Plato: Mathematics

This paper takes a detailed look at the Republic’s Divided Line analogy and considers how we should respond to its most contentious implication: that pistis and dianoia have the same degree of ‘clarity’ (σαφήνεια). It argues that we must take this implication at face value and that doing so allows us to better understand both the analogy and the nature of dianoia.

In the first argument of Metaphysics Μ.2 against the Platonist introduction of separate mathematical objects, Aristotle purports to show that positing separate geometrical objects to explain geometrical facts generates an ‘absurd accumulation’ of geometrical objects. Interpretations of the argument have varied widely. I distinguish between two types of interpretation, corrective and non-corrective interpretations. Here I defend a new, and more systematic, non-corrective interpretation that takes the argument as a serious and very interesting challenge to the Platonist.

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 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 consider three issues: the Platonic theory of “two worlds”, the method of building a mathematical structure, and the ontology of mathematics. (shrink)

The digression of Plato’s Theaetetus (172c2–177c2) is as celebrated as it is controversial. A particularly knotty question has been what status we should ascribe to the ideal of philosophy it presents, an ideal centered on the conception that true virtue consists in assimilating oneself as much as possible to god. For the ideal may seem difficult to reconcile with a Socratic conception of philosophy, and several scholars have accordingly suggested that it should be read as ironic and directed only at (...) the dramatic character Theodorus. When interpreted with due attention to its dramatic context, however, the digression reveals that the ideal of godlikeness, while being directed at Theodorus, is essentially Socratic. The function of the passage is to introduce a contemplative aspect of the life of philosophy into the dialogue that contrasts radically with the political-practical orientation characteristic of Protagoras, an aspect Socrates is able to isolate as such precisely because he is conversing with the mathematician Theodorus. (shrink)

Este trabajo pretende ser una referencia útil para los estudiosos de la filosofía de Platón. Aporta un enfoque original a la investigación de los procesos lógicos que condicionan que unas formas participen de otras. Con la introducción del concepto de participación relacional, abre una posible vía de solución a las distintas versiones del argumento del «tercer hombre». Puede resultar de interés asimismo el método de generación de los números a partir de lo par y lo impar, propuesto en la interpretación (...) de Parménides 143e-144a. El escrito tiene continuidad en un segundo artículo acerca del ser y la verdad en Platón. (shrink)

This paper presents a new interpretation of the objects of dianoia in Plato’s divided line, contending that they are mental images of the Forms hypothesized by the dianoetic reasoner. The paper is divided into two parts. A survey of the contemporary debate over the identity of the objects of dianoia yields three criteria a successful interpretation should meet. Then, it is argued that the mental images interpretation, in addition to proving consistent with key passages in the middle books of the (...) Republic, better meets those criteria than do any of the three main positions. (shrink)

This paper concerns the two arts of measurement discussed at Statesman 283-287b. In particular, it argues against the standard interpretation of the first art of measurement, according to which the various branches of mathematics are instances of the first art. Having argued against this standard view, this paper then supplies a more accurate interpretation in its place. Furthermore, it discusses the consequences of this interpretive disagreement for how we understand the relationship between the Statesman's art of measurement and Aristotle's doctrine (...) of the mean. (shrink)

In this essay, we consider the formal and ontological implications of one specific and intensely contested dialectical context from which Deleuze’s thinking about structural ideal genesis visibly arises. This is the formal/ontological dualism between the principles, ἀρχαί, of the One (ἕν) and the Indefinite/Unlimited Dyad (ἀόριστος δυάς), which is arguably the culminating achievement of the later Plato’s development of a mathematical dialectic.3 Following commentators including Lautman, Oskar Becker, and Kenneth M. Sayre, we argue that the duality of the One and (...) the Indefinite Dyad provides, in the later Plato, a unitary theoretical formalism accounting, by means of an iterated mixing without synthesis, for the structural origin and genesis of both supersensible Ideas and the sensible particulars which participate in them. As these commentators also argue, this duality furthermore provides a maximally general answer to the problem of temporal becoming that runs through Plato’s corpus: that of the relationship of the flux of sensory experiences to the fixity and order of what is thinkable in itself. Additionally, it provides a basis for understanding some of the famously puzzling claims about forms, numbers, and the principled genesis of both attributed to Plato by Aristotle in the Metaphysics, and plausibly underlies the late Plato’s deep considerations of the structural paradoxes of temporal change and becoming in the Parmenides, the Sophist, and the Philebus. After extracting this structure of duality and developing some of its formal, ontological, and metalogical features, we consider some of its specific implications for a thinking of time and ideality that follows Deleuze in a formally unitary genetic understanding of structural difference. These implications of Plato’s duality include not only those of the constitution of specific theoretical domains and problematics, but also implicate the reflexive problematic of the ideal determinants of the form of a unitary theory as such. We argue that the consequences of the underlying duality on the level of content are ultimately such as to raise, on the level of form, the broader reflexive problem of the basis for its own formal or meta-theoretical employment. We conclude by arguing for the decisive and substantive presence of a proper “Platonism” of the Idea in Deleuze, and weighing the potential for a substantive recuperation of Plato’s duality in the context of a dialectical affirmation of what Deleuze recognizes as the “only” ontological proposition that has ever been uttered. This is the proposition of the univocity of Being, whereby “being is said in the same sense, everywhere and always,” but is said (both problematically and decisively) of difference itself. (shrink)

One of the questions regarding the Parmenides is whether Plato was committed to any of the arguments developed in the second part of the dialogue. This paper argues for considering at least one of the arguments from the second part of the Parmenides, namely the argument of the generation of numbers, as being platonically genuine. I argue that the argument at 142b-144b, which discusses the generation of numbers, is not deployed for the sake of dialectical argumentation alone, but it rather (...) demonstrates key platonic features, such as the use of the greatest kinds and the generation principle. The connection between the argument for the generation of numbers and Plato’s philosophy of mathematics is strengthened by the exploration of a possible reference in Aristotle’s Metaphysics A6. Taken as a genuine platonic theory, the argument could have significant impact on how we understand Plato’s philosophy of mathematics in particular, and the ontology of the late dialogues in general – that numbers can be reduced to more basic entities, i.e the greatest kinds, in a way similar to the role the greatest kinds are assigned in the Sophist. (shrink)

In this paper, I reconsider the commonly held position that the early moral education of the Republic is arational since the youths of the Kallipolis do not yet have the capacity for reason. I argue that, because they receive an extensive mathematical education alongside their moral education, the youths not only have a capacity for reason but that capacity is being developed in their early education. If this is so, though, then we must rethink why the early moral education is (...) arational. I argue that the reason is rooted in the nature of moral explanations. These sorts of explanations are rooted in the Forms and thus one can only understand those explanations when they have knowledge of the Forms. But this requires preparation – the very sort of preparation that is provided by both the mathematical and moral educations. (shrink)

It is well known that the largest philosophers differently explain the origin of mathematics. This question was investigated in antiquity, a substantial and decisive role in this respect was played by the Platonic doctrine. Therefore, discussing this issue the problem of interaction of philosophy and mathematics in the teachings of Plato should be taken into consideration. Many mathematicians believe that abstract mathematical objects belong in a certain sense to the world of ideas and that consistency of objects and theories really (...) describes mathematical reality, as Plato quite clearly expressed his views on math, according to which mathematical concepts objectively exist as distinct entities between the world of ideas and the world of material things. In the context of foundations of mathematics, so called ‘Gödel’s Platonism‘ is of particular interest. It is shown in the article how Platonic objectification of mathematical concepts contributes to the development of modern mathematics by revealing philosophical understanding of the nature of abstraction. To substantiate his point of view, the author draws the works of contemporary experts in the field of philosophy of mathematics. (shrink)

What is mathematics about? And if it is about some sort of mathematical reality, how can we have access to it? This is the problem raised by Plato, which still today is the subject of lively philosophical disputes. This book traces the history of the problem, from its origins to its contemporary treatment. It discusses the answers given by Aristotle, Proclus and Kant, through Frege's and Russell's versions of logicism, Hilbert's formalism, Gödel's platonism, up to the the current debate on (...) Benacerraf's dilemma and the indispensability argument. Through the considerations of themes in the philosophy of language, ontology, and the philosophy of science, the book aims at offering an historically-informed introduction to the philosophy of mathematics, approached through the lenses of its most fundamental problem. (shrink)

Many have argued against the claim that Plato posited the mathematical objects that are the subjects of Metaphysics M and N. This paper shifts the burden of proof onto these objectors to show that Plato did not posit these entities. It does so by making two claims: first, that Plato should posit the mathematical Intermediates because Forms and physical objects are ill suited in comparison to Intermediates to serve as the objects of mathematics; second, that their utility, combined with Aristotle’s (...) commentary on Plato’s posit of Intermediates, provides good reason to conclude that Plato did posit them. (shrink)

I argue against a formidable interpretation of Plato’s Divided Line image according to which dianoetic correctly applies the same method as dialectic. The difference between the dianoetic and dialectic sections of the Line is not methodological, but ontological. I maintain that while this interpretation correctly identifies the mathematical method with dialectic, ( i.e. , the method of philosophy), it incorrectly identifies the mathematical method with dianoetic. Rather, Plato takes dianoetic to be a misapplication of the mathematical method by a subset (...) of practicing mathematicians. Thus, Plato’s critique of dianoetic is a not a critique of mathematics, as such, but of mathematicians. (shrink)

The second highest level of the divided line in Plato’s Republic appears to be about the entities of mathematics—entities such as particular triangles. It differs from the highest level in two respects. It involves reasoning from hypotheses, and it uses visible images. This article defends the traditional view that the passage is indeed about these mathematical ‘intermediates’; and tries to show how the apparently different features of the second level are related, by focussing on Plato’s need to give an account (...) of how we can speak of many particulars of the same kind, without assuming that they are imperfect copies, in the way sensible things can be imperfect copies, of forms. (shrink)

I argue that recollection, in Plato's Meno , should not be taken as a method, and, if it is taken as a myth, it should not be taken as a mere myth. Neither should it be taken as a truth, a priori or metaphorical. In contrast to such views, I argue that recollection ought to be taken as an hypothesis for learning. Thus, the only methods demonstrated in the Meno are the elenchus and the hypothetical, or mathematical, method. What Plato's (...) Meno demonstrates, then, is that we cannot be philosophers if we fail to make use of the mathematician's hypothetical method. (shrink)

The unifying theme of this issue is Plato’s dialectical view of mathematical progress and hypotheses. Besides provisional propositions, he calls concepts and goals also hypotheses. He knew mathematicians create new concepts and goals as well as theorems, and abandon many along the way, and erase the creative process from their proofs. So the hypotheses of mathematics necessarily change through use — unless Benson is correct that Plato believed mathematics could reach the unhypothetical goals of dialectic. Landry discusses Plato on mathematical (...) and philosophical problem solving. Arsen and White relate Plato’s philosophy to mathematics in his time, and to Aristotle. (shrink)

Questo testo nasce da alcune indagini sul nesso tra matematica e filosofia in ambiente “accademico”. È interessante notare che l'esplorazione di tale nesso costituisce un felice tratto di continuità tra gli studi più classici e ...

This paper aims to show that—and how—Plato’s notion of the receptacle in the Timaeus provides the conditions for developing a mathematical as well as a physical space without itself being space. In response to the debate whether Plato’s receptacle is a conception of space or of matter, I suggest employing criteria from topology and the theory of metric spaces as the most basic ones available. I show that the receptacle fulfils its main task–allowing the elements qua images of the Forms (...) to exist as sensible things by being that in which the elements appear, change and move–in virtue of being pure continuity. All further qualifications required for a full notion of space are derived solely from the content of the receptacle. (shrink)

Burt C. Hopkins presents the first in-depth study of the work of Edmund Husserl and Jacob Klein on the philosophical foundations of the logic of modern symbolic mathematics. Accounts of the philosophical origins of formalized concepts—especially mathematical concepts and the process of mathematical abstraction that generates them—have been paramount to the development of phenomenology. Both Husserl and Klein independently concluded that it is impossible to separate the historical origin of the thought that generates the basic concepts of mathematics from their (...) philosophical meanings. Hopkins explores how Husserl and Klein arrived at their conclusion and its philosophical implications for the modern project of formalizing all knowledge. (shrink)

At 435c-d and 504b ff., Socrates indicates that there is a "longer and fuller way" that one must take in order to get "the best possible view" of the soul and its virtues. But Plato does not have him take this "longer way." Instead Socrates restricts himself to an indirect indication of its goals by his images of sun, line, and cave and to a programmatic outline of its first phase, the five mathematical studies. Doesn't this pointed restraint function as (...) a provocation, moving us to want to begin the "longer way" and to make use of its conceptual resources to rethink Socrates' images? I begin by finding a double movement in the complex trajectory of the five studies: they both guide the soul in the "turn away from what is coming to be ... [to] what is" (518c) and, at the same time, lead the soul back, albeit in the medium of pure intelligibility, to the sensible world; for the pure figures and ratios that they disclose constitute the core structures of sensible things. I then draw on what Socrates says about geometry and harmonics to address three fundamental questions that he leaves open: the nature of the Good in its responsibility for truth and for the being of the forms; the relations of forms, mathematicals, and sensibles as these are disclosed by dialectic; and the bearing of the philosopher's discovery of the Good on his disposition towards his community and the task of ruling. I close by marking six sets of further questions that these reflections bequeath for dialogues to come. (shrink)

The compulsion of proofs is an ancient idea, which plays an important role in Plato’s dialogues. The reader perhaps recalls Socrates’ question to the slave boy in the Meno: “If the side of a square A is 2 feet, and the corresponding area is 4, how long is the side of a square whose area is double, i.e. 8?”. The slave answers: “Obviously, Socrates, it will be twice the length” (cf. Me 82-85). A straightforward analogy: if the area is double, (...) the side is double. Nevertheless, the answer is wrong. Socrates wants to lead the slave to the right conclusion. The boy should reach the truth through steps that are all “his own”, performed with full conviction. To this aim, Socrates addresses a series of short pressing questions to the slave boy. Simple questions provoke equally simple replies, though the boy is sometimes puzzled and surprised by the answers he feels compelled to give. In the first part of the exchange the boy is gradually forced to admit that a square B with double side (i.e. side of length 4) has an area which is not double, but four times as big, i. e. 16. In the second part the boy hazards the guess that the square with double area has a side of length 3, since 3 is between 2 and 4. But Socrates easily drives him to acknowledging that if the side is 3, the area of the resulting square C is 9, not 8. The boy can only exclaim: “By Zeus, Socrates, I do not know!”. In the third part, at Socrates’ prompting, the diagonal of the original square A is drawn within the second fourfold square B. Responding to Socrates’ questions, the boy is led to conclude that the square D whose side is the diagonal of A is precisely the required square: its area is 8. (shrink)

Glaucon in Plato's _Republic_ fails to grasp intermediates. He confuses pursuing a goal with achieving it, and so he adopts ‘mathematical platonism’. He says mathematical objects are eternal. Socrates urges a seriously debatable, and seriously defensible, alternative centered on the destruction of hypotheses. He offers his version of geometry and astronomy as refuting the charge that he impiously ‘ponders things up in the sky and investigates things under the earth and makes the weaker argument the stronger’. We relate his account (...) briefly to mathematical developments by Plato's associates Theaetetus and Eudoxus, and then to the past 200 years' developments in geometry. (shrink)

Annotations to the Speech of the Muses (Plato Republic 546b-c).

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)

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)

Anyone who has read Plato’s Republic knows it has a lot to say about mathematics. But why? I shall not be satisfied with the answer that the future rulers of the ideal city are to be educated in mathematics, so Plato is bound to give some space to the subject. I want to know why the rulers are to be educated in mathematics. More pointedly, why are they required to study so much mathematics, for so long?

A close reading of the five mathematical studies Socrates proposes for the philosopher-to-be in Republic VII, arguing that (1) each study proposes an object the thought of which turns the soul towards pure intelligibility and that (2) the sequence of studies involves both a departure from the sensible and a return to it in its intelligible structure.

A close reading of the five mathematical studies Socrates proposes for the philosopher-to-be in Republic VII, arguing that (1) each study proposes an object the thought of which turns the soul towards pure intelligibility and that (2) the sequence of studies involves both a departure from the sensible and a return to it in its intelligible structure.

The figure of the cordial host of the Academy, who invited the most gifted mathematicians and cultivated pure research, whose keen intellect was able if not to solve the particular problem then at least to show the method for its solution: this figure is quite familiar to students of Greek science. But was the Academy as such a center of scientific research, and did Plato really set for mathematicians and astronomers the problems they should study and methods they should use? (...) Our sources tell about Plato's friendship or at least acquaintance with many brilliant mathematicians of his day (Theodorus, Archytas, Theaetetus), but they were never his pupils, rather vice versa -- he learned much from them and actively used this knowledge in developing his philosophy. There is no reliable evidence that Eudoxus, Menaechmus, Dinostratus, Theudius, and others, whom many scholars unite into the group of so-called "Academic mathematicians," ever were his pupils or close associates. Our analysis of the relevant passages (Eratosthenes' Platonicus, Sosigenes ap. Simplicius, Proclus' "Catalogue of geometers", and Philodemus' "History of the Academy", etc.) shows that the very tendency of portraying Plato as the architect of science goes back to the early Academy and is born out of interpretations of his dialogues. (shrink)

Available from UMI in association with The British Library. ;Plato's philosophy of mathematics must be a philosophy of 4th century B.C. Greek mathematics, and cannot be understood if one is not aware that the notions involved in this mathematics differ radically from our own notions; particularly, the notion of arithmos is quite different from our notion of number. The development of the post-Renaissance notion of number brought with it a different conception of what mathematics is, and we must be able (...) to see the differences before we can begin to appreciate Plato's views, which must never be seen as applying to a revolutionary notion of mathematics . For this reason the first part of this dissertation must comprise a detailed analysis of these differences, along with some criticisms of those of Plato's modern interpreters who have not taken this point. Much of what will be found here can be found in Jacob Klein's Greek Mathematics and the Origin of Algebra, which will be extensively quoted and interpreted. This book is often quoted, but very seldom understood; or at least, those lessons which should be taken from it are rarely applied to Platonic exegesis. In the second part I attempt a better explanation of Plato's mathematical ontology, beginning with the most extensive passage in the dialogues, the triple image of Sun, Line and Cave in Republic. This reveals a relation between sensibles and forms which is at odds with the traditional view that mathematical objects are never precisely accurate instances of their forms--but this view can be refuted. Plato's view of the nature of mathematical objects will be seen to be a metaphorical way of saying what Aristotle says logically, and is entirely appropriate to the aims and methods of his contemporary mathematicians. Aristotle's criticism of Plato's views is examined and found to make sense and to be appropriate to views which Plato could consistently have held. (shrink)

An account of how the mathematical sciences turn the mind away from becoming and towards being. There are four main conclusions. 1. The study of numbers, when treated independently of the other sciences, uses a particular conception of the nature of numbers to detach the mind from the influence of perceptible objects. 2. The study of ratios and proportions, explicitly the core of Plato's harmonics, is fundamental also to plane and solid geometry and astronomy. 3. Ratios and proportion form the (...) basis for the synoptic vision of all the mathematical sciences. This vision is an essential preparation for dialectic because it makes mathematicians become aware of the limitations of the mathematics they have been engaged in but it also shows them how they can begin to overcome those limitations. 4. The main purpose of the study of numbers, treated finally in relation to the other sciences, is to calculate ratios. (shrink)

We examine the "Theaetetus" in the light of its juxtaposition of philosophical, mathematical and sophistical approaches to knowledge, which we show to be a prominent feature of the drama. We suggest that clarifying the nature of philosophy supersedes the question of knowledge as the main ambition of the "Theaetetus". Socrates shows Theaetetus that philosophy is not a demonstrative science, like geometry, but it is also not mere word-play, like sophistry. The nature of philosophy is revealed in Socrates' activity of examination (...) and his refusal to deny his ignorance about knowledge. (shrink)