Abstract
This chapter deals with the philosophy of space and the theory of geometry developed by the Renaissance philosopher Francesco Patrizi da Cherso. Patrizi’s metaphysics of space shares several common features with other similar constructions (by Bruno, Campanella, and others) aimed at radically rethinking the notions of space and place present in Aristotelian traditions. The uniqueness of Patrizi’s proposal, however, is to be found in his attempt to ground geometry in this new conception of space, thus claiming for the first time in history that geometry is the science of space rather than the “science of continuous magnitudes” as it had been conceived from Antiquity to his day and age.
An Italian version of this essay has been published in the volume Locus-Spatium of the Lessico Intellettuale Europeo, edited by Delfina Giovannozzi and Marco Veneziani (Firenze, Olschki 2014). I would like to thank Marco Santi for providing me with a draft of the English translation of it, and Jonathan Regier and Alexander Reynolds for their suggestions and careful editing of the text. I also thank the participants to the conference Spaces, Knots and Bonds for their valuable comments.
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Notes
- 1.
This declaration was originally formulated in the following terms: “Le Matematiche tutte, e principali, e subalterne, ne si astraggono dalle cose naturali, ne sono nella fantasia, ne nella dianea, ma lo spazio è generale lor subietto” (Nuova geometria, p. 2). Patrizi states the novelty of his project in the preface to the work, stressing that he has discovered a “royal road” to geometry which “è del tutto nuova, e da niuno antico ne moderno, che si sappia, non tentata”.
- 2.
Plotinus discusses the notion of ὄγκος principally in Enn. II 4 [12], 11. I refer to Plotinus rather than to other Middle- or Neo-Platonic authors because, first and foremost, it seems to me that his strong anti-Aristotelianism, and his stature as an original thinker, contributed to the genuine establishment of this concept of quantified matter. Another reason is that (as we will shortly see) his personal views exerted a direct influence on the formation of the theories of mathematical space developed in the Renaissance. On Plotinian ὄγκος see Brisson (2000); for a more extensive treatment of the role of this concept in modern theories of space, see De Risi (2012b).
- 3.
- 4.
The most extensive survey of spatial theories in the Middle Ages is still Grant (1981).
- 5.
On Patrizi and the Hermetic tradition, see at least Leijenhorst (1997).
- 6.
Henry (1979), Grant (1981), and Edelheit (2009) provide general treatments of Patrizi’s theory of space. Vedrine (1996) is a French translation of De spacio physico & mathematico, with an introduction and annotations, and Brickman (1943) is a (partial) English translation with a short presentation. Cassirer’s discussion of Patrizi in the first volume of Das Erkenntnisproblem is still very useful. There exist almost no studies on Patrizi’s mathematics: besides the above-mentioned essays by Vedrine and Edelheit, and a short (but excellent) treatment in Cassirer, the reader might also usefully consult Muccillo (1993b).
- 7.
The first volume of the Discussiones was published in 1571. However, from a philosophical standpoint, it is the remaining three volumes, especially the fourth, that are particularly relevant; these latter were all published in 1581. On the genesis of the first edition of the work see Artese (1986); on the contents of the first volume, see Antonaci (1984). This was not Patrizi’s first philosophical work. In his youth, he had published several booklets in Italian propounding Platonic and Ficinian philosophical views (especially on love). At any rate, the Discussiones remain Patrizi’s first important and authentically original work.
- 8.
The frontispiece bears the date of 1587, but we know from Patrizi’s correspondence with the mathematician Giambattista Benedetti that, as early as July 1586, he was in a position to send him the first printed pages, and that the whole book had appeared by early December of that year (Patrizi, Lettere, pp. 42–44; the December letter had already been edited, though erroneously dated, in Claretta 1862). It is remarkable, however, that Patrizi affirms in the treatise on geometry that he presupposes therein some principles which he has, so he claims, already proven in his two books on space, since the latter had probably not yet appeared in print by the end of 1586. We may assume, therefore, that he had already written his metaphysical essays by the time of the composition of the Nuova geometria. Conversely, in the final sentence of the two booklets De spacio physico & mathematico, Patrizi noted that the Italian treatise Della nuova geometria was intended as the continuation of the essays on space. This corresponds to the arrangement of the material in the synthesis of the Nova philosophia. We can also note that on p. 210 of the Nuova geometria, where he introduces the concluding book on the geometry of triangles, Patrizi states that he intends afterward to deal with the geometry of curves: he must, therefore, have conceived of a sequel for his geometrical work—which, however, never appeared.
- 9.
We know for certain that Patrizi had originally conceived of his books “de spacio” as the beginning of a larger work, since he explicitly states this in his letter to Tebalducci of 29 June 1587 (Lettere, p. 57), and repeats it in the incipit of De spacio physico (p. 2r). The alterations to the two treatises on space for their inclusion in the Nova philosophia amount to some marginal corrections and to a certain radicalization of the anti-Aristotelian positions. For instance, in Chap. 5 of De spacio physico, Patrizi criticizes certain ancient philosophers, not mentioned by name, whereas in the corresponding locus of the 1591 work their place is taken by “Aristotle”. The only relevant addition is a philosophical introduction (Nova philosophia, pp. 61r–61v) which serves the function of welding these two books more tightly together in the context of the new work. The 227 pages of Nuova geometria, on the other hand, are so reduced in the Latin version that they only make up 5 sheets of the Nova philosophia. Their subject, in fact, largely exceeded that of a treatise on the metaphysics of light. Although the three works, on space and on geometry, had been conceived as a single unitary work (cf. the previous footnote), we possess a letter by Patrizi from 1590 which implies that, a mere few months before the appearance of the Nova philosophia, the work’s outline was much different from the published version, and included only one generic book De spacio and probably nothing on mathematics; see Purnell (1978). Yet other arrangements of the book’s matter are given as an appendix to Patrizi’s Lettere (pp. 550–552). The work in its final form is divided into four sections titled Panaugia, Panarchia, Panpsychia, and Pancosmia, the latter of which includes the works on space and geometry as its Books 1–3.
- 10.
Patrizi dedicated one section of the Pancosmia to Cardinal Ippolito Aldobrandini. Just a few months later, Aldobrandini was elected Pope, taking the name of Clement VIII, and summoned Patrizi to Rome, where a new Chair in Platonic Philosophy was instituted at the Sapienza with the aim of fighting the dangerous heresies of radical Aristotelianism. Patrizi’s lectures in Rome enjoyed a broad resonance and success and were attended by many prominent personalities (among whom was Torquato Tasso, whom Patrizi found at the time “smagrito e smagato e incanutito”, Lettere, p. 88). However, precisely the novelty of his thought and his fervent anti-Aristotelianism caused suspicions to arise against him, so that by the end of 1592 his major work was under examination by the Holy Office, which found in it many theses worthy of censoring. Patrizi’s defense consisted of an Apologia and certain Declarationes (now in Gregory 1955), and he also partially retracted his theses in an Emendatio in libros suos novae philosophiae (now in Kristeller 1970), but this did not prevent his book from being eventually, in 1596, added to the Index, after Toletus had also approved of this in 1594. On the condemnation, see also Firpo (1950) and Rotondo (1982). After news of the condemnation became public, the publisher, Meietti republished in 1594 in Venice some copies of the work bearing a fake frontispiece dated 1593 (that is, before the negative judgment passed by the Church), which are sometimes erroneously indicated as stemming from the work’s second edition (whereas, in fact, no such edition appeared). Patrizi died a few months after the definitive condemnation in 1596, and in 1597 Roberto Bellarmino proposed the abolition of the Roman chair in Platonic Philosophy, judging the latter doctrine (at least in the reading of it propagated by Patrizi) to be even more dangerous than radical Aristotelianism. Clement refused to abolish the chair, but assigned it to the semi-concordist Jacopo Mazzoni, who professed Platonic views leaning towards Aristotelianism and would publish in the same year a study on the agreement of ideas between the two great Greek philosophers (In universam Platonis et Aristotelis philosophiam praeludia). Mazzoni, as is well known, had been Galileo’s philosophy teacher and was still in contact with him (cf. Footnote 56 below). He had fought several intellectual disputes with Patrizi, especially of a literary character, as is attested by some booklets and by Patrizi’s correspondence (see Lettere, pp. 54–56).
- 11.
This revision was certainly due to the inclusion of the work in the Index of prohibited books, from which Patrizi hoped thereby to rescue it. Already Garin (1953) published the first part of a reworking of the Panarchia (the second section of the Nova philosophia), and Gregory (1955) mentioned the existence of a manuscript with a revision of the Panaugia (the first section). More recently, Puliafito (1993) published all the extant materials of Patrizi’s rewriting of the Nova philosophia. Nothing was found concerning the books on space and geometry. On the other hand, according to the ecclesiastical materials about the work’s condemnation which have survived (which are, however, very incomplete), it does not seem that the Inquisition had ever objected to Patrizi’s theses on space. Therefore, he would have most probably kept the contents of De spacio physico & mathematico as they were. Finally, there is a Patrizian manuscript from 1594 containing a treatise in Pythagorean numerology, De numerorum mysteriis, which is still unpublished to this day (see Muccillo 1993b).
- 12.
This attitude is in fact usually to be noted in Patrizi’s other works as well, as he always boasted that his views were certainly new (as is indicated by such titles as the Nova philosophia or the Nuova geometria), but at the same time age-old and already propounded by the ancient Pythagoreans. For instance, when arguing in De spacio physico, p. 9v (=Nova philosophia, p. 63v) for the existence of extramundane space, Patrizi rejects the principle of authority, but also thinks he can oppose an older authority to that of Aristotle, who had denied such space. On other occasions as well he engages in a competition with his opponents (whoever they may be) to find the most ancient source and thus, as it seems, the one closest to the spring of wisdom. Patrizi’s attitude was certainly also due to prudence, and most of the defenses he mounted against the Church’s charges of heresy for his Nova philosophia consisted in arguing that it was at the same time a perennis philosophia, so that a great many of his positions could find support in the authority of the Church Fathers, or in that of ancient philosophers in general. Thus, in his declarationes to the Holy Office, he claims that his philosophy is genuinely new only insofar as it is a whole, that is, insofar as he had intended to reunite into a coherent system the fragments of Trismegistus, Ocellus, Archytas, and Timaeus which the corrosion of time had made barely comprehensible in their own right. It is quite remarkable, however, that Patrizi excluded from this strategy precisely the foundations of his doctrine of space, especially the position that it holds within itself every being (as we will shortly see), which latter theses, he claims, “propria nostra sunt”. He thus possessed the awareness and the pride of having at least offered a new theory of space. On the declarationes against the censors, see Gregory (1955, pp. 422–423). On the concept of perennis philosophia see (among others) Schmitt (1966) and Muccillo (1988). An extensive and elaborate examination of Patrizi’s judgments on Aristotle and pre-Socratic philosophy in the Discussiones is provided by Muccillo (1975, 1981).
- 13.
The discussion of the Aristotelian theory of place is contained chiefly in Discussiones II, VI (pp. 246–248). It must be noted that the Platonic theory of place, which Patrizi here and there opposes to the the Aristotelian one of Phys. 1–5, only amounts to some few mentions of place from the Parmenides, since Patrizi, like the whole tradition of ancient and Renaissance Platonism, construed Plato’s χώρα as prime matter rather than space.
- 14.
The discussions conducted in this work seem to suggest that Patrizi subscribed to a theory of place as the three-dimensional extension of the universe, which he probably shared with Philoponus. In the letter to Tarquinia Molza of 13 November 1577 (thus a few years before the Discussiones) he presented in fact a metaphysics of extended but finite space, which seems perfectly to accord with Philoponus’s views; cf. Lettere, p. 15.
- 15.
Discussiones II, IV, pp. 221–224. On this locus see especially Muccillo (1993b).
- 16.
See esp. Discussiones II, IV, pp. 224–225. Mathematical abstractionism is, of course, of unequivocally Aristotelian origin (although Aristotle certainly provided a much more complex version of it than is to be found in Patrizi’s Discussiones). The reference to imagination, however, is, properly speaking, only Neoplatonic, not genuinely Aristotelian. The tradition linking the object of geometry to φαντασία takes its starting point, as a whole, from Proclus’s commentary on Euclid’s Elements, but the Renaissance had received it through other sources as well (although Proclus’s work had been published in Greek in 1533, together with the editio princeps of Euclid, and translated into Latin by Francesco Barozzi in 1560). Proclus, of course, was by no means an abstractionist, and in fact fiercely opposed Aristotle on this point. On Aristotelian-Neoplatonic eclecticism, and its characterization of mathematical imagination as an abstractive act, see Footnote 57 below, as well as Vasoli (1989) that comments on this passage of the Discussiones. It should also be noted that a form of geometrical abstractionism (without reference to imagination) is also presented in Patrizi’s essay De’ corpi, which probably dates from 1577 to 1578 (in Lettere, p. 171).
- 17.
It is perhaps worth recalling that the dispute on the legitimacy of employing motion in geometry was quite lively in the second half of the Sixteenth Century, at least since Jacques Peletier had fiercely criticized such use in his Demonstrationum in Euclidis elementa geometrica libri sex of 1557, and various mathematicians (including Clavius) had felt compelled to respond to him. At any rate, everyone in that era was convinced that Euclid and the other ancient mathematicians had admitted, rightly or wrongly, the use of motion in geometry (something which we, today, may have reason to doubt). It is certainly possible that Patrizi was aware of this discussion, and that he intended to take part in it in some way. At any rate, here he argues for the movability of the objects of geometry not for foundational reasons in this science, but rather simply in order to criticize the Aristotelian definition of mathematics as the science of non-separable and immovable objects (given in Metaph. Ε 1; but the immovability of μαθηματικά is restated in many other loci). See Footnote 60 below.
- 18.
Patrizi’s treatment of the status of sciences takes up the whole Book Four of Discussiones III; the part mostly concerned with mathematics is on p. 318, where Patrizi denies that the Aristotelian definition of science can be applied to the demonstrative method of mathematics. These are opinions which were widely diffused in the Sixteenth Century, which had actually had their origin in Late Antiquity, had developed during the Scholastic Middle Ages, and had been revived by Alessandro Piccolomini’s aggressive discussion in his De certitudine mathematicarum of 1547. On Patrizi’s relationship with Aristotelian logic, see Deitz (2007).
- 19.
On prime matter, see Discussiones II, VI (pp. 236–238), then chiefly Discussiones IV, III. The distinction between Aristotle’s genuine position and that of the “Senatus Populusque Peripateticus” about the quantification of prime matter can be found on pp. 392–94. Patrizi bases his belief that Aristotle construed prime matter as a non-sensible body (σῶμα οὐκ αἰσθητόν) on an incorrect reading of a passage in De gen. et corr. Α 5, 320b1–2, which calls (dialectically, moreover) the void, and not prime matter, “non-sensible body”; based on this, Patrizi interprets “non-sensible” as “non-qualified”, according to a conception of qualities which is much more Neoplatonic (Plotinian) than Aristotelian, thus ascribing to Aristotle the idea of a quantified prime matter—which is, in fact, a much later theoretical construction. On the concept of prime matter in the Discussiones, see Deitz (1997).
- 20.
Patrizi had cordial relations with, and certainly admiration for, Telesio. In 1572 Patrizi sent to the philosopher a thorough critical analysis of the first book of his De rerum natura, where he principally criticized him for allowing the existence of prime matter: as it is not perceptible to the senses it should, in fact, he argued, be rejected as a Scholastic chimera according to Telesio’s own sensualistic and naturalistic standards (at least as Patrizi interpreted them). Patrizi’s objections to Telesio and Telesio’s reply can be read in Fiorentino (1872–1874, vol. 2, pp. 375–398; the criticism of prime matter is to be found on p. 382); on this subject, see also Garin (1949) and Aquilecchia (1993).
- 21.
On the primacy of space in Patrizi see the short introduction to De spacio physico (p. 2r) or the different introduction to the corresponding book of the Nova philosophia (for instance: “Id (scil. spacium) enim ante alia omnia necesse est esse, quo posito, alia poni possunt omnia; quo ablato, alia omnia tollantur”, p. 61r). And see especially Chap. 8 of the same book, which is entirely devoted to this subject, and ends with these words: “Est ergo spacium sui natura mundo prius, primumque rerum omnium mundanarum; ante quod nihil fuit, & post quod omnia fuere” (De spacio physico, p. 14v; Nova philosophia, p. 65r).
- 22.
On the supersubstantiality of Patrizian space see for instance this passage: “Si substantia est, quae aliis substat, spacium maxime omnium substantia est; omnibus enim substat aliis naturae rebus. … Hisce ergo rationibus omnibus patuit clarissime, spacium maxime omnium substantiam esse, sed non est categoriae substantia illa. Sed alia quaedam extra categoriam substantia est” (De spacio physico, p. 15v; Nova philosophia, p. 65r). We ought to remember that Aristotle himself had remarked that such a conception of space as the foundation of all bodies (which he attributed to Hesiod’s χάος) was actually incompatible with his metaphysics, and that he had, therefore, rejected it; cf. Phys. 1, 208b34–209a3: “If such a thing is true, then the power of place will be a remarkable one, and prior to all things, since that, without which no other thing is, but which itself is without the others, must be first. For place does not perish when the things in it cease to be” (trans. Hussey). Patrizi quotes this passage from the Physics approvingly (in De spacio physico, p.14r; Nova philosophia, p. 64v), and one could even suppose that Aristotle himself had provided him with the suggestion of how to overthrow Aristotelian philosophy. It can also be noted that the same train of thought had been expressed in the pseudo-Aristotelian work De Melisso, Xenophane, Gorgia (976b17–18), which, somewhat ironically, Patrizi had claimed to be perhaps the only genuinely Aristotelian work among those transmitted under his name (Discussiones peripateticae, I, III; p. 26).
- 23.
The subject is discussed at length in Chaps. 4 and 5 of De spacio physico (=Nova philosophia, pp. 63r–64r), which admit all three sorts of void. These are, firstly, the disseminated void, i.e. the microscopic void which separates the particles from one another and originates the phenomena of rarefaction and condensation (Patrizi thinks that bodies are, in themselves, completely impenetrable and inelastic, and could not be compressed in the absence of a void); secondly, the coacervated void, that of perceptible size (here Patrizi presents the reader with some experiments with clepsydrae); and thirdly, the extracosmic void, beyond the stars. The clepsydra experiment, a rather classical one, is probably taken from Philoponus, In phys. 569. On the metaphysical rather than experimental character of Patrizi’s theories on the void, see Schmitt (1967).
- 24.
Patrizi’s arguments for the infinity of space are found mainly in Chap. 6 of De spacio physico (pp. 11r–12v = Nova philosophia, p. 64r). In Book Eight of the Pancosmia, Patrizi was to try to conclude that the world itself is infinite, but his discussion does not actually concern the infinity of corporeal matter; rather, it concerns, once again, only the infinity of space and of light (Nova philosophia, pp. 82v–83v).
- 25.
Here is one of Patrizi’s most explicit passages: “Cum ergo, nec corporis terminis, nec spacii alterius, nec suis, nec incorporeis finiatur, necessario concluditur, spacium illum a mundo recedens, in infinitum recedere, & infinitum esse … Nos spacium illud, actu infinitum esse concludemus” (De spacio physico, pp. 12r–12v; Nova philosophia, p. 64r).
- 26.
Patrizi devotes to his theory of minima the whole second chapter of De spacio physico, which by itself makes up the largest part of the work. Here he discusses first of all the classical Aristotelian arguments in Phys. Ζ on the continuum, opposing to them equally classical rejoinders (such as that if two lines of different lengths were both infinitely divisible, there would be greater and lesser infinites, which is absurd). It is evident, at any rate, that the theoretical principle of the dispute is precisely that space is divided in actu, since it is in no case (as, by contrast, had been true of matter) divided in potentia: hence Patrizi cannot accept the Aristotelian theory of the (material) continuum, which rests entirely upon the distinction between potential and actual parts. On the other hand, if the spatial continuum must be composed out of actual parts, these cannot be unextended. Patrizi believes he can prove geometrically that many unextended items (the points) cannot possibly compose something extended. The proof is given in the Nuova geometria, Book II, Prop. 2 (pp. 19–20), which, to be sure, proves nothing more than that two unextended points, taken together, do not occupy an extended space; to which a “Corellario” adds, without offering the least argument to this effect, that if this holds for two points, then it must hold for an infinity of points—which unfortunately is precisely what needed to be proven in the first place. Patrizi concludes from this that there must be minimum parts of extension, that is, indivisible lines (De spacio mathematico, pp. 20r–24r; Nova philosophia, pp. 66v–68r).
- 27.
The idea of extended minima was quite widespread in Renaissance philosophy, and was professed by a great many authors; it certainly had medieval origins (see Grellard and Robert 2009), but its proponents could support it with (alleged) classical theories. Bruno, an important representative of this current, was perhaps inspired by some remarks to this effect which are to be found in Cusanus (see, among others, Bönker-Vallon 1995, Seidengart 2000, De Bernart 2002 and Omodeo 2013; Vedrine 1976 also discusses Patrizi to some extent), but Patrizi’s theory of indivisible lines does not appear to share the same sources (Patrizi mentions Cusanus in the Preface to the Nuova geometria, but does not seem to build on his work), and appears rather to be based almost uniquely on the pseudo-Aristotelian treatise De lineis insecabilibus, which presents (and refutes) a view on the indivisibility of minimal lines attributed to Plato and Xenocrates (but cf. also Metaph Α 9, 992a19–24; De gen. et corr. Α 8, 325b24–29). The view that Plato had postulated indivisible lines, or at least indivisible surfaces, had been espoused by Philoponus as well (see In gen. et corr. 27), although with arguments very different from Patrizi’s; it is thus unlikely, but not impossible, that Patrizi also had in mind a revision of Philoponus’s metaphysics in this field. He believed himself, at any rate, to be rendering a good service to Platonism (and to Pythagoreanism as well) by fending off Aristotle’s attacks on that theory.
- 28.
To this extent, the fundamental step in Patrizi is the one stating that although the height of a minimal equilateral triangle (i.e., with sides of minimal size) cannot bisect the base, this fact does not bring the whole of geometry to ruin. It will, he argues, suffice to consider a bigger triangle, and there the theorem will be found to be valid. Here is the text: “Quòd si in minimo isopleuro, non possit minima cathetus basin secare, non ideo pernicies inde Geometriae creatur universae. Quin etiam ex pluribus minimis simul iunctis, maiorem lineam fieri, nihil vetat” (De spacio mathematico, p. 23r; Nova philosophia, p. 67v). This theory is important because it was to be shared by the proponents of sensible minima in the Eighteenth Century, and Berkeley was to offer a similar argument about the possibility of carrying out geometrical demonstrations on larger figures.
- 29.
Regarding Patrizi’s astronomical and cosmological hypotheses, the most extensive discussion remains that included in Rossi (1977). The fundamental element in these hypotheses (besides the affirmation of the Earth’s rotation) was the denial of celestial spheres and the decided affirmation of a fluid universe, which is certainly connected with Patrizi’s spatial theories, although it is not their product. This cosmological view, indeed, was already present in Patrizi’s aforementioned criticisms of Telesio from 1572, and thus predates by far the elaboration of a full-blown metaphysics of space of Neoplatonic spirit. Patrizi’s anti-Aristotelian polemic had led him to deny celestial spheres before Brahe (though on the basis of metaphysical, rather than empirical, considerations), and he even failed to notice that the great Danish astronomer had arrived (and with stronger reasons) at the same opinion as him; thus, he criticized the latter for admitting the spheres, to Brahe’s great resentment. The echo of this controversy (Patrizi apologized in vain for his misunderstanding) reached even Kepler, many years after Patrizi’s death; although Kepler shared with the Italian philosopher a common Neoplatonic metaphysical framework, it seems that he did not esteem him very highly: the Astronomia nova voiced the harsh opinion that Patrizi had become “lucidly insane” (cum ratione insanire; cf. KGW III, p. 62).
- 30.
On the motion, augmentation or contraction of the universe, see De spacio physico, p. 11r; Nova philosophia, pp. 63v–64r. On annihilation, see De spacio physico, p. 14v; Nova philosophia, p. 65r. The hypothesis (certainly regarded as counterfactual) of a finite cosmos moving through infinite empty space precedes, of course, Patrizi’s speculations, as does that of a universe changing its size. These doctrines, however, show in the clearest of ways the dependence of Newton’s spatial philosophy (or of that of the tradition which inspired Newton) on Patrizi’s work; these are precisely the theses which Leibniz, more than a century later, was to fiercely attack (against Clarke) to defend his own spatial theory, which was struggling to take the place of the one whose birth we are currently examining.
- 31.
In the Discussiones IV, IX (p. 463), Patrizi had claimed that God is not in space (see Vasoli 2006). In De spacio physico & mathematico we only find very few brief references to God. On the other hand, in the Panarchia section of Nova philosophia, so rich in theological determinations, we can even find a book (number Twenty) completely devoted to the relations between God and space, where Patrizi argues (mainly by negative theology) that God is not, properly speaking, in any place (since he is immaterial); however, he is everywhere, at least by his power (and indeed “omnia ex Deo, ut ex loco; omnia in Deo, ut in loco”); whereas it could not be said that he is only somewhere, i.e., in the heavens, which is the position of the much-hated Aristotle (the whole discussion is on pp. 43r–44v). Finally, in the Pancosmia version of De spacio physico, Patrizi seems (at least dialectically) to concede the possibility of the opinion on God’s celestial localization as well, and writes “Si divinitas universa indivisibilis sit, ut est, in spacio erit indivisibili, & a divisibili spacio circumque erit obvoluta. Si nullibi item sit, sine spacio non cogitatur, si sit alicubi, vel in coeli culmine, vel supra coelum, in spacio certe erit. Si vero sit ubique, in spacio non esse nequit” (Nova philosophia, pp. 61r–61v). In the same paragraph, Patrizi also argues for the soul’s localization on the basis of the various ontological hypotheses regarding this latter (“Sin vero ratio, & mens animae corpus informet … Sin vero anima sit in corpore, non ut forma sed ut formatrix … Sin vero corpus sit in anima … ipsa quoque erit in spacio”), and thus concludes that “sunt ergo entia cuncta, & ea quae supra entia sunt, in spacio” (Nova philosophia, p. 61r).
- 32.
Patrizi ascribes to Plato the idea that all beings as such are located in place, on the basis of one Timaeus passage (52B) where the ancient philosopher does present this opinion, but in fact only to refute it (since the Ideas, for Plato, are in no place). The thesis which Plato intended to criticize, and which Patrizi embraces, should probably be attributed to Zeno and is ascribed to him in a passage of the pseudo-Aristotelian De Melisso, Xenophane, Gorgia (979b25–26), a work which Patrizi, as we have seen, considered authentic. The thesis of the transcendentality of space, however, was held in Patrizi’s age by other philosophers as well, and even by a Scholastic thinker like Fonseca, who advances it rather cautiously (In metaphysicam V, XV, q. 9, s. 3; vol. 2, pp. 768–769). Since that theory is not present in Patrizi’s work from 1587 but rather first appears in 1591, it could in fact even be conjectured that it had come to Patrizi through Fonseca, whose positions had been published in 1589. Although it does not seem impossible that Patrizi had read Fonseca (in the Discussiones peripateticae, he had commented on Aristotle and was probably interested in every new exegetical effort in this field), the hypothesis is not strictly necessary, given the importance which space possessed in Patrizi’s ontology already in earlier years. We can note here that Bruno, by contrast, considered space to be the place of bodies alone (De immenso I, VIII; Opera, I 1, p. 231).
- 33.
At times, Patrizi appears to liken space to the other corporeal elements (see Footnote 52 below), and in this sense it seems that the former would have to be created, just like the latter. Patrizi says in the introduction added to the discussion of space in the Nova philosophia (missing in the original De spacio physico) that space is the first thing God produced extra se; this could also suggest a creative act in the proper sense. However, in other places he apparently claims that God created both cosmic matter and light in space, which is the condition for their creation and appears itself to be uncreated. See for instance this passage: “At si iterum, coelos novos, terramque novam, Deus reficiat, … spacium praeexistit, quod novos capere possit coelos, terramque novam” (De spacio physico, p. 14v; Nova philosophia, p. 65r). We should bear in mind that since the Middle Ages the creation of an infinite being was generally held to be impossible (even for God); this was one of the reasons why space was more easily construed as a divine attribute than as one of his creations. On Patrizi’s emanationism in the Panarchia, see Puliafito (1988) and Muccillo (2003).
- 34.
The statement that God is place is from the Panarchia: “At ipse (scil. Deus) locus est, in ipso enim omnia sunt” (Nova philosophia, p. 42r). That God is the place of creatures is actually a quite common statement in the Middle Ages (in Eckhart, for example, or then in Weigel), or in the tradition (stretching into the seventeenth century up to More, Cudworth, and Newton himself) which saw in “makom”, the Hebrew word for “place”, the least inappropriate among divine names.
- 35.
Bruno’s writings dealing more consistently with the formulation of a new theory of space are De l’infinito, universo e mondi from 1584, which thus predates by a few years Patrizi’s works on space, and De immenso et innumerabilibus, whose composition probably began just after said Italian dialogue but which was completed and published in 1591, after Patrizi’s De spacio physico & mathematico and simultaneously with his Nova philosophia. It seems that Patrizi never mentioned Bruno’s works, and we do not know whether he knew them and what he thought of them. Bruno, for his part, in the 1584 De la causa, principio e uno passes a famous and tremendously negative judgment on Patrizi (“sterco di pedanti”), whose Discussiones peripateticae were at this time the only work known to Bruno; he believes that Patrizi had understood Aristotle “né bene né male: ma che l’abbia letto e riletto, cucito scucito, e conferito con mill’altri greci autori amici e nemici di quello; et al. fine fatta una grandissima fatica, non solo senza profitto alcuno, ma etiam con un grandissimo sprofitto: di sorte che chi vuol vedere in quanta pazzia e presuntuosa vanità può precipitar e profondare un abito pedantesco, veda quel sol libro, prima che se ne perda la somenza” (Oeuvres, III, pp. 165–167). Bruno may perhaps have moderated his opinion of Patrizi after reading his Nova philosophia, and when he knew that Patrizi had been appointed by the Pope to the Roman chair precisely because of that work, Bruno thought that perhaps he himself could venture to return to Italy, since his views must have seemed to him not so dissimilar to those of Patrizi (whom he even believed to be an atheist, see Mercati 1942 and Yates 1964, p. 345). As we know, Clement VIII’s liberality soon dissolved; the Roman fortune of the Nova philosophia lasted only a short time, and Bruno’s an even shorter.
- 36.
Patrizi’s aforementioned criticisms of Telesio were leveled against the second edition (1570) of De rerum natura, which does not contain extensive treatments of space. A discussion on the subject appears in the third edition from 1586, which thus precedes the publication of De spacio physico & mathematico by a mere few months (and follows that of Bruno’s De l’infinito); thus, it is by no means apparent that Patrizi and Bruno could have been influenced by the book. Telesio’s spatial theory, moreover, was rather similar to Philoponus’s, so that the very same ideas were already in circulation independently of his work. On Telesio’s theory of place, see first of all Chaps. 25–28 of Book One of the 1586 De rerum natura (pp. 36–42); on his theory of geometry, the objects of which are the magnitudes and shapes of bodies (and which has no relation to space), see Chap. 4 of Book Eight (pp. 316–318).
- 37.
On substance and accident, and the categories in general, see for instance this passage: “Sed sunto categoriae in mundanis bene positae, spacium de mundanis non est; aliud quàm mundus est, Nulli mundanae rei accidit, sive ea corpus est, sive non corpus, sive substantia, sive accidens, omnia haec antecedit; omnia illi uti accedunt, sic etiam accidunt; ita ut non solum quae in categoriis numerantur accidentia, verum etiam quae ibi est substantia, illi sunt accidentia. Itaque aliter de eo philosophandum, quam ex categoriis” (De spacio physico, p. 15v; Nova philosophia, p. 65r). A few lines further on, Patrizi asserts that space is neither form nor matter, and that it is not a genus because it is not predicated of either the species or the individual. Very similar arguments can be read in Bruno, especially in De immenso, I, VIII, where he claims that space is neither matter nor form (Opera, I 1, p. 232), neither substance nor accident (pp. 232–233), and neither genus nor species (p. 233); see also the Articuli adversus peripateticos (nn. 28–30) and the Acrotismus Camoeracensis (Opera, I 1, pp. 123–228), where Bruno states, inter alia, that place is a fifth kind of cause. On Bruno’s spatial metaphysics see the article by Seidengart in this volume. It should be noted, however, that such a frontal assault on the Aristotelian ontology was not a prerogative of the Italian novatores, because already the Medieval notion of imaginary space, which in the Renaissance was commonly employed in the School, seemed by far to exceed the Peripatetic categories. Thus, we find that Fonseca himself, in the treatment of this concept of space (which he certainly accepts and employs), is obliged to conclude that it is neither substance nor accident, neither matter nor form, etc. (In metaphysicam, V, XIII, q. 7, s. 1; vol. 2, p. 604). Toletus, by contrast (who, as we have seen, played a role in Patrizi’s condemnation) discusses at length the alternatives to Aristotle’s conception of space, and seems to incline towards a notion of place as three-dimensional extension (less radical, of course, than Patrizi’s own, and simply of Philoponian or Avicennian descent); but at the end of his extensive discussion, so rich in objections to Aristotle, he nevertheless concludes that such extension would be neither substance nor accident; this prompts him to reject this concept and to choose to keep rather to the Aristotelian definition, in spite of all its difficulties (see the Commentaria in libros de physica auscultatione, p. 116r).
- 38.
I think that the whole De spacio physico is rich in echoes of Plotinian thoughts about the genera of being, taken in the first place from Enn. VI 1–3 [42–44]; these intend to show the inadequacy of Aristotle’s category theory for treating extramundane beings, and therefore employ the Aristotelian categories in a negative or especially emphatic form when predicating them of such beings. To be sure, neither Plotinus nor Patrizi really wish to accept Aristotle’s natural philosophy as a correct description of the sensible world, and if Plotinus proposes his original system of sensible genera, Patrizi in the Nova philosophia also resorts to his non-Aristotelian elements (lux, calor, fluor) and to a cosmology completely different from the Peripatetic one; in any case, both philosophers insist on the insufficiency of the Aristotelian ontology for the proper description and discussion of the supersensible world. On the ancient sources of Patrizi’s anti-Aristotelian polemic, see Leinkauf (1990); Plotinus’s influence on the spatial doctrines had already been stressed by Kristeller (1964).
- 39.
The definition of space as hypostatic extension occurs in De spacio physico, p. 15v; Nova philosophia, p. 65r.
- 40.
In the classical terminology, place (τόπος, locus) always designates a place occupied by some body and is, as such, opposed to the void; thus, by definition, there are no empty places (which of course is part and parcel of the consideration of place as an accident of a corporeal substance). Classical Antiquity also tried to articulate a concept of spatiality capable of being indifferent to plenitude and the void, and of representing the common genus of both notions: but this concept, merely sketched by some philosophers, remained so vague that it was not even given a name (see Sext. Emp. Adv. Phys. Β 2). The Latin spatium, on the other hand, usually translates the Greek διάστημα (interval) which directly designates an extension (spatial or temporal), but lacks, in itself, any local character, and is more closely connected to the category of quantity than to place (though in Phys. 2, 209b6–13, Aristotle somehow accommodates διάστημα in the broader discussion on the essence of τόπος, arguing at length precisely that it is completely inadequate to represent a principle of localization). Patrizi initially defines space as (hypostatic) extension, rather than as place (De spacio physico, p. 2v; Nova philosophia, p. 61v), and then intends to prove that the place of a body, to whose essence all Classical theories ascribe certain attributes (such as immovability and separability; Patrizi certainly has in mind the famous Aristotelian passage in Phys. Γ 4, 210b34–211a6), must also be extension, and thus space according to his definition (De spacio physico, pp. 6r–6v: Nova philosophia, p. 62v). In fact, he concludes the treatise (De spacio physico, p. 17v; Nova philosophia, p. 65v) precisely with a discussion of the various powers of space, the most important of which are that of conferring location on the bodies and of being an environment for them (vis locandi et ambiendi).
- 41.
It must be borne in mind that Aristotle himself seems at times to construe place as the tridimensional extension of the located body, considered separately from it. Some interpreters had believed themselves to recognize this theory in a rather obscure passage of Cat. 6, 5a8–14, which seems to run counter Aristotle’s explicit doctrines in the Physics. It is very doubtful, however, that in this case there actually is a discrepancy between the two works (but see at any rate Mendell 1987), and none of the ancient commentators took notice of it. In the Middle Ages, however, the difficulty was explicitly noted, and many attempts, at times rather elaborate ones, were made to reconcile the two accounts. In the Fourteenth Century, however, an Averroist idea (not directly argued for by Ibn Rushd) gained ground: the claim that the Categories did not represent Aristotle’s genuine account, but merely the common opinion. This is, for instance, the position of Buridan (see Grant 1981 and Algra 1994), but it was also held, precisely in the age of Patrizi, by Fonseca, In metaphysicam V, XIII, q. 7, s. 1 (vol. 2, p. 604). At any rate, it was no absurdity, in the Renaissance, to state that Aristotle himself had embraced, seriously or as a kind of joke, a theory of place as three-dimensional extension.
- 42.
It seems to me that such a theory of place as the three-dimensional extension proper to each object can be found, for instance, in Scaliger’s natural philosophy (see De subtilitate, V, 3). In fact, I would regard these doctrines as quite common and certainly not revolutionary.
- 43.
On Philoponus’s spatial theories, which are principally to be found in the Corollarium de loco (or Digressio de loco, as it was called in the Sixteenth Century) of his commentary on Aristotle’s Physics, see at least Sorabji (1988) and De Haas (1997). The editio princeps of Philoponus’s commentary had been published in 1535, and translated into Latin in 1539. Both Bruno and Patrizi knew Philoponus’s works very well, and Patrizi even translated his commentary on the Metaphysics (which, however, is probably spurious) in 1583, precisely in the years between the Discussiones peripateticae and the new theories of De spacio physico & mathematico. Bruno makes approving mention of Philoponus’s account of space, especially with regard to the concept of a void (which, as we shall see, he accepts without reserve, with its strengths and weaknesses), in De immenso, I, VIII (Opera, I 1, p. 231). In any case, Philoponus’s commentary on the Physics had a considerable influence in the Sixteenth Century; his anti-Aristotelian theories of space found a first application in Gianfrancesco Pico’s Examen vanitatis from 1520 (that is, even before Philoponus’s printed edition; cf. Book VI, pp. 176–179 in Pico), and continue to be mentioned and discussed (among others) by Cardano, Vimercati, Pandasio, and many Scholastic thinkers. It may well be that Telesio depended on them as well. On the spatial doctrines of Pico, who may have been an important source for Patrizi’s Discussiones, see Schmitt (1968, pp. 138–144). On the diffusion of Philoponus’s commentary in the Renaissance, see also the excellent article by Schmitt (1987), who however does not consider this doctrine to have exerted a major influence on Patrizi. In Footnote 14 above, I have already mentioned a passage in which the young Patrizi seemed to adopt Philoponian ideas.
- 44.
The theory of place as extension in Philoponus descends at any rate from his theory of quantified matter, that is, from the idea that there can be a quantitative material extension independent of substance. This independence of extension from substance justifies in Philoponus’s view the concept of extended space as well. But in no case is this space really distinct from matter and substance, and in fact it depends ontologically (though not logically) on the latter. On the relation between space and quantified matter, see Philop. In phys. 578–579; on the finiteness of space, due to that of the world, see In phys. 582–583.
- 45.
In the discussion on the primacy of space in De spacio physico, Patrizi makes this point very clearly and espouses, so to speak, Philoponus’s doctrine only provisionally and in order to radicalize it. He starts off, namely, by showing that the place of a body (understood as its three-dimensional extension) is ontologically prior to the body itself and its accidents (“Corpora ergo qualitatibus suis sunt priora. Corporibus vero priores sunt loci”, De spacio physico, p. 14r; Nova philosophia, p. 64v), and that a tree, for instance, can only exist if its place already exists. Now, this might count as a good portrait of the anti-Aristotelian theory characteristic of the Neoplatonism of late antiquity; but Patrizi then adds that the world (in its entirety) is prior to everything which it contains, that is, bodies, their accidents, but also their places; and concludes by adding that space is prior to the world itself (something which Philoponus would not have allowed), and thereby to places and to bodies too: “At cum entia nulla alia in natura sint praeter haec quatuor, spacium, locus, corpus, qualitas; corpus autem qualitate prius est; & corpore locus; & loco spacium, spacium nimirum rerum omnium primum est. Mundus itidem prior omnibus quae in eo sunt, locis, corporibus, qualitatibus est. Spacium autem ante quam mundus est: spacium nimirum mundanorum omnium primum erit” (De spacio physico, p. 14v; Nova philosophia, p. 65r).
- 46.
The claim that space is the origin of sensible quantity is of such importance that it appears immediately after its definition as a hypostasis: “Spacium ergo extensio est hypostatica, per se substans, nulli inhaerens. Non est quantitas. Et si quantitas est, non est illa categoriarum, sed ante eam, eiusque fons et origo” (De spacio physico, p. 15v; Nova philosophia, p. 65r).
- 47.
On the penetrability of space, see De spacio physico, p. 13r; Nova philosophia, p. 64v. The definition of body as three-dimensional extension endowed with ἀντιτυπία is a very classical one (though not Aristotelian), and probably dates back to Democritus. It is attested to, for instance, in Sext. Emp. Pyrrh. Hyp. Γ 39 and Adv. Phys. Β 257, or in Hero, Def. 11; the notion is also extensively discussed in Plot. Enn. II 6 [17], 2 and Enn. VI 1 [42], 26, and it is not unlikely that Patrizi took it from there. In Patrizi’s natural philosophy, the impenetrability of bodies is a consequence of the elementary mixture of light and fluor. Later doctrines on the penetrability or impenetrability of space were, by contrast, to be rather varied; and several authors who certainly shared ideas similar to Patrizi’s were to vigorously argue that space (as hypostasis and foundation of the sensible world) is impenetrable, and itself penetrates all bodies without its parts ever being separated. See for instance Bruno, De immenso I, VIII (Opera, I 1, p. 232); and in later ages Henry More, Joseph Raphson and many others.
- 48.
The statement that the essence of a body consists in impenetrability alone, whereas its dimensions are only accidentally acquired by virtue of its being in space, is explicit at least in this passage: “Proprium enim corpori naturali, qua corpus naturale est antitypia illa est, & quam vocant anteresim. Hoc est resistentia, & renitentia. Quae renitentia, trinon illo spacio opus habuit, ut subsisteret. Qua ratione tres distantiae, sunt fere corpori, tam alienae, quàm fuerant corporeis. Locus verò ita propriae sunt, ut ei non accidant, aut ei aliunde accidant, sed ipse locus, non sit aliud, quàm distantiae illae. Et spacium, verus sit locus. Et locus, verum sit spacium” (De spacio physico, p. 6v; Nova philosophia, p. 62v). Patrizi also restates the idea in the final revision of the Nova philosophia written for the Inquisition (see Puliafito 1993, p. 38).
- 49.
The quantification of space is a result of the late Neoplatonic speculations of Philoponus and Simplicius, which had left no trace in the Medieval theories of ecological place or of imaginary space; it was also usually attacked by the Second Scholasticism. So, for instance, Fonseca in the aforementioned passage In metaphysicam V, XIII, q. 7, s. 1 (vol. 2, pp. 604–606) where he mounts a refutation of Philoponian space precisely because he wishes to affirm that imaginary space, being pure negation, cannot fall under the category of quantity. Note that Fonseca believed that, although he had to accept the Aristotelian notion of place as containing surface in the Physics, he could at the same time complement it with a (much more elaborate) doctrine of space, construed as the spatium imaginarium of the Medieval tradition, which is in fact independent of the existence of bodies and therefore shares (at least prima facie) some of the properties of Patrizi’s “modern” space: though not quantification.
- 50.
Cf. one of the principles of the Nuova geometria: “Lo spazio è estensione, e la estensione è spazio” (p. 2).
- 51.
See for instance De l’infinito, where the ambiguity clearly appears in the following passage, according to which space is (at least) similar to matter: “Lascio che il luogo, spacio et inane ha similitudine con la materia, se pur non è la materia istessa; come forse non senza caggione tal volta par che voglia Platone e tutti quelli che definiscono il luogo come certo spacio…” (Oeuvres, IV, p. 113; but cf. p. 69, also on the Platonists; and the Acrotismus in Opera, I 1, pp. 126–128). Other passages where Bruno seems to construct space as matter or aether occur at De immenso, IV, XIV: “Aether vero idem est quod caelum, inane, spacium absolutum” (Opera, I 2, p. 78). On the subject, cf. Amato (1997) and Giudice (2001).
- 52.
It must be admitted, however, that some fluctuation is still present in Patrizi as well, and that (as was noted by Grant 1981, pp. 386–387, n. 139) in the Pancosmia he reverts to a Neoplatonic emanationist account according to which the perfection and originality of a being is inversely proportional to its corporeality, so that space becomes the least corporeal and the rarissimum of elements, followed by the slightly more corporeal light, then heat, fluor, the aether, and so on to air and the other common elements. However, it seems to me that we should not take this scale of corporeal densities too seriously, since light is straightforwardly said to be incorporeal in other texts, and body is defined precisely as spatial extension endowed with the resistance to penetration. Still, it is a fact that Patrizi, even in the more cautious De spacio mathematico (p. 25v; Nova philosophia, p. 68r), paradoxically affirms that space is a corpus incorporeum. But there, as we shall shortly see, he needs the argument to prove that mathematics, the object of which is incorporeal (space as an ideal being), nevertheless applies to bodies, because they are in space. Moreover, Patrizi certainly owed the oxymoron of ‘incorporeal body’ to his Platonic studies, namely to the analogous characterization of χώρα offered by Calcidius (In Tim. §§ 319–320), as well as to Plotinus’s incorporeal ὄγκος which is however at the same time σῶμα μαθηματικόν. We should finally observe that some interpreters (starting with Henry 1979; Deitz 1999 insisted on this point) have noted the affinity between Patrizi’s views on Nature, especially his metaphysics of light, and Proclus’s philosophy of space, which identified place with an immaterial body spread across the universe and probably identical with light (we owe this information to Simpl. In phys. 612; published in Latin in Venice in 1566, pp. 221–222). There is no doubt that Patrizi knew the work of Proclus very well: he had translated his Elements on physics and theology in 1583, and explicitly mentions many of his other works (the most relevant for our purposes being the commentary on Euclid); certainly Patrizi’s emanationistic system retains some similarities to Proclus’s, but not a great deal, since the latter is interwoven with Plato’s and Ficino’s: see Kristeller (1987). However, the actual similarities between the two concepts of space are rather scarce, and Proclus’s meditations appear to extend no further than that very notion of corporeal (and finite) space which Patrizi aspires to transcend. On space in Proclus see Schrenk (1994) and Sorabji (1987).
- 53.
On Ficino’s translation, see first of all Garin (1975), Wolters (1986), Saffrey (1996), Chiaradonna (2006). Ficino’s edition of the Enneads was read and studied throughout the Renaissance and beyond. The editio princeps of the Greek text only appeared in 1580 (and was still presented along with Ficino’s translation), so it is evident that most scholars continued, for many years, to rely on the only available translation. On the other hand, we know that Patrizi was in a position to inspect the Greek original before the appearance of the printed edition, because he owned a manuscript codex of the Enneads copied in 1563 (cf. Jacobs 1908; Muccillo 1993a). Ficino’s commentary refers to Plotinus’s crucial passages on ὄγκος in Enn. II 4 [12], 11, and in that passage he only uses the term spatium (Ficino, Opera, vol. 2, pp. 1949–1950). Even in translating the Greek original, on at least one occasion Ficino complements Plotinus’s elliptic text with the alien notion of space; cf. the incipit of Ficino’s translation of Chap. 12 in the above-mentioned treatise (ed. Perna, p. 166; it should be borne in mind that the subdivision into chapters of the treatises in the Enneads, still in use today, originates with Ficino), where he conjectures that the missing subject in the sentence in question is ὄγκος from the preceding discussion (it is the concluding word of Chap. 11), instead of ὕλη as is usually (and perhaps wrongly) assumed by modern translators: he thus decides to take advantage of the freedom offered by the text and to translate the term in question as spatia. Elsewhere, again, in Enn. II 4 [12], 11, Ficino renders with spatium the Greek word διάστημα. Patrizi never concealed his admiration for Ficino, and he even stated that his reading of the Theologia platonica had paved his way to philosophy (indeed, all his earliest works are variations on themes from Ficino): see the letter to Baccio Valori of 12 January 1587 (Lettere, p. 47); on Patrizi’s Ficinianism, see Muccillo (1986). Furthermore and chiefly, as we have noted several times, the whole Nova philosophia abounds with genuinely Enneadic elements, especially when it comes to quantified matter or to quantity itself. As regards Bruno, it is very well known that his works display a widespread presence of Plotinian themes (see Chiaradonna 2011, who also gives some examples of Bruno’s dependency on Ficino’s interpretation—which is not the case, however, on the subject of matter); it should be noted, however, that in De la causa, principio e uno (Oeuvres, III, p. 237) Bruno mentions explicitly precisely the treatise on matter in Enn. II 4 [12], the one which contains the notion of ὄγκος and Ficino’s remarks on space. A little later (pp. 249–251), he connects Plotinus’s theory of ὄγκος to that of dimensiones indeterminatae (which in fact originates with later Neoplatonism, or even with Averroes). We can also note that in De umbris idearum (1582) Bruno not only made ample use of Plotinus, but also approved of his interpretation of matter as a mirror (which also played some role in the formation of the modern concept of space).
- 54.
For instance, we find that the Filosofia naturale (1560) by Alessandro Piccolomini, an author of major significance for Renaissance mathematical epistemology, discusses the theory of dimensiones indeterminatae (which Piccolomini attributes to Averroes) stating that matter can receive “spatio, & misura di quantità” independently of substantial form. Evidently, “space” here means simply extension and interval, and the text contains no theory of space as place (Piccolomini in fact espouses a perfectly Aristotelian doctrine of place), nor a theory of space as the object of geometry (Piccolomini in fact takes rather intelligible matter to be said object). However, it does contain the explicit contiguity of spatiality and quantified matter which may have precipitated the later theories of Patrizi and other contemporaries.
- 55.
We will discuss here only Patrizi’s geometrical theories, not the arithmetical ones (which he, moreover, did not develop in much detail, and which he perhaps intended to work out completely in De numerorum mysteriis). It must, however, be borne in mind that space is the origin of quantity in general, continuous as well as discrete, and consequently the source of both extension and numbers (in terms of the Classical, Aristotelian division of the genus quantity). Patrizi thus claims that geometrical magnitudes and numbers are equally primitive and neither can be founded on the other (De spacio mathematico, p. 19v; Nova philosophia, p. 66v); moreover, he affirms (as we have seen) that “le matematiche tutte”, and not only geometry, are founded on space (Nuova geometria, p. 2). He also claims, however, that geometry has a privileged connection with space; on this basis he affirms that arithmetic derives from geometry, which is the first mathematical science (De spacio mathematico, p. 25v; Nova philosophia, p. 68r). This is opposed to Aristotle’s view of the primacy of arithmetic, but also opposed to the Pythagorean position.
- 56.
I think that the best example of what I call here the Platonic, but not Neoplatonic, theory of the mathematizability of the world can be found, in Patrizi’s time, precisely in the work of his successor on the Roman chair, Jacopo Mazzoni. He was a proponent of a form of Platonism which insisted on the applicability of mathematics to the world and on the superiority of a quantitative study of this world (on the example of Archimedes, Ptolemy or the modern mathematician Giambattista Benedetti) over Aristotelian qualitative physics. Mazzoni, indeed, rejected that variety of Renaissance Platonism (championed for instance by Francesco Barozzi) that, on the basis of the epistemology of the Republic, defended the pure character of mathematics and its non-applicability; he based, instead, his interpretation of Plato on the Timaeus and on the mathematical structure of the cosmic elements which this work advocated. Mazzoni in fact regarded as his greatest exegetical rival precisely Plotinus, who had separated, with his new theory of matter, Plato from Aristotle and thereby from natural inquiry and had inaugurated the drift towards the radical, anti-“concordist” Neoplatonism which was eventually to lead to Patrizi’s thought; for the same reason, Mazzoni deems it appropriate to reject the Neoplatonic Plato of Ficino’s interpretation. These theoretical (and philological) moves were very common at the time, and if Giovanni Pico had already criticized Plotinus for his anti-concordism, Perna similarly undertook his Greek edition of Plotinus in 1580 because he wanted to free this latter writer from the interpretative superimpositions of Ficino’s Latin version; still others attempted to make Plotinus, Plato and Aristotle consistent with one another (such as Gabriele Buratelli in 1573: see Muccillo 1994). However, as we have seen, it was precisely the Plotinian concept of a quantified matter (and ὄγκος) that gave birth to the long intellectual tradition which finally arrived at the assertion that quantity is the essence of all bodies, and consequently paved the way for a (different, but very effective) epistemology of natural philosophy. It seems, at any rate, quite beyond doubt that Galileo’s Saggiatore, with its “book of Nature” written in triangles and circles, descends to some extent from Mazzoni (and ultimately from the Timaeus); it is no surprise that Galileo was never greatly to appreciate the alternative attempt at justifying a mathematical knowledge of Nature carried out in his time by Campanella, on the same Neoplatonic and spatial foundations as Patrizi’s. On Mazzoni and the Renaissance philosophy of mathematics, see Purnell (1972), who also discusses his relationship with Benedetti; and De Pace (2006a), who deals with the limitations of Mazzoni’s concordism and his openings towards skepticism. By the same author, see also De Pace (1993), that suggests that Mazzoni (whose Praeludia date from 1597) might have been acquainted with Kepler’s Mysterium cosmographicum, which had been published in the preceding year and immediately sent to Galileo in Italy, and which also contains an epistemology modeled on the Timaeus (and hostile, as we have seen, to Patrizi and his cosmological hypotheses); on the influence of Patrizi’s metaphysics of light on Kepler, see Lindberg (1986). Finally, see De Pace (2006b), who, arguing against Hankins (2000), also finds an important difference between Galileo’s Platonic epistemology and Ficino’s Neoplatonism—even though along different lines than those sketched here.
- 57.
This devaluation of the cognitive value of mathematics was already marginally present in the late Greek commentators on Aristotle, that is, Philoponus and Simplicius, and is to be ascribed to the eclectic attitude of the latter authors, who inserted into the Neoplatonic conception of quantified prime matter the Aristotelian doctrine of mathematics as a product of abstraction; with the consequence that quantity, this primitive ontological determination of entities, instead of being considered as its foundation was regarded as its poorest and basest abstraction. From this it followed that mathematics could be applied to the world of corporeal substance only in the guise of a hypothesis, whereas only (qualitative) natural philosophy, which alone was concerned with substantial forms, could aspire to a non-hypothetical knowledge of Nature. These tenets, though variously extended and modified, had nonetheless reached and been absorbed by Renaissance Aristotelianism, and had been embraced by Piccolomini in De certitudine mathematicarum; in a period closer to Patrizi’s, however, they were also propounded in Pereira’s De communis omnium rerum naturalium principiis, published in Rome in 1576 and reprinted several times (for instance in 1585, around the very time that Patrizi composed his works). Patrizi vocally opposes these epistemologies of the world’s (non)-mathematizability, and regards the attribution of quantity to space rather than to matter as the ontological foundation necessary to overcome the abstractionist theory and its consequences. In cosmology as well, in fact, he was a lively opponent of the view that astronomy deals with simple mathematical hypotheses rather than with the actual motions of the celestial bodies. Toletus, in fact, had already defended the nobility of mathematics against physics, and some years later (in 1615) Biancani was to claim that mathematics is more certain than natural philosophy since an accident is easier to know than a substance (thus, with an argument that Patrizi could not possibly have endorsed): see Biancani, De mathematicarum natura, p. 26.
- 58.
Patrizi thus revives the picture of mathematics as an intermediate science between metaphysics and the study of appearances, which can be, to some extent, traced back to Plato’s Republic; he connects the intermediate position he assigns to mathematics with the analogous character of space, which is intermediate between the natures of (ideal) incorporeal objects and (phenomenal) corporeal things: “Eandem hanc rationem consequitur, ut mathematica anterior sit quam physiologia. Media quoque est, inter incorporeum omnino, & corporeum omnino, non qua ratione veteres dixere, per abstractionem a rebus naturalibus corpoream quasi fieri, sed quia revera spacium sit corpus incorporeum, & incorporeum corpus” (De spacio mathematico, p. 25v; Nova philosophia, p. 68r).
- 59.
I believe that this is the principal thesis of De spacio mathematico, which articulates in better detail the claim from the Nuova geometria (quoted above in Footnote 1) that the object of mathematics is neither abstractive nor imaginative: “Mentem nostram finita sibi in opus sumere, quae spaciis mundanorum corporum possint accommodari. A quibus corporibus non per abstractionem, mens ea separat, ut quidam contenderunt. Quoniam ea spacia non sunt primo, & per se in mundanis corporibus. Sed sunt ante corpora, actu in primo spacio. Neque etiam in phantasia, aut dianea nostra (ut quidam alij viri admirabiles tradiderunt) veluti in subjecto, dimensiones illae & quae inde formant reliqua, subsistentiam habent. Sed mens è spacio illo primo vi sua, ea partes desecat, quae sibi sunt, vel contemplationi, vel operi, usui futura” (De spacio mathematico, p. 24v; Nova philosophia, p. 68r). We ought to note that in Patrizi’s vocabulary, “mens” is always equivalent to “intellect”. A useful contrastive example is provided by comparing Patrizi’s desecare with designare in Campanella, who was, a few years later, to propose another theory of the geometry of space while at the same time readopting the productive imagination of the Neoplatonic tradition: “Basis enim intrinseca sustentans omnia corpora est spatium incorporeum, in quo mens imaginatrix seu ideatrix omnia mathematica designat” (Campanella, Mathematica, p. 32).
- 60.
It is very important not to confuse this doctrine of ῥύσις as the flowing of geometrical objects with the theory of their (rigid) motion. The two concepts, of flow and of rigid motion, had already parted ways in Classical Antiquity, and the notion of ῥύσις was commonly employed in Neoplatonic ontology, where it had also usually been characterized in terms of a movement of imagination. In the modern era, the difference is clear in the case of Jacques Peletier: although he had criticized the use of “mechanical” motion in a pure science like geometry, he had nothing to object against defining a line as the continual flow of a point. Patrizi overthrows the usual positions, and admits rigid motion (as we have seen), while rejecting the flow: “Non ergo motus attingit punctum. Si non motus, nec productio. Si non productio, nec principium lineae punctus erit. Quid ergo linea a puncto non producitur, linea non erit? Erit sanè, sed non producta. Nec erit linea punctus fluens; quod veteres aliqui autumarunt. Quid ergo linea est? Pars ea spacij quae inter duo puncta interiacet” (De spacio mathematico, p. 19r; Nova philosophia, pp. 66r–66v). It is possible, however, that at the time of the Nova philosophia Patrizi had already moved beyond the opinions concerning rigid motion that he had endorsed in the Discussiones.
- 61.
The thesis of the constructivism of Greek geometry was put forward by the Neo-Kantian historiography of science in the second half of the Nineteenth Century, and certainly mirrored the epistemological concerns of the Critical Philosophy. To be sure, it cannot be doubted that Kant himself was inspired, as regards his philosophy of mathematics, mainly by the Neoplatonic tradition of productive (or projective) imagination which derived from Proclus’s commentary on the Elements. A criticism of the idea of the constructive character of Greek mathematics is expressed in the now classical essay Knorr (1983); on the epistemological motivations behind Proclus’s constructivist interpretation, see Harari (2008).
- 62.
Galileo devoted to the theory of proportions the (posthumous) Fifth Day of his Two New Sciences. For a general overview of the significance of that theory for Seventeenth-Century mechanics, and of the role of the existential assumption of the fourth proportional, see Giusti (1993). References to other non-constructive solutions in modern geometry can be found, among others, in Bos (2001). We can note in passing that Bruno accepts as an axiom the proposition of Elements V, 18, in which Euclid makes use of the fourth proportional; therefore he does not need to discuss the question in general (cf. Articuli adversos mathematicos; Opera, I 3, p. 10).
- 63.
What I mean is that the existential assumptions which were becoming more and more common in the late Renaissance were still completely lacking an epistemology and an ontology which could support and justify them. In Clavius’s commentary on the Elements, for instance, the Jesuit mathematician states that the existence of the fourth proportional can simply be admitted because it does not contradict the other Euclidean principles (Euclidis, p. 221). This statement seems to identify mathematical existence with logical consistency tout court, but a general theory is lacking. It must in any case be noted that Patrizi’s ontology could at most justify the existence of geometrical entities, whereas a relevant part of Renaissance and modern disputes revolved around the applicability of the theory of proportions (and its existential requirements) to magnitudes of any kind whatsoever, such as speed or force. In this sense, Patrizi’s epistemology encouraged the geometrization of space and the corporeal world, but did not yet fully deal with the question of the applicability of mathematics to dynamic Nature as a whole (that is, to the study of motion).
- 64.
For instance, compare the text by Patrizi in Footnote 59 above with this famous passage from Newton’s De gravitatione (which sounds, as it were, like an excerpt from De spacio mathematico) on the actual existence of figures in space, which are delineatae (that is, desecatae), but not produced, by the geometer: “Et hinc ubique sunt omnia figurarum genera, ubique sphaerae, ubique cubi, ubique triangula, ubique lineae rectae, ubique circulares, Ellipticae, Parabolicae, ceteraeque omnes, idque omnium formarum et magnitudinum, etiamsi non ad visum delineatae. Nam materiali delineatio figurae alicujus non est istius figurae quoad spatium nova productio, sed tantum corporea representatione ejus ut jam sensibus apparet esse quae prius fuit insensibilis in spatio” (De gravitatione, p. 100). But also Lambert’s theory of geometry as the anatomy of space, that is, as the discipline concerned with “cutting out” geometrical figures from the ambient space assumed as existing, is certainly a late offspring of Patrizi’s spatial conception.
- 65.
Patrizi believes himself to have proven the existence of the straight line between any two points of space in Proposition 11 of Book III (Nuova geometria, p. 34); and in Propositions 23–25 of the same Book, he proves its infinite extendibility (Euclid’s Second Postulate). However, even this extendibility must be simply understood as the actual infinite extension of the straight line, rather than as the capacity to be prolonged in thinking or in drawing: “Lineam enim negamus, a nostra mente, aut arte in infinitum posse produci: attamen eam quae punctis finita est, iis liberata infinitudinem sui natura, fatemur subire” (De spacio mathematico, p. 20r; Nova philosophia, p. 66v). Let me note that this idea of a line infinite in nature but constrained by its endpoints in a bounded segment is also present in Proclus (In Euclidis 101), and this is certainly where Patrizi derived it from; in Proclus, however, the statement had an altogether different meaning, namely, that the line’s boundaries (its endpoints) determine it as a definite magnitude, and in the absence of those boundaries it is mere potentiality, that is an indefinite (and certainly not: actually infinite) magnitude. See also In Euclidis 86; and, for a plain reading of these passages of Proclus in the late Renaissance, also Biancani, De mathematicarum natura, pp. 5–7. The change of perspective in Patrizi is very significant, and behind the same turn of phrase there lies hidden an altogether new mathematical epistemology. It is common to contrast the existential structure of Hilbertian axioms with the constructive one of Euclidean axioms: but as we see here, Hilbert’s idea has a longer and more complex history.
- 66.
The complete system of principles of the Nuova geometria is actually slightly more complex than we have indicated. Patrizi lists five Supposizioni, eight Diffinizioni, and six Axioms. The suppositions, he says, have been proven in De spacio mathematico, and state that geometry is the science of space, that space is a quantitative extension with minima and maxima, and that it has three dimensions. The definitions, upon which the whole actual deductive procedure of the book rests, characterize the minimal, median and maximal space, then the point, the line, the surface, the (solid) body and the angle. In the course of his proofs, then, Patrizi assumes a few other definitions, such as that of parallel lines or that of a triangle, but he does not seem to regard them as genuine principles. The reason for this seems to be that he introduces these definitions only after having proven the existence of their object: that is, Patrizi first proves, for instance, that there can be three straight lines enclosing a two-dimensional space (Nuova geometria, Book XV, Propositions 1–7, pp. 211–216), and then calls this space a “triangle”. On the other hand, the existence of minima and maxima, points, lines and surfaces, solids and angles, is never proven, but is rather assumed along with the existence of space and as a consequence of the metaphysical reflections on it (namely, those Supposizioni that characterized it as three-dimensional, and allowed for the existence of minima as shown in the De spacio mathematico). Patrizi’s axioms, in turn, are nothing else than a version of Euclid’s common notions, usually phrased in the form of other definitions (for instance: “Tutto, è quello che ha parti”), and concern general mereological and quantitative, but not strictly geometrical, concepts. They find actual application in the course of Patrizi’s proofs, not unlike proper definitions. The system of Patrizi’s geometrical principles thus ultimately comes down to the definitions of a few elementary objects (geometrical, as well as mereological or set-theoretical) and the assumption of their existence, while no operation on these objects is actually defined—as we could expect from a theory of demonstration incapable of dealing with a logic of relations. On science as the “diffinizione dell’essenza”, also see the Nuova geometria, p. 1.
- 67.
The definition of geometry as a purely intellectual science was, of course, very widespread in the Renaissance, and had originated in Plato (but had then been accepted by Aristotle, the Neoplatonists, and in the end by virtually everyone). The idea that geometrical axioms could be proven on the basis of definitions was a rather widely-held idea too. The novelty of Patrizi’s approach consists therefore simply in the radicalization of this assumption, to the point of resolving the whole discipline into a chain of a priori reasoning free from any recourse to intuition.
- 68.
The principal result of this logical attitude towards geometry were the Analyseis geometricae by Herlinus and Dasypodius, published in 1566, which attempted a reduction of the first six Books of the Elements to syllogistic demonstrations. The underlying logic of the Analyseis also comprised other types of logical inference, including, of course, propositional logic and some principles on equality which, together with the usual diagrammatic inferences of classical mathematics, could lend to their endeavour at least a hope of success. There is no evidence, I believe, that Patrizi knew this work, and perhaps, if he had known it, he would have mentioned it; at any rate, it rested upon a Neoplatonic philosophy of mathematics (with a theory of magnitudes completely independent of space, and an extensive use of productive imagination) which Patrizi could not approve of. Herlinus’s and Dasypodius’s attempt, however, enjoyed wide resonance, and is deemed perfectly valid by Clavius (whose edition of the Elements enjoyed an enormous diffusion) and many other mathematicians after him (including Wolff, almost two centuries later).
- 69.
The division of the proof of a geometrical statement into six parts is given by Proclus, In Euclidis 203, and soon became the common property of mathematical epistemology.
- 70.
Leibniz struggled all his life to devise an adequate formalization of elementary geometry, for which he also designed peculiar symbols and appropriate logical axioms; this project went by the general name of characteristica geometrica. The ultimate aim of Leibniz’s analyses, however, also came down to proving all of geometry by means of simple logic, starting from a general definition of space (which was very different from Patrizi’s), while space itself was (as in Patrizi) assumed to exist on the basis of metaphysical arguments.
- 71.
Patrizi’s deductive procedures are in fact completely inane, and one could criticize them in the same terms as those adopted in Lambert’s later attack on Wolff’s (similar) geometry, and say that it is superdefinitory and consists in nothing else than proceeding from names to other names, without ever coming into real contact with the concept. For instance, at the beginning of the First Book of the Nuova geometria Patrizi proves that, since the point is a minimum (according to his definition), it is also simple; and since it is simple, it is also indivisible; and since it is indivisible, it is not a whole; hence it has no parts; therefore it is not partible; hence it is not divisible; and in conclusion it has no quantity. Each of these passages consists of one or two theorems, which merely swap synonymous terms with each other, and it is very difficult to understand what their real advantage could be. The style of the demonstration is something of this sort: “Il punto, perché è semplice, è anche indivisibile. Dimostrazione: Perché se il punto non fosse indivisibile, sarebbe divisibile ne’ suoi componenti; e perciò saria composto, e non semplice. Ma dimostrato s’sè, per la precedente, che il punto è semplice. Adunque il punto, perché è semplice, è anche indivisibile” (Nuova geometria, p. 5).
- 72.
Here the Leibnizian quote: “Je ne me souviens pas maintenant d’avoir veu un philosophe demonstrateur du siecle passé, si ce n’est que Tartaglia a fait quelque chose sur le mouvement, et Cardan parlant des proportions, et Franciscus Patritius, qui estoit un homme de belles veues, mais qui manquoit de lumieres necessaires pour les poursuivre. Il voulut redresser les façons de demonstrer des Geometres, il avoit veu en effect qu’il leur manque quelque chose, et il voulut faire autant dans la Metaphysique, mais les forces lui manquerent; la preface est admirable de sa Nouvelle Geometrie dediée au Duc de Ferrare, mais le dedans fait pitié” (Projet et essais pour avancer l’art d’inventer, in A VI, 4, n. 205, p. 966).
- 73.
In most cases, moreover, the positional concepts did not become a stable part of the definitions of the geometrical objects of Classical Greek mathematics; as a consequence, although in Euclid’s Data, for instance, a geometrical point can be “given in position” (Def. 4: τῇ θέσει δεδόσθαι λέγονται σημεῖα…), it is not defined in terms of this latter, and it is only a non-spatial object to which we later ascribe a certain position in a given configuration. There was, certainly, a genuine definition of point as a unity endowed with position, μονὰς θέσιν ἔχουσα, which Aristotle attributes to the Pythagoreans (De an. Α 4, 409a5–7; Metaph. Δ 6, 1016b24–29; also cf. An. post. Α 27, 87a35–37, Metaph. Μ 8, 1084b26–27). This definition was discussed throughout antiquity, but the addition of the point’s positional character must have soon begun to appear as a philosophical whim devoid of mathematical significance, since no relevant consequence followed from it. Therefore Proclus (In Euclidis, 95–96) believed the Euclidean definition which makes no reference to θέσις to be perfectly correct. I think that this was also the usual attitude in the Renaissance, when geometers often stated the Pythagorean (or Aristotelian) definition for the sake of completeness, but then made no use of it.
- 74.
- 75.
A significant part of Leibniz’s studies on the characteristica geometrica were in fact aimed at producing a combinatorial theory of the dispositions of points or lines; this is the main reason why some historians believed (a bit too generously, however) that they recognized in that Leibnizian theory the origin of topology—the same that was said of Euler’s later theorems on polyhedra. We can note that Patrizi explicitly includes the combinatorial art in the number of the mathematical disciplines in De spacio mathematico (p. 26r = Nova philosophia, p. 68r), and that he seems to resort to arguments of this sort to justify the three-dimensionality of space (De spacio mathematico, p. 19v; Nova philosophia, p. 66v). In the latter case, these were probably Pythagorean arguments (briefly discussed by Aristotle); Leibniz himself discussed them at length in his own theory of dimensions (see again De Risi 2007).
- 76.
Patrizi defines the point as the minimum in space (Nuova geometria, p. 3), and then characterizes it on the basis of the local predicates which derive from that definition: “il punto, perché è nello spazio, ha sito. Il punto, perché sito hà, ha anche posizione, o positura nello spazio. Il punto, poiché positura ha, havrà anche riguardo ad altri uno, o più, e punti, e linee, e angoli, e superficie, e corpi, che nello spazio sono” (Nuova geometria, pp. 14–15). From these premises, Patrizi develops the whole theory of Book One, which is the science of the point which the “ancients” lacked (p. 16). It does not seem to me that any of the Classical authors actually claimed that there cannot be a science of the point, although this certainly followed from the definition of geometry as the science of magnitudes (since the point is not a magnitude); but the thesis was in fact maintained in the Middle Ages, and we find it, for instance, in a manuscript of quaestiones on mathematics, which states that “de puncto nulla passio probatur in tota geometria” (see Dell’anna 1992, p. 132; cf. pp. 141–49). The idea of the point’s not being an object of science will have been transmitted to Renaissance Aristotelianism in this form. From a lexical standpoint, we can also note that θέσις did not necessarily have a local connotation and Aristotle claims, for instance, that mathematical objects can have θέσις without being in a τόπος. This mathematical θέσις was generally translated as positio. On the other hand, the term situs often had a geographical or cosmographical import, and was related to the regions of space (to indicate the different climatic zones of the world, and therefore sometimes even the customs of the peoples inhabiting them); the same was still to be true of Campanella, although he, for his part, was open to the notion of a geometry of position. Patrizi’s terminological choice may therefore mirror the intent of distinguishing his own geometry of space from the simple geometry of position which had been conceived (and soon abandoned) in antiquity.
- 77.
The definition of the point in terms of situation became more and more usual during the Seventeenth Century, and precisely in those authors who were most engaged in the edification of a geometry of space. To mention a few, this is present in Campanella: “Punctum est unitas situalis, cuius ergo nulla est pars” (Mathematica, p. 37); then in Pascal: “Les points ne diffèrent que de situation” (Introduction à la géométrie, p. 87); and finally in Leibniz: “Punctum est cujus pars nulla est. Addendum est, situm habens. Alioqui et temporis instans, et Anima punctum foret” (In Euclidis πρῶτα, in GM V, p. 183), who also adds (just as Patrizi before him) that: “Ea est natura situs, ut omnia quae habent situm, habeant etiam situm inter se” (Prima Geometriae Principia, C 541).
- 78.
Book Three of the Nuova geometria is entirely devoted to the theory of lines, and in particular of indivisible lines. Patrizi tries to show there that it is possible that the reciprocal situation of two points may be such that their distance is the smallest of all, and that, posited in this way, they enclose a minimal space, which is an indivisible straight line. This is certainly one of the cases in which the work’s deductive form lends an appearance of rigour to common-sense lines of reasoning (chiefly in Proposition 3, which proves that two points can enclose a minimal space simply because they can also enclose a medium or a maximal space), which then turn out to be fatal to the mathematical procedures. It is true, however, that the existence of indivisible lines, which taken all together constitute the continuum, should be founded on the situational relations of points. Since, however, the Nuova geometria confines its treatment to linear magnitudes, we do not know how Patrizi would have dealt with plane or solid minima (the existence of which he affirms in De spacio mathematico, p. 24r; Nova philosophia, p. 67v), nor how indivisible lines are supposed to constitute a tridimensional space. However, if we assume that he was simply following the Platonic tradition of the De lineis insecabilibus, the answer could be that he would have then theorized minimal triangles, and perhaps even the Platonic solids of the Timaeus, which consist of those triangles. If this were the solution, tridimensional extension would also arise from the combinatorics of situational relations between elementary objects.
- 79.
Leibniz’s conception of space as the order of situations, that is, as a relational structure, was completely lacking in Patrizi. The fundamental idea of Leibnizian geometry is that of constructing the extension and quantity of space not as its principal attribute (as in Patrizi), but as a property which can be deduced from the set of situational relations. In this way, Leibniz could count on a pure concept of spatiality, whereas Patrizi was still employing a hybrid of the quantitative definition of space and the local properties which are attributed to it subsequently by virtue of its also being place (in the Classical sense). On the construction of extension on the basis of situational relations in Leibniz, see De Risi (2007).
- 80.
The modern age’s various attempts at defining the straight line represent one of the major foundational efforts in the history of geometry. Its definition as the shortest line between two points was widely accepted, but everyone would have preferred to simply take it as a characterization or a property of that line, and define the latter on the basis of formal and figural, rather than metric, properties (as Archimedes himself would most probably have understood this topic). Patrizi devotes the whole of Book Four of the Nuova geometria to the definition and characterizations of a straight line, and makes, in fact, some progress in this direction because he can count on positional relations. Thus, he tries to define the straight line through its situation in space (as a line that does not determine a plane or the whole space, but only, as we would say today, a one-dimensional subspace), and tries to deduce from this that it is the shortest (Proposition 1), that it lies evenly with the points on itself (the very obscure Euclidean definition; Proposition 2), and, most importantly, that it is similar to itself in all its parts (Proposition 8), that it is uniform (Proposition 11), and that any one of its parts is congruent with any other (Proposition 21). All these characterizations, in fact, were to be found once again in Leibniz’s situational geometry, which was to take pains to define the straight line as an axis of rotation (that is, by way of its situation in the ambient space), as linea brevissima, as linea sibi similis, as uniform line, and finally as self-congruent line. The comparison of Leibniz’s studies with Patrizi’s certainly shows the wide abyss dividing the mathematician from the philosopher daring to take his first steps in a field foreign to him; and it makes clear how Patrizi’s characterizations remain, for the most part, simply nominal (since he, for instance, never defines what similarity or uniformity are, and therefore moves from one definition to the other on a merely intuitive basis). However, it is still remarkable that the simple transition from a geometry of magnitudes to a geometry of space and situation can carry with it a whole world of new definitions and characterizations of the straight line (as well as of other geometrical objects) which can be found in the same terms, at least nominally, in these two authors. On the Leibnizian theory of the straight line see De Risi (2007, pp. 226–264); there is no trace, however, of a direct influence of the Nuova geometria on the development of Leibniz’s analysis situs.
- 81.
The theory of the reciprocal situation of straight lines in the plane, and thus of parallelism, takes up the greatest part of the Nuova geometria (Books VI through XIV), and represents in fact a rather admirable attempt at a (non-metric) geometry of intersection. Precisely the definition of the parallels as equidistant straight lines at the beginning of Book VI (p. 82), however, perturbs the overall construction, and, in substance, assumes surreptitiously what would have to be proven in a demonstration of the Parallel Postulate (which eventually closes Book XIV). Patrizi’s mathematical incompetence is most clear from Book VII, however, in which the usual logical procedure in forma once again conceals simple common-sense arguments which should not be admitted in geometry. Thus, the most delicate question in the whole parallel theory is whether two straight lines that approach each other must eventually meet, or whether there can be (as in hyperbolic geometry) asymptotic straight lines; a problem which, in fact, had been already formulated in Classical antiquity, and is attested to in Proclus (In Euclidis 192; 364–365). Patrizi does not appear to see the difficulty, and thus in Proposition 11 of Book VII, after having proven that certain pairs of straight lines approach each other more and more, he concludes simply that they meet, with an argument of this kind: “Perché se piu anche allungandosi, non concorressero finalmente, non sempre più s’appresserebbono. Ma per la precedente dimostrato si è, che allungandosi sempre piu s’appressano. Adunque due o piu rette linee inclinate, perche dalla parte ove meno distano allungandosi, sempre piu s’appreßano, finalmente concorreranno” (p. 98). Here we can only note, in partial defense of Patrizi’s naiveté, that the possibility of asymptotic curves was still a matter of debate in the Renaissance, and was considered an “admirable” difficulty; by the time of the Nuova geometria it had been discussed by several authors, such as Peletier and Finé in France, and also in a very well-known booklet by Francesco Barozzi (whose contribution to Sixteenth-Century Platonic philosophy of mathematics is so remarkable), published in the same year 1586. Bruno was another author who denied the possibility of asymptotic curves, in the De minimo; on this point see Maierù (2012). Moreover, we must bear in mind that the general debate on the theory of parallels in the Renaissance starts, in substance, with the second edition of Clavius’s commentary on the Elements, in which the great German mathematician extensively discusses the fallacious attempt at a demonstration of the Parallel Postulate, exposing to the eye all the usual paralogisms (including that of asymptotic straight lines); this second edition of Clavius’s Euclidis was first to appear in 1589, and thus follows the Nuova geometria by two or three years.
- 82.
We have already mentioned the fact that Patrizi deemed it necessary to discuss the geometry of curves in the sequel to this work, which, in the end, did not actually write. It is not immediately clear how his doctrine of minimal (but extended) straight lines would have fitted together with a theory of curves, and it seems that Patrizi would have had to change something in the general structure of his new geometry to accommodate this latter.
- 83.
Campanella’s main views on space and mathematics are to be found in his Metaphysics, published in 1638; but his essay on Mathematica, where he puts them to a concrete use in actual geometry (not unlike Patrizi’s Nuova geometria) remained unpublished until a few years ago.
- 84.
On this clandestine edition of the Nova philosophia, see Zambelli (1967).
- 85.
On Patrizi’s influence on Sennert, and on Jessenius’s book and its diffusion in German universities, see Zanier (2007).
- 86.
Gassendi’s views on space are exhaustively expounded in the Physica, I, II, 1 and 6 (Syntagma, pp. 179–184 and 216–220; the intermediate chapters are devoted to the void). Here we can find the by now usual assertions that space is neither substance nor accident, that it is a three-dimensional and infinite quantitative extension which precedes bodies and can be empty, and the like. On the reception of Patrizi in Gassendi see Muccillo (2010); on Gassendi’s spatial theory and its fortunes in the Seventeenth Century (and thus on Patrizi’s indirect legacy in that century), see Lennon (1993). Moreover, Deitz (1997) also mentions a volume by L. Crasso, Elogii d’huomini letterati (1666), reprinted in French in 1715, that mentions Patrizi as a forerunner of both Descartes’s and Gassendi’s notions of extension.
- 87.
In his Quaestiones celeberrimae in Genesim, cc. 739–741; here Mersenne does not discuss space, however, but chiefly Patrizi’s theory of light, which he intends to criticize (but Mersenne maintained a certain consideration for Patrizi’s theories, and will recommend his works in the 1648 Optique).
- 88.
On the English reception of Patrizi, see Henry (1979). On the other hand, Henry (1982) points out that two copies of the Nuova geometria were in Henry Percy’s library, and it is thus very likely that the work was known by Harriot and the circle of the Wizard Earl; on Patrizi’s influence on Warner in particular, see Prins (1994). Henry More discusses the metaphysics of space in the Enchiridium metaphysicum of 1667; but he had also addressed the question in his The immortality of the soul (1659) which was certainly known to Newton, and especially in his correspondence with Descartes, which was published in 1655 but had enjoyed a certain manuscript circulation in earlier years (we know that Newton had already read it in 1654). Hobbes’s conceptions seem rather different from Patrizi’s, but see at any rate Schuhmann (1986). It is remarkable, however, that Hobbes possessed a copy of the Nuova geometria (that is, not only of Patrizi’s treatises on spatial metaphysics). Note also that Berkeley still knows and quotes Patrizi with some admiration.
- 89.
On Barrow’s theories on space, see Lecture Ten in his Lectiones mathematicae, where however he repeats that space is not in itself quantity (and only acquires quantity through the bodies existing in it), so that geometry remains a science of magnitudes rather than of space.
- 90.
On Leibniz’ opinion about the Nuova geometria see above Footnote 72. It is to be noted, however, that this is a text from 1688 to 1690, when Leibniz was already quite advanced in his own geometrical endeavors, and that there is not to be found any previous reference to Patrizi’s geometry among his papers. We should also note that Leibniz was rather critical of Patrizi’s metaphysics as well, as he considered himself to be a good Platonist and regarded this Late Renaissance Neoplatonism as a corruption of Plato’s original philosophy: “Itaque saepe miratus sum nondum extitisse quendam qui systema philosophiae Platonicae sanioris daret, nam Franciscus Patritius, non contemnendi vir ingenii Pseudo-Platonicorum lectione animum praecorruperat” (Ad constitutionem scientiae generalis, in A VI, 4, n. 114, p. 479).
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De Risi, V. (2016). Francesco Patrizi and the New Geometry of Space. In: Vermeir, K., Regier, J. (eds) Boundaries, Extents and Circulations. Studies in History and Philosophy of Science, vol 41. Springer, Cham. https://doi.org/10.1007/978-3-319-41075-3_3
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