I argue here that a properly Platonic theory of the nature of number is still viable today. By properly Platonic, I mean one consistent with Plato's own theory, with appropriate extensions to take into account subsequent developments in mathematics. At Parmenides 143a-4a the existence of numbers is proven from our capacity to count, whereby I establish as Plato's the theory that numbers are originally ordinal, a sequence of forms differentiated by position. I defend and interpret Aristotle's report of (...) a Platonic distinction between form and mathematical numbers, arguing that mathematical numbers alone are cardinals, by reference to certain non-technical features of a set-theoretical approach and other considerations in philosophy of mathematics. Finally I respond to the objections that such a conception of number was unavailable in antiquity and that this theory is contradicted by Aristotle's report in Metaph . XIII that Platonic numbers are collections of units. I argue that Aristotle reveals his own misinterpretation of the terms in which Plato's theory was expressed. (shrink)

Rather than reading Plato's philosophy of arithmetic ‘charitably’, it is better to try to explain its failure to generate any fruitful ideas. Prominent in the explanation is Plato's focus on predicates assigning cardinalities and on ‘groups’ falling under them. This focus left Plato unable to envisage the possibility, emerging in Dedekind and Frege but which arithmetic in Plato's time would not easily have suggested, of regarding numbers as objects essentially ranged in the structure of a progression.

In defense of anti-essentialism, pragmatist Richard Rorty holds that we may think of all objects as if they were numbers. I find that Rorty’s metaphysics hinges on two rather weak arguments against the essences of numbers. In contrast, Plato’s metaphysics offers a plausible definition of essentiality by which numbers do have essential properties. Further, I argue that Rorty’s argumentative mistake is mischaracterizing Plato’s definition. I conclude that Plato’s definition of “essential” is a robust one which implies that many properties, beyond (...) those we might intuitively think of, can count as essential properties of objects. (shrink)

This paper analyzes and explains certain parts of Syrianus’s Commentary on book M of Aristotle’s Metaphysics, which details Syrianus’s response to Aristotle’s attack against the Platonic position of the separate existence of numbers. Syrianus defends the separate existence not only of eidetic but also of mathematical numbers, following a line of argumentation which involves a hylomorphic approach to the latter. He proceeds with an analysis of the mathematical number into matter and form, but his interpretation entails that form is (...) the constituent of number, which has the status and role of a Platonic Form. This solution allows him not only to explain and justify the unity of number, but also to apply the Platonic thesis of the separate existence of numbers, to the mathematical or monadic numbers themselves. It also betrays its tendency to combine theses of the Platonic Ontology with fundamental Aristotelian doctrines. (shrink)

In defense of anti-essentialism, pragmatist Richard Rorty holds that we may think of all objects as if they were numbers. I find that Rorty’s metaphysics hinges on two rather weak arguments against the essences of numbers. In contrast, Plato’s metaphysics offers a plausible definition of essentiality by which numbers do have essential properties. Further, I argue that Rorty’s argumentative mistake is mischaracterizing Plato’s definition. I conclude that Plato’s definition of “essential” is a robust one which implies that many properties, beyond (...) those we might intuitively think of, can count as essential properties of objects. (shrink)

Plato's political thought gave rise to a number of concepts and issues - such as the idea of a normative theory, the philosophical foundation of politics, the philosopher-kings, the standard of utopian theory - which have played a significant role on Western political and philosophical thought. -/- This volume aspires to bring out Plato's concept of efficacy in a normative theory. -/- By efficacy, the author means the way in which the theory conceives of its practical realization. If in the (...) Republic this issue is particularly problematic, in the Laws, instead, the claim of realizability becomes the main concern. -/- By defining the concept of efficacy and going through the main works of Plato, this conceptual and textual analysis aims to shed a new light on Plato's political thought, which is neither a utopia, nor a short term political program, but the first normative theory encompassing a theory of efficacy. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:160 HISTORY OF PHILOSOPHY A PLATONIC PARALLEL IN THE DISSOI LOGOI The Dissoi Logoi or Two-/old Arguments (Diels-Kranz, II, 405-416) is an anonymous sophistic treatise written in literary Doric at some time subsequent to the end of the Peloponnesian War in 404-403.1 As early as 1911, A. E. Taylor wrote that the treatise "must be seriously reckoned with in any attempt to reconstruct the history of Greek thought (...) in the generation immediately anterior to Plato." 2 But scholars have been slow to take up Taylor's challenge. From those writing in English we have had little beyond the summary in Kathleen Freeman's Companion a and the papers of Adolfo Levi4 and E. S. Ramage." There is not even a full English translation in print, s In this paper I shall call attention to one brief passage which seems to me to be of exceptional interest since it offers a solution to a type of sophistical argument which occurs in Plato's Euthydemus. In so doing, I hope to arouse interest in a document much neglected by historians of ancient thought. Dzssoz Loaox, V, 15. With respect to the assertion that the same man both is and is not, I put the following question: "Does he exist with respect to some particular thing, or just in general?" Then if someone denies that the man exists, he is mistaken, because he is treating (the particular and) universal senses as the same. Because everything exists in some sense. GENI~I~AL REMAI~KS.The nature of the solution offered is clearly based on a distinction similar to that made by Aristotle in On Sophistical Re]utations (166b 37 ft.) between expressions used "in a certain respect" and "absolutely." The formulation of the fallacy to which the distinction is intended to be applied, however, is considerably less clear. Is it supposed that the opponent has put forward an argument purporting to demonstrate (a) that the same man both is and is not, or (b) that a particular man, thought to exist, does not in fact exist? This question can best be answered by an analysis of the Euthydemus passage, to which I now turn. EUTttYDgMUS 283b-e: The sophist Dionysodorus inquires of Socrates and his friends whether they genuinely wish their young prot4g6, Cleinias, to become wise. On gaining their assurance that they do, he then proceeds as follows: 1) You wish him to become wise and not to be ignorant (283d 1). 1This terminus post quem is derived from I, 8, in which the author lists a number of victories and speaks of that won by the Spartans over the Athenians as being "the most recent." "The 6,~ao~X6-tot,"Varia Socratica, Fizst Series (Oxford, 1911), p. 128. aCompanion to the Pre-Socratic Philosophers, (Oxford, 1946),pp. 417-423. ""On 'Two-fold Statements,'" American Journal o/Philology, LXI (1940), 292-~06. 5"An Early Trace of Socratic Dialogue," American Journal of Philology, LXXXII (1961), 418-424. 6The present writer has cempleted a translation which is to be published in Mind in the near future. NOTES AND DISCUSSIONS 161 2) You wish him to become what he is not, and no longer to be what he is now (literally: what he is now, no longer to be [283d 2-3]). 3) You wish for his death, since you wish him no longer to be (283d 5-6). The obvious way of dealing with this argument is to make precisely the distinction made by the author of the Dissoi Logoi at V, 15: we should say to Dionysodorus, "Your conclusion that the friends of Cleinias wish for his death is the result of an illegitimate step: you went from 'you wish him not to be ignorant' to 'you wish him not to be'. To have made this step is to have ignored the distinction between existence with respect to some particular thing, i.e., ignorance, and existence in general." (A later critic might point out to Dionysodorus that he has confused the copulative and existential uses of the verb "to be.") It now becomes possible to return to the question, which of the two formulations, (a) or (b... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:160 HISTORY OF PHILOSOPHY A PLATONIC PARALLEL IN THE DISSOI LOGOI The Dissoi Logoi or Two-/old Arguments (Diels-Kranz, II, 405-416) is an anonymous sophistic treatise written in literary Doric at some time subsequent to the end of the Peloponnesian War in 404-403.1 As early as 1911, A. E. Taylor wrote that the treatise "must be seriously reckoned with in any attempt to reconstruct the history of Greek thought (...) in the generation immediately anterior to Plato." 2 But scholars have been slow to take up Taylor's challenge. From those writing in English we have had little beyond the summary in Kathleen Freeman's Companion a and the papers of Adolfo Levi4 and E. S. Ramage." There is not even a full English translation in print, s In this paper I shall call attention to one brief passage which seems to me to be of exceptional interest since it offers a solution to a type of sophistical argument which occurs in Plato's Euthydemus. In so doing, I hope to arouse interest in a document much neglected by historians of ancient thought. Dzssoz Loaox, V, 15. With respect to the assertion that the same man both is and is not, I put the following question: "Does he exist with respect to some particular thing, or just in general?" Then if someone denies that the man exists, he is mistaken, because he is treating (the particular and) universal senses as the same. Because everything exists in some sense. GENI~I~AL REMAI~KS.The nature of the solution offered is clearly based on a distinction similar to that made by Aristotle in On Sophistical Re]utations (166b 37 ft.) between expressions used "in a certain respect" and "absolutely." The formulation of the fallacy to which the distinction is intended to be applied, however, is considerably less clear. Is it supposed that the opponent has put forward an argument purporting to demonstrate (a) that the same man both is and is not, or (b) that a particular man, thought to exist, does not in fact exist? This question can best be answered by an analysis of the Euthydemus passage, to which I now turn. EUTttYDgMUS 283b-e: The sophist Dionysodorus inquires of Socrates and his friends whether they genuinely wish their young prot4g6, Cleinias, to become wise. On gaining their assurance that they do, he then proceeds as follows: 1) You wish him to become wise and not to be ignorant (283d 1). 1This terminus post quem is derived from I, 8, in which the author lists a number of victories and speaks of that won by the Spartans over the Athenians as being "the most recent." "The 6,~ao~X6-tot,"Varia Socratica, Fizst Series (Oxford, 1911), p. 128. aCompanion to the Pre-Socratic Philosophers, (Oxford, 1946),pp. 417-423. ""On 'Two-fold Statements,'" American Journal o/Philology, LXI (1940), 292-~06. 5"An Early Trace of Socratic Dialogue," American Journal of Philology, LXXXII (1961), 418-424. 6The present writer has cempleted a translation which is to be published in Mind in the near future. NOTES AND DISCUSSIONS 161 2) You wish him to become what he is not, and no longer to be what he is now (literally: what he is now, no longer to be [283d 2-3]). 3) You wish for his death, since you wish him no longer to be (283d 5-6). The obvious way of dealing with this argument is to make precisely the distinction made by the author of the Dissoi Logoi at V, 15: we should say to Dionysodorus, "Your conclusion that the friends of Cleinias wish for his death is the result of an illegitimate step: you went from 'you wish him not to be ignorant' to 'you wish him not to be'. To have made this step is to have ignored the distinction between existence with respect to some particular thing, i.e., ignorance, and existence in general." (A later critic might point out to Dionysodorus that he has confused the copulative and existential uses of the verb "to be.") It now becomes possible to return to the question, which of the two formulations, (a) or (b... (shrink)

Die sogenannte ‚Hochzeitszahl’ aus Platons Staat gilt wohl zu Recht als eines der bedeutendsten, noch immer ungelösten Rätsel der Antike. In einem kunstvoll verschlüsselten Text deutet Platons Sokrates an, auf welche Weise der Idealstaat als gerechteste und glücklichste Form einer menschlichen Gemeinschaft seine politische Kontinuität erhalten kann: Es genügt nicht nur, dass die amtierende Führungsschicht von bestqualifizierten, für sich selbst aber besitzlosen Philosophinnen und Philosophen ihre potenziellen Nachfolger aufs Beste ausbildet, nein, gemäß pythagoreischer Lehre muss bereits bei deren Zeugung darauf (...) geachtet werden, dass die Paarung ihrer idealen Eltern zu einem Zeitpunkt erfolgt, der auch vom kosmischen Lebensrhythmus des Universums her geeignet ist, der künftigen Elite das optimale Rüstzeug für ihre göttliche Aufgabe schon vorgeburtlich mitzugeben. Diese Geheimformel stellt ein faszinierendes Beispiel für das ganzheitliche Denken der alten Pythagoreer dar, in welchem heilige Mathematik, naturwissenschaftliche Himmelsmechanik und die schöpferische Liebesfunktion eines lebendigen, von naturgesetzlicher Harmonie und Weisheit erfüllten Universums noch eine untrennbare Einheit bilden. The so-called “wedding number” in Plato’s Republic is rightly considered one of the most important unsolved riddles of antiquity. This secret formula is a fascinating example of the holistic thought of the ancient Pythagoreans, in which divine mathematics, scientific celestial astronomy, and the creative function of love in a living universe full of naturally ordained harmony and wisdom still form an indivisible whole. (shrink)

Edward Feser defends the ‘Neo-Platonic proof ’ for the existence of the God of classical theism. After articulating the argument and a number of preliminaries, I first argue that premise three of Feser’s argument—the causal principle that every composite object requires a sustaining efficient cause to combine its parts—is both unjustified and dialectically ill-situated. I then argue that the Neo-Platonic proof fails to deliver the mindedness of the absolutely simple being and instead militates against its mindedness. Finally, I (...) uncover two tensions between Trinitarianism and the Neo-Platonic proof. (shrink)

This volume contains outstanding studies by some of the best scholars in ancient Greek Philosophy on key topics in Socratic, Platonic, and Aristotelian thought. These studies provide rigorous analyses of arguments and texts and often advance original interpretations. The essays in the volume range over a number of central themes in ancient philosophy, such as Socratic and Platonic conceptions of philosophical method; the Socratic paradoxes; Plato's view on justice; the nature of Platonic Forms, especially the Form of (...) the Good; Aristotle's views on the faculties of the soul; Aristotle's functionalist account of the human good; Socratic, Platonic, and Aristotelian views on the nature of desire and its object. The volume will be of interest to students and scholars of ancient philosophy and classics. (shrink)

La question au nombre irrationnel et de l'irrationnel mathématique en général, tient une part discrète dans l'oeuvre de Platon, mais elle est comme cette "pierre délaissée par les architectes" et qui est pourtant "la pierre angulaire". Elle concentre toutes les questions de l'être et du non-être, du possible et de l'impossible, du fini et de l'infini et ouvre la voie à la liberté pleine et entière de l'homme en quête de vérité. En elle, convergent pensée mathématique et spéculation philosophique, en (...) une harmonie riche de conséquences inestimables. C'est cette harmonie que révèle Imre Toth dans un essai brillant et rigoureux, le dernier qu'il ait écrit avant sa brusque disparition en mai 2010. "Les Editions de l'éclat viennent de contribuer à réparer ce qui apparaît comme une lacune de l'édition française en publiant deux textes récents d'Imre Toth" - Françoise Balibar (Critique). "Le livre de Toth est extrêmement séduisant, croisant la polémique politique la plus vive avec l'initiation limpide aux problèmes fondamentaux de la philosophie des mathématiques" - Jean-Paul Thomas (Le Monde). "Philosophes comme historiens des sciences pourront tirer profit de ce regard enflammé sur le processus mathématique, qui place la liberté au coeur de la création, mais va au-delà en cherchant à comprendre son fonctionnement, précisément en mathématique" - Jean-Michel Kantor (La Quinzaine littéraire). (shrink)

As prose dramatic texts Plato's dialogues would have been read by their original audience as an alternative type of theatrical composition. The 'paradox' of the dialogue form is explained by his appropriation of the discourse of theatre, the dominant public mode of communication of his time. The oral performance of his works is suggested both by the pragmatics of the publication of literary texts in the classical period and by his original role as a Sokratic dialogue-writer and the creator of (...) a fourth dramatic genre. Support comes from a number of pieces of evidence, from a statue of Sokrates in the Academy to a mosaic of Sokrates in Mytilene, which point to a centuries-old tradition of treating the dialogues in the context of performance literature and testify to the significance of the image of 'Plato the prose dramatist' for his original and subsequent audiences. (shrink)

Aristotle's syllogistic theory, as developed in his Prior Analytics, is often regarded as the birth of logic in Western philosophy. Over the past century, scholars have tried to identify important precursors to this theory. I argue that Platonic division, a method which aims to give accounts of essences of natural kinds by progressively narrowing down from a genus, influenced Aristotle's logical theory in a number of crucial respects. To see exactly how, I analyze the method of division as it (...) was originally conceived by Plato and received by Aristotle. I argue that, while Plato allowed that some divisions fail to rigorously investigate the essence, he began a program continued by Aristotle (and others in antiquity and the middle ages) of seeking norms for division that would apply in any domain whatsoever. This idea of a rigorous, general method was taken up and developed by Aristotle in his syllogistic. Aristotle also used Plato's conception of predication as parthood in his semantics for syllogistic propositions. As part of my argument, I prove that a semantics based on Platonic divisional structures is sound and complete for the deduction system used in the literature to model Aristotle's syllogistic. (shrink)

This book discusses how Plato, one the fiercest legal critics in ancient Greece, became – in the longue durée – its most influential legislator. Making use of a vast scholarly literature, and offering original readings of a number of dialogues, it argues that the need for legal critique and the desire for legal permanence set the long arc of Plato’s corpus—from the Apology to the Laws. Modern philosophers and legal historians have tended to overlook the fact that Plato was the (...) most prolific legislator in ancient Greece. In the pages of his Republic and Laws, he drafted more than 700 statutes. This is more legal material than can be credited to the archetypal Greek legislators—Lycurgus, Draco, and Solon. The status of Plato’s laws is unique, since he composed them for purely hypothetical cities. And remarkably, he introduced this new genre by writing hard-hitting critiques of the Greek ideal of the sovereignty of law. Writing in the milieu in which immutable divine law vied for the first time with volatile democratic law, Plato rejected both sources of law, and sought to derive his laws from what he called ‘political technique’. At the core of this technique is the question of how the idea of justice relates to legal and institutional change. Filled with sharp observations and bold claims, Platonic Legislations shows that it is possible to see Plato—and our own legal culture—in a new light “In this provocative, intelligent, and elegant work D. L. Dusenbury has posed crucial questions not only as regards Plato’s thought in the making, but also as regards our contemporaneity.”—Giorgio Camassa, University of Udine “There is a tension in Greek law, and in Greek legal thinking, between an understanding of law as unchangeable and authoritative, and a recognition that formal rules are often insufficient for the interpretation of reality, and need to be constantly revised to match it. Dusenbury’s book illuminates the sophistication of Plato’s legal thought in its engagement with this tension, and explores the potential of Plato’s reflection for modern legal theory.”—Mirko Canevaro, The University of Edinburgh. (shrink)

The complete, or almost complete, absence from the Theaetetus of any unequivocal reference to Platonic Forms is a problem, the solution of which appeared to many scholars to have been found and convincingly presented in the late Professor Gornford's book Plato's Theory of Knowledge, published in 1935. Put briefly, his contention was that the main purpose of the dialogue is to show that no acceptable definition of knowledge can be reached if the Forms are left out of account, that (...) there are a number of passages in which any reader of the middle dialogues would see that they were deliberately being left out of account, and one or two, namely in the famous digression on the philosophic life, and in the discussion of ‘common terms’ , where they are actually brought in, though not indeed under the name of. (shrink)

In this paper, I defend traditional Platonic mathematical realism from its contemporary detractors, arguing that numbers, understood as abstract, non-physical objects of rational intuition, are indispensable for the act of counting.

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Literature 26.1 (2002) 84-106 [Access article in PDF] Irony in the Platonic Dialogues Charles L. Griswold, Jr. I INTERPRETERS OF PLATO have arrived at a general consensus to the effect that there exists a problem of interpretation when we read Plato, and that the solution to the problem must in some way incorporate what has tendentiously been called the "literary" and the "philosophical" sides of Plato's (...) writing. The problem is created by the fact that Plato wrote in dialogue form, indeed a specific type of dialogue form. The solution must somehow combine into a coherent theory of the "meaning" of the dialogues, the ways in which these texts work (such as the use of imagery, metaphor, myth, allusion, irony, as well as argument). At the same time, there exists enormous disagreement about how one ought to move from these general observations to interpretation of a particular text. 1The problem of interpretation lies not merely in the fact that Plato wrote dialogues. That alone would not necessarily present any hermeneutical issues of unusual difficulty. Plato's distinctive use of the dialogue form creates the difficulties. The genre that we might call "the Platonic dialogue" is distinguished by several relevant characteristics.First, there is no character called "Plato" who speaks in any of the dialogues; indeed, "Plato" is mentioned twice in the entire corpus, once as being absent (Pho. 59b10), and once as being present (Apol. 38b6). Authorial anonymity is thus an important feature of the dialogues. 2 At least ab initio, we are not justified in identifying Plato with any one of his characters. Indeed, there are positive reasons why he cannot be identified even with Socrates, as I will discuss below. 3 Plato is absent from his own texts; no simple act of reading them will allow us to ascertain what his views are. 4 It does not follow that Plato's views are [End Page 84] entirely absent from the dialogues—say, in the form of some mysterious "esoteric teaching"—and that they cannot be elicited by a complex act of reading. Just how Plato's views are to be extracted has been, again, the subject of a very long debate. 5 I myself think that Plato's views can be so elicited. 6 But the fact of his anonymity as author means that "Plato's meaning" is not ascertainable in the way that, say, "Kant's meaning" may be ascertainable in the Critique of Pure Reason.A second feature of Plato's dialogues contributes to the interpretive puzzle. The dialogues are clearly fictional in character. They are not—they could not be—transcriptions of conversations which took place. Some dialogues could not have taken place even when the interlocutors were historical figures (for example, at the Phaedrus' ostensible dramatic date the real Phaedrus was not in Athens). Not only are many of the characters entirely fictional, there is clear evidence that even "Socrates" is very much a fictionalized version of the historical character. I am referring not just to the famous statement in the (possibly inauthentic) Second Letter, 7 but to the fact that the Socrates of Plato's dialogues is presented as being a super-human character. This is evident from Alcibiades' description of him in the Symposium as well as from a number of subtle fictions. For example, Plato has Socrates narrate the entirety of the Republic from memory, without a single moment of hesitation or (so far as the reader can tell) any lapse of memory. This fantastic feat is complemented by Socrates' equally fantastic stamina in conversation. The lengthy Protagoras, for example, is narrated in its entirety by Socrates immediately after the dialogue itself has occurred. Socrates never shows any fatigue of any sort. It is not just hyperbole to say that Plato's Socrates is heroic; he is carefully compared, in passages I cannot examine here, to both Achilles and Odysseus. Socrates' absolute calm and self-control before death also evidence his superhuman character. The relation between the historical Socrates and the Platonic Socrates is the source of much fruitful controversy. It suffices for... (shrink)

As the youngest work in the Platonic corpus, the Axiochus interacts with other texts in the corpus as well as with its contemporary philosophical milieu. How it does so, however, and what the purpose of the work is, is still unclear. This paper proposes a new theoretical approach to this text, arguing that the Axiochus anchors a number of innovations. It discusses three innovations in particular: the introduction of philosophical therapy in Platonism, the use of Epicurean arguments in Academic (...) philosophy, and a renovated Platonism on the contemporary philosophical scene. The Axiochus aims, so this paper argues, to make these innovations acceptable to different audiences by anchoring them in the Socratic dialogue and the therapeutic paradigm of philosophy, respectively. (shrink)

In 2020, Daniele Molinini published a paper outlining two types of mathematical objectivity. One could say that with this paper Molinini not only separated two mathematical concepts in terms of terminology and content, but also contrasted two mathematical-philosophical contexts, the traditional-idealistic and the modern-practical. Since the first context was the theoretical basis for a large number of analyses that we find in the framework of the philosophy of mathematics, the space was now offered to re-examine such analyses in the second (...) context. More specifically, with respect to the second context, we will analyse the strength of one of the few explicit Platonic arguments that seeks to justify the ontological status of mathematical objects and the central claim of mathematical Platonism about the existence of mathematical objects. (shrink)

Author: Ziemiński Ireneusz Title: PHILOSOPHY AS MEDITATIO MORTIS (PLATO AND MONTAIGNE) (Filozofia jako meditatio mortis (Platon – Montaigne)) Source: Filo-Sofija year: 2006, vol:.6, number: 2006/1, pages: 43-58 Keywords: MEDITATIO MORTIS, DEATH, PLATO, MONTAIGNE Discipline: PHILOSOPHY Language: POLISH Document type: ARTICLE Publication order reference (Primary author’s office address): E-mail: www:The idea of philosophy as meditatio mortis is illustrated with the examples of Plato’s and Montaigne’s views. According to Plato, life within body is a kind of evil, and death is the way (...) of release and return to divine life. Philosophy regarded as seeking the truth is at the same time an exercise in dying because it consists in taking off reason from body and senses. Philosophy as meditatio mortis is then preparing to true and eternal life. According to Montaigne, death is the necessary end of human existence, which we should accept without reservation and fear. Philosophy lies in preparing to death as a natural biological event, common to all living creatures. Human dying should be liberated from all ceremonies and cultural rituals because they are the main reasons of our fear and prevent us from accepting death as a natural event. (shrink)

Some people think that numbers and other mathematical entities exist. They believe in a platonic heaven of ideal mathematical objects, as some people like to put it. This may seem a very strange thing to believe in: after all, we cannot see numbers, nor touch them, nor smell them. So why should one believe that they exist? Because, as Putnam and Quine used to say, numbers are indispensable to science: it seems almost impossible to state our best scientific theories (...) without mentioning numbers or other mathematical objects. (shrink)

This paper proposes a new interpretation of Gadamer’s problematic appropriation of Platonic metaphysics. It argues that Gadamer, attempting to respond to the challenge posed by Heidegger’s interpretation of Platonic metaphysics and of its role in the history of Being (Seinsgeschichte), downplayed the transcendence of Platonic Forms. Gadamer achieves a reconfiguration of this transcendence and its transposition into what I call here a plane of immanence through two hermeneutic gestures: 1) interpreting Forms in light of Greek mathematics and (...) especially in light of the structure of the sum-number; 2) introducing temporality and historicity in the Medieval doctrine of transcendentals by giving priority to the Beautiful over the Good. I contend that, Platonically speaking, this amounts to the rejection of νοήσις in favor of διάνοια, and that this raises issues concerning the problem of finitude and the potential limits of linguisticality (Sprachlichkeit). (shrink)

Ancient Greek Philosophy routinely relied upon concepts of number to explain the tangible order of the universe. Plotinus' contribution to this tradition, however, has been often omitted, if not ignored. The main reason for this, at first glance, is the Plotinus does not treat the subject of number in the Enneads as pervasively as the Neopythagoreans or even his own successors Lamblichus, Syrianus, and Proclus. Nevertheless, a close examination of the Enneads reveals that Plotinus systematically discusses number in relation to (...) each of his underlying principles of existence--the One, Intellect, and Soul. Plotinus on Number offers the first comprehensive analysis of Plotinus' concept of number, beginning with its origins in Plato and the Neopythagoreans and ending with its influence on Porphyry's arrangement of the Enneads. It's main argument is that Plotinus adapts Plato's and the Neopythagoreans' cosmology to place number in the foundation of the intelligible realm and in the construction of the universe. Through Plotinus' defense of Plato's Ideal Numbers from Aristotle's criticism, Svetla Slaveva-Griffin reveals the founder of Neoplatonism as the first post-Platonic philosopher who purposefully and systematically develops what we may call a theory of number, distinguishing between number in the intelligible realm and number in the quantitative, mathematical realm. Finally, the book draws attention to Plotinus' concept as a necesscary and fundamental linke between Platonic and late Neoplatonic schools of philosophy. (shrink)

It is argued that Plato views forms as the proper objects of mathematical research, in contrast to what Aristotle says about the ontologically intermediate state of math?matiká in Platonism. Plato’s particularistic conception of ideas is compared with the nowadays customary mathematical practice of studying types of structures by examining canonical representatives. The case is illustrated by considering the shift from a universalistic conception of natural numbers, in the Frege-Russell-tradition, to a particularistic conception, as in von Neumann. Finally, the characterization of (...) dialectical thinking ventured by Plato at the end of Politeia VI is related, by evaluating Gödel’s incompleteness result, to the Gödelian model of what can be recognized as a partial transcending of the limits of axiomatic systems. (shrink)

Resumen Platón proyecta en la República un programa cultural que supone la redefinición del papel de la poesía tradicional en razón de su asociación con los regímenes democrático y tiránico. Esto, según pretendo mostrar, puede vincularse de manera legítima con la polémica anticirenaica de Platón contra Aristipo. Para ello, por un lado, exploraré los rasgos del biotipo tiránico y su régimen concomitante en la República VIII-IX y, por otro, analizaré sus vínculos con los planteamientos anticirenaicos en el Gorgias. Este examen (...) permitirá ofrecer claves de análisis para dos pasajes muy discutidos de la República IX: el símil del ascensor y el número del tirano, que cobran una nueva luz a partir de la consideración de esta polémica intrasocrática.Plato put forward in Republic a cultural program which implies the redefinition of the role of traditional poetry due to its association with the democratic and tyrannical regimes. I intend to show that this point is connected with the anti-cyrenaic arguments that Plato raises against Aristippus. On the one hand, I will study the features of the tyrannical biotype and its correspondent regime in Republic VIII-IX, and, on the other, I will analyze its links with the anti-cyrenaic arguments in Gorgias. This exam will allow us to offer some clues to better understand two widely debated passages in Republic IX: the analogy of the lift and the tyrant’s number. Both acquire a new perspective when this intra-socratic polemic is taken into account. (shrink)

From the 1st century BC onwards followers of Plato began to systematize Plato's thought. These attempts went in various directions and were subjected to all kinds of philosophical influences, especially Aristotelian, Stoic, and Pythagorean. The result was a broad variety of Platonisms without orthodoxy. That would only change with Plotinus. This volume, being the fruit of the collaboration among leading scholars in the field, addresses a number of aspects of this period of system building with substantial contributions on Antiochus and (...) Alcinous and their relation to Stoicism; on Pythagoreanising tendencies in Platonism; on Eudorus and the tradition of commentaries on Aristotle's Categories; on the creationism of the Jewish Platonist Philo of Alexandria; on Ammonius, the Egyptian teacher of Plutarch; on Plutarch's discussion of Socrates' guardian spirit. The contributions are in English, French, Italian and German. (shrink)

Who is Socrates? While most readers know him as the central figure in Plato's work, he is hard to characterize. In this book, S. Montgomery Ewegen opens this long-standing and difficult question once again. Reading Socrates against a number of Platonic texts, Ewegen sets out to understand the way of Socrates. Taking on the nuances and contours of the Socrates that emerges from the dramatic and philosophical contexts of Plato's works, Ewegen considers questions of withdrawal, retreat, powerlessness, poverty, concealment, (...) and release and how they construct a new view of Socrates. For Ewegen, Socrates is a powerful but strange and uncanny figure. Ewegen's withdrawn Socrates forever evades rigid interpretation and must instead remain a deep and insoluble question. (shrink)

Why is the number seven lucky--even holy--in almost every culture? Why do we speak of the four corners of the earth? Why do cats have nine lives? From literature to folklore to private superstitions, numbers play a conspicuous role in our daily lives. But in this fascinating book, Annemarie Schimmel shows that numbers have been filled with mystery and meaning since the earliest times, and across every society. In The Mystery of Numbers Annemarie Schimmel conducts an illuminating tour of the (...) mysteries attributed to numbers over the centuries. She begins with an informative and often surprising introduction to the origins of number systems: pre-Roman Europeans, for example, may have had one based on twenty, not ten, while the Mayans had a system more sophisticated than our own. Schimmel also reveals how our fascination with numbers has led to a rich cross-fertilization of mathematical knowledge: "Arabic" numerals, for instance, were picked up by Europe from the Arabs, who had earlier adopted them from Indian sources. But the heart of the book is an engrossing guide to the symbolism of numbers. Number symbolism, she shows, has deep roots in Western culture, from the philosophy of the Pythagoreans and Platonists, to the religious mysticism of the Cabala and the Islamic Brethren of Purity, to Kepler's belief that the laws of planetary motion should be mathematically elegant, to the unlucky thirteen. After exploring the sources of number symbolism, Schimmel examines individual numbers ranging from one to ten thousand, discussing the meanings they have had for Judaic, Christian, and Islamic traditions, with examples from Indian, Chinese, and Native American cultures as well. Two, for instance, has widely been seen as a number of contradiction and polarity, a number of discord and antithesis. And six, according to ancient and neo-platonic thinking, is the most perfect number because it is both the sum and the product of its parts. Using examples ranging from the Bible to the Mayans to Shakespeare, she shows how numbers have been considered feminine and masculine, holy and evil, lucky and unlucky. A highly respected scholar of Islamic culture, Annemarie Schimmel draws on her vast knowledge to paint a rich, cross-cultural portrait of the many meanings of numbers. Engaging and accessible, her account uncovers the roots of a phenomenon we all feel every Friday the thirteenth. (shrink)

The aim of this paper is to show that John Wyclif’s theory of space is at once an interpretation of the Platonic theory of place and a Neopythagorean conception of magnitudes and numbers. The result is an original form of mathematical atomism in which atoms are point-like entities with a particular situation in space. If the core of this view comes from Boethius’ De arithmetica, John Wyclif is also influenced by Robert Grosseteste’s metaphysics, which includes the Boethian number theory (...) within the Christian tale of the creation of the world ex nihilo. John Wyclif, however, adds some novelty to this theory concerning the epistemological status of this hypothetical description of the creation of the world out of atoms. First, according to Wyclif, whereas geometry is concerned with sensible and imaginable beings, arithmetic, which is purely intellectual, has access to the deep mathematical structure of the universe. He then suggests a subordination of geometry under arithmetic, which he considers the most solid basis for his metaphysics. As a result, with the attribution of numbers and units to every level of reality, it becomes possible to reform our natural imagination, so that it can imagine the atomic structure of matter and space. (shrink)

The study of Plato's dialogues has traditionally oscillated between two paradigms: one that portrays the dialogues as treatises expounding doctrines and one that sees them as purely skeptical, rhetorical, or literary. This collection of new essays by twelve noted Plato scholars illustrates the fruitfulness of breaking away from those paradigms, which have divided Platonic scholarship and led it to a number of dead ends. While the essays are diverse in their approaches, each seeks to find a 'third way' to (...) understand Plato, reading him as neither a dogmatist nor a skeptic but as a philosopher capable of reconciling the content and form of his writings. (shrink)

At first russell thought (p) that whatever a proposition is about must be a constituent of it. Then, Around 1900, He discovered denoting concepts and realized that a proposition could be about something and have only its denoting concept as constituent. However, A number of remarks that he made through the years can only be understood as inspired by (p). In particular, The arguments offered in "on denoting" against the doctrine of denotation of "principles" are grounded on (p).

In one of the most famous but equally obscure passages in the Timaeus, Plato describes the generation of time and the heavens. The “moving image of eternity” (37d5) is commonly read as Plato’s most general characterisation of time. Rémi Brague famously challenged the traditional interpretation on linguistic grounds by claiming that Plato actually did not conceive of time as an image (εἰκών) but rather as a number (ἀριθμός). In this paper, I shall claim that this controversy is by no means (...) a modern one. The traditional interpretation is mostly owed to Plato’s most prominent reader, Plotinus, who famously conceives of time in relation to eternity (Enn. III.7.13.24-25). Brague’s alternative reading, however, is anticipated by Simplicius’ attempt to refute the Plotinian interpretation, as I shall show. According to my reconstruction, Simplicius’ reading of the Timaeus not only shows why the traditional interpretation falls short, but it also offers a systematic argument that bolsters Brague’s alternative reading. Finally, I shall show that this is consistent with Plato’s text. It shall become clear that current interpretative problems are essentially prefigured in the late ancient debate. (shrink)

One of this century's most important philosophers here focuses on Plato's Protagoras, Phaedo, Republic, and Philebus and on Aristotle's three moral treatises to show the essential continuity of Platonic and Aristotelian reflection on the nature of the good. "Well translated and usefully annotated by P. Christopher Smith.... Gadamer's book exhibits a broad and grand vision as well as a great love for the Greek thinkers."--Alexander Nehemas, New York Times Book Review "The translation is highly readable. The translator's introduction and (...) frequent annotation provide special elucidation on points of doctrinal complexity, giving ample references to other works and rival interpretations."--Choice "This book is an important addition to the steadily growing number of Gadamer's works available in English. In it, we see Gadamer at his best, that is, engaged in the practice of interpreting important texts from the philosophical tradition, and also at his most controversial.... I enthusiastically recommend this...challenging book as one that rewards all efforts to understand the important claims it makes on its readers."--Francis J. Ambrosio, International Philosophy Quarterly Hans-Georg Gadamer is professor of philosophy emeritus at the University of Heidelberg. He is the author of numerous books, including two others translated by Smith: Dialogue and Dialectic: Eight Hermeneutical Studies on Plato and Hegel's Dialectic: Five Hermeneutical Studies. (shrink)

The paper deals with the ontological status of number. The authors are convinced that it is useful to discuss the concept of number within the framework of the Aristotelian division of being into substance and accident. Number can thus be taken as ens in se or ens in alio. Aristotle and Thomas Aquinas believed that number is an accident and their concept is explained in the first part of the paper. In the second part it is shown that the Aristotelian (...) concept is not correct. However, if number is not an accident then it seems that it must be identified with a Platonic entity. In the third part the authors reject this Platonic conclusion that Frege seems to have defended. In the final part, a possible solution is shown: From the logical point of view number is an object but from the ontological point of view it is an entity that depends on linguistic structure. (shrink)

On the Cratylus dialogue, much has been written. However, is it possible to say that the beauty and the intelligence of that Platonic dialogue has been fully explained and justified? First of all, the Cratylus continues to be considered a work that is not fully successful artistically: its very long etimological review has yet to find an explanation that justifies, from a dramatic point of view, such an excessively prolix section. Secondly, the Cratylus is a dialogue that is doctrinally (...) empty. Plato seems to have dedicated an entire work to the issue of the theory of names, only to demonstrate in the end that considering such a question is not in fact useful to the understanding of the truth; to understand the truth, one must look not to language but to the things themselves. This article, other than indicating the status questionis regarding the Cratylus, suggests a new approach which will resolve both the issues raised above. It intends to demonstrate that to formulate an interpretation worthy of Plato from both the speculative and dramatic points of view, one must undertake an ontological and mystagogical reading of the Cratylus, an approach not yet made. The ontological reading complements the anti-ideological interpretations already developed by a number of scholars; the mystagogical reading is explicitly alternative to the esoteric interpretation of the dialogue proposed by Szlezák, the authoritative representative of the Tubingen School. (shrink)

In this thesis, I examine the perceptibility of the Platonic Ideas in the thought of Arthur Schopenhauer. The work is divided into four chapters, each focusing and building upon a specific aspect related to this question. The first chapter (“"Plato and the Primacy of Intellect"”) deals with Schopenhauer’s interpretation specific to Platonic thought. I there address the question of why it is that Schopenhauer should consider Plato to have interpreted the Ideas as 'perceptible', particularly in view of evidence (...) which seems to testify to the contrary. Does Schopenhauer misinterpret Plato, or are there sufficient grounds for considering his interpretation consistent? This is important in light of Schopenhauer’s reinterpretation of the Platonic Idea on the basis of the primacy of the will in contradistinction to Plato’'s own emphasis upon the intelligible. In effect, for Schopenhauer, Plato confuses the nature of imperceptible concepts with the more perceptually grounded Ideas. In the second chapter (“"On the Direct Path to Knowledge"”), I explore Schopenhauer’'s methodological approach on the basis of the will. I there discuss the difference between Schopenhauer and Kant with respect to Transcendental Idealism, particularly in terms of their separate analysis and derivation of the thing-in-itself. From there, I turn to a consideration of Schopenhauer'’s discussion of the nature of formal and empirical intuitions, of abstraction and the nature of universal concepts, and finally of the intuitively grounded nature of mathematical demonstration which serves as an analogy for the manner in which Schopenhauer deals with knowledge of the Ideas. In the third chapter ("“The Perceptibility of the Ideas”"), I discuss the manner in which the Ideas become accessible as the manifestation of the Will, arising through the representation of perception. There a number of distinctions are made between Ideas and concepts, as well as the nature of contemplation through Genius. The Ideas are found to be perceptible as aesthetic intuitions which relate not only to the in-itself nature inherent to phenomena, but serve also as the essential basis and foundation for all genuine forms of art. In the fourth and final chapter (“"Critical Discussion of Schopenhauer’s Ideas"”), I consider and outline a number of potential problems with respect to Schopenhauer’s analysis. In particular, Schopenhauer encounters difficulties on the basis of the simultaneous immanence and transcendence of the Ideas; of the knowledge of the Ideas characterized as intuitive while yet having a strangely abstract basis; of the unity of the will in face of the plurality of Ideas and phenomena; and finally of the annihilation of the will which creates essential problems in terms of the desideratum of knowledge itself. I here also attempt to offer various solutions to these problems where relevant. The conclusion which I arrive at in this work is that any radical reorientation of ontology must bear an essential effect upon the character and methodology of knowledge itself. In other words, Schopenhauer’s interpretation of the perceptibility of the Ideas is a direct consequence of hi s analysis of the will as the metaphysical thing-in-itself. In fact, this essential change colors his entire epistemological approach, from science and mathematics, to logic and the Ideas—from the ground the up. (shrink)

One of this century’s most important philosophers here focuses on Plato’s _Protagoras, Phaedo, Republic, _and _Philebus_ and on Aristotle’s three moral treatises to show the essential continuity of Platonic and Aristotelian reflection on the nature of the good. “Well translated and usefully annotated by P. Christopher Smith…. Gadamer’s book exhibits a broad and grand vision as well as a great love for the Greek thinkers.”—Alexander Nehemas, _New York__ Times Book Review_ “The translation is highly readable. The translator’s introduction and (...) frequent annotation provide special elucidation on points of doctrinal complexity, giving ample references to other works and rival interpretations.”—_Choice_ “This book is an important addition to the steadily growing number of Gadamer’s works available in English. In it, we see Gadamer at his best, that is, engaged in the practice of interpreting important texts from the philosophical tradition, and also at his most controversial…. I enthusiastically recommend this…challenging book as one that rewards all efforts to understand the important claims it makes on its readers.”—Francis J. Ambrosio, _International Philosophy Quarterly_ Hans-Georg Gadamer is professor of philosophy emeritus at the University of Heidelberg. He is the author of numerous books, including two others translated by Smith: _Dialogue and Dialectic: Eight Hermeneutical Studies on Plato _and _Hegel’s Dialectic: Five Hermeneutical Studies._. (shrink)

‘Is being one only one? – The Argument for the Uniqueness of Platonic Forms’ Abstract: Each Form is unique in number; no two numerically distinct Forms can share the same nature. Plato argues for this claim in Republic X. I identify the metaphysical principles Plato presupposes in the premises of the argument, by examining the reasoning behind them, and offer a reconstruction of the argument showing the principles in use. I argue that the metaphysical significance of the argument’s conclusion (...) is to establish that if a Form F were not unique, if there were many Forms F, their nature would alter along with their number: a Form cannot recur without change in its constitution. This is why there can be only one Form for each character in the world. (shrink)

One of this century’s most important philosophers here focuses on Plato’s Protagoras, Phaedo, Republic, and Philebus and on Aristotle’s three moral treatises to show the essential continuity of Platonic and Aristotelian reflection on the nature of the good.“Well translated and usefully annotated by P. Christopher Smith.... Gadamer’s book exhibits a broad and grand vision as well as a great love for the Greek thinkers.”-Alexander Nehemas, New York Times Book Review“The translation is highly readable. The translator’s introduction and frequent annotation (...) provide special elucidation on points of doctrinal complexity, giving ample references to other works and rival interpretations.”-Choice“This book is an important addition to the steadily growing number of Gadamer’s works available in English. In it, we see Gadamer at his best, that is, engaged in the practice of interpreting important texts from the philosophical tradition, and also at his most controversial.... I enthusiastically recommend this...challenging book as one that rewards all efforts to understand the important claims it makes on its readers.”-Francis J. Ambrosio, International Philosophy QuarterlyHans-Georg Gadamer is professor of philosophy emeritus at the University of Heidelberg. He is the author of numerous books, including two others translated by Smith: Dialogue and Dialectic: Eight Hermeneutical Studies on Plato and Hegel’s Dialectic: Five Hermeneutical Studies. (shrink)

The main conclusion of Nigel Hiscock's important but ill‐arranged book is that the ground plans of abbeys and cathedrals of the tenth and eleventh centuries incorporate Platonic wisdom—hence the “wise” in the title catchwords, which come from Paul's first letter to the Corinthians . There Paul likens himself to a sapiens architectus who lays the foundations on which others erect the building. In three of the four translations in The Complete Parallel Bible, however, Paul does not declare himself wise (...) but, rather, “skilled” or “trained.” The discrepancy makes an emblem for Hiscock's investigation: Did the architects deliberately express geometrical ratios significant in Platonic philosophy in their designs or did they proceed by applying their practical skill and training to the lie of the land and the resources available? A subsidiary question is also symbolized in Paul's verse: Was the designer a wise cleric who set the proportions and left the heavy work to someone else?Hiscock builds his answers in two ways. For one, he delivers a chrestomathy of Platonic snippets known in the tenth century and a few telling quotations from wise clerics—the latter concern the expression of significant numbers in the church fabric: a baptistery is octagonal because of an association between creation and the number eight, and a church with a half‐dozen altars is complete because of the perfection of the number six. But, Hiscock emphasizes, numerology does not account for floor plans, and to test the wisdom of the designers he scrutinizes plans of two dozen churches, especially those associated with monastic reformers. He expresses his results in color‐coded lines superposed on the plans, some one hundred of which comprise a valuable appendix.An example will indicate the interest of the findings and the riskiness of the business. The crossing of the transept at St. Michael's Abbey, Hildesheim, makes a square ABCD, proceeding clockwise from A in the northeast corner. AD and BC prolonged give the line of nave piers, AB and DC prolonged that of the arms of the transept. Lines drawn from B at angles of 54° and 60° with BC cut CD extended in E and F, respectively, locating the end piers in the side aisles. Now 60°, as the invariable angle of a three‐sided figure all of whose parts are equal, carries a cornucopia of Platonic and Christian symbolism. Also 54°, because derivable from a pentagon, has the pregnant significance of the figure five, which is the sum of the first male and first female numbers. Repetition of the transept square fixes the nave piers. Then lines between piers, pillars, and pilasters make the special angles with the north‐south and east‐west lines of the grid all over the plan.How much of this construction is mere coincidence? Requiring the intervals between piers in the same line to be equal, we have DE = FE = FD/2. And so it is: / = 0.51. Otherwise put, if angle FBC is 60° and E divides FD equally, angle EBC must be 54° without reference to a pentagon. Against such arguments, Hiscock observes that his analysis has uncovered previously unknown relations between churches confirmed by documents and that his technique, when applied to large nineteenth‐century buildings with repeated bays such as the Crystal Palace, picks out relatively few structurally significant elements.Hiscock is senior lecturer on the design, theory, and history of architecture at Oxford Brookes University and a master craftsman. His scrupulously prepared color‐coded drawings contain much of interest and pleasure for a geometer. Has he definitively answered the old vexed question whether the sapiens architectus intended to express ancient wisdom in his plans? If the architect thus proposed, did his builder faithfully dispose? Paul again may give guidance. “The Lord knows the thoughts of the wise, that they are futile”. (shrink)