As we will show in the present work, the historical development of pure geometry did not arise as a direct “transition” from practical geometry into pure geometry, at least as these are usually understood. We can discern four phases related to this evolution. Initially, we have practical geometry as applied in ancient Greece and other ancient civilizations. This surveyors’ practical geometry was somewhat transformed in “didactic” contexts when applied to problem-solving. This not-so-practical geometry is the direct antecedent of the first (...) form of pure geometry, that of Hippocrates of Chios. Only afterward did pure geometry, as we usually understand it, emerge. The form of pure geometry that we find in Euclid’s Elements. (shrink)

We present an approach in which ancient Greek mathematical proofs by Hippocrates of Chios and Euclid are addressed as a form of (guided) intentional reasoning. Schematically, in a proof, we start with a sentence that works as a premise; this sentence is followed by another, the conclusion of what we might take to be an inferential step. That goes on until the last conclusion is reached. Guided by the text, we go through small inferential steps; in each one, (...) we go through an autonomous reasoning process linking the premise to the conclusion. The reasoning process is accompanied by a metareasoning process. Metareasoning gives rise to a feeling-knowing of correctness. In each step/cycle of the proof, we have a feeling-knowing of correctness. Overall, we reach a feeling of correctness for the whole proof. We suggest that this approach allows us to address the issues of how a proof functions, for us, as an enabler to ascertain the correctness of its argument and how we ascertain this correctness. (shrink)

The cognitive basis of geometry is still poorly understood, even the ‘simpler’ issue of what kind of representation of geometric objects we have. In this work, we set forward a tentative model of the neural representation of geometric objects for the case of the pure geometry of Euclid. To arrive at a coherent model, we found it necessary to consider earlier forms of geometry. We start by developing models of the neural representation of the geometric figures of ancient Greek practical (...) geometry. Then, we propose a related model for the earliest form of pure geometry – that of Hippocrates of Chios. Finally, we develop the model of the neural representation of the geometric objects of Euclidean geometry. The models are based on the hub-and-spoke theory. In our view, the existence of specific models opens the possibility of addressing the relationship between geometric figures and geometric objects, in a novel way, in terms of their neural representation. (shrink)

The aim of this article is to show that intuition plays no role in mathematics. That intuition plays a role in mathematics is mainly associated to the view that the method of mathematics is the axiomatic method. It is assumed that axioms are directly (Gödel) or indirectly (Hilbert) justified by intuition. This article argues that all attempts to justify axioms in terms of intuition fail. As an alternative, the article supports the view that the method of mathematics is the analytic (...) method, a method originally used by the mathematician Hippocrates of Chios and the physician Hippocrates of Cos, and first explicitly formulated by Plato. The article examines the main features of the analytic method, and argues that, while intuition plays an essential role in the axiomatic method, it plays no role in the analytic method, either in the discovery or in the justification of hypotheses. That the method of mathematics is the analytic method involves that mathematical knowledge is not absolutely certain but only plausible, but the article argues that this is the best we can achieve. (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)

In this paper, we propose two agent-based simulation models for the evolution of language in the framework of evolutionary language games. The theory of evolutionary language games arose from the union of evolutionary game theory, introduced by the English biologist John Maynard Smith, and language games, developed by the Austrian philosopher Ludwig Wittgenstein. The first model proposed is based on Martin Nowak’s work and is designed to reproduce and verify the results Nowak obtained in his simplest mathematical model. For the (...) second model, we extend the previous one with the introduction of a world where the languages live and evolve, and which influences interactions among individuals. The main goal of this research is to present a model which shows how the presence of a topological structure influences the communication among individuals and contributes to the emergence of clusters of different languages. (shrink)

In the Vita Aristotelis of Diogenes Laertius and elsewhere we come across an epigram of Theocritus of Chios directed against Aristotle. I cite the poem in the form in which it has most recently been published by D. L. Page.

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Theopompus of Chios: History and Rhetoric in the Fourth Century B.C.John BucklerMichael A. Flower. Theopompus of Chios: History and Rhetoric in the Fourth Century B.C.Oxford: Clarendon Press, 1994. xii + 252 pp. Cloth, $49.95.Theopompos is a historian fully worthy of the attention of Michael A. Flower's new study of him. The results, unfortunately, are for the most part disappointing. F.'s most important contribution to an understanding (...) of Theopompos' method and goals is his incontrovertible conclusion that Isokrates' influence [End Page 495] on Theopompos is overrated and indeed misunderstood. F. successfully rejects the notion that Isokrates created a fourth–century genre of historical writing or that he produced from his school a cadre of historians. Despite the evidence of TT 2 and 5, the testimonia indicate only that Theopompos and Ephoros were students of Isokrates. Nothing in the evidence proves that Isokrates gave them any historical training. Isokrates was a rhetorician who educated others in oratory. He admits (Ant. 45–46; Panath. 1–2) that he himself had decided not to write history. His goal in life was to teach rhetoric that could be used for a variety of literary purposes. Although Isokrates obviously uses historical examples in his own writings, they make only a rhetorical effect. F. is surely right to say that Isokrates' influence on Theopompos was predominantly cosmetic—that he taught Theopompos the style but not the essence of his histories. The undue emphasis on Isokrates' influence on Theopompos obscures the historical genius of the latter.In chapter 4 F. examines the moral and political philosophy of Theopompos. He easily refutes (63ff.) the opinion of R. Lane Fox that Theopompos was merely a moralist without political experience and without practical criteria by which to assess people and politics. No one, however, can doubt that Theopompos was indeed a moralist. He criticizes the deleterious effects of luxury on various individuals and peoples, effects rendering them dissolute and lazy. He thinks instead that a well–ordered life of individuals and communities leads to security and prosperity. He is also a champion of ordinary and traditional religion. He is likewise opposed to democracy because he feels that it led to a dissolute life. Yet F. fails to explain the connection between personal virtue or vice and public affairs. So did Theopompos. Two examples will suffice. Although Theopompos considers Lysandros and Agesilaos good men and the Spartan constitution a good one for its moral and abstemious way of life, the efforts of both leaders led to defeat and disaster for Sparta. Conversely, Theopompos considers Philip II and the Macedonians decadent and debased because of their sexual and alcoholic excesses. Yet these degenerates won control of Greece. Without serious examination of this point, F. (117–18, 127) falls back upon Theopompos' explanation: if Philip was base, the Athenians were even more so. Perhaps Philip was just lucky, this founded in no small part upon the evidence of Demosthenes, who had himself fled from the battlefield of Chaironeia. F. never explains in historical terms why the Athenians were unable successfully to oppose Philip. In short, neither F. nor perhaps Theopompos succeed in demonstrating how good moral fibre ensures actual political success. When F. most effectively touches upon the point, he seems (114) to contradict his other conclusions.F. is surely right to endorse W. R. Connor's conclusion that Theopompos was no admirer of Philip. In chapter 5 F. examines Polybios' use of Theopompos, the latter's portrayal of Philip's private life, and the king's political and military abilities. The study of Polybios and Theopompos is needlessly speculative, and could have profited greatly by a careful reading of K. Meister, Historische Kritik [End Page 496] bei Polybios (1975), which is never cited. Theopompos and Demosthenes, neither a friend of Philip, describe Philip and his officers as hard drinkers and given to homosexual orgies. Again, no one can reasonably doubt that Philip and his Macedonians were guilty of both charges. But to repeat; neither moral defect impeded their political and military success, and in strictly historical terms the point is insignificant. The obvious truth is that Philip triumphed because he was a brilliant... (shrink)

χαιρέτω ἡμέτερος βασιλεὺς σωτήρ τε πατήρ τε·ἡμῖν δὲ κρητῆρ’ οἰνοχόοι θέραπεςκιρνάντων προχύταισιν ἐν ἀργυρέοις· †ὁ δὲ χρυσὸςοἶνον ἔχων χειρῶν νιζέτω εἰς ἔδαφος.†σπένδοντες δ’ ἁγνῶς Ἡρακλεῖ τ’ Ἀλκμήνηι τε,Προκλεῖ Περσείδαις τ’ ἐκ Διὸς ἀρχόμενοιπίνωμεν, παίζωμεν· ἴτω διὰ νυκτὸς ἀοιδή,ὀρχείσθω τις· ἑκὼν δ’ ἄρχε φιλοφροσύνης.ὅντινα δ’ εὐειδὴς μίμνει θήλεια πάρευνος,κεῖνος τῶν ἄλλων κυδρότερον πίεται.May our king rejoice, our saviour and father; let the attendant cup-bearers mix for us a crater from silver urns; †Let the golden one with wine in his hands wash (...) to the base† Pouring libations piously to Heracles and Alcmene, Procles and the sons of Perseus and Zeus first of all, let us drink, let us play, let our song rise through the night. Dance someone, willingly begin the festivities. And anyone who has a fair girl waiting to share his bed will drink more like a man than all the others.(Ion, fr. 27 West = fr. 90 Leurini). (shrink)

In the Armenian tradition of Ailios Theon’s Progymnasmata two otherwise unknown historians occur. By examining the context of the passage and comparing it with authors like Philostephanos, it is made plausible that the ‘unknown’ historians are Phileas of Athens and Skymnos of Chios, whose names are misspelled in the Armenian text.

For the life of the poet Ion we have more certain dates than for most of the Other writers of the fifth century. He produced his first tragedy in the 82nd Olympiad, 452after the death of Perikles and when the revolt of Lesbos was imminent7dgr; 'Aa fact which seems significant of the position which Athens had won for herself by the foundation of the Delian League. At the same time this fact clearly indicates that his father Orthomenes, whom they called (...) Xuthos, had whole-heartedly espoused the cause of Athens. He bequeathed his loyalty to his son Ion, who adhered to Athens through all vicissitudes of fortune, and he in his turn bequeathed it to his son Tydeus, who paid for it with his life. (shrink)

When analyzing complex longitudinal data, especially data from different educational settings, researchers generally focus only on the mean part (i.e., the regression coefficients), ignoring the equally important random part (i.e., the random effect variances) of the model. By using Project English Language and Literacy Acquisition (ELLA) data, we demonstrated the importance of taking the complex data structure into account by carefully specifying the random part of the model, showing that not only can it affect the variance estimates, the standard errors, (...) and the tests of significance of the regression coefficients, it also can offer different perspectives of the data, such as information related to the developmental process. We used xxM (Mehta, 2013), which can flexibly estimate different grade-level variances separately and the potential carryover effect from each grade factor to the later time measures. Implications of the findings and limitations of the study are discussed. (shrink)

T. A. Cavanaugh's Hippocrates' Oath and Asclepius' Snake: The Birth of the Medical Profession articulates the Oath as establishing the medical profession's unique internal medical ethic - in its most basic and least controvertible form, this ethic mandates that physicians help and not harm the sick. Relying on Greek myth, drama, and medical experience (e.g., homeopathy), the book shows how this medical ethic arose from reflection on the most vexing medical-ethical problem -- injury caused by a physician -- and (...) argues that deliberate iatrogenic harm, especially the harm of a doctor choosing to kill (physician assisted suicide, euthanasia, abortion, and involvement in capital punishment), amounts to an abandonment of medicine as an exclusively therapeutic profession. The book argues that medicine as a profession necessarily involves stating before others what one stands for: the good one seeks and the bad one seeks to avoid on behalf of the sick, and rejects the view that medicine is purely a technique lacking its own unique internal ethic. It concludes noting that medical promising (as found in the White Coat Ceremony through which U. S. medical students matriculate) implicates medical autonomy which in turn merits respect, including honoring professional conscientious objections. (shrink)

Hippocrates, celebrated as the Father of Medicine, emphasized the importance of observation in diagnosis and prognosis. In so doing, he argued that the observant physician could draw on both senses and logic in interpreting clinical findings for the benefit of the patient. Among his many writings is a collection of aphorisms that remain highly relevant to the practice of medicine to this day. The first of these is the best known: which can be translated as: Deceptively simple in structure, (...) this aphorism offers many levels of meaning and interpretation that, as I will show, remain highly relevant to the contemporary practice of medicine. Containing a richness of... (shrink)

The author argues that an analysis ofthe texts ofthe Collection of Hippocrates leads to the conclusion that long before the methodological genius of Aristotle there existed a highly analytical culture among medical professionals. The differences in understanding of the value and objectives of a valid inference in Hippocrates and Aristotle are explained in terms of the characteristics of the discourse that each of them used. Aristotle is argued to have been using a social-dialectical discourse, whereas, in medical practice, (...) a combination of skills and reasoning had the highest value because what was expected from a doctor was the right diagnosis and the right treatment. The author argues that rational medical tradition of antiquity did not reach the logical reflection. According to her, the doctrine of the necessary connection between the logical attribute and the object of inferences had been developed in the Buddhist logic, which, as the author argues, can be considered a rational component of education in Tibetan medicine. (shrink)

The author argues that an analysis ofthe texts ofthe Collection of Hippocrates leads to the conclusion that long before the methodological genius of Aristotle there existed a highly analytical culture among medical professionals. The differences in understanding of the value and objectives of a valid inference in Hippocrates and Aristotle are explained in terms of the characteristics of the discourse that each of them used. Aristotle is argued to have been using a social-dialectical discourse, whereas, in medical practice, (...) a combination of skills and reasoning had the highest value because what was expected from a doctor was the right diagnosis and the right treatment. The author argues that rational medical tradition of antiquity did not reach the logical reflection. According to her, the doctrine of the necessary connection between the logical attribute and the object of inferences had been developed in the Buddhist logic, which, as the author argues, can be considered a rational component of education in Tibetan medicine. (shrink)

There is a certain disconnect between the airiness with which modern research handles the person of Cynaethus of Chios and the scarcity of ancient information on him, the unreliability of which is at the heart of this essay. Cynaethus of Chios occurs only twice in ancient literature, in two anonymous undatable scholia on Pindar, N. 2, 1, the first of which not only satisfies the need for attributions of names on the part of modern philologists, but presumably satisfied (...) already the same need on the part of ancient philologists. In order to illustrate why it might indeed make sense to assume that there was a certain Cynaethus of unknown origin, but in fact not necessarily a Cynaethus of Chios, said scholion is thoroughly re-examined here. (shrink)

Several of the treatises and lectures that make up the Hippocratic corpus begin with more or less extended statements about the physical composition and operation of the world at large, and approach the study of human physiology from this angle. We see this, for example, in De Natwra Hominis, De Flatibus, De Carnibus, De Victu; it was the approach of Alcmaeon of Croton, Diogenes of Apollonia, and according to Plato of Hippocrates himself. The work known as De Hebdomadibus would (...) appear to be a prime example of the type. The first twelve chapters are cosmological. They are dominated by two ideas: that everything in nature is arranged in groups of seven, and that the human body is constructed on the same pattern as the whole world. In the later part of the book we pass to the subject of fevers, their causation and treatment. But as Roscher observed, the cosmology and the pathology do not belong together. (shrink)

In what follows, I reply to critical appraisals of my book entitled Hippocrates’ Oath and Asclepius’ Snake: The Birth of the Medical Profession. Professors Tollefsen, McPherson, and Potts separately offer these thoughtful critiques. Professor Tollefsen approaches the work from the standpoint of the physician-patient relationship. Professors McPherson and Potts both address it in terms of virtues. Potts treats the theme of virtue generally while McPherson focuses on the virtue of piety. Since virtues attend relationships, in what follows, I discuss, (...) first, the physician-patient relationship ; second, virtue generally ; and third, and finally, the virtue of piety specifically. (shrink)

After reading carefully the essay which, in his recently-published Varia Socratica, Part I., Prof. A. E. Taylor has written on the use of the words and in the Greek literature of the Socratic and Platonic periods, I find myself on the one hand in agreement with him as to the importance of such linguistic investigations for the understanding of Plato, and on the other in frequent disagreement with him as to the meaning of the words in the passages he cites, (...) and the inferences he draws from them. Thinking that Prof. Taylor's analysis of the use of the terms in Hippocrates is very far from final, I offer in this paper a further contribution to their interpretation, confining myself as far as possible to the question, What do and mean in the Hippocratean writings? (shrink)

This is a condensed description of the contents and overarching argument found in Hippocrates’ Oath and Asclepius’ Snake: The Birth of the Medical Profession. In that work, I maintain that the basic medical ethical problem concerns iatrogenic harm. I focus particularly on what I refer to as ‘role-conflation’. This most egregious form of iatrogenic harm occurs when a physician deliberately adopts the role of wounder. A contemporary practice such as physician-assisted suicide exemplifies a doctor’s deliberate wounding. I argue that (...) the author of the Hippocratic Oath – aware of the chronic risk of role-conflation – responds to this threat by having the oath-taker specifically forswear signal acts that wound the sick, such as giving a deadly drug even if asked, giving an abortifacient, sexual predation, and gossiping. I argue that by means of the Oath, Hippocrates founds medicine as a profession, a practice intimately involving promising. (shrink)

In Hippocrates’ Oath and Asclepius’ Snake: The Birth of the Medical Profession, Thomas Cavanaugh focuses on performative aspects of the taking of the oath which bear upon the formation of that community we identify as the medical profession. In this paper, I suggest that we can go further than Cavanaugh does in identifying what the Hippocratic oath makes possible. Given its particular content and what it communicates, the oath makes possible, to a degree few other oaths could, and in (...) a way which might be significantly impaired by the absence of an oath, the form of community we have come to call the physician-patient relationship. In Section I, I will discuss Cavanaugh’s treatment of the oath’s relationship to the community of medical professionals. In Section II, I discuss at a relatively abstract level the relationship between communication and community. In Section III I show how the oath can additionally constitute the primordial moment in the existence of the physician-patient relationship. (shrink)

In Ancient Concepts of the Hippocratic, Lesley Dean-Jones and Ralph Rosen have gathered 19 international authorities in ancient medicine to identify commonalities among the treatises of the Hippocratic Corpus which led scholars of antiquity to group them under one name.

At Plato’s Phaedrus 270c, Socrates asks whether one can know souls without knowing ‘the whole.’ Phaedrus answers that ‘according to Hippocrates’ the same demand on knowing the whole applies to bodies. What parallel is intended between soul-knowledge and body-knowledge and which medical passages illustrate the analogy have been much debated. Three dominant interpretations read ‘the whole’ as respectively (1) environment, (2) kosmos, and (3) individual soul or body; and adduce supporting Hippocratic passages. But none of these interpretations accounts for (...) the Phaedrus’ rhetorical method. A better reading sees the whole as the genos ‘soul,’ as the Phaedrus’ taxonomies divide that genus. (shrink)