The Socratic maieutic or obstetric method of philosophical didactic, the Socratic method of refutation (ἔλεγχος), along with Plato’s methods of Hypothesis, dialectic and division were seminal achievements of theoretical philosophy. The Platonic approach focuses on the priority of knowledge over presumption (δόξα) or phenomenon. Plato explained the relation between things and Ideas (Forms) with the notion of ‘participation’ (μέθεξις). The Forms exist separately (χωριστά) from all the particulars and each Form is the standard particular of the kind in question or the model (παράδειγμα) to which other particulars approximate. Valid inference, according to Plato, is necessary connexion that holds between Forms (εἴδη).
The requirements for indisputability, argumentation and development of criteria had shaped the starting points of the pre-Socratic philosophy of Diogenes of Apollonia and Parmenides. By demanding consistency, philosophers started reflecting on methodology, while the word method emerged firstly in Plato and Aristotle. Plato formulated his style influenced by the Socratic maieutic or obstetric method of didactic. The Platonic philosophy distinguished disputes and rivalries that perforate through scientific research and ethics. The emphasis was given on critique, disagreement, alteration, discontinuity; in other words, upon defining substances by differentiating. At the same framework, apart from the Socratic method of refutation (elenchus), Plato introduced the method of Hypothesis to acquire the knowledge of an answer to a specific question, when no one who already possesses that knowledge is to be found.1
The Platonic method was basically the dialectic, while Aristotle introduced the theories of argumentation and predication. Definitions, statements, dialectics and predications were gradually developed in the history of philosophy, especially after the outstanding intervention of Socrates, as Aristotle holds. The introduction of the Ideas was a direct consequence of Plato’s inquiry on the topic of definitions. The Ideas or Forms are the causes of all things, while the One is the cause of the Forms, according to Plato. Although Pythagoreans conjectured that the relation between numbers and things bears on imitation, Plato explained the relation between things and Ideas with the notion of ‘participation’ (μέθεξις). Plato placed mathematical objects in the middle space between Ideas and things.
Gilles Deleuze suggested that an overthrowing of Platonism originated with a Nietzschean conception of crisis in philosophy, which had been proposed as “the twilight of the idols” that would turn into stressing the differences rather than resemblance. According to Deleuze, the deeper meaning of the platonic method of division is the conscious pursue of dispute and rivalry, in order to discover the ideal and authentic forms. The Phaedrus, the Statesman and the Sophist are the most significant dialogues that exemplify the method of division. For instance, the division between icon and simulacra is explained in Sophist (236b, 264c), based on the internal and spiritual correspondence of the icon to the Idea.
The method of division is for Plato an important research tool, useful to clarify our understanding of the good (ἀγαθόν), which is the prominent subject of platonic philosophy. However, the original meaning of goodness cannot be grounded as a pharmakon (φάρμακον) of incompleteness or remedy of the lack of something. Plato suggests in Lysis that the exemplification of the idea of the Good in the relation of friendship is founded on the notion of familiarity (οἰκεῖον), which pertains to the household, to the οἶκος.2
Under the same notion of intimate sameness, habitual and unexceptional familiarity, Aristotle defines in the Categories the symmetric relations, like friendship, contrary to the asymmetric ones, which cannot interchange their terms reciprocally. It is essential to his argument that the movement of a ship is violent, since it can be manipulated by a rudder or with other external, alternate means. Therefore, the relation of a rudder to a ship is not an intimate, symmetric, familiar one. Aristotle, in fact, followed Plato in this thematic of monadic properties.
Plato introduces his theory of plural-partaking in Forms in his early dialogue Hippias Major. According to this theory, individuals acquire relational qualifications either by sharing a monadic property instance in symmetric cases, or a pair of property instances in asymmetric cases, as Scaltsas (2016) supports. Only the symmetric cases are qualified by Forms, because the nature of the Forms is monoeidic, uniform.
Then they [the fine things] have some thing that itself makes them be fine, that common thing [i.e. the Form of Fine] that belongs to both of them in common, and to each privately. Because I don’t suppose there’s any other way they would both and each be fine (300a9–b2).
Socrates confronts the sophistic argument of Hippias, by distinguishing between plural and distributive predication. A telling example that Socrates gives is that two colours taken separately may be attractive, but juxtaposed together unattractive. If ‘that attribute adheres in both, but not in each … then that’s not what makes each of them… ; it doesn’t adhere in each’ (302e5–10). Plato did not introduce relational Forms, but only plural predication through monadic Forms and in forms of Opposites, as Scaltsas emphasises.
Plato seems to hold that a sentence is true if the arrangement of its parts reflects or corresponds to a connexion between Forms. Discourse is possible only on the supposition that there is connexion between Forms (διὰ γὰρ τὴν ἀλλήλων τῶν εἰδῶν συμπλοκὴν ὁ λόγος γέγονεν ἡμῖν, Sophist, 259E) and true discourse speaks of realities as they are (λέγει τὰ ὄντα ὡς ἔστι, op. cit. 263B). Valid inference, according to Plato, is neither the tracing of necessary connexions between sentences, nor between judgements. According to him necessary connexion holds between Forms (εἴδη). Forms correspond in part at least to what later philosophers have called ‘universals’. The Forms exist separately (χωριστά) from all the particulars and each Form is the standard particular of the kind in question or the model (παράδειγμα) to which other particulars approximate.
Observational evidence was subservient to intellectual illumination in Platonic philosophy, since pure knowledge was considered as superior to the disrupted and inattentive perceptions of the senses. This contradiction between evidence and reason was steadily effective from Democritus to Plotinus and from Porphyry to the Nominalists. On the contrary, the Christian tradition from Augustine to Aquinas defended the correspondence between evidence and reason. Although Platonism opened the way to philosophical and religious innovations, it did not allow for any priority to empirical evidence rather than reason, but simply argued that the cognitive subject is able to stand in an appropriate position for the acquisition of evidence.
The cognitive subject could find and assess evidence through natural dialectical investigation. Dialectic was introduced by Socrates and developed by his student Plato, who wrote dialogues that hung around the middle-ground between poetry and prose. Diogenes Laertius (1921a) stressed that Plato’s dialectic is the art of discussion, through two main modes of presentation, the instruction and the investigation. The instruction is subdivided to theoretical (either Physics or Logic) and practical (either Ethics or Politics). The investigation is also subdivided to rehearsing (either thought aiming or checking out) and quarrelling (either demonstrating or contradicting).
The empirical-nominalist turn of anti-Platonism
In the Platonic dialogue Timaios the universe was conceived as created by the Demiurge, according to two overarching principles: reason and necessity.3 The philosophers, thereafter, arguing on cause, chance, and necessity, admitting or rejecting divine providence and fate, will produce many different models of causality, wavering from free action to determinism under different combinations of modal premises. Aristotle was one of the first empiricists that criticised Platonism. Their differences get clear in the field of kinematics. The prime origin of any kind of motion is, according to Plato, movement per se, spontaneous and self-moving, identified as life. Without any external cause, self-movement is indestructible and perpetual. It excludes any distinction between mover and driven, between action and passion. Self-movement is purely spiritual, out of space and senses. Every imparted movement depends on self-motion, as Plato believed, and is ordered less or more under the perfection of its participation to the sensible world of the ideas through spiritual motion. The soul leading the spirit to the contemplation of the form is the perfect representation of self-motion. Its least imperfect image would be axial rotation:
If we described them both as moving regularly and uniformly in the same spot, round the same things and in relation to the same things, according to one rule and system – reason, namely, and the motion that spins in one place (likened to the spinning of a turned globe), – we should never be in danger of being deemed unskillful in the construction of fair images by speech (Plato, Laws, 10, 898 a-b).
Aristotle, however, contended that νόησις νοήσεως, the self-contemplation of the Prime Mover is not movement at all. There are four species of movement – locomotion, alteration, diminution, growth. Movement is sensible, and time is the measure of movement. The perfection of movement depends on the sort of matter included. Self-movement is impossible, because every motion originates from another mover. The most perfect movement is the movement that is “purely local and foreign to all annoyance, this movement must be continuous, uniform (without change in quantity and quality) and without rest (without change in the moving substantial). Here it is the kinematic principle of primacy, of simplicity, and the perfection of the circular movement” (Vuillemin, 2005: p. 307).
Augustine and nominalists were respectively critical to Platonism either occasionally or fundamentally. Nominalists had shown that the descriptive expressions also acquire their meaning according to rules. The general predicate expressions, according to nominalists, do not have any platonic type correspondents, but they must be defined either as “general names,” such as in the Middle Ages, or as names without a name-bearer, or as names without existence-entitlement. Moreover, overthrowing Platonism could be understood as overthrowing Cartesian thought, as well. The Belgian priest and philosopher Mercier4 had argued that the philosophers, which were inspired by the Cartesian thought, represented the mind as a receptive mirror that reflects reality, conforming thus the subject to the object of consciousness. Similar was Plato’s error, as Augustine stressed, to uphold that things obtain in subjects the same nature as in reality. Consciousness yet is the fruit of two principles, the thing to know and the subject that knows. It is a palpable contradiction to demand that representation reproduces without any change, in an absolute and true manner, the nature of the thing reproduced, independently of the nature of the subject.
Free logic, intuitionist logic and mathematics, became responsible to a recent threat to Platonism. They disputed what Plato had enunciated as a number of logical principles incidentally. One famous example is his formulation of the Law of Contradiction in the course of his argument in the Republic to prove that the soul has independent parts (Rep. 436B: “It is obvious that the same thing will never do or suffer opposites in the same respect in relation to the same thing and at the same time”).
Where Plato and Aristotle agree
The enlightened philosophy of religion and the humanistic education are the central points of agreement between Plato and Aristotle. The Platonic Symposium was written to explain the outstanding meaning of love. The educated peers exercise their minds in an appropriate manner to resist transitory passion and petty ambitions. The task is, therefore, that the poets, the artists, the educators, more urgently, who will shape with their knowledge the perfect citizen (Laws, 643d) and the legislators, should direct young people into virtue. The pedagogical instruction of children, as reflected in the Republic and Laws (which may be studied in parallel), may amplify the spirituality of the people, establish ethical principles and nurture creativity. Children should neither become self-indulgent nor humiliated by punishments, but instructed towards idealization. By this way, takes place the metamorphosis of love from the individual body and personal soul to the love of beauty itself. This refinement and idealization of love is possible only through intensive mental training and inspired by the vision of Good.
The role of God in Peripatetic philosophy was shaped through the influence of the seminal text of Plato’s Republic. The higher task of the ideal republic, the education of youth in order to become conscious citizens and guardians, should be based on the rational critique of traditional Greek religion and poetry. Philosophical education must represent God as a reality that is not responsible for the misfortunes of human affairs; God is good and not hurtful, because God does not act in an evil manner and does not deceive us, but his beneficial reality is the cause of well-being:
Nor yet… is it proper to say in any case -what is indeed untrue- that gods wage war against gods, and intrigue and fight among themselves; that is, if the future guardians of our state are to deem it a most disgraceful thing to quarrel lightly with another…But if there is any possibility of persuading them, that to quarrel with one’s fellow is a sin of which no member of the state was ever guilty, such ought to be rather the language held to our children from the first, by old men and old women, and all elderly persons… For a child cannot discriminate what is allegory and what is not; and whatever at that age is adopted as a matter of belief, tends to become fixed and indelible; and therefore, we ought to esteem it of the greatest importance that the fictions which children first hear should be adapted in the most perfect manner to the promotion of virtue… Do you think that God is a wizard, and likely to appear for special purposes in different forms at various times, sometimes actually assuming such forms, and altering his own person into a variety of shapes, and sometimes deceiving us and making us believe that a transformation has taken place… (Plato, Republic, II 378-380).
Moreover, the postulation of an Unmoved Mover is associated with the revised platonic and peripatetic conception of God, as beauty and goodness that “abides simply and without variation in his own form” (Plato, Republic, II 381). The related cosmological argument is an a posteriori argument, as it contains an existential premise. Its conclusion is based either on the principle of the sufficient reason for the world or on the prerequisite for a first cause of the universe. On the other side, the prime mover argument does not seek for a cause of the world’s existence but of the unbegotten, indestructible, first cause of all motion, including the motion of the astronomical system of spheres. Plato attempted “to prove that immortal soul is the ἀρχὴ κινήσεως and that this is God” (Craig, 1980: p. 4; Plato, Phaedrus 245e-246: “that which moves itself is nothing else than the soul, – then the soul would necessarily be ungenerated and immortal”). Further on, in the tenth chapter of the Platonic Laws we find the teleological argument for the existence of the Gods. The counterargument adopts that the universe is merely the product of regularity and chance.
The cosmological and mathematical knowledge is possible, because God relates to evidence of the Good according to Plato (“is visible only to the mind, the pilot of the soul… the divine intelligence… it is nurtured on mind and pure knowledge,” Phaedrus 247 d 2ff.; “the fourth kind of madness, which causes him to be regarded as mad… when he sees the beauty on earth, remembering the true beauty,” 249 d ff.).
Calcidius5 discusses the Platonic idea that providence precedes and fate follows. In Timaeus 41d-e Plato suggests that God divided the souls equal in number with the stars and provided one soul for each star, pointed to the nature of the universe and revealed the universal series of fates. Calcidius adds that Plato presents fate as an inevitable decree (Phaedrus), as the laws of the movement of the celestial bodies (Timaeus), and as the speech of Lachesis (Republic). Plato, as Calcidius insists, did not suggest that all things occur from providence:
thus, some things are from providence only, others by decree, some from our will, some also from the variability of fortune, and many by chance and occurring as chance has it (Calcidius, 145).
The divine, the intelligible things and those that are found nearest to them are coming from providence exclusively, while the natural and the corporeal ones are in accordance with fate. Similarly, there are other kinds of things that their origin is voluntary, fortuitous or by chance. In this framework, as Calcidius (147) adds, “although fate is indeed from providence, nevertheless providence is not from fate.” As the platonic Timaeus asserts and Calcidius repeats, the fate of the human soul is depended upon its conformity with God. The soul remains unharmed as far as, throughout celestial revolutions, it beholds any of the things that truly are.
The synthetical work of the accomplishment of the Neoplatonist natural theology would be undertaken by John Scotus Eriugena (c. 815 – c. 877 CE), who defined God as Nature which creates and is not created, being the first-principle, the medium, and the final cause. In his otherness, God manifests himself as a theophany in all creatures through grace and divine love (O’Meara, 1988; Carabine, 2000).
Realistic strongholds
In Platonic dialogues realism and idealism are evenly important and entangled. Concerning this mind-involving philosophical approach, Rescher (1988) distinguished ontological idealism (“to be real is to be recognized as such by a real mind”) from conceptual idealism (“to be real is to be recognizable as such by a possible mind”). The restoration of Platonism by Plotinus, Porphyry, Iamblichus, Syrianus, Proclus and Ammonius, cultivated the ground for the commentaries of Simplicius and John Philoponus and shaped Neoplatonic philosophy. Iamblichus revived also Pythagorean philosophy, with his work On Pythagoreanism. The strong relationships between Platonic and Pythagorean philosophy concern bold associations of ideas with numbers, according to O’Meara (1990). The conceptualisations upon numbers, forms, ideas and regular geometrical objects were warranted by the background distinction between primary and secondary qualities, which was introduced by Democritus. Furthermore, Platonic and Pythagorean philosophers emphasized the method of analogy and the importance of robust reasoning.
The problem of realism and the belief in the existence of abstract mathematical objects were focal topics in Platonic philosophy. The ideal of a logical foundation of mathematics was already present in Plato’s Phaedrus, where dialectical method consisted of four stages: analysis, definition, division, and probative. Famous natural scientists, as Kepler, embraced mathematical realist theories, based on the Platonic solids and the Pythagorean harmonies of the world. However, the disagreements were never diminished. A modern critic of mathematical Platonism, Paul Benacerraf (1973), based his allegations against Platonic idealizations on a so-called causal theory of knowledge: since abstract mathematical objects exist outside of spacetime, mathematical knowledge and Platonism are incompatible to each other; there is no causal connection between them.
On the contrary, Penrose (1989), Walker (2000), and Chopra & Kafatos (2015) turn to Platonism through philosophy of cosmology: is it more than a coincidence that intelligent life inhabits the present earth and cosmos? Intelligence uncovers itself though an isomorphism or map that gives a one-one platonic correspondence between the equations and the real world.
In the same direction, Floris Cohen (1994) pointed to the important contribution of Alexandre Koyré (1939; 1957; 1968) in underscoring the consequences of the significant influence and “integration” of Neo-Platonism in the scientific revolution. More specifically by undertaking the defence of Plato in the same context. Even in the educated classical Athens, the apparent daily motion of the sun from east to the west it was difficult or impossible to be regarded as a mere appearance, misleading yet. At least with the means of those times. Plato’s insistence on the dismantling of phenomena uncovered the planetary movements also as a problem. By insisting on the cyclical nature of the movement, Plato highlighted the value and the need of mathematical conceptualisation. This approach focuses on the priority of knowledge over presumption (δόξα), namely the phenomenon. Therefore, Plato directed a challenge to the mathematicians, to examine this problem methodically. Eudoxus accepted this calling of his master. It is no coincidence that the alleged mathematical error of Aristarchus, to consider the center and the surface of the sphere as proportional sizes, had undermined the cogency of his thesis, as it was suggested by Archimedes in The Sand Reckoner.
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