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)

This essay examines the quantitative aspects of Greco-Roman science, represented by a group of established disci¬plines, which since the fourth century BC were called mathēmata or mathē¬ma¬tikai epistē¬mai. In the group of mathēmata that in Antiquity normally comprised mathematics, mathematical astronomy, harmonics, mechanics and optics, we have also included geography. Using a dataset based on The Encyclopaedia of Ancient Natural Scientists, our essay considers a community of mathēmatikoi (as they called themselves), or ancient scientists (as they are (...) defined for the purposes of the present paper) from a sociological point of view, focusing on the size of the scientific population known to us and its disciplinary, temporal and geographical distribution. A diachronic comparison of neighboring and partly overlapping communities, ancient scientists and philosophers, allows the pattern of their interrelationship to be traced. An examination of centers of science throughout ancient history reveals that there were five major centers – Athens, Alexandria, Rhodes, Rome and Byzantium/Constantinople – that appear and replace one another in succession as leaders. These conclusions serve to reopen the issue of the place of mathēmata and mathēmatikoi in ancient society. (shrink)

in this paper we return to Marshall Clagett’s view about the existence of an ancient Greek geometry of motion. It can be read in two ways. As a basic presentation of ancient Greek geometry of motion, followed by some aspects of its further development in landmark works by Galileo and Newton. Conversely, it can be read as a basic presentation of aspects of Galileo’s and Newton’s mathematics that can be considered as developments of a geometry (...) of motion that was first conceived by ancient Greek mathematicians. (shrink)

The ancient Greek philosophers – like Parmenides – reasoned that observable reality cannot exist by itself. It has to be a creation of an underlying reality. An all-in­clusive existence that has a structure because observable reality shows structure at every scale size. Although observable reality is involved in a continuous transformation too. If our concept about the relation between phenomenological reality and the creating underlying reality is correct, the unification of the properties of phenomenological reality is part of (...) an enveloping mathematical model. DOI: 10.5281/zenodo.5457368. (shrink)

This short article pairs the realms of “Mathematics”, “Philosophy”, and “Poetry”, presenting some corners of intersection of this type of scientocreativity. Poetry have long been following mathematical patterns expressed by stern formal restrictions, as the strong metrical structure of ancient Greek heroic epic, or the consistent meter with standardized rhyme scheme and a “volta” of Italian sonnets. Poetry was always connected to Philosophy, and further on, notable mathematicians, like the inventor of quaternions, William Rowan Hamilton, or Ion (...) Barbu, the creator of the Barbilian spaces, have written appreciated poems. We will focus here on an avant-garde movement in literature, art, philosophy, and science, called Paradoxism, founded in Romania in 1980 by a mathematician, philosopher and poet, and on the laboured writing exercises of the Oulipo group, founded in Paris in 1960 by mathematicians and poets, both of them still in act. (shrink)

The purpose of this work is to address the relation existing between ancient Greek practical geometry and ancient Greek pure geometry. In the first part of the work, we will consider practical and pure geometry and how pure geometry can be seen, in some respects, as arising from an idealization of practical geometry. From an analysis of relevant extant texts, we will make explicit the idealizations at play in pure geometry in relation to practical geometry, some (...) of which are basically explicit in definitions, like that of segments in Euclid’s Elements. Then, we will address how in pure geometry we, so to speak, “refer back” to practical geometry. This occurs in two ways. One, in the propositions of pure geometry. The other, when applying pure geometry. In this case, geometrical objects can represent practical figures like, e.g., a practical segment. (shrink)

We have reached the peculiar situation where the advance of mainstream science has required us to dismiss as unreal our own existence as free, creative agents, the very condition of there being science at all. Efforts to free science from this dead-end and to give a place to creative becoming in the world have been hampered by unexamined assumptions about what science should be, assumptions which presuppose that if creative becoming is explained, it will be explained away as an illusion. (...) In this paper it is shown that this problem has permeated the whole of European civilization from the Ancient Greeks onwards, leading to a radical disjunction between cosmology which aims at a grasp of the universe through mathematics and history which aims to comprehend human action through stories. By going back to the Ancient Greeks and tracing the evolution of the denial of creative becoming, I trace the layers of assumptions that must in some way be transcended if we are to develop a truly post-Egyptian science consistent with the forms of understanding and explanation that have evolved within history. (shrink)

At least since Socrates, philosophy has been understood as the desire for acquiring a special kind of knowledge, namely wisdom, a kind of knowledge that human beings ordinarily do not possess. According to ancient thinkers this desire may result from a variety of causes: wonder or astonishment, the bothersome or even painful realization that one lacks wisdom, or encountering certain hard perplexities or aporiai. As a result of this basic understanding of philosophy, Greek thinkers tended to regard philosophy (...) as an activity of inquiry (zētēsis) rather than as a specific discipline. Discussions concerning the right manner of engaging in philosophical inquiry – what methodoi or routes of inquiry were best suited to lead one to wisdom – became an integral part of ancient philosophy, as did the question how such manners or modes of inquiry are related to, and differ from, other types of inquiry, for instance medical or mathematical. In this special issue of History of Philosophy & Logical Analysis, we wish to concentrate in particular on ancient modes of inquiry. (shrink)

The discussions of conatus – force, tendency, effort, and striving – in early modern metaphysics have roots in Aristotle’s understanding of life as an internal experience of living force. This paper examines the ways that Spinoza’s conatus is consonant with Aristotle on effort. By tracking effort from his psychology and ethics to aesthetics, I show there is a conatus at the heart of the activity of the ψυχή that involves an intensification of power in a way which anticipates many of (...) the central insights of early modern and 20th century European philosophy. The first section outlines how Aristotle’s developmental conception of the soul as geometrically ordered lays the foundation for his understanding of effort. The developmental series of powers of the soul are analogous to the series of shapes in mathematics. The second section links the striving of the soul to the gradual acquisition of virtues as a directed activity unifying multiplicity. The third examines the paradigm of self-awareness that Aristotelian effort involves. In the final section I show how ancient Greek theories of music were founded on the experience of striving. The “nature” of music is defined by Aristoxenus, and Theophrastus, in relation to the passion and intentionality of the soul. The geometrical order, as a synthesis of elements in geometry, music, or ethics, is a generative process in which past elements are retained and reintegrated in later stages of development. It requires effort to think geometrically, and the progress of knowledge itself is an integral aspect of all effort. Effort is the lived and self-aware cause which, moving step by step in an orderly and deliberate way, grows and advances upon itself. For both Spinoza and Aristotle, effort is the immanent intelligence which accomplishes what is in the purview of its understanding. Thus, will, in this conception of effort, is not something we already possess innately, but emerges gradually by an effort aimed at improvement. (shrink)

Most historians and philosophers of philosophy and history of mathematics hold one interpretation or the other of the nature of method of analysis and synthesis in itself and in its historical development. In this paper, I am trying to prove – through three points – that, in fact, there were two understandings of that method in Greek mathematics and philosophy, and which were reflected in Arabic mathematical science and philosophy; this reflection is considered as proof also of (...) this double nature of that method. Thus, we have to rethink the nature of Arabic philosophy systems. (shrink)

First, I offer a short overview on the classical occidental philosophy as propounded by the ancient Greeks and the natural philosophies of the last 2000 years until the dawn of the empiricist logic of science in the twentieth century, which wanted to delimitate classical metaphysics from empirical sciences. In contrast to metaphysical concepts which didn’t reflect on the language with which they tried to explain the whole realm of entities empiricist logic of science initiated the end of metaphysical theories (...) by reflecting on the preconditions for foundation and justification of sentences about objects of investigation, i.e. a coherent definition of language in general, which was not the aim of classical metaphysics. Unexpectedly empiricist logic of science in the linguistic turn failed in the physical and mathematical reductionism of language and its use in communication, as will be discussed below in further detail. Nevertheless, such reflection on language and communication also introduced this vocabulary into biology. Manfred Eigen and bioinformatics, later on biolinguistics, used ‘language’ applied linguistic turn thinking to biology coherent to the logic of science and its formalisable aims. This changed significantly with the birth of biosemiotics and biohermeneutics. At the end of this introduction it will be outlined why and how all these approaches reproduced the deficiencies of the logic of science and why the biocommunicative approach avoids their abstractive fallacies. (shrink)

John Corcoran. 1979 Review of Hintikka and Remes. The Method of Analysis (Reidel, 1974). Mathematical Reviews 58 3202 #21388. -/- The “method of analysis” is a technique used by ancient Greek mathematicians (and perhaps by Descartes, Newton, and others) in connection with discovery of proofs of difficult theorems and in connection with discovery of constructions of elusive geometric figures. Although this method was originally applied in geometry, its later application to number played an important role in the early (...) development of algebra [Jacob Klein, English translation, Greek mathematical thought and the origin of algebra, especially pp. 154–157, M.I.T. Press, Cambridge, Mass., 1968]. -/- It is universally agreed that the method of analysis begins by “assuming the thing sought after” (e.g., in geometry, the truth of the proposition to be proved or the existence of the geometric figure to be constructed). Aside from this, little else can be taken for granted. There is disagreement concerning the “direction of analysis”, i.e. whether one is to seek implications of the assumption or whether one is to seek implicants of it. There is also disagreement concerning what is to be “anatomized” (analyzed), i.e., whether one analyzes mathematical objects (figures), mathematical propositions (the axioms, known theorems, and analytic assumption) or an imagined proof (of the analytic assumption from axioms and known theorems). (shrink)

Greek philosophical tradition, not only the Aristotelian one, is strongly associated with proportion (Eco, 1993: 90). This principle of symmetry is generalisable; forasmuch as it is used as a normative rule in figurative arts. Nonetheless, the proportion for Ancient Greeks does not only describe a mathematical relation, but also represents a metaphysical principle. Thus, beauty is the measurement of the elements of the external form (in the case of tragedy, the meter, the symmetry of the parts, the number (...) of syllables, the number of verses) as well as the aesthetic harmony which the play transmits as total to the spectators (in the same case, the content must be neither superior nor inferior to the chosen form; otherwise the form would give the impression of grotesque or inadequate). (shrink)

The strings of physics’ string theory are the binary digits of 1 and 0 used in computers and electronics. The digits are constantly switching between their representations of the “on” and “off” states. This switching is usually referred to as a flow or current. Currents in the two 2-dimensional programs called Mobius loops are connected into a four-dimensional figure-8 Klein bottle by the infinitely-long irrational and transcendental numbers. Such an infinite connection translates - via bosons being ultimately composed of 1’s (...) and 0’s depicting pi, e, √2 etc.; and fermions being given mass by bosons interacting in matter particles’ “wave packets” – into an infinite number of 8-Kleins. Each Klein 1) is one of the universe’s subuniverses (our own is 13.7 billion years old), 2) is made flexible through its binary digits which seamlessly, or almost seamlessly, join it to surrounding subuniverses and eliminate its central hole, and 3) possesses warped time and space because its foundation is the programmed curves in its mathematical Mobius loops (along with the twists they generate [p.7]). The universe functions according to the rules of fractal geometry. So the Mobius does not exist only at the cosmic level. It also manifests at the quantum scale, giving us photons and protons etc. Space and time are no longer separate, but are an indivisible space-time. So if space and the universe are infinite, how can time not be eternal? The past and the future must both extend forever (the idea of time being finite arises from confusion of our subuniverse with the one infinite universe). -/- BITS (Binary digiTS) only suggest existence of the divine if time is linear. Although a non-supernatural God is proposed via the inverse-square law coupled with eternal quantum entanglement, Einstein taught us that time is warped. Warped time is nonlinear, making it at least possible that the BITS composing space-time and all particles originate from the computer science of humans. -/- I suspect many readers will be content with reading this abstract. While there are more details, and mathematics, in the content; my natural style of writing is to avoid jargon and maths. I also tend to get philosophical. While I personally feel that there’s a lot of precious information in the content, I realize it won’t all be to everyone’s liking. Other subjects dealt with in this article are - the “Pioneer anomaly”, refinement of gravitational physics, dark energy and dark matter, quantum phenomena like mass and electric charge and quantum spin, Kepler’s laws of planetary motion, deflection of starlight by the sun, tides, falling bodies, Earth’s orbit, ancient Greek philosophers, Newton, Kepler, Galileo, Aristotle, Parmenides, Zeno of Elea, time travel into the past as well as the future, the elimination of distances in space, humanity’s construction of this universe we live in, The Law of Conservation of Matter-Energy, and support for the science-fiction-like idea of the electronic binary digits of 1 and 0 being the building blocks of our universe. (shrink)

There was a time in my school years when I have learned about Achilles and Tortoise “paradox” originated from Zeno. It was then clear that the ancient Greeks were arguing about this problem but contemporary science has clarified the issue. Yet to my surprise the problem is still debated over and over, despite the fact there exist mathematical proofs. I feel like reminding myself why this is not a paradox beyond reasonable doubt. This is a draft to a section (...) of a future publication. (shrink)

A model is proposed for interpreting the course of Western Science’s conception of mathematics from the time of the ancient Greeks to the present day. According to this model, philosophy of science, in general, has traced a horseshoe-shaped curve through time. The ‘horseshoe’ emerges with Pythagoras and other Greek scientists and has curved ‘back’—but not quite back—towards modern trends in philosophy of science, as for example espoused by Bas van Fraassen. Two features of a horseshoe are pertinent (...) to this metaphor: (1) The horseshoe’s semicircular shape models the emergence of science from monism towards dualism, and its modern return towards unity. (2) The logical ‘horseshoe’ operator (“if…then”) makes possible the logical rule ‘modus ponens’--so central to Western science’s deductive approach (which is reaching its limits). Descartes fits approximately at the turning point of the curve. ‘Where might the curve go next?’ is posed to the reader. (shrink)

The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. Next, the (...) geometrical meaning of the double contradiction is considered as follows. A tangent at a point on a convex curve is iterated. Then, the slope of the tangent at the point is sandwiched by two kinds of lines. The first kind of line crosses the curve at the original point and a point to the right of it. The second kind of line crosses the curve at the original point and a point to the left of it. Then, the double contradiction can be applied, and the slope of the tangent is determined as a single value. Finally, the meaning of this method for the foundation of mathematics is considered. We reflect on Dehaene’s notion that the foundation of mathematics is based on the intuitions, which evolve independently. Hence, there may be gaps between intuitions. In fact, the Ancient Greeks identified inconsistency between arithmetic and geometry. However, Eudoxus developed the theory of proportion, which is equivalent to the Dedekind Cut. This allows the iteration of an irrational number by rational numbers as precisely as desired. Simultaneously, we can define the irrational number by the double contradiction, although its existence is not guaranteed. Further, an area of a curved figure is iterated and defined by rectilinear figures using the double contradiction. (shrink)

The explanation of nature in theoretical terms was first postulated and initiated by Ancient Greek philosophers. With the rise of monotheistic religions, however, curiosity about our transient world was widely regarded as contributing nothing to salvation. There was a decline in natural philosophy, which lasted for several centuries and was then reversed both in Islamic philosophy and in Christian theology in the Middle Ages. At this point, the "Book of Nature" was recognized as a complement to the Book (...) of Revelation. Originally, Aristotelian philosophy played the leading part in this process, but only after Aristotelian physics was substituted by experimental mathematical mechanics did it become possible for modern science to develop. General laws of physics form the basis of the explanation of events in space and time, including life processes. However, modern science has also revealed its own limitations. The scope and limits of scientific knowledge allows for different philosophical, cultural and religious interpretations of human beings and the universe. The history of science demonstrates erratic and persistent developments, but the long-term retrospective view discredits extreme forms of relativism and structuralism. Science appears as a construct of the human mind, and yet it is capable of generating valid explanations of nature. In the book more attention than usual is given to Cusanus (Nikolas de Cusa). -/- . (shrink)

I demonstrate that Deleuze's identification of Aion as an empty form offers a fascinating model of temporality that prioritises variation. First, I suggest that Deleuze's identification of time as an empty form is supported by ancient Greek and Gnostic concepts of the relation of Aion and Chronos. From Plato, through Aristotle, to Plotinus the concept of time undergoes substantive revision, in the sense that temporal measurement becomes removed from the measurement of existent entities. This gradual untethering of time (...) from movement gives rise to the development of the concept of eternity as an ontologically comprehensive mode of time that is devoid of content. It is here, with Deleuze's reading of the Platonic cosmology, that we see the first hints of the suggestion that Aion is involved with ontogenesis. Eternity is characterised as: a temporal ‘all’ that is non-reducible to the determinacy implied by any particular temporally localised existent or temporal series ; that which tends towards a diversity of possible states of affairs. Perhaps one of the most interesting aspects in the long history of Aion is that – in the ancient world – it was used in magical incantations. For the Gnostics and Oracles, Aion was a deity, and a potent one at that. From the Gnostic papyri, we get a vision of Aion as a force which enjoys eternal realisation. I suggest that the papyri conjure an image of Aion as a deity that is liberated from time, in the sense that it enjoys a neutrality with respect to the movements of any particular entity or group of entities – a form, in the most general sense of the term. Then, to clarify this mercurial aspect of an empty form of time, I elaborate on a complex analogy between Aion and Deleuze's concept of an ‘ideal game’ – an analogy that Deleuze specifies through reference to Fitzgerald and Borges. The claim is that Aion is an analogue of an ideal game in the sense that both share essential properties. Both Aion and the ideal game involve the multiplication of chance. Finally, I suggest that the differential aspects of Aion imply that it is pure variability; something which can be illustrated through a differential equation. These yield the suggestion that Aion enjoys realisation as an ontogenetic force. I further claim that ontogenetic forces may enjoy expression as the content of literature and mathematics. In concrete terms, temporality is involved in the creation of existent entities, which may be illustrated as various types of continuous multiplicity. (shrink)

The conceptualisation of numbers is culturally bound. This may seem like a counterintuitive claim, but one illustration thereof is the limitations of the resemblance of the ancient Greek concept of number to that in modern mathematics.

The concept of discrete space can be termed as “the ex­ternal mathematical reality hypothesis”. The concept was already known among the ancient Greek philosophers (≈ 500 BC). Unfortunately the phenomenological point of view has dominated science during more than 2000 years and it is only recently that the concept of discrete space gets “tangible” attention again in philosophy and theoretical physics. Although the model de­scribes the existence of the universal conservation laws, constants and principles in a convincing way, (...) the re­lation between phenomenological reality and the geo­metrical description of discrete space is difficult to ima­gine for everyone who is only familiar with phenomeno­logical reality. The purpose of this paper is to describe some easy to imagine properties of discrete space. DOI: 10.5281/zenodo.5498120. (shrink)

Origin of life studies have presented one of the most serious challenges to the mechanistic conception that life can be explained scientifically as a mere product of chemistry and physics. Hypotheses about the origin of life can be divided into two categories: (1) biogenesis – life comes from life, and (2) abiogenesis – life comes from non-living matter. The theory of the spontaneous generation of life from inanimate matter had been held even by the ancient Greeks and by numerous (...) scientists well into the 19th century. The theory of abiogenesis poses many problems for understanding the origin of life on Earth, and the appearance of life early in Earth’s history. Numerous chemical, mathematical and informational problems arise which make random mechanical processes of cellular formation and function unlikely. Fossil evidence contradicts a gradualist evolutionary mechanism of development of life, especially the well-known Cambrian explosion, in which highly developed metazoan species suddenly appear in the geological column without intermediate predecessors. But the physical conundrums that mechanistic theories of chemistry and physics face are only one side of the problem. Along with a rising chorus of philosophers, Thomas Nagel, an atheist philosopher, has protested that essential questions about the origin of life, and features such as mind, intelligence and morality are completely left unexplained by mechanistic evolutionary theories. In Mind and Cosmos: Why the Materialist Neo-Darwinian Conception of Nature Is Almost Certainly False, Nagel plainly lays out his argument that the modern materialist approach to life has conspicuously failed to explain something so integral to nature as mind or consciousness, thereby threatening to unravel the entire naturalistic world picture of biology, evolutionary theory and cosmology. [13] As an alternative he argues that at least natural teleological principles must be admitted to play a role in our view of science. He writes: “Each of our lives is a part of the lengthy process of the universe gradually waking up and becoming aware of itself.” The Vedantic view of the Absolute as sentient conceives of Bhagavan as the conscious and consequently personal source of the universe. This view holds that life is fundamental, and not merely coextensive with matter. It is thus consistent with the law of biogenesis which is scientifically established in agreement with empirical evidence. Life is the basis of Nature, not matter, and Nature is a system in which the different species are nodes or niches, each possessed of variety and adaptability. Evolution is of consciousness, not of the bodies of organisms. The sedimentary fossils are the result of catastrophic deposits, and are thus not indicative of gradual evolution which is concluded only on the questionable assumption of uniformitarianism. (shrink)

A recently developed computational methodology for executing numerical calculations with infinities and infinitesimals is described in this paper. The approach developed has a pronounced applied character and is based on the principle “The part is less than the whole” introduced by the ancient Greeks. This principle is applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The point of view on infinities and infinitesimals (and in general, on Mathematics) presented in (...) this paper uses strongly physical ideas emphasizing interrelations that hold between a mathematical object under observation and the tools used for this observation. It is shown how a new numeral system allowing one to express different infinite and infinitesimal quantities in a unique framework can be used for theoretical and computational purposes. Numerous examples dealing with infinite sets, divergent series, limits, and probability theory are given. (shrink)

“Being” or “being and beings” might be the meta-notion when philosophy and science were still sharing the same mind or minds in ancient Greek and Europe. To find its meaning and to become a part of the meaning are however more and more vital nowadays, not for philosophy but for science, especially for physics and biology. The author proposes in this essay that this concept may mean to us nothing more or less than the unity or unification of (...) Parmenides’ idea “being is one” and Heraclitus’ idea “all things are in flux and nothing may remain the same”. That is just as to say that a being and an entity may never be identified with the same boundary. In other words, nothing or none should ever be understood as a being. This understanding is conceptualized and linguisticalized by the author as the compound symbol “OC”. And the OC represents a synthetic and also analytical understanding of both our universe and us together. And its value and strength may be found more obvious in his concise discussion, in the last part of the essay, on the OC’s meanings to ontology, epistemology, axiology, logic, mathematics, physics, biology, cosmogony and more. (shrink)

Cannabis in the Ancient Greek and Roman World explores the use of cannabis and hemp in medicine, religion, and recreation in the classical period. This work surveys the plant in Greek and Roman literature and provides a compendium of primary sources discussing hemp through the Middle Ages.

Ancient Greek comedy takes interesting approaches to mythological narrative. This article analyzes one excerpt and eight fragments of ancient Greek Old, Middle, and New Comedy. It attempts to show a comic rationalizing approach to mythology. Poets analyzed include Aristophanes, Cratinus, Anaxilas, Timocles, Antiphanes, Anaxandrides, Philemon, Athenion, and Comic Papyrus. Comparisons are made to known rationalizing approaches as found in the mythographers Palaephatus and Heraclitus the Paradoxographer. Ancient comedy tends to make jokes about the ludicrous aspects (...) of myth. Early Greek myth rationalization and mythography share a similar approach to comedy in that they attempt to rationalize the improbable parts of myth narrative. (shrink)

This chapter aims to illuminate ways in which hope was significant in the philosophy of classical Greece. Although ancient Greek philosophies contain few dedicated and systematic expositions on the nature of hope, they nevertheless include important remarks relating hope to the good life, to reason and deliberation, and to psychological phenomena such as memory, imagination, fear, motivation, and pleasure. After an introductory discussion of Hesiod and Heraclitus, the chapter focuses on Plato and Aristotle. Consideration is given both to (...) Plato’s direct comments on hope and to the narrative contexts of his dialogues, with analysis of Plato’s positive and negative representations of hope, hope’s relationship to reason, and Plato’s more psychological approach in the Philebus, where hope finds a place among memory, recollection, pleasure, and pain. The chapter then reviews Aristotle’s discussions of confidence, hope, and courage, observing that although Aristotle does not mention hope as a virtue, he does note its importance to human agency and deliberation and as a foundation for the further development of virtue. The chapter concludes that discussions surrounding hope in ancient Greek philosophy are rich and challenging and can serve as a lively stimulus to further exploration of the concept of hope. (shrink)

This paper is about the history of a question in ancient Greek philosophy and medicine: what holds the parts of a whole together? The idea that there is a single cause responsible for cohesion is usually associated with the Stoics. They refer to it as the synectic cause (αἴτιον συνεκτικόν), a term variously translated as ‘cohesive cause,’ ‘containing cause’ or ‘sustaining cause.’ The Stoics, however, are neither the first nor the only thinkers to raise this question or to (...) propose a single answer. Many earlier thinkers offer their own candidates for what actively binds parts together, with differing implications not only for why we are wholes rather than heaps, but also why our bodies inevitably become diseased and fall apart. This paper assembles, up to the time of the Stoics, one part of the history of such a cause: what is called ‘the synechon’ (τὸ συνέχον) – that which holds things together. Starting with our earliest evidence from Anaximenes (sixth century BCE), the paper looks at different candidates and especially the models and metaphors for thinking about causes of cohesion which were proposed by different philosophers and doctors including Empedocles, early Greek doctors, Diogenes of Apollonia, Plato and Aristotle. My goal is to explore why these candidates and models were proposed and how later philosophical objections to them led to changes in how causes of cohesion were understood. (shrink)

Good governance, first appeared in the nineties within the United Nations, the World Bank and International Monetary Fund refers to describe how public organizations best conduct public affairs and deliver public goods and services. Today, about three decades later good governance seems to be still popular since there are still many challenges ahead for many governments especially in less-developed and developing countries. Hence the notion of good governance was emerged as a normative commencement of the principles, values and ethics to (...) realise the acts of governance. Some thoughts on good governance and related topics advocated by three great ancient philosophers, who appear to be relevant at the present time are considered herein. One of the said philosophers is Kautilya (also known as Chanakya and Vishnugupta), who lived in India around 150AD. The others are the two great Greek philosophers Plato (427- 347BC) and Aristotle (384-322 BC), who stand with Socrates as the shapers of the whole intellectual tradition of the West. The thoughts are extracted from that detailed manual of state-craft and the science of living known as the Arthashastra written by one of classical India’s greatest minds Kautilya, Plato’s great work known as ‘The Republic’ and Aristotle’s treatise titled ‘The Politics”. The objective of this paper is to study the thoughts of ancient Greek philosophers who lived many centuries ago and their ideas are still relevant today. (shrink)

From famous figures in the history of philosophy to questions in religious theology to the relationship between knowledge and power, The Handy Western Philosophy Answer Book: Ancient Greek to Its Influence on Philosophy Today takes the sometimes esoteric ideas and the jumble of names and makes them easy to understand, enriching readers' lives and answering the question "What do the ancient Greek philosophers have to teach us about contemporary culture?".

Phenomenology, broadly construed, is the study of the meaningful structure of human experience. It is a philosophical tradition that begins with Edmund Husserl, develops with thinkers like Martin H...

In this second of two essays on the topic of ancient Greek inscriptions, I will briefly explore and discuss the textuality of ritual norms or, 'sacred laws', by looking 1) at the reasons for these ritual norms to have been written down in the first place and 2) how these norms/laws/decrees were able to get their observers to adhere to them. Throughout the essay I have made use of J.L. Austin's Speech Act Theory to better contextualize the meaning (...) of the inscriptions dealt with. (shrink)

A new method of adult and children mental development is investigated – the trilogy of mind, which basic operation is the logic operation of comparison. This method was created considering Aristotle's understanding of philosophy as “the science about the first reasons and origins” of cognition, the beginning of presocratic cognition of the surrounding world. This new method is an effective mean of mental thinking development that is oriented at the understanding of natural and social processes with the help of general (...) scientific comparative notions – kinds of opposition, worked out in ancient times. (shrink)

This book offers inter alia a systematic investigation of the actual argumentative strategy of Socratic conversation and explorations of Socratic and Platonic morality including an examination ofeudaimonia and the mental conception of health in the Republic as self-control, with a view to the relation of individual health/happiness to social order. The essays cover a period from 1968 to 2012. Some of them are now published for the first time. Self-motion in the later dialogues involves tripartition and tripartition in turn involves (...) embodiment. The Philebus psychology too anticipates Aristotle. The Forms of the Timaeus are patterns, but the two-world picture is abandoned: there is one world constituted by Forms and Place. The Epinomis is arguably genuine. More generally, denying that Plato develops, e.g. exegetically and psychologically, is absurd. There are too many contradictions in the Corpus. The dialogues are artistic wholes and the author's message must be interpreted accordingly: hence in a sense every character is Plato's mouthpiece. Aristotle's idea of the human good or quality of life as optimal mental activity according to the special human capabilities is the root of the modern selfactualization projects. Panaetius (free reason) and Posidonius (science) mark the end of the older Stoa's hard-core materialism and the beginning of a new more 'modern' era. -/- . (shrink)

This paper reconsiders Antigone’s role in the ancient Greek polis in the framework of Hegel’s concept of Sittlichkeit, as developed in the Phenomenology of Spirit. My main hypothesis is that Antigone appears to challenge both the Greek androcentric order and Hegel’s hypotheses on subjectivity. I prove this by reevaluating Hegel’s notion of the Ethical act (sittliche Handlung). Finally, I identify the endowment of Sittlichkeit on natural sexual distinction as the real reason for its collapse and point out (...) the problematic consequences of such endowment for further development of the Phenomenology of Spirit. (shrink)

There seems to be confusion and disagreement among scholars about the meaning of interpersonal forgiveness. In this essay we shall venture to clarify the meaning of forgiveness by examining various literary works. In particular, we shall discuss instances of forgiveness from Homer’s The Iliad, Euripides’ Hippolytus, and Aristotle’s Nicomachean Ethics and we shall focus on the changes that the concept of forgiveness has gone through throughout the centuries, in the hope of being able to understand, and therefore, of being able (...) to use more accurately, contemporary notions of forgiveness. We shall also explore the relationship between forgiveness and concepts that are closely associated with it, such as anger/resentment, hurt, clemency, desert/merit, excuse, etc. (shrink)

The question on the essence of man and his relationship to nature is certainly one of the most important themes in the philosophy of Hans Jonas. One of the ways by which Jonas approaches the issue consists in a comparison between the contemporary interpretation of man and forms of wisdom such as those conveyed by ancient Greek philosophy and the Jewish tradition. The reconstruction and discussion of these frameworks play a fundamental role in Jonas’s critique of the modern (...) mind. In the first section I introduce the anthropological problem in Hans Jonas’s oeuvre. Moreover, I clarify why it becomes essential for Jonas to resort to different forms of traditional wisdom. In the second and third sections I try to give an account (as complete as possible) of the two generalisations which Jonas shapes in order to criticise the modern concepts of man and nature. In the last section I show how Jonas links these generalisations to his own philosophical assessment of modernity. Finally, I focus on his methodology, which exemplifies how critical thinking may arise from a reconsideration of traditional contents. (shrink)

This paper seeks to explore the way Giovanni Pico della Mirandola treated the Orphics and the Pythagoreans in his Conclusiones nongentae, his early and most ambitious work, so that he formulates his own philosophy. I do not intend to present and analyze the sum of Pico’s references to Orphics and Pythagoreans, since such an attempt is beyond the scope of this paper. Rather, I aim to highlight certain Pico’s aphorisms that allow readers to understand and evaluate his syncretic method and (...) his goals. In addition, I attempt to trace Pico’s sources and evaluate his proper knowledge, understanding and treatment of the Orphics and the Pythagoreans. Pico resorts to the Orphics and the Pythagoreans because he wants to give a practical and applied dimension to his philosophy. He attempted the revival of the original wisdom that underlies the traditions he combined. He was not a fan of the Aristotelian θεωρίης ἢνεκεν. Philosophy does not aim at proper knowledge, but is the key for the manipulation of the cosmos, physically and metaphysically speaking. Pico probably thinks of himself as the modern Orpheus: he does not intend to reveal the paths which cross the sensible and the intelligible, but he aspires to tread them. (shrink)

In the chapter Spirit of the book "Phenomenology of spirit" in a section called "True spirit, ethical Life", Hegel looks into the happy state of "ethical life" in Greece. The concept of ethical life is a very crucial concept because it formulates Hegel's fundamental political and social ideal, which is to establish synthesis between the community and the individual. In this research, we study the ethical life of people who are unreasonably immersed in the customs and laws of a certain (...) society. In the following, we see that this world is divided and the unity of laws in it is shaken. For this reason, Hegel discusses the literary genres of epic and tragedy in order to show that in them, a person progressively becomes himself and becomes aware of his individual character. Hegel destroys his primitive ethical substance dialectically through essential conflicts, with forced actions and untangling choices of people with will and as a result people are deprived of their social identities. In this dialectical process, the "living unity" of the polis is fragmented and becomes a mass of separate atoms. (shrink)

Bios Philosophos. Philosophy in Ancient Greek Biography (Brepols, 2016), organized by Mauro Bonazzi and Stefan Schorn, delivers deep and wide tours through the philosophical aspects of Greek biographical production. On the one hand, it does not concentrate only on the later periods of Greek philosophy, when biographical production abounded; instead, it goes all the way back to the fourth century BCE, when biographical texts were fragmentary and mingled with other styles. On the other, it tries to (...) unveil the philosophical motives in authors' works who tend to be disregarded as historians, biographers, hagiographers, or even as mere fans of the most prominent figures of their own schools. -/- In our review, we will attempt to give a brief account of the ten articles that make up this volume, which, in turn, will hopefully provide an overview of the different connections between the biographies and biographers and their philosophical motives. (shrink)

Since the dawn of philosophy, the paradoxical interconnection between the continuous and the discrete plays a central rôle in attempts to understand the ontology of the world, while defying all attempts at consistent formulation. I investigate the relation between (classical) logic and concepts of “space” and “time” in physical and metaphysical theories, starting with the Greeks. An important part of my research consists in exploring the strong connections between paradoxes as they appear and are dealt with in ancient philosophy, (...) and their re-appearance in early modern natural philosophy, as well as in the foundations of contemporary science and mathematics. The way paradoxes are dealt with sheds light on a theory’s hidden metaphysical assumptions, especially with respect to matter, space, time and causation, it defines its ontological signature. This conclusion led me to the in-depth study of early modern natural philosophy, the origin of natural science, especially the influences ancient thinkers had on the new conceptions of space, time and causation developed by Newton, Huygens and Leibniz. (shrink)

The article deals with the doctrines of Orpheus and Pythagoras about the immortality of the soul in the context of the birth of philosophy in ancient Greece. Orpheus demonstrated the closeness of heavenly (divine) and earthly (human) worlds, and Pythagoras mathematically proved their fundamental identity. Greek philosophy was “an investment in the afterlife future”, being the product of the mystical (Orpheus) and rationalist (Pythagoras) theology.

'Reason, Causation and Compatibility with the Phenomena' strives to give answers to the philosophical problem of the interplay between realism, explanation and experience. This book is a compilation of essays that recollect significant conceptions of rival terms such as determinism and freedom, reason and appearance, power and knowledge. This title discusses the progress made in epistemology and natural philosophy, especially the steps that led from the ancient theory of atomism to the modern quantum theory, and from mathematization to analytic (...) philosophy. Moreover, it provides possible gateways from modern deadlocks of theory either through approaches to consciousness or through historical critique of intellectual authorities. This work will be of interest to those either researching or studying in colleges and universities, especially in the departments of philosophy, history of science, philosophy of science, philosophy of physics and quantum mechanics, history of ideas and culture. Greek and Latin Literature students and instructors may also find this book to be both a fascinating and valuable point of reference. (shrink)

This chapter contains sections titled: Going Beyond Nature in Order to Explain it Technē, epistēmē and alēthēs doxa Mathematics, pure and applied Observation and Experimental Verification Bibliography.

The author investigates greek ontologies that apparently rely on a conflation of "binary" (x is f) and "monadic" (x is) uses of 'is'. He uses Aristotelian and other texts to support his proposal that these ontologies are explained by the Greeks using two alternative semantic analyses for 'x is F'. The first views it as asserting a relation between x and F, the second as asserting that a "predicative complex" exists, where a predicative complex is a complex consisting of (...) x and F. The article concludes by indicating some uses for the novel concept of predicative complex in aristotelian interpretation. (shrink)

It has been argued in relation to Old Babylonian mathematical procedure texts that their validity or correctness is self-evident. One “sees” that the procedure is correct without it having, or being accompanied by, any explicit arguments for the correctness of the procedure. Even when agreeing with this view, one might still ask about how is the correctness of a procedure articulated? In this work, we present an articulation of the correctness of ancient Egyptian and Old Babylonian mathematical procedure texts (...) – mathematical texts presenting the solution of problems. We endeavor to make explicit and explain how and why the procedures are reliable over and above the fact that their correctness is intuitive. (shrink)

The demand of philosophizing in Africa has faced a history of criticism that has been particularly Eurocentric and strongly biased. However, that trend is changing with the emergence of core philosophical thinking in Africa. This paper is an attempt to articulate a singular issue in this evolution— the originality of African philosophy, through ancient Egypt and its influence on Greek philosophy. The paper sets about this task by first exposing the historical debate on the early beginnings of the (...) philosophical enterprise, with a view to establishing the possibility of philosophical influences in Africa.It then goes ahead to posit the three hypotheses that link Greek philosophy to have developed from the cultural materiality of Ancient Egypt, and the Eurocentric travesty of history in recognizing influences of philosophy as from Europe alone, apart from Egypt. (shrink)

This book treats ancient logic: the logic that originated in Greece by Aristotle and the Stoics, mainly in the hundred year period beginning about 350 BCE. Ancient logic was never completely ignored by modern logic from its Boolean origin in the middle 1800s: it was prominent in Boole’s writings and it was mentioned by Frege and by Hilbert. Nevertheless, the first century of mathematical logic did not take it seriously enough to study the ancient logic texts. A (...) renaissance in ancient logic studies occurred in the early 1950s with the publication of the landmark Aristotle’s Syllogistic by Jan Łukasiewicz, Oxford UP 1951, 2nd ed. 1957. Despite its title, it treats the logic of the Stoics as well as that of Aristotle. Łukasiewicz was a distinguished mathematical logician. He had created many-valued logic and the parenthesis-free prefix notation known as Polish notation. He co-authored with Alfred Tarski’s an important paper on metatheory of propositional logic and he was one of Tarski’s the three main teachers at the University of Warsaw. Łukasiewicz’s stature was just short of that of the giants: Aristotle, Boole, Frege, Tarski and Gödel. No mathematical logician of his caliber had ever before quoted the actual teachings of ancient logicians. -/- Not only did Łukasiewicz inject fresh hypotheses, new concepts, and imaginative modern perspectives into the field, his enormous prestige and that of the Warsaw School of Logic reflected on the whole field of ancient logic studies. Suddenly, this previously somewhat dormant and obscure field became active and gained in respectability and importance in the eyes of logicians, mathematicians, linguists, analytic philosophers, and historians. Next to Aristotle himself and perhaps the Stoic logician Chrysippus, Łukasiewicz is the most prominent figure in ancient logic studies. A huge literature traces its origins to Łukasiewicz. -/- This Ancient Logic and Its Modern Interpretations, is based on the 1973 Buffalo Symposium on Modernist Interpretations of Ancient Logic, the first conference devoted entirely to critical assessment of the state of ancient logic studies. (shrink)

This chapter analyses Soren Kierkegaard's thoughts and opinions about ancient Greek philosophy. It examines the significance of Kierkegaard's references to Greek philosophy in his writings and suggests that his use of classical thought was part of his effort to define his own intellectual project. The chapter investigates how Greek philosophy influenced Kierkegaard's works and views about ethics, existential thought, Socratic faith, love, and virtue, and also considers what Kierkegaard believed was the legacy of ancient (...) class='Hi'>Greek philosophy. (shrink)