This paper examines two ways in which the ‘privileged coordinates’ of a geometric space might have significance. First, the structure of the space might be ‘determined by its privileged coordinates’. Second, the space might be presentable using ‘Kleinian methods’. We examine the geometric spaces for which these two conditions hold. Along the way, we investigate the relationship between these two conditions.

Marsilio Ficino (1433-1499) was one of the luminaries of the Florentine Renaissance and the scholar responsible for the revival of Platonism. The translator and interpreter of the works of both Plato and Plotinus as well as of various Hermetic and Neoplatonic texts, Ficino was also a musician, priest, magus and psychotherapist, an original philosopher and the author of a vast and important correspondence with the intellectual figures of his day including Lorenzo the Magnificent. Professor Allen has become the foremost interpreter (...) of Ficino's metaphysics and mythology, and the ancient sources they draw upon; and this collection of essays assembles his work on Ficino's complex interrogation of Platonic 'theology' as not only a preparation for Christianity but as an enduring medium for intellectuals to explore and to express Christian truths. (shrink)

Fifteen of these essays by one of the leading authorities on Renaissance Platonism explore the complex philosophical, hermeneutical, and mythological issues addressed by the Florentine, Marsilio Ficino (1433-99). Ficino was the pre-eminent Platonist of his time and a distinguished philosopher, scholar and magus who had an enormous influence on the intellectual and cultural life of two and a half centuries, and who is one of the most important witnesses to the preoccupations of his age, above all to its fascination with (...) ancient poetry and philosophy and their uneasy accommodation as an ancient "theology" with Christianity. Two further essays treat of cognate themes taken up by Ficino's younger friend and rival, the dazzling prince of Concordia, Giovanni Pico della Mirandola (1463-94), who was fascinated by Platonism in his youth but also by other philosophical legacies from the past, including Cabala and the Scholastic Aristotelianism of the Middle Ages. This volume's initial essay serves as an introduction to the comprehensive phenomenon of Renaissance Platonism. (shrink)

The duality between congruence lattices of semilattices, and algebraic subsets of an algebraic lattice, is extended to include semilattices with operators. For a set G of operators on a semilattice S, we have \ \cong^{d} {{\rm S}_{p}}}\), where L is the ideal lattice of S, and H is a corresponding set of adjoint maps on L. This duality is used to find some representations of lattices as congruence lattices of semilattices with operators. It is also shown that these congruence lattices (...) satisfy the Jónsson–Kiefer property. (shrink)