This chapter is introductory. I first survey, in a quick sweep, mathematics before Greece. This is followed by the historical context for the rise of mathematics in Greece itself (a discussion heavy with historiographical problems because so much is speculative). Finally, I conclude with a picture of the earliest known Greek mathematics.

Platonic and Plotinian conceptions of beauty played a role in Ficino’s understanding of beauty in De amore and subsequently in Leone Ebreo’s (Judah Abravanel’s) understanding of beauty in Dialoghi d’amore. Plato’s Symposium is an inspiration to both but does not determine the structure of either author. This essay contrasts Ficino’s presentation of seven friends’ viewpoints on love and Leone’s dialogue between Philo and Sophia, who together create PhiloSophy. While Ficino’s letters to Cavalcanti, to whom he dedicates the work, provide further documentation of Ficino’s intent to experience a higher form of love, Leone’s view of Aristophanes’ androgyne as based on a pre-Talmudic Hebraic concept of Adam as a hermaphrodite indicates his subordination of the Symposium to the Hebraic tradition.

The essay shows the influence of Plotinus’ idea of love of intellect alone for the heavenly Venus and the influence of the Middle Stoics and Middle Academics on Ficino’s belief that there are generative seeds of virtue and knowledge in the human soul. Alberti’s concept of concinnitas and measure are at the root of Ficino’s notions of harmony of sounds and harmony of sights. Leone, like Ficino, values the senses of hearing and sight and disparages the senses of taste, smell and touch.

Both authors correlate beauty and virtue. Ficino is a major source for the Renaissance tendency to view the picture of a beautiful woman as a picture of a virtuous woman. Sophia and Philo both grasp the relationship between beauty and virtue and that the highest goal is a love relationship with the Divine Creator; nevertheless, Philo resorts to lecherous longing for Sophia despite recognizing that coppulazione is the moment of a human’s intellectual vision of God.

Marsilio Ficino adopted the concept of love from Plato and the Neoplatonists in order to elaborate his philosophy of humanity and divinity. In commenting on Plato’s Symposium, he employed amor/eros for his philosophical theology according to which God and the universe are united: love is that which binds all levels together. As such a mediator and power, it engages humans as a species, as individuals, and in societies. Love is the formula that makes the metaphysical reality intelligible and teachable. Love divides the good from the abject and sets the standard that is attainable and binding for all.

A one-volume history of Greek mathematics has to be selective and compressed. All the more so, a one-chapter history of Greek astronomy. We follow an entire trajectory—from the beginnings to Ptolemy (second century ce). Chapter 3, “The Generation of Archimedes,” was a rondo with Archimedes as its recurrent theme. Here, we progress through three subchapters for our three main cast members:

1. I. Rising: Greek Astronomy to Eudoxus

2. II. Conjunction: Greece, Babylon, and Hipparchus

3. III. Culmination: Ptolemy

In a world of ambitious mathematicians, each vying to become the new Archimedes, Evangelista Torricelli, born in Faenza, Italy, in 1608, was determined to out-Archimedes them all. The one book he published in his short life (he died at age thirty-nine) is the Geometrical Works, a series of mathematical marvels, some reproducing results from Archimedes, some extending the spirit of Archimedes by measuring new curved figures. Torricelli may have been the first to measure the area of the cycloid, a figure originally proposed by Galileo. (The seventeenth century was full of Archimedes-like challenges; unlike in the third century bce, such challenges tended to generate many responses, with many ensuing fights over priority.) He invented and measured his own—paradoxical—figure. Consider a hyperbola, one of its asymptotes, and some perpendicular to that asymptote cutting the hyperbola. The three taken together define an infinitely long figure that is widest at its base (the perpendicular) and keeps narrowing—without ever reaching the zero width of a point—as it moves away from that base. Rotate this figure around the asymptote, and you get an infinitely long solid, shaped rather like an infinitely long, curved funnel. Torricelli measured the volume of this infinite solid—which turned out to be, incredibly, finite. Infinite surface, finite volume! A new Archimedes, right there!

This essay offers an account of Thomas Aquinas’s very complex understanding of love (amor) and related concepts, such as delight (delectatio) and divinely infused charity. It acknowledges the many Platonic influences on Aquinas’s thought in these regards, influences that come to him especially through Augustine of Hippo and Pseudo-Dionysius – both of whom he occasionally criticizes, however, for adhering too closely to Platonic doctrines. Aquinas is intent upon tying all human love in some sense to nature, acknowledging that this entails that in many regards human love is necessitated. To the extent, however, that the rational appetite (the will) is involved, love is not necessitated. The divinely infused virtue of charity brings in further complications, for it is a love that is not natural but comes to a person from above: it is supernatural. And yet it too, according to Aquinas, is not wholly necessitating but affects man in a way that leaves his will free.

This chapter explores the different treatments the topic of love received in the Symposium and the Phaedrus. The Symposium expressly addresses the question of the character and benefits of eros – and the question of its nature. It is a philosophical enquiry into this phenomenon, conducted by a set of mature men (and the memory of one exceptional woman) who have personal experience in this topic and have thought about it. In the Phaedrus, by contrast, the topic is addressed from the point of view of a would-be lover trying, through a speech addressed to the young and inexperienced man, to persuade him to submit to his overtures and desires. For Plato, eros is crucial to the practice of philosophy, a force which can take us in two separate directions, towards the good and towards the bad.

Elite friendship discourse in the Renaissance was shaped by a set of commonplaces inherited from classical antiquity according to which friends were virtuous, male, and few in number, and their relationships egalitarian and non-sexual. Neoplatonic love had the power to disrupt many of these received ideas. Ficino’s account of male friendship in his Lysis commentary emphasized the importance of spiritual desire in initiating relationships and foregrounded a pedagogical dimension more in keeping with a chaste version of Greek pederasty than the non-hierarchical models of friendship inherited from Aristotle and Cicero. In a poem on the Platonic androgyne, Antoine Héroët used the language of friendship to describe heterosexual unions as offering a potential step towards union with God. Bonaventure des Périers warned instead of the dangers of earthly erotic entanglements in a verse commentary to his translation of Plato’s Lysis, thereby concurring with the beliefs of his benefactor Marguerite de Navarre while suggesting that female community might offer the soul some solace before death provided the possibility of joining with God. Finally, Montaigne’s unorthodox account of his relationship with his deceased friend La Boétie engaged with the Neoplatonic tradition while eschewing the possibility it might facilitate spiritual ascent.

Texts that warn of the dangers of passionate or excessive love have a history in Western culture going back to antiquity. Writings in this contra-amorem tradition typically characterize obsessive love or lovesickness as a disease and then offer remedies for the sufferer. When interest in Marsilio Ficino’s doctrine of Platonic love began to spread from Florentine philosophical circles to aristocratic courts throughout Italy in the late fifteenth century, some authors writing in the contra-amorem tradition responded directly to the new enthusiasm for Ficino’s ideas. A comparison of two contra-amorem texts – Bartolomeo Platina’s ‘pre-Ficinian’ On Love (c. 1466) and Battista Fregoso’s ‘post-Ficinian’ Anteros (1496) – will illustrate the ways in which the later text directed its arguments against Ficino’s doctrine, and did so with an audience of aristocratic young men particularly in mind. It is noteworthy that Anteros predates the first vernacular popularizations of Platonic love in Pietro Bembo’s Asolans (1505) and Baldassare Castiglione’s Book of the Courtier (1528), and also that Castiglione’s Courtier responds, in turn, to Anteros by assimilating some elements from that work into its own treatment of Platonic love.

Augustine is rightly regarded as one of the major figures of Christianity. Through him Platonist and Neo-Platonist philosophy was appropriated in the new religious context and there is hardly another thinker of the early Christian tradition who can better illustrate Nietzsche’s remark on Christianity as a Platonism for the people than Augustine. Things are, however, more complicated than Nietzsche suggests. As Étienne Gilson has showed, there is a strong tension between medieval Aristotelianism and the necessity to respect the early foundational authority, which Saint Augustine incontestably was. Aquinas has in this sense a quite complicated relation to his predecessor. In order to understand what is at stake in Augustine’s understanding of the biblical message of love, heavily influenced by Paul, it is important not only to describe the Neoplatonist roots of Augustine’s thought, but also to take the wider ancient context into account – and in this way arrive at a more complete picture of the historical and intellectual setting. The modification Augustine brings to the Aristotelian conception of the soul is hereby particularly revealing. A closer study of the relation between desire and love can shed some light on this highly significant constellation.

This essay undertakes a study of the views of Pico della Mirandola on Platonic love, stimulated as they are by the publication of a poem on the subject published in the mid-1480s by Girolamo Benivieni, a friend of his and of Marsilio Ficino. In a discussion of the Renaissance background, the essay emphasizes the importance of defusing the homosexual element of Platonic love by substituting maidens for boys. It then provides an extended discussion of Pico’s commentary on Benivieni’s poem, in which he draws on, not only the Symposium and Phaedrus of Plato, but also Plotinus’ Ennead III 5: On Love, and Hermeias’ commentary on the Phaedrus. A number of passages from the poem itself are also quoted and discussed.

In the Sophist, Plato asks how there can be false statements. A false statement (logos) says what is not. But a statement cannot say nothing – there must be something that it says. Plato tries to solve the puzzle by investigating the notion of not-being and the notion of a logos. To understand not-being, we must first understand being. That in turn requires understanding how something can be called by many different names. Plato’s solution to these problems rests on the distinction between what something is of itself and what it is because it is related to other things. It also rests on an understanding of the notion of difference. The class of not-beautiful things, for example, is a real group: they are different from the beautiful things. Similarly, not-beings are a real group. The upshot is this: “Theaetetus is flying” is false, in that flying is different from what is with reference to Theaetetus. Theaetetus has being. Flying has being. But “Theaetetus is flying” says with reference to him what is not.

Throughout his political works, Plato takes the aim of politics to be the virtue and happiness of the citizens and the unity of the city. This paper examines the roles played by law in promoting individual virtue and civic unity in the Republic, Statesman, and Laws. Section 1 argues that in the Republic, laws regulate important institutions, such as education, property, and family, and thereby creating a way of life that conduces to virtue and unity. Section 2 argues that in the Statesman, the political expert determines the mean between extremes and communicates it to citizens through laws that guide their judgment and conduct, so that they become virtuous themselves and the city is unified; this account of the role of law suggest how even non-expert legislation can contribute to virtue and unity. Section 3 argues that the Laws affirms and develops the idea that citizens should know and accept the laws to become virtuous themselves and to unify the city, and explains how the persuasive preludes and the sanction for violation attached to laws contribute to citizen virtue and civic unity.

Plato’s dialoguesespecially the Republiclead us to wonder what the objects of mathematics are. For Plato, no perceptible three is unqualifiedly three, a necessary condition for being an object of knowledge. Aristotle controversially ascribes to Plato the view that mathematical objects are “intermediates,” between perceptibles and Forms: multiple but also eternal, lacking change, and separate from perceptibles. The hunt for or against intermediates in Plato’s dialogues has depended on two ways of understanding Plato on scientific claims, a Form-centric approach and a subject-centric (semantic) approach. Although Socrates does not present intermediates in the Republic, it is difficult to see how the units of the expert arithmetician or motions of the real astronomer could be simply Forms or perceptibles. The standard over-reading of the Divided Line, where the middle sections are equal, further obscures our understanding. The Phaedo and the Timaeus provide candidates for mathematical objects, although these have only some of the attributes ascribed to intermediates. We are left with no clear answer, but exploring options may be exactly what Plato wants.