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  1. Meaning of "aporetic" in the English dictionary
  2. Oxford Studies in Ancient Philosophy
  3. Committed educator and deep thinker
  4. Translation of «aporetic» into 25 languages

Meaning of "aporetic" in the English dictionary

Draw- ing on reports of Plato's lecture on the Good, one might interpret it as a principle of unity both for each Form and for the whole system of Forms. In order to clarify the sun simile, Plato uses the simile of the di- vided line to capture the distinction between the sensible realm over which the sun 'reigns' and the intelligible realm, governed by the Form of the Good D.

Such a differentiation of realms is com- pared to a line that is divided into two unequal parts, each of which is divided again in the same ratio. In proportion theory this implies that the highest level of the sensible realm is equivalent to the lowest level of the intelligible e. Adam4 takes this to be 'a slight though unavoidable defect' in the line analogy because these two parts are not equal with respect to clearness. But, since Plato was no mathe- matical slouch, we should reconsider Adam's verdict in terms of what each section represents:' The lowest section of the visible is held to consist of images, such as shadows and appearances in water and other smooth or bright materials.

They are inferior in clarity to the contents of the second section; i. Thus the analogical relationship goes as follows 51 OA : just as the object of opinion is to the object of knowledge with respect to clarity, so the image stands to the origi- nal which it imitates. Since the division so far has been made in terms of characteristic objects, one would expect a corresponding. In the higher section, by contrast, the noetic soul is held SlOB to proceed from hypotheses 'up' to an unhypothetical first principle, without using the images of the other section, while moving within the realm of Forms.

Thus the division of the intelligible realm is made in terms of two distinct methods of inquiry rather than with reference to their characteristic objects and, as Bumyeat notes, Plato has not specified whether there are distinct kinds of objects corresponding to different modes of cognition.

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The dianoetic activity of the soul is characterized by the use of sensible images and the necessity of moving from hypotheses to con- clusions. The second feature is illustrated 51 OC-D with reference to mathematicians who begin from hypotheses such as the odd and the even, without giving any account of them, and reach consistent con- clusions about their subject of inquiry. In view of Plato's repeated claim that mathematicians are forced to begin from hypotheses, Bumyeat claims that he cannot have intended it as a critical com- ment on the shortcomings of mathematical method.

While Bumyeat may be right about Plato not rejecting the hypo- thetical method of mathematicians, he can hardly use Euclid's Ele- ments to show that this was the on! J way in which they could have proceeded, since that is to ignore the older tradition of empirical mathematics. So it is with some reservations that I accept Bumyeat's suggestion that the problem being raised by Plato about mathematics concerns its ontology rather than its method. He conjectures that the reason why the divisions within the intelligible realm are delineated in terms of method rather than of subject-matter is that Plato had become aware of an unresolved problem about the ontological status of mathematical objects.

While I am indebted to Bumyeat for these helpful suggestions, I am not convinced that they make complete sense of passages in the Republic and elsewhere which discuss the subject-matter and method of mathematics. In the present passage, for instance, why should Plato emphasize that mathematicians are forced to begin from hypotheses, if he only means to acknowledge a fact about their actual practice?

Furthermore, what is the connection if any between the hypotheti- cal method and the use of images by mathematicians, and especially by geometers? Since these two features of mathematical method are frequently mentioned in tandem, one might look for a closer link between them. He claims that they are giving their accounts 'for the sake of' the square itself and the diameter itself but talk about those figures that they draw. Ap- pealing to a previous analogy, he explains 51 OE that such constructed figures correspond to the originals of which the shadows in water are images, though they are now being used as images to inquire into those things that cannot be seen except in thought.

Along with throw- ing light on the equality between two sections of the line, this expla- nation also suggests a direct connection between the reliance on images and the necessity of using hypotheses to investigate the contents of the intelligible realm. Such a connection is confirmed by a subsequent 5llA recapitu- lation dealing with the intelligible genus which the soul studies when it is forced to make use of hypotheses that are not grounded in a first principle.

Here the word order suggests that the soul is thus constrained because it uses as images those very things which in the sensible realm are regarded as originals. Once again, the equation of two sections of the divided line implies that the method of hypoth- esis and the use of images are related procedures, though it is un- clear what sense is to be made of this connection. My suggestion is that the reliance of geometers upon sensible diagrams necessitates the use of hypotheses as starting-points. In the sensible world there is nothing which is simply a triangle, for instance, and so this intel- ligible entity functions as a hypothesis under which the diagram serves as an image of Triangle Itself.

Perhaps one can clarifY this point indirectly by means of Plato's description 5llB of the noetic activity of the soul which does not treat hypotheses as starting-points but rather as stepping-stones for. Indulging in some word-play, Socrates stresses that the postulates of the mathe- maticians are not ultimate 'beginnings' but literally 'hypotheses,' since they lack the grounding provided by dialectic. It is significant that, when describing such grounding in terms of an unhypothetical prin- ciple, he insists that a dialectical account makes no use of anything sensible but rather confines itself to formal implications between Forms themselves.

Indirectly, this confirms the connection between the use of sensible images by mathematicians with their method of positing ungrounded hypotheses and of drawing conclusions from them. Such a link is also suggested by the Phaedo lOOA where the method of hypothesis is introduced to guard against possible confusion aris- ing from the fact that sensible things often present contrary appear- ances. For instance, one may become confused about whether two is generated by addition or division unless one somehow keeps Twoness separate from what participates in it. The suggested way of avoiding such confusion is to set down some hypothesis and to explore its consequences from the point of view of their mutual consistency.

If someone challenges the hypothesis itself, however, one can defend it by going up to a more adequate hypothesis. This is similar to Plato's distinction in the Republic between descending from hypotheses and ascending to the unhypothetical. While the latter approach belongs to dialectic, the former is typical of mathematics, which makes use of ungrounded hypotheses and relies upon images in the form of dia- grams.

In the Phaedo passage, Plato warns against the confusion that may result from treating mathematical objects in a quasi-physical manner, and this is consistent with his distrust of constructive lan- guage and of an excessive reliance on geometrical instruments. The mathematician must avoid the confusion of conflicting predicates by determining one predicate through an hypothesis and by fixing it in a visible diagram. Bumyeat argues that, since Plato continually emphasizes that mathematicians are forced to adopt their particular procedure, he cannot be criticising them for their method qua mathematicians.

While this sounds plausible from the point of view of Aristotle's division of the theoretical sciences, there is little evidence that Plato made such a division. So, even if his criticism of mathematical method was based only on its neglect of philosophical groundwork, this would still mean for him that there is something wrong with mathematics as a theoreti- cal science. My interpretation is based on a different reading of Plato's insis- tence that mathematicians are forced to employ hypotheses; namely, that their reliance upon sensible images brings with it the threat of confusion, so that they are forced to take refuge in the intelligible realm by means of hypotheses.

Although there is an internal tension between these two basic procedures, yet it is precisely this feature of mathematics that makes it suitable for Plato's purpose of 'converting' the soul to the intelligible realm. I find support for my interpretation in Republic VII, which uses that memorable image of the cave as an allegory for the human condition. Given their parallel distinctions between originals and images, it is fairly easy to line up the correspondences between the cave and the sensible section of the divided line, though it is more difficult to do so for the intelligible section and the world beyond the cave.

Socrates hints at such a correspondence when he says B-C that this Form is the. Such a reading indirecdy supports my interpretation. But this seems to be merely a passing remark, since it is not recalled or developed at A-B where the correspondences are explicidy discussed. Perhaps Plato's silence here is an indication of an unresolved problem.

In the sensible world the Good is said to produce both light and the sun itself, while in the intelligible world it is directly responsible for 'truth and intelligence. But when one compares the account of the ascent from the cave with the analogy of the divided line, there arises a problem about what kind of objects are studied by the mathematical sciences.

One can address this problem within the context of Plato's educational curriculum by asking why he gives B to mathematics the crucial role of turning the soul around m:puxynv from the sensible to the intelligible world. The leading question here D is about which of the disciplines has the power to draw the soul from the realm of Becoming into the realm of Being. Mter considering the traditional discipline of music, which he had earlier made part of the basic training of the guardians, Socrates rejects it as unsuitable for his present purposes because it does not lead towards knowledge of Being.

Instead he proposes to Glaucon that they look for the common thing that is used by all the crafts and sciences, and which everyone must learn at the beginning. This is the deceptively simple ability to distinguish and count things, which is called number and calculation C. For purposes of clarification, Socrates introduces A-B an important distinction among sense perceptions between those which do not call upon intelligence and those which do. The explicit grounds for this distinction is that the first type of sense impression is ad- equately judged by perception, whereas the second type yields conflict-.

Plato, the first seems to refer to numbers in themselves, whereas the second consid- ers the relations between numbers. This is the point of the reference D to Palamedes' mockery of Agamemnon as a general who was unable to count. By way of clarification, Socrates refers to the per- ception of three fingers side by side on a single hand, since this can be used to illustrate both kinds of perception.

Taken by itself, the perception of a finger raises no doubts in the soul, whatever its situ- ation or color or size in relation to other fingers. Thus Socrates declares D that the soul of the many 12 is not forced avaYJCal;e'tat to call upon intelligence to decide what is a finger, because sight never indicates to it that the finger is at the same time its opposite.

Oxford Studies in Ancient Philosophy

By contrast, sight does not have a sufficiently stable perception of the size of a finger such that it makes no difference whether it is situated in the middle or at the extremities. For instance, when compared with the thumb the index finger appears to be large but, at the same time, it appears to be small compared with the middle finger.

Simi- larly Socrates finds that the same thing often appears to be both hard and soft, and so he concludes that the senses by their very nature must deal with such opposites. With reference to the ascent of the soul, therefore, it is only such sensory impressions that provoke the soul to reflect on the true na- ture of the things perceived. In the case of touch, for instance, Socrates says A that the soul must be puzzled a1tope'iv as to what per- ception means by hard if the same thing also appears to be soft.

It is noteworthy that Plato identifies the apparent relativity of sensory opposites as a stimulus for the soul to engage in theoretical reflec- tion. In fact, his claim is much stronger than this when he repeat- edly says that the soul is forced to call upon intelligence to clarifY what the senses mean by these opposites that they find in the same thing. In fact, the construction emphasizes that the soul, puzzled by contrary perceptions, is.

My reading leaves open the possibility that Plato is critical of ordinary mathematicians for failing to give any further account of the intelligible foundations of their disciplines. But, more significantly, Socrates says B that the first step towards theoretical reflection is taken when the soul calls upon thinking to determine whether the opposing perceptions are one or two in number.

Committed educator and deep thinker

The older Pythagoreans thought that number is given clearly in perception, as when one sets out rows of figures in different configu- rations such as the well-known tetractys. By contrast, Plato empha- sizes C that the one and the many are commingled in sense perception, just like the large and the small in the perceptible finger. Thus, when the soul responds to puzzling phenomena by distinguishing between largeness and smallness as two distinct things, it has taken its first step into the realm of the intelligible with the aid of calculation.

Socrates explicitly identifies C a state of puzzlement as the original source for the leading question about what is largeness and smallness. He explains that such a question is at the root of the general distinction between the sensible and intelligible realms, and that this is what he was trying to get at through his previous distinc- tion between perceptions that call upon intelligence and those which do not. But the strategy of his argument becomes clearer when Glaucon is asked to which class the perceptions of number and of unit belong.

Everything hinges on the answer to this question because if the unit "in and by itself" is adequately perceived by the senses then it would not serve as a track 14 towards intelligible reality. On the other hand, if the one always appears along with its opposite i. Since Glaucon agrees A-B that we always see the one together with the indefinitely many, it follows that the study of the one i. Having decided that arithmetic belongs among the studies sought for the 'conversion' of the soul, Socrates proposes C that those guardians who are to rule should study arithmetic until they are able to contemplate the nature of numbers in themselves.

This proposal. This simile reinforces the impression that the soul is under some duress in its ascent to the intelligible realm; cf. In this connection, Socrates refers to the behavior of contemporary experts in arithmetic who ridicule anyone who thinks that the unit can be divided. Thus, he says D-E , if you divide their unit they will multiply it, thereby rejecting any suggestion that it may have parts. The point of the question is to get at the hidden assumptions that lie behind the practice of con- temporary arithmeticians who are imagined A as answering that these units can be grasped only by thinking and not in any other way.

Thus, pace Adam, 17 Plato does not here pro- vide an account of mathematical objects as intermediate entities between Forms and sensible things. While such an account may be reconstructed from Aristotle's reports on the so-called 'unwritten doctrines,' all one finds in the Platonic dialogues are vague hints. Adam ii, See Aristotle's Metaplrysics 1.

Since that goal is promoted by whatever compels avaYKasn the soul to orient itself towards true Being, he adopts E5 this as a criterion for any discipline to quality as propaideutic. Thus Socrates describes A5 such talk as 'quite absurd' and, like the banausic uses of arithmetic, as 'rather vulgar. Here we find an implied criticism of contemporary practitioners of geometry for their failure to recognize. Triangle, since that is neither scalene nor isosceles nor equilateral.

It is a moot question, however, whether he is talking about 'intermediates' or about sensible things considered in a certain way. Epinomis C: yeA. Denniston When read in this natural way, the whole clause provides little support for Burnyeat's claim that Socrates is accepting constructive language as being 'unavoidable,' given the nature of Greek geometry.

Proclus, in Eucl. Perhaps inspired by the work of Theaetetus, some mathematicians within the Academy compiled Elements that were later incorporated into the work of Euclid. Although most pre-Academic geometry has been lost due to this Platonic reform, one may conjecture that it was quite empirical in its procedure and proofs; i. He draws attention to the question about the real purpose of as- tronomy by having Socrates upbraid D-E Glaucon for wishing to make this discipline seem useful in order to satisfY ordinary people who would not accept the most important reason for pursuing as- tronomy; namely, rekindling an intellectual fire that has been quenched through banausic pursuits.

Instead of trying to persuade such people, Socrates advises Glaucon to pursue these studies for the sake of purifYing his own soul. So this whole discussion about the theoretical purposes of inquiry leads to the subsequent proposals for the reform of contemporary astronomy. In the meantime, however, the order of discussion is deliberately interrupted A-B to correct a mistake in the previous order of inquiry.

Socrates says that they were wrong to move directly from plane geometry to astronomy, which involves the solid in motion, without considering the solid in itself. Instead they should have stud- ied the 'third increase' after the second; i. Greek mathematics does not square with the tradition that he objected to the use of all mechanical devices in geometry, except for the straightedge and compass; c Plutarch, Quaestiones Convivales e- Such an objection fits with his critical remarks here about the absurd constructive language of geometers, given the theoretical goals of the discipline.

Adam thinks however that the problems of stereometry had not yet been discovered or solved, and he cites the Delian problem as a leading example of such unsolved problems. But the solution to that problem, involving the discovery of two mean proportionals, is presupposed in the Timaeus for the continued geometrical proportion between the four elements within the World-Body.

Therefore, one should interpret this digression on stereometry in the light of Plato's proposal to reform the mathematical sciences from the viewpoint of their theoretical foundations. The lack of support for stereometry among the ordinary citizens is consistent with their low estimation of all theoretical disciplines, but this one is particu- larly neglected because it does not appear to have any of the prac- tical uses of arithmetic and geometry.

The absence of leadership that is cited as a reason for the feeble state of stereometry should not be taken as a historical remark about any lack of outstanding research- ers but rather as a comment about the lack of proper direction in the field. What is needed is someone with a vision of the true theoretical nature of the discipline who will pose the right questions and reori- ent the whole field of research.

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Once again, a rather comic ex- change with Glaucon exposes a misunderstanding of the proper. On the contrary, says Socrates, those who try to lead us up to philosophy make us look downwards on account of the way they practice astronomy. The implied criticism of contem- porary astronomers becomes obvious when he compares them to people gaping up at ornaments on a ceiling A-B. Just like Glaucon, these practitioners mistakenly think that they can acquire knowledge with their eyes rather than with their minds. There is no knowledge of such sensible things and therefore the empirical astro- nomer is looking downwards rather than upwards B-C.

What this statement suggests is that Plato himself sees the contemporary practice of astronomy as seriously misdirected when it concentrates on observing the sensible heavens. If this is the case then it has more general implications for his views about the reform of the mathema- tical sciences and about their appropriate objects. The proposed reforms of astronomy are introduced C so that the discipline will function better in orienting the soul towards intel- ligible reality. The first step seems to be the most difficult because it involves persuading practicing astronomers that the real objects of their inquiry are not heavenly bodies but intelligible i.

In this case Socrates does not mention any sensible opposites that would force their souls to seek help from the intellect, as in the case of arithmetic and geometry. Such realities are the motions by which real speed and real slowness express true number and true shapes. According to a tradition relayed by Simplicius in Gael.

The persuasive force of this proposal depends on the paral- lel with geometry and its use of sensible diagrams. Socrates says A that if an experienced geometer were to see even the most beauti- fully crafted plans, he would admire them but would think it absurd to seek in them the truth about the equal or the double or any other symmetry. Similarly, when the real astronomer observes the move- ments of the stars, he will view them as the beautiful workmanship of the Demiurge but he will refuse to accept that the length of days, months, and years never deviates anywhere, since they are material and visible.

So he will try to discover the truth by studying astronomy through problems, just as the real geometer does, and 'let things in the sky go' C 1. As with geometry, the proposed reform of as- tronomy is designed to make the naturally intelligent part of the soul useful for theoretical activity. Although this is a very controversial passage, I will not rehearse the many interpretations it has been given.

The issue has been admirably treated by Vlastos , and I only want to take issue with him on a minor point. If this was what Plato really meant, he argues, then it would be unlikely that Eudoxus could have been so influenced by him and still construct a system of homocentric spheres to 'save the phenomena. But yet Plato's proposal for reforming the discipline with respect to its axiomatic foundations may have led Eudoxus to adopt a more theoretical line of inquiry.

On this basis Vlastos argues that Eudoxus could not have been an associate if Plato's attitude was so radically anti-empiricist as some commentators suggest. Even though geometers begin by constructing a visible diagram, they cannot base their proof on observation if they are to move beyond true opinion to knowl- edge. In order to do this, they must 'put aside' the diagram and find the geometrical principles from which to derive the required conclu- sion as a deductive consequence. Vlastos submits that the same is true for Plato's proposal that the visible heavens be used as a model for the sake of understanding the real motions of the heavenly bod- ies.

So the accumulated empirical data about the diurnal and annual motions of the sun can serve as a preamble to a genuinely scientific inquiry about its 'true' motions. But if the methodological parallel with geometry is to be taken seriously, the problem facing a scientific astronomer will be to iden- tifY a set of assumptions from which to deduce consequences that will accord with the phenomena.

In the Timaeus, for instance, the sun's complex motion is analysed into two perfectly circular motions that are combined in a closed spiral and this is consistent with most of the observable phenomena. However, unlike modern scientific conjectures, these hypotheses are not held to be refuted by a direct appeal to incompatible phenomena.

The likely target is the school of Harmonists, who used the quarter-. By contrast, Dicks sees him as making a proposal for the systematization of astro- nomical knowledge which does not exclude observations, as long as these are inte- grated into the system. This means that observations played the secondary role of illustrations, by contrast with their primary role as tests in modern astronomy.

According to Socrates such people are laboring in vain, just like the empirical astronomers, because they measure heard sounds and harmonies against each other rather than against ideal harmonies. The criticism applies not only to practicing musicians, who torture their instruments and dispute about whether the quarter-tone is the smallest discernible interval, but even to those Harmonists who search for numbers in the consonances that they hear B-C.

Their mistake is that they do not make their way up to problems, which would involve asking about which numbers are consonant and why they are so. In general, Plato's argument for a theoretical approach in all these dis- ciplines is that only in this way can we gain access to intelligible entities as the ultimate causes of sensible things.

Dialectic as the mistress-science While the propaedeutic function of mathematics with respect to dia- lectic is clear, the same cannot be said for the ontological relation- ship between their respective objects. One seeks in vain for further clarification of this matter in Republic VI-VII, where Socrates talks about the propaideutic function of the mathematical disciplines.

At D, for instance, he says that if the inquiry about these sciences grasps their intercourse and kinship with one another and if they are considered in so far as they are akin, then their investigation does promote the kind of subject-matter that is sought and the labor will not be in vain. Socrates emphasizes D-E that this mighty labor is a necessary preamble to the science of dialectic.

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  • It seems however that this projected science is differentiated from the mathematical sciences in terms of its methodological approach. For example, by proving that the semi-tone of the perfect fourth is not strictly equal to half the perfect whole-tone, he showed that mathematics contradicts not only the testimony of the ear but also the assumption of the Harmonists that a perfect half-tone must exist. The problem of the relation between mathematics and reality was discussed by Archytas of Tarentum in Elements rif Music, parts of which make up the so-called Section rif the Canon.

    For instance, math- ematicians are said A to be unable to give a reasoned account of their objects and so they do not really know what they ought to know, though this is the task which dialectic fulfils. Dialectic stimu- lates the intellectual capacity of the soul to find the true reality of each thing without help from the senses, and not to give up before grasping the Good Itself.

    Socrates draws an explicit parallel here between the allegorical journey up out of the cave and the intellec- tual path through dialectic towards a vision of the Good. Since this is familiar territory, let us focus on just one detail that bears on our guiding question about the ontological relationship between the ob- jects of mathematics and of dialectic. When describing the ascent from the cave into the daylight, Socrates says C that the prisoner is unable to look at the original living creatures but only at the 'divine images' and shadows of things in water.

    Then he concludes C that the mathematical sci- ences as previously discussed have the power to lead the best part of the soul upwards to the contemplation of what is best in reality, just as the clearest sense in the body presumably sight gazes on the brightest thing in the visible world. It seems that here a parallel is being suggested between the objects of mathematics and the 'divine images' in water, especially since the use of images was previously associated with mathematical method.

    Yet, since Plato does not draw any such parallel, we must confine ourselves to his subsequent elabo- ration on the methodological differences between dialectic and the other disciplines. When Glaucon demands A to be shown the power of dialec- tic itself, Socrates doubts his ability to comply because it no longer involves images but rather the truth itself.

    On the other hand, how- ever, he insists that this power is available only to someone who is already trained in the mathematical disciplines. While underlining the role of mathematics as propaideutic, Socrates also emphasizes B that dialectic must be differentiated by the method that it uses to grasp what each thing really is. By contrast, all the other crafts concern themselves with the opinions of men or with other natural things that are equally transient.

    Even the mathematical dis- ciplines which have some purchase on intelligible reality, Socrates describes C as 'dreaming about being' so long as they use hypotheses without reflection and fail to give a reasoned account of them. Pace Burnyeat, the crucial point of the criticism seems to be that the failure of mathematicians to reflect upon the foundations of their discipline undermines their status as theoretical sciences.

    In support of his criticism, Socrates asserts C-D that dialectic is the only discipline to follow the route of 'destroying' hypotheses by going beyond them to a principle, so that they may be firmly grounded. This science gently draws the eye of the soul, which in fact is buried in a kind of barbaric bog, 37 and leads it upwards by using the mathematical disciplines as means of spiritual conversion.

    Mter noting the propaideutic character of these disciplines, Socrates insists that they have only been called 'sciences' out of force of habit and that some other name is needed to indicate that they are clearer than opinion, though more obscure than science. While warning against any dispute about names, he recalls as suitable the term ouivota that was used previously to demarcate such disciplines. Again we should notice that the division is made in terms of the respective faculties of the soul rather than their corresponding objects.

    He also repeats the propor- tions of the divided line with reference to the faculties of the soul, while pointedly refusing to spell out the proportion between their respective objects in case it should involve many more arguments. Thus Burnyeat rightly sees in this whole passage Plato's im- plicit recognition of a problem about the ontological status of math- ematical objects, which could have been the topic for an 'unwritten chapter. From this we may infer that the mathematician can still be described as having understanding ouivota of his subject-matter.

    But another clear implication is that such a person is not a scientist in the strict sense because he cannot defend his hypotheses by appealing to higher principles, such as the One and the Good. The dialectician, on the other hand, can resist all attempted refutations because he argues according to the real nature of things rather than according to opin- ion.

    Translation of «aporetic» into 25 languages

    Since the unphilosophical mathematician cannot do this, the impli- cation is that he only manages to grasp an image of reality through opinion, and again the dream metaphor is used C-D to char- acterize the epistemological condition of such a person. Some further light can be thrown on this condition by contrasting it with the state of knowledge of a dialectician, who considers the first principles of reality. At Philebus 23C, for instance, with reference to the question about the best way of life, Plato introduces a fourfold division of the universe which is presumably the sort of thing that a dialectician would know.

    This division consists of the familiar Pythagorean principles of Limit and Unlimited, along with a mixture of these two and a cause of that mixture. Socrates illustrates 24A the nature of the Unlimited in terms of comparatives like hotter and colder where the more and the less range in an indefinite continuum.

    Being boundless in this way means being absolutely unlimited both in quality and in quantity. However, once a definite limit has been set to the continuum, it loses its more-or-less character and becomes a definite mixture. In linguistic terms, therefore, the Unlimited is reflected in adverbs like strongly or slightly, whereas the Limit is expressed by terms like equal, double, or indeed any ratio. Hence, as Socrates puts it 25E , the mixture of Limit and Unlimited in- volves some ratio that puts an end to the conflict of opposites, and makes them well-proportioned and harmonious through the intro- duction of number.

    For instance, in the case of the high and low in pitch, or the swift and slow in motion the Unlimited , numerical ratios introduce limit and thereby establish the whole art of music. But perhaps he is moving beyond the Pythagorean tradition when he introduces as an additional principle the cause or maker of that mixture, and gives it priority in being on the grounds that what makes something is prior to what is made. The dialogical purpose of intro- ducing the fourfold division was to classify the lives of pleasure, rea- son and the mixture of both.

    Obviously, the mixed life falls under mixture, and Philebus readily agrees that pleasure belongs to the class of the Unlimited, since he does not wish to have pleasure limited in any way. This leaves open the possibility that reason belongs either to the class of Limit or to that of cause. The matter is decided in- directly through a discussion of cosmology where reason is described 28C as the king of heaven and earth. In this regard, the crucial question 28D is whether the universe is controlled by an irrational and blind power, which operates by mere chance, or whether it is governed by a wondrous regulating intelligence.

    Protarchus finds it blasphemous to suggest that the world is ruled by chance. The argument against this Democritean view of the world draws 29E on the following parallel between the microcosm and the macrocosm: just as the animal body is composed of the four ele- ments, so is the body of the universe. But it is the cosmic elements that nourish and sustain the particular elements that compose the human body. Similarly, our bodies could not have a soul if the body of the universe did not also have a soul, since it would be absurd to suppose that the cause of soul in us should fail to provide soul for the whole universe.

    Here 30A-B we have the bare bones of an argument for the existence of the World-Soul, which plays such a prominent role in the Timaeus. But, since wisdom and reason presuppose soul, the highest divinity tradi- tionally called Zeus has a royal soul and a royal reason by virtue of the power of the cause i. All of these hints point us towards Plato's cosmology in the Timaeus dialogue. In order to clarify the function of mathematics in Plato's cosmology, I will treat the Timaeus as a rudimentary physical inquiry which antic- ipates Aristode's teleological approach.

    While paying attention to Plato's appropriation of earlier tradi- tions of physical inquiry, I will focus mainly on the mathematical aspects of the Timaeus. For instance, his account of the 'works of Reason' seems to reflect a Pythagorean cosmology that is dominated by the search for abstract numerical relations as the principles of order. By contrast, the account of 'things that happen of Necessity' seems indebted to the views of physiologoi like Empedocles and the Atomists. But Plato outstrips his predecessors by showing how their partial views can be integrated into a complete cosmos that is guided by principles of order and harmony.

    My general hermeneutical strat- egy, therefore, is to read the dialogue as a dialectical synthesis in which conflicting opinions are reviewed in the search for fundamen- tal principles that will 'save the phenomena. The demand for teleological explanation At Phaedo 96A the question of whether human souls are really inde- structible and immortal is said to require an account of the causes of.

    While I have benefited greatly from their remarks, they are not responsible for any deficiencies in my discussion. But the experience of grappling unsuccessfully with these ques- tions left Socrates totally confused. For instance, with regard to the question of what causes a person's growth, it had always seemed obvious that he grew because he ate and drank, with the result that flesh was added to flesh, bone to bone, and so on for the other parts of the body.

    Signifi- cantly enough, he chooses an arithmetical example to illustrate his puzzlement; i. The puzzling point here 97 AS is that, when the units were apart from each other, each was 1 and there was as yet no 2, but as soon as they approached each other there seemed to be a cause for 2; namely, the union in which they were put next to each other. On the other hand, if we divide the unit, should we say that the division is the reason for the generation of 2? But this would involve us in the contradiction of explaining the same thing by means of two opposite causes; namely, bringing together as opposed to sepa- rating.

    Such contradictions undermine any physical explanation of how a unit comes into being or perishes or even exists. So this pas- sage implies that the empirical approach to mathematics should be replaced by the hypothetical method that uses only intelligible or ideal entities. This view has been disputed by Vlastos who thinks that Plato himself bears a heavy but unacknowledged debt to Democritus.

    But this sophistical way of putting things leads to confusion. Socrates approved 97B-C of Mind being the cause of everything because it must do all its order- ing in the way that is best for each individual thing. Hence, if one wanted to discover the cause of anything coming into being or per- ishing or existing, the right question to ask was how it was best for that thing to exist or to act or. Since these are iden- tical with questions posed earlier 96A , Plato is implying that it was Socrates rather than Anaxagoras who anticipated the teleological approach to physical inquiry.

    This is confirmed by the inference which Socrates draws from a mere hint of Anaxagoras' theory that Mind is the cause of the cos- mos. According to such an account, he reasons 97D3 , the only thing a man had to think about was what is the highest good. It is no accident, however, that he infers that one must also know what is bad, since knowledge is always of opposites. The crucial point is that Socrates had adopted a teleological perspective bifore he read Anaxa- goras, given that he always inquired about good and evil.

    For in- stance, Socrates says 97D that such calculating led him to believe that he had found in Anaxagoras a teacher about the cause of things that corresponded to his own thinking. Here are two of the ques- tions he wants 97E to ask: 1 Whether the earth is flat or round, and why is it better for it to be as it is?

    Nicholas Rescher - - Review of Metaphysics 41 2 - Poetry and Access to Knowledge I. Aristotle on the Many Senses of Priority. On the Terminology of 'Abstraction'in Aristotle. Cleary - - Phronesis 30 1 Cleary - - Phronesis 30 1 - Why Stars Have No Feet. Mariska Leunissen - - In A. Wildberg eds. Added to PP index Total views 15 , of 2,, Recent downloads 6 months 6 , of 2,, How can I increase my downloads? Sign in to use this feature.

    This article has no associated abstract. No keywords specified fix it. History of Science in General Philosophy of Science categorize this paper. Applied ethics. History of Western Philosophy. Normative ethics. Philosophy of biology. The Concept of Potency in Plato and Aristotle. Proclus Elaborate Defence of Platonic Ideas. Proclus as a Reader of Platos Timaeus. Proclus and Hegel. Platos Philebus as a Gadamerian Conversation? Index of Modern Authors. Besides his main publications "Aristotle on the Many Senses of Priority," , and "Aristotle and Mathematics: Aporetic Method in Cosmology and Metaphysics," he wrote widely on ancient philosophy, philosophy of mathematics, and theories of education.

    He taught at Berkeley until , when he was appointed to the Regius Professorship of Greek at Trinity College Dublin, where he remained until his retirement in He is the author or editor of over 30 books in Greek Philosophy, in particular the history of the Platonic tradition. He wrote his Ph. He is currently working on the theme of self-knowledge in Sophocles and Plato.