Mathematical reasoning in Plato’s Epistemology (original) (raw)
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Plato, Mathematics, and the Forms: The Perceptual Foundation of Platonic Theory
That Plato was fascinated with mathematics and saw in it the basis for the theory of the Forms is a truism. Most commentators argue that this linkage was based on a rational deduction discovered through argument or dialectic. In other words, the philosopher-kings were able to apprehend the Forms or Ideas as a result of their ability to reason. The Forms themselves were simply the product of a rather mechanical process which the philosophers were taught. This exposes Plato to severe criticism, however, since it is not clear that such a process cannot be widely taught. If that is true, then his reservation of political authority to a select group of philosophers has struck many as fairly arbitrary and likely a mask for more sinister motives—most famously for Karl Popper, making Plato the originator of the totalitarian impulse. I wish to suggest that Plato’s Forms were based, not on a process of rational reflection, but on a kind of perception that I shall term “mathematical visualization.” Instead of being a product of a reflective process, they are akin to what mathematicians and others “see” when they consider problems and relationships. I consider the implications of this suggestion for Plato’s justification for philosophic rule in the Republic.
The theory of ideas and Plato’s philosophy of mathematics
2019
In this article I analyze the issue of many levels of reality that are studied by natural sciences. Particularly interesting is the level of mathematics and the question of the relationship between mathematics and the structure of the real world. The mathematical nature of the world has been considered since ancient times and is the subject of ongoing research for philosophers of science to this day. One of the viewpoints in this field is mathematical Platonism. In contemporary philosophy it is widely accepted that according to Plato mathematics is the domain of ideal beings (ideas) that are eternal and unalterable and exist independently from the subject’s beliefs and decisions. Two issues seem to be important here. The first issue concerns the question: was Plato really a proponent of present-day mathematical Platonism? The second one is of greater importance: how mathematics influences our understanding of the nature of the world on its many ontological levels? In the article I c...
Abstract: The purpose of this paper is to reassess some mathematical examples in Plato’s dialogues which at a first glance may appear to be nothing more than trivial puzzles. In order to provide the necessary background for this analysis, I shall begin by sketching a brief overview of Plato’s mathematical passages and discuss the criteria for aptly selecting them. Second, I shall explain what I mean by ‘mathematical examples,’ and reflect on their function in light of the discussion on παραδείγματα outlined in the Sophist and the Statesman. Against this framework, I shall move to a close examination of specific examples drawn from the Theaetetus (154c–55d), the Republic (523c–24d), and the Phaedo (96d–97b, 100e–101d). By placing these examples in the broader context of pre-Euclidean mathematics, I shall show that their mathematical content is often less elementary than might appear at first sight. Moreover, by placing emphasis on the specific philosophical concerns that motivated their introduction, I shall argue that the examples are not merely nonsensical jokes. Even if their illustrative purpose is not always immediately clear, and even if they can sound playful and bizarre, they in fact fulfil an important function: by virtue of their power to generate wonder or confusion, they serve as exercises and act as a trigger with respect to the difficult philosophical issues they are intended to clarify.
In Defense of Plato on Mathematical Ideas: A Commentary on Aristotle's Metaphysics XIII 6-8
Where Plato had robustly conceived of numbers as Mathematical Ideas generated by the supreme Principles and multiply instantiated in numerically distinct sensible objects, Aristotle rejects Mathematical Ideas and thinly re-conceives of numbers as no more than abstract concepts generalized by the intellect from quantities of numerical distinct sensible substances. Aristotle’s many criticisms of Plato’s theory of Mathematical Ideas are, however, an ignorant argument (ignorationes elenchi) that, not only disregards the eidetic generation of numbers from the supreme Principles, but may only plausibly succeed against his own forced re-conception of eidetic numbers as mathematical numbers. The many absurdities that Aristotle purports to derive from Plato's theory of Mathematical Ideas are thus the consequence of his own, rather than Plato's, conception of mathematical objects. The following commentary will (§I) describe how Aristotle re-conceives of Plato's Mathematical Ideas of eidetic numbers; (§II) defend Plato's theory of Mathematical Ideas against Aristotle's criticisms in Metaphysics XIII 6-8; and (§III) prosecute the case for Plato's transcendental argument for eidetic numbers against Aristotle's abstraction theory for mathematical numbers.
Plato's Use of Geometrical Logic
2003
Socrates’ brief mention of a complex problem in geometrical analysis at Meno (86d-87c) remains today a persistent mystery. The ostensible reason for the reference is to provide an analogy for the method of hypothesis from the use of hypotheses in analytic geometry. Both methods begin by assuming what is to be demonstrated and then show that the assumption leads to a well-founded truth father than something known to be false. But why did Plato pick this particular problem in analysis and why at this particular place in the inquiry? For those of us who view the dialogues as pedagogical puzzles for readers of all time to “scour” out the subtle and complicated details, this is an unquiet mystery that demands further examination. In this paper I will defend the claim that Plato had developed a powerful new heuristic method for the clarification and resolution of a broad range of philosophical problems. This method, based on the techniques of inquiry used in geometry, was a kind of concep...
2024
This thesis examines the nature and purpose of the Greek sciences ἀριθμητική, λογιστική and γεωμετρία in the texts of Plato. The statements of some other ancient authors are also mentioned, and the relevant modern research is consulted. ἀριθμός is at any instance, as Klein has already noted, 'a definite number of definite objects'. In Plato's philosophical ἀριθμητική, ἀριθμός seems to always consist of 'the odd and even', or it is the 'multitude of the μονάδων/units', just as in Euclid. Many of the key concepts of Plato's mathematics appear to have a hierarchical order, or a duality (perhaps later called 'προποδισμός' process, progression). Plato seems to employ a peculiar 'oracular/religious vocabulary' which is only recognized in the original Greek sources. There is an obvious form of 'spirituality' in the entire philosophy of Plato's mathematics. The source of the mathematical concepts, and of the 'Forms', is from a god (Prometheus?). The concept of the soul's purification and 'σωτηρία' (salvation) is probably one of the ultimate purposes of Plato's mathematics, along with the aim of reaching the 'Good' and 'Being'. ἀριθμητική, λογιστική and γεωμετρία draw the soul towards 'Truth' ("πρὸς ἀλήθειαν"), and this is one of their purposes and an oft-mentioned theme by Plato. It is concluded that Plato's mathematics is in its broadest extent an all-encompassing study of the very things (τῶν ὄντων) of nature and existence, in the background of a spiritual philosophy.
Méthexis , 1996
The word theologia is attested for the first time in Plato’s Republic II, 379a4: Hoi tupoi peri theologias. According to Werner Jaeger (The Theology of the Early Greek Philosophers, Oxford 1947, 4-13), Plato coined the word to support the introduction of a new doctrine which resulted from a conflict between the mythical and the natural (rational) approach to the problem of God. For Jaeger, the word theologia designates what Aristotle was later to call theologikê or “first philosophy (hê protê philosophia) – whence his translation of hoi tupoi peri theologias by “outlines of theology.” Victor Goldschmidt, for his part, in an illuminating article entitled “Theologia” (in Questions Platoniciennes, Paris, 1970, 141-72) will have nothing to do with such a contention. He argues that the word theologia here used by Plato means nothing more than a species of muthologia. While the principal lexicons agree with Jaeger, that is, that theologia bears the sense of “science of divine things,” the majority of contemporary translators follow Goldschmidt in taking theologia as an equivalent to muthologia or a species of it. In view of the importance of the concept of theologia in the Western tradition, I believe it merits another analysis. The aim of this paper is to show that the word theologia in this passage of the Republic can mean “science of divine things,” contrary to the claim of Goldschmidt and his followers, but not in the context of natural philosophy as Jaeger seems to imply. The most important thing is to determine whether the element logia should be translated as “science” or “speech,” that is, whether Plato is making a value judgement about theos. I argue that he does, and this is something that contemporary translators continue to miss.