Scholars ubiquitously refer to the method εξ υποθεσεως, introduced at Meno 86e1-87d8, as a method of hypothesis. In contrast, this paper argues that the method εξ υποθεσεως in Meno is not a hypothetical method. On the contrary, in the Meno passage, υποθεσις means “postulate”, that is, cognitively secure proposition. Furthermore, the method εξ υποθεσεως is derived from the method of geometrical analysis. More precisely, it is derived from the use of geometrical analysis to achieve reduction, that is, reduction of a (...) less tractable problem to a more tractable problem. As such, the method εξ υποθεσεως does not by itself serve to solve problems. (shrink)

ExtractDuring the second half of the twentieth century an attention of historians of mathematics shifted to mathematics of the Late Antiquity and its subsequent development by mathematicians of the Arabic world. Many critical editions of works of mathematicians of the Hellenistic era have made their appearance, giving rise to a new, more detailed historical picture. Among these are the critical editions of the works of Diophantus, Apollonius, Archimedes, Pappus, Diocles, and others.Send article to KindleTo send this article to your Kindle, (...) first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.The Readings of Apollonius' On the Cutting off of a RatioVolume 22, Issue 1Ioannis M. Vandoulakis DOI: https://doi.org/10.1017/S0957423911000130Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. The Readings of Apollonius' On the Cutting off of a RatioVolume 22, Issue 1Ioannis M. Vandoulakis DOI: https://doi.org/10.1017/S0957423911000130Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. The Readings of Apollonius' On the Cutting off of a RatioVolume 22, Issue 1Ioannis M. Vandoulakis DOI: https://doi.org/10.1017/S0957423911000130Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation Request permission. (shrink)

Some philosophers hold that mathematics depends on idealising assumptions. While these thinkers typically emphasise the role of idealisation in set theory, Edmund Husserl argues that idealisation is constitutive of the early Greek geometry that is codified by Euclid. This paper takes up Husserl's idea by investigating three major developments of Greek geometry: Thalean analogical idealisation, Hippocratean dynamic idealisation, and Archimedean mechanical idealisation. I argue that these idealisations are not, as Husserl held, primarily a matter of ‘smoothing out’ sensory reality to (...) produce ideal, ‘perfect’ figures. Rather, Greek geometry depends on assuming some falsehoods – equidistance from a source of illumination, a perfectly unwavering hand, or a machine that weights abstract objects – in order to make complex problems tractable. Although these idealisations were rarely discussed explicitly in antiquity, they can be systematically reconstructed from our extant sources. (shrink)

The business ethics field contains a number of explanations for the imagination’s influence on decision-making. This has benefited moral theorizing because approaches that utilize the imagination tend to acknowledge important biological and psychological forces that influence the way we understand situations, develop strategies for problem-solving, and choose courses of action. But, I argue, the broad range of approaches has also served as an obstacle to theory development in the field. Given the variety of theoretical and disciplinary approaches, coupled with the (...) diversity of applications, it would be fair to judge the current state of theory as fractured. To bring focus to theory development, this conceptual study of the moral imagination is grounded in the work of one discipline, one theorist, one text: Plato’s Republic. The primary outcome of this study is the demonstration of the conditions under which the imagination serves to augment and support rationality rather than serving as an impediment. The systematic nature of Plato’s theory aids in the formation of more coherent conceptual grounding than currently available in the field. A final contribution of this study is the positioning of Plato as a proper beginning or foundation to any further theory development in the moral imagination. (shrink)

This paper re-constructs Plato's ‘philosophy of geometry’ by arguing that he uses a geometrical method of hypothesis in his account of the cosmos’ generation in the Timaeus. Commentators on Plato's philosophy of mathematics often start from Aristotle's report in the Metaphysics that Plato admitted the existence of mathematical objects in-between ( metaxu) Forms and sensible particulars ( Meta. 1.6, 987b14–18). I argue, however, that Plato's interest in mathematics was centred on its methodological usefulness for philosophical inquiry, rather than on questions (...) of mathematical ontology. My key passage of interest is Timaeus’ account of the generation of the primary bodies in the cosmos, i.e. fire, air, water and earth ( Tim. 48b–c, 53b–56c). Timaeus explains the primary bodies’ origin by hypothesising two right-angled triangles as their starting-point ( arkhê) and describing their individual geometrical constitution. This hypothetical operation recalls the hypothetical method which Socrates introduces in the Meno (86e–87b), as well as the use of hypotheses by mathematicians which is described in the Republic (510b–c). Throughout the passage, Timaeus is focussed on explicating the bodies in terms of their formal structure, without however considering the ontological status of the triangles in relation to the physical world. (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 Regress: Knowledge, we like to suppose, is essentially a rational thing: if I claim to know something, I must be prepared to back up my claim by statingmy reasons for making it;and if my claim is to be upheld, my reasons must begood reasons. Now suppose I know that Q; and let my reasons be conjunctively contained in the proposition that R. Clearly, I must believe that R ;equally clearly, I must know that R . Thus if I know (...) that Q, I know that R. But if I know that R, then I must have my reasons, R' for holding R; and, by the same argument, I mustknow that R'. And if R', then R”; and so on, ad infinitum. (shrink)

The Regress: Knowledge, we like to suppose, is essentially a rational thing: if I claim to know something, I must be prepared to back up my claim by statingmy reasons for making it;and if my claim is to be upheld, my reasons must begood reasons. Now suppose I know that Q; and let my reasons be conjunctively contained in the proposition that R. Clearly, I must believe that R ;equally clearly, I must know that R. Thus if I know that (...) Q, I know that R. But if I know that R, then I must have my reasons, R' for holding R; and, by the same argument, I mustknow that R'. And if R', then R”; and so on, ad infinitum. (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)