19 Apart from this, he kept his belief in universal harmony, and worked hard to reconcile his geometrical hypotheses with Brahe’s precise observations. The most relevant result of this effort was his Harmonices Mundi, published in 1619.

## Kepler’s Epistemology

**According to Kepler,** astronomy cannot do without cosmological assumptions. Thus, he opposed the mathematic-conventionalist interpretation of Copernicus’s work (generally referred to by historians of science as the "Wittenberg interpretation").20 At that time, most European mathematicians shared the conventional point of view, first endorsed by German scholars, that restricted astronomical theory to its predictive capability, regardless of its physical tenability. In order to defend against the apparent inconsequentially of this approach, they alleged the logical assumption (necessitas syllogistica) that false hypotheses might lead to true conclusions. The theologian Osiander had already presented this argument in the anonymous preface of the first edition of Copernicus’s De revolutionibus, in order to preserve "mathematical" astronomy from conflict with peripatetic physics and Biblical exegesis. Legitimized by conventionalism, many post-Copernican mathematicians (e.g. Wittich from Breslau, and the Scotsman Liddel, who taught in Rostock and Helmstedt) demonstrated the geometrical equivalence of different planetary models, regardless of their physical reality.21

**In Astronomia nova** Kepler drew anti-conventionalist arguments from the French philosopher Petrus Ramus.22 Yet he did not share Ramus’s general idea of freeing astronomy from all hypotheses. Besides Ramus, the Italian Neoplatonic philosopher Francesco Patrizi also maintained that geometrical hypotheses are useless, because celestial bodies move irregularly, and only God’s Providence accounts for their motions. In the first topic of Apologia Tychonis, Kepler refuted Patrizi’s cosmology. He accepted both the free motion of planets in space ("Primum hoc illi [Patricii] facile concessero, solidos orbes nullos esse"), and intelligent universal design ("Neque nego, planetarum circuitus ratione summa administrari"),23 but considered hypotheses necessary, because he held that God’s creation realizes uniform and perfectly circular motions ("ut uniformem et quam fieri potest regolarissimum circulum describanf’).24 Kepler was in disagreement with the Italian philosopher also on another cosmological issue: the infinite space beyond the fixed stars. In fact, according to Kepler, geometrical perfection implied cosmological proportion and finiteness.

**In Apologia Tychonis,** he went deeply into the epistemological status of hypotheses. In the section Quid sit hypothesis astronomica, he criticized the imperial mathematician Ursus for embracing conventionalism. Against such a "vain" approach to astronomy, Kepler affirmed astronomical hypotheses to be both useful and true. In fact, false hypotheses, like lies, generate innumerable errors, especially in physics. However, Kepler distinguishes various meanings of "hypothesis." First of all, in geometry, it means the starting point of a demonstration ("certum quodam initium"), similar to the foundations of a building in architecture ("fundamenta domus"). Secondly, in Aristotle’s logic, "hypothesis" means the premise of a syllogism. Thirdly, in astronomy, there are two meanings of "hypothesis." In origin, this term referred to empirical data, on which theory is based. The meaning of "general conception" ("summam quodam conceptionum Celebris alicuius artificis, ex quibus totam ille rationem motuum coelestium demonstrat’)25 became then usual. According to Kepler, conventionalism brings out an incorrect analogy between hypotheses in astronomy, suppositiones in geometry (where different presuppositions can demonstrate the same thesis), and premises in logic (where it is true that F^T).

**Thus conventionalism entails an equivocation:** the confusion of geometrical, logical, and astronomical hypotheses. Unlike geometrical models, astronomical systems cannot be equivalent, because they imply different physical consequences ("Nam si in geometricis duabus hypothesibus conclusiones coincidant, in physicis tamen qualibet habebit suam peculiarem appendicewT).26 For instance, a Ptolemaic astronomer like Magini, though using Copernicus’s tables, must consider the parallax of Mars greater than the sun’s—an incorrect consequence of geocentrism.27 Another example is taken from a comparison between Copernicus and Brahe. In the absence of an observable stellar parallax, the distance of the stars is much greater according to heliocentrism. According to this theory, in fact, their distance is deduced from the diameter of the earth’s orbit around the sun, instead of the earth’s diameter only, as is the case with geocentrism and geo-heliocentrism. As a matter of fact, Kepler remarked that all cosmological hypotheses bear important and problematic consequences, such as the earth’s motion and the immensity of the sky according to Copernicus; the troublesome rotation of the planets around a rotating sun according to Brahe; and the uttermost rapidity of daily stellar motion according to Ptolemy (and Brahe).

**Kepler encouraged astronomers to discover the** "true" motions, which agree with theory ("vias vero veras invenire, opus esse astronomiae contemplative").28 It was his belief that hypotheses must be "true in every respect" ("et proinde hypothesibus hoc estproprium … utsint undiquaque verae"), as was the case with Copernicus, pauculis mutatis.29

## Benedetti’s Mathematical Philosophy

**Perhaps one way to sum** up Kepler’s position would be to say that he held that astronomy ought to be considered part of natural philosophy—a discipline on which peripatetic philosophers claimed the monopoly. In Apologia Tychonis, he wrote explicitly: "The astronomer ought not to be excluded from the community of philosophers who inquire into the nature of things."30 A similar opinion was expressed by Giovanni Battista Benedetti, who was mathematician to the Court of Savoy in Turin, and was regarded by Kepler as one of the few Italians who were not "asleep."31

**In Diversarum speculationum mathematicarum et physicarum liber (Turin, 1585),** Benedetti asserted that mathematicians can legitimately deal with natural issues. His purpose was, indeed, to mathematize physics. In a letter to the Venetian patrician Domenico Pisani, included with the title De philosophia mathematica in the already mentioned Diversarum speculationum liber, Benedetti reaffirmed the philosophical status of his discipline, at the same rank as physics, metaphysics, and morals, by reason of the certainty of its demonstrations ("certitudo suarum conclusionum").32 Like his teacher Niccolo Tartaglia and his correspondent Pietro Catena, a professor at the University of Padua,33 Benedetti opposed Alessandro Piccolomini’s and Benedict Pereira’s Aristotelian refutation of the possibility of explaining nature by the means of mathematics.34 As a direct consequence of this epistemology, he dismissed the traditional distinction between physics and mathematics also in cosmology. That is, he refused to divide the investigation of "causes" from that of calculation.35 This anti-conventionalist opinion was combined with adherence to the Copernican system.36

**Benedetti corresponded** with Patrizi,37 agreeing with him that space is infinite above the fixed stars, with the difference that he conceived of the planetary system as heliocentric. In a letter to the Savoian court historian Pingone, Benedetti maintained that space is boundless: "it is not necessary that the place of fixed stars be terminated by any convex-concave surface."38 Benedetti’s argument is aprioristic, in that he deduces the reality of infinite space from its mere possibility. In fact, he appears to believe that, once the rational possibility is ascertained, it is not necessary to demonstrate that the universe is infinite, but rather that it is limited. Moreover, according to Benedetti, the sky is fluid, so the earth and the other planets move in a motionless aer.

**The title of another letter,** De … infinito spacio extra coelum, coelique figura, points out the distinction between infinite space (spacium) and finite heaven (coelum). The world, plunged in cosmological infinity, is spherical by reason of an aprioristic consideration of the "economy" of this solid figure, "because no body can be defined by a limit more briefly than by a sphere."39 Yet Benedetti’s Pythagorean philosophy is remarkably different from Kepler’s. Benedetti was convinced, unlike Kepler, that there is an ontological hiatus between ideal perfection and its concrete realization. He kept closer to Proclus’s doctrine, reflected in Renaissance Neoplatonist theory, of the ontological-epistemological medietas of mathematical beings, placed between imperfection and perfection, sensible and intelligible reality—or better, between creation and God’s mind.

**Benedetti worked on calendar** reform and ephemerides long enough to notice that there were no absolutely reliable predictions, but only more-or-less exact calculations and tables. Nonetheless, he wrote an apology for ephemerides, Defensio ephemeridum, in which he defended the validity of astronomical calculations in spite of their intrinsic limits.40 He tended to explain the incomplete regularity of celestial motions through the Platonic doctrine of natural imperfection, in a way similar to cardinal Cusanus, who, in the fifteenth century, argued that the only absolute equality is that of God with himself ("praecisam aequalitatem solum deo convenire").41

**Benedetti shared Kepler’s** aprioristic assumption of celestial harmony. The Italian mathematician reflected thereupon in a section of his Diversarum speculationum, in which he refuted peripatetic physics, Disputationes de quibusdamplacitisArist[otelis]. In topic XXXIII, Pythagoreorum opinionem de sonitu corporum coelestium non fuisse ab Aristotele sublatam, Benedetti denied that the "sound of celestial bodies" is the material production of any sounds. Unlike Kepler, he thought there was no harmonic proportion among planetary motions, as there was no perfect astronomical geometry. Rather, he reduced the Pythagorean doctrine of world harmony to Divine Providence.42

**It is remarkable how different** are the conclusions that Benedetti and Kepler reached from the same philosophical starting-point. They were both convinced of the geometrical-rational structure of the universe first taught by Pythagoras; but the former believed that Divine Providence must manifest itself through infinite space, whereas, according to the latter, God’s creation must be finite, harmonic, and proportional. Furthermore, Benedetti did not suppose that nature can fully realize mathematical perfection, whereas Kepler held the opposite view.

## Bruno’s Infinite and Homogeneous Universe

**Benedetti was rather elusive about** the metaphysical reasons for cosmological infinity. In contrast, his contemporary Giordano Bruno was explicit and exhaustive on this issue.43 In the Italian dialogue De I’infinito, universo e mondi (1584) and in the Latin poem De immenso et innumerabilibus (1591), Bruno analyzed, discussed, and refuted Aristotle’s arguments against the infinite universe. Moreover, in the eighth topic of De immenso, he criticized the infinite cosmology of Palingenius’s Zodiacus Vitae (1534). He mocked this author as one of those "sleepers with the rabble who realize that they are dreaming, and try to shake off their sleep; but presently they dream that they are awake, having merely changed the vision of sleep, not driven it away."44 Palingenius, like the later Patrizi and Benedetti, believed space to be infinite, but the world finite, surrounded by incorporeal light. In contrast, Bruno stated that God cannot create a finite world and a heterogeneous universe, as a consequence of his wisdom, power, love, supremacy, glory, and life.45 A God of infinite power but finite creative action would be infinitely "jealous" and finitely good. There follows Bruno’s infinite and homogeneous cosmology, as an application of the so-called "principle of plenitude."46 One of the main sources of Bruno’s cosmology is cardinal Cusanus’s De docta ignorantia, from which he derived arguments for the infinite worldly "sphere," as well as for the earth’s motion. It is notable that Bruno often coupled the names of Cusanus and Copernicus.47

**Bruno distinguishes the "infinite sphere**" of the world from the "finite spheres" of celestial bodies. These are either "fires"/suns, or "waters"/earths.48 According to him not only is the universe infinite, but it also includes innumerable worlds, or better, heliocentric planetary systems ("synodis ex mundis").A9

**The difference between** Bruno’s conception and those of Kepler and Benedetti is noteworthy. It emerges clearly from his Articuli … adversus huius tempestatis mathematicos atque philosophos (Prague, 1588), article 142, which rejects every possible limitation of the coelum: "For us, neither similitude to archetypes, nor convenience of capacity, nor necessity of distinction, nor the impossibility of the vacuum, nor the unsuitability of the penetration of bodies means that the heavens are spherical."50

## Nature and Mathematics in Bruno

**In natural philosophy,** Bruno always preferred physical to mathematical explanation. Notably, he criticized Copernicus’s "excess" of mathematics, at the expense of philosophical and cosmological speculation. In his conception, the universe is similar to an immense animal, living in every part.51 A vital impulse ("vis animalis") permits planetary motions ("omnium principium motus intrinsecus est animalis appulsus, atque spiritus universum exagitans").52 Planets have, in fact, a sensible and intellectual soul in order to perform their functions. Universal vitalism implies, according to Bruno, the becoming of all beings, called vicissitudo or, in Italian, vicissitudine5 This doctrine is closely related to atomism. Atoms are what persist in natural transformations: the ephemeral life of compounds is caused by the movement of their never-perishing constituents. Furthermore, Bruno reduces all changes to local (atomic) motion, an idea explicitly drawn from Democritus, Epicurus, and Lucretius.54

**In Articuli adversus mathematicos**, and above all in De triplici minimo et mensura (1591), Bruno regards mathematics as a doctrine dealing with bodily properties. Thus, the finite divisibility of the bodies (atoms are, indeed, the last indivisible components of matter) prohibits dividing numbers to infinity. Irrational numbers are banished. For the same reason, Bruno also denies the possibility of squaring the circle, and regards geometrical figures as irreducible and qualitatively different, "because a polygonal figure and a circular one cannot be composed of the same number of parts."55 It is not possible to compare "either the square with the circle, nor the square with the pentagon, nor the triangle with the square, nor a figure of a species with a figure of another species."56 Moreover, since geometrical figures reveal the properties of atomic compounds, Bruno is convinced that reflection on geometrical figures, and on numerical sequences expressing their augmentation, leads to comprehension of bodily basilar characteristics. Hence the complex numerology he expounds in De triplici minimo. In his opinion, the possibility of a mathematical natural science depends on the atomic structure of reality. Its validity is strictly related to Democritean physics. Bruno writes that scholars of geometry violate the laws of nature, if they maintain infinite numerical divisibility, because they neglect atomism: "Thus, the surveyor who divides to infinity what has a precise quantity is wrong, does not follow nature [naturae vestigia], and never grasps it, nor agrees with it in any respect."57

**The minimum is only intelligible**, that is, it is conceivable but not given to the senses. In both physics and mathematics the five senses are therefore useless. On this basis, Bruno distinguishes primary (true) properties from secondary (apparent) properties of natural beings: "With the eyes we perceive light, colour and motion, but with them we cannot grasp the truth of that colour and that light which we perceive, nor distinguish it from appearances of the same species."58 Furthermore,the becoming of all compounds implies that macroscopic phenomena (such as planetary motions) are not perfectly mathematical. Their measurement is extremely difficult, as absolute exactness is revealed to be impossible:

**Nam rerum numeros alios momenta reportant Singula**, quae celeri nulla virtute coirent; Organa qui poterit reputari exacta dedisse Heic ubi nec fluxus eadem est dimensio, ut inde Terminus a reliquo aeque absistat, vel semel unus?

[Singular aspects of things, which slowly change, express different numbers; who can hold that instruments provide accurate measuring, since the becoming of things does not keep a constant rhythm and no limit remains at the same distance from another?]59

**This consideration does not lead to skepticism.** Bruno explains that, for practical purposes such as measuring, one should be satisfied with approximation (" aequale magis suscipere atque minus”").60 Approximation is necessary in the quantification of all natural phenomena, including the astronomical.61

**To sum up,** Bruno’s view of mathematics and mathematical explanation directly derives from his vitalist and atomist natural philosophy. In particular, his atomistic foundation diverges from the Platonic (and Proclean) conception. According to that tradition, in fact, the mathematical order of nature relies on ideal geometrical entities, and on an ontological hierarchy of reality. This viewpoint was shared by Kepler and Benedetti, though they were in disagreement as to the degree of mathematical perfection of the material world. In contrast, Bruno’s mathematics is based on the so-called "minimum," the mathematical indivisible point, counterpart of the physical atom.

## Kepler-Galilei-Bruno

About 1610 Kepler seriously confronted Bruno’s cosmology and philosophical approach. It happened after the publication of Galilei’s Sydereus Nuncius (1610), which described new telescopic discoveries: the mountainous surface of the moon, a large number of newly observed stars and, above all, some satellites of Jupiter. Galilei had already declared his adherence to the Copernican system in the dedication to Cosimo De’ Medici, suggesting that the discovery of the

Jovian satellites proved that the moon was not the only planetary satellite of the heliocentric system.62 In fact, the exception of the moon had been regarded as an inconvenient aspect of Copernicus’s theory.

**Kepler was informed of Galilei’s discoveries** by his friend Wackher von Wackenfels, who was an imperial functionary in Prague. Wackenfels, an adherent of Bruno’s natural philosophy, believed that the recently-discovered planets could prove cosmological infinity. As he was not well informed, he thought that the new planets rotated around some fixed star, whose cosmological necessity had been maintained by Bruno.63 Furthermore, Bruno’s cosmology was endorsed also by another acquaintance of Kepler’s, the English mathematician and botanist Edmund Bruce, who, between 1599 and 1605, was in epistolary communication with him from Italy, acting as an intermediary between him and Galilei.64

**Kepler’s initial concern about the new** planets was that this discovery could invalidate the cosmic order presented in the Mysterium. As an answer to Galilei he wrote Dissertatio cum Nuncio Sydereo (1610), where he declared his satisfaction that the new planets rotated around a planet, and not around a star: "If you had discovered any planets revolving around one of the fixed stars, there would now be waiting for me chains and a prison amid Bruno’s innumerabilities, I should rather say, exile to his infinite space."65 As it was, Kepler rejoiced at Galilei’s telescopic observations, agreeing with him that his discoveries reinforced the heliocentric system, showing other satellites besides the moon.

**In the Dissertatio Kepler also meant**, however, to give Galilei a lesson on epistemology. Sustaining the superiority of an aprioristic approach, he did not hide his conviction of the pre-eminence not only of his work, but also of Bruno’s, towards Galilei’s more empirical astronomy. He celebrated, among the ancients, Pythagoras, Plato, and Euclid, who were guided by the light of their reason only, and held that the universe respects a divine and geometrical law of harmony. Copernicus, revealing the true planetary motions and the centrality of the sun, established "mere" facts, whereas Kepler took credit for the discovery of the "secret causes" of the world’s construction:

**For the Glory of the Creator [Architectus]** of this world is greater than that of the student of the world, however ingenious. The former brought forth the structural design from within himself, whereas the latter, despite strenuous efforts,scarcely perceives the plan embodied in the structure. Surely those thinkers who intellectually grasp the causes of phenomena, before these are revealed to the senses, resemble the Creator more closely than others who speculate about the causes after the phenomena have been seen.66

Kepler firmly believed that good science is a priori. On this matter he clearly agreed with Bruno and Benedetti. Moreover, insofar as Kepler committed himself, and subsequent astronomy, to the discovery of hidden laws—the secrets of the universe—he did so precisely on this a priori basis.

## Conclusion and Future Research

**In the late Renaissance,** as we have seen, the most influential post-Copernican astronomers founded their views on metaphysical speculations about the perfection of the universe, though differently conceived. Despite the fact that empirical astronomy—developed primarily by Brahe—permitted better predictions, the a posteriori approach began to be regarded as less important than rational cosmology. This is clearly revealed, for instance, by Kepler’s judgment on his predecessors and Galilei. Notably, the emerging cosmological debate was not so much a criticism of empirical astronomy as an internal confrontation among concurring aprioristic epistemologies and philosophies of nature.

**Despite their common background**—a Pythagorean and Platonic anti-Aristotelian philosophy, implying cosmic perfection and universal harmony— Kepler, Benedetti, and Bruno proposed very different philosophical foundations for nature and astronomy: a strictly geometrical science (Kepler); a more moderate, slightly skeptical, mathematical one (Benedetti); and a vitalist, infinitely multiform, and anti-geometrical view of nature (Bruno). The differences among these three thinkers reveal the importance of epistemological and aprioristic (theological, natural, metaphysical, and cosmological) reflections for the development of the modern scientific world view. According to them, indeed, the defense of the new science could not be separated from strong ontological and epistemological convictions.

**Besides,** their disagreement on cosmology, perfection of the world and mathematics reveals that, between the sixteenth and the seventeenth centuries, there was no standard Copernicanism, despite the simplified image, later promoted by Galilei, of a struggle between two chief world systems, Copernican and Ptolemaic. In fact, the concurring cosmologies were more than two, even within the heliocentric framework. Copernicus’s work did not lead to a shared conception, or to a unified natural science, but rather to a plurality of possible "Copernicanisms." Instead of a standard view of nature and method, early-modern science gave birth to a pluralistic and dubitative attitude, expressed through an intense and eclectic philosophical debate.