Mathematics as Paideia in Proclus
ABSTRACT: I examine one aspect of the central role which mathematics plays in Proclus's ontology and epistemology, with particular reference to his Elements of Theology. I focus on his peculiar views about the ontological status of mathematical objects and the special faculties of the soul that are involved in understanding them. If they are merely abstract objects that are "stripped away" from sensible things, then they are unlikely to reorient the mind towards the intelligible realm, as envisioned by Plato in the Republic. Thus, in order to defend the function of mathematics as a prodaideutic to dialectic, Proclus rejects Aristotelian abstractionism in favor of an elaborate account in terms of Nous projecting images of its Forms through the medium of the imagination. In metaphorical terms, he replaces the Aristotelian image of the soul as a blank tablet with that of a tablet that has always been inscribed and is always writing itself, while also being written on by Nous. The mediating function of mathematics for understanding the higher realities is grounded in the fact that its central principles of Limit and Unlimited have a universal provenance in Proclus's whole system of reality.
Introduction
Alexander of Aphrodisias established abstractionism as an Aristotelian dogma about mathematical objects, but for later Neoplatonists this proved difficult to reconcile with the educational function of mathematics in Plato's philosophical curriculum. Thus Proclus, for example, rejected abstractionism as a basis for the ascent to the realm of Forms, and proposed an alternative view based on the typical Neoplatonic hierarchy of Nous, Soul, and Nature. At the highest noetic level, geometrical Forms are unextended and indivisible, so that only at the level of Soul can they become available for study by the geometer when they are embodied in the intelligible matter supplied by the imagination. Proclus also accepted that geometrical forms can be embodied in sensible matter, though they never have the exactitude necessary for science, nor could they ever acquire it through abstraction. Thus the diagrams used by the geometer are products of the imagination, which are really projections by the higher intellect onto a lower level so as to facilitate the study of geometrical objects. Proclus seems to accept that the human intellect can never attain the Platonic goal of studying geometrical Forms in their pure and unextended form as paradigms. Although such a goal can be achieved only by divine Nous, yet it becomes for Proclus the guiding rationale for his whole system.
Samuels, Charles T. "The Greatness of ‘Gatsby'." Fitzgerald's The Great Gatsby: The Novel, The Critics, The Background. Ed. Henry D. Piper. Charles Schribner's Sons, New York: 1970.
Trask, David F. "A Note on Fitzgerald's The Great Gatsby." University Review 33.3 (Mar. 1967): 197-202. Rpt. in Novels for Students. Ed. Diane Telgen. Vol. 2. Detroit: Gale, 1998. Literature Resources from Gale. Web. 15 Jan. 2014.
ABSTRACT: In this paper, I will show the deep roots of dialogue in Plato’s thought, in order to examine the validity of the so-called ‘esoteric Plato’. The confrontation between dialogicity and unwritten doctrines is the main theme of this article. These two views — Hermeneutics and Tübingen School — are not far away on concrete contents, with more or less variations. But it must be noticed that both conceptions of Platonic thinking are contradictory and that is reflected in their explanations of Plato’s own philosophical project.
In order to substantiate Orthodox triadology, there was an emergent necessity for St. Basil to develop a clear and understandable terminology. The most plausible of historical and philosophical sources that the saint used in the doctrine of differentiating between “essence” and “hypostases” by the so-called generic principle, is the “Introduction” of Porphyry of Tyre, who was a Neoplatonic philosopher, and “Categories” of Aristotle. In the understanding of “essence” (as opposed to the terms Aristotle used, while this term was used by Gregory of Nyssa), there is a place for orthodox stoic character. St. Basil characterized God’s essence using the ideas of community, identity, unity and simplicity, and yet God’s essence is still not comprehensible. The prelate says that the Father’s “hypostasis” has a distinctive characteristic—“fatherhood,” Son—“sonship,” and Holy Spirit—“...
One of the key tenants of Plato's teachings is The Forms. Plato attempts to explain t...
To recall another relic of ancient Greece, Plato had strong opinions on artwork, even that which was created during his time. Plato believed tha...
Euclid’s Elements are predominantly the most fundamental concepts of mathematics, but his perspective on geometry was the model for over two millennia. He is believed by many to be the leading mathematics teacher of all time. However, little is known about his life outside of mathematics, or even when he was born or when he died. According to a passage written by Proclus, Euclid probably lived after Ptolemy and the pupils of Plato, but came before Archimedes and Eratosthenes. This places his existence sometime around 300 B.C. Euclid is most famous for having set the guidelines for geometry and arithmetic written in Euclid’s Elements, a series of thirteen books in which Euclid states definitions, postulates, and theorems for mathematical concepts that are still used today. What is most remarkable about the Elements is the simple, rational, and very logical structure in which Euclid presents the accumulated geometrical knowledge from the past several centuries of Greek mathematicians. The manner in which the propositions have been derived is considered to be the prime model of the axiomatic method. (Hartshorne 296).
ABSTRACT: At issue is the reliability of Heidegger’s contention that Greek thinking, especially Plato’s, was constricted by an unthought "pre-ontology." "The meaning of being" supposedly guiding and controlling Greek ontology is "Being = presence." This made "the question of the meaning of ousia itself" inaccessible to the Greeks. Heidegger’s Plato’s Sophist is his most extensive treatment of a single dialogue. To test his own reliability, he proposes "to demonstrate, by the success of an actual interpretation of [the Gigantomachia], that this sense of Being [as presence] in fact guided [Plato’s] ontological questioning . . .". I will show Heidegger’s strategy in connecting what he takes to be Plato’s naive pre-ontology — Being = Presence — to the ontology of the Gigantomachia — Being = Power. I will show that Heidegger blatantly misreads the text to make the connection: he completely misses the distinction between bodies and bodiless things. The text makes sense, I will show, if and only if its explicit ontology — Being = Power — is its implicit pre-ontology. Plato wrote his text not to discuss, but to exemplify, Heidegger’s ontology-preontology distinction. He wrote the Gigantomachia for Heidegger, but Heidegger missed it.
Plato was born in Athens, Greece around 427 B.C. He was always interested in politics, until he witnessed his mentor and teacher, Socrates, death. After learning of the callousness of politics, Plato changed his mind and eventually opened up The Academy, which is considered if not the first, one of the first Universities. Students at the Academy studied many different fields of science, including biological and astronomical. The students also studied many other fields, such as math. Plato developed many views that were mathematical in nature. He expressed these views through his writings. According to Dr. Calkins of Andrew University, "Timaeus is probably the most renowned of Plato's thirty-five dialogues. [In it] Plato expresses that he believes that the heavenly bodies are arranged in perfect geometric form. He said that because the heavens are perfect, the various heavenly bodies move in exact circles." (Calkins 1). Of course that is a much summarized view of what Plato discusses in Timaeus, but still a solid view on Plato's beliefs about cosmology. Cosmology can be loosely defined as everything being explained and in its place or beautiful. The cosmos is beautiful because everything is perfect. Plato understood that when he defined the most perfect geometric design as the circle. In a circle one line is always equidistance from one point. In Plato's universe there are two realms, eternity and time. The factor that creates "time" out of the chaos of "eternity" is the Demiurge. Plato's Demiurge can be defined as an architect creator theological entity. The importance of the Demiurge in this paper is to compare and contrast him with Boethius's God in The Consolation of Philosophy.
Fitzgerald, F. Scott, and Matthew J. Bruccoli. The Great Gatsby. New York, NY: Scribner, 1996. Print
I will particularly concentrate on the role of the masculine Demiurge as the author of matter and how his creation is replicated by the pantheon of Greek gods, who are said to create humans, and how this account transgresses conventional ones that focus on the earth and physical nature. As part of these considerations, I will also engage with Judith Butler and Luce Irigarary’s readings of this theory of materiality. In the second section, I will consider the implication of the Demiurgic model of creation and how it is manifested in the earlier tale that consolidates the ancient Athenians’ association with their patron, Athena. I will show that Athena is understood to be strongly connected to the Demiurge and his creation, and she imparts this association on the ancient Athenians, who demonstrate the reality and power of the demiurgic model of creation.This is especially demonstrated through the ancient Athenian victory over the Atlanteans, who are associated with a more conventional and what Plato understands to be an incorrect model of creation. Thirdly, I will show how the perseverance of this tale and the character of the ancient Athenians through history and epic poetry through a revered figure like Solon shows not only Plato’s concern with the cultural and political impact that his theory of matter has upon everyday life but is a means of consolidating the theory in a large cultural milieu. Ultimately, the cultural underpinning of the theory of materiality, which underlies the practices and characters of the ancient Athenians that contemporary Athenians should model all serve to institute the political structures that are advocated in the Republic and restated at the outset of the Timaeus. Plato’s metaphysical theories and politics thus, have a symbiotic relationship where the political practices that are instituted lead to
Sutton, Brian. "Fitzgerald's The Great Gatsby." Explicator 59.1 (Fall 2000): 37-39. Rpt. in Twentieth-Century Literary Criticism. Ed. Linda Pavlovski. Vol. 157. Detroit: Gale, 2005. Literature Resource Center. Web. 24 Feb. 2011.
In this essay I will be discussing the thought-provoking theory of universals and be asking whether this idea of an invisible yet prominent realm of reality can claim to have a place in existence. I shall firstly examine Plato's dialogue of Parmendides and see if a partition can be drawn between the forms and universals. I shall then move on to the opposing argument which invariably denies such dimensions in reality before reaching my conclusion.
...er is the Allegory of the Cave, in which he explains a scenario where people, from birth, are chained by their necks and ankles with the view of a wall in this cave that ultimately limit their experiences of life until freed. This extended metaphor contrasts the way we perceive and believe what reality is. The thesis behind his allegory is that all we perceive are imperfect "reflections" of the ultimate Forms, which subsequently represent truth and reality. In Plato’s Metaphor of the Divided Line, he goes into detail about the levels of knowledge that can be obtained through a hierarchy with the most superficial type of knowledge being the lowest while perfect intelligence is the highest. Plato's most outstanding student was Aristotle, perhaps the first truly systematic philosopher. Aristotelian logic was the first type of logic to attempt to categorize every valid