Sophism. The term originated from Greek σόφισμα, sophisma, from σοφίζω, sophizo "I am wise"; confer σοφιστής, sophistēs, meaning "wise-ist, one who does wisdom," and σοφός, sophós means "wise man". Etymology[edit] The Greek word sophist (sophistēs) derives from the words sophia, and sophos, meaning "wisdom" or “wise” since the time of Homer and was originally used to describe expertise in a particular knowledge or craft.[1] Gradually, however, the word also came to denote general wisdom and especially wisdom about human affairs (for example, in politics, ethics, or household management). This was the meaning ascribed to the Greek Seven Sages of 7th and 6th century BC (such as Solon and Thales), and it was the meaning that appeared in the histories of Herodotus. Richard Martin refers to the seven sages as "performers of political poetry Sophists of ancient Greece[edit] Protagoras was one of the most well-known and successful teachers.
Gorgias is another well-known Sophist. Modern usage[edit] Socrates. Socrates (/ˈsɒkrətiːz/;[2] Greek: Σωκράτης [sɔːkrátɛːs], Sōkrátēs; 470/469 – 399 BC)[1] was a classical Greek (Athenian) philosopher credited as one of the founders of Western philosophy. He is an enigmatic figure known chiefly through the accounts of classical writers, especially the writings of his students Plato and Xenophon and the plays of his contemporary Aristophanes. Plato's dialogues are among the most comprehensive accounts of Socrates to survive from antiquity, though it is unclear the degree to which Socrates himself is "hidden behind his 'best disciple', Plato".[3] Through his portrayal in Plato's dialogues, Socrates has become renowned for his contribution to the field of ethics, and it is this Platonic Socrates who lends his name to the concepts of Socratic irony and the Socratic method, or elenchus.
Socratic problem Nothing written by Socrates remains extant. Socrates as a figure Socrates as a philosopher Biography Early life Military service Arrest of Leon Trial and death Notes. Plato. Plato (/ˈpleɪtoʊ/; Greek: Πλάτων Plátōn "broad"pronounced [plá.tɔːn] in Classical Attic; 428/427 or 424/423 – 348/347 BCE) was a philosopher, as well as mathematician, in Classical Greece. He is considered an essential figure in the development of philosophy, especially the Western tradition, and he founded the Academy in Athens, the first institution of higher learning in the Western world. Along with his teacher Socrates and his most famous student, Aristotle, Plato laid the foundations of Western philosophy and science.[2] Alfred North Whitehead once noted: "the safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato.
" Plato's dialogues have been used to teach a range of subjects, including philosophy, logic, ethics, rhetoric, religion and mathematics. His lasting themes include Platonic love, the theory of forms, the five regimes, innate knowledge, among others. Biography Early life Birth and family Name Education. Aristotle. Aristotle's views on physical science profoundly shaped medieval scholarship. Their influence extended into the Renaissance and were not replaced systematically until the Enlightenment and theories such as classical mechanics.
Some of Aristotle's zoological observations, such as on the hectocotyl (reproductive) arm of the octopus, were not confirmed or refuted until the 19th century. His works contain the earliest known formal study of logic, which was incorporated in the late 19th century into modern formal logic. His ethics, though always influential, gained renewed interest with the modern advent of virtue ethics. All aspects of Aristotle's philosophy continue to be the object of active academic study today. Though Aristotle wrote many elegant treatises and dialogues – Cicero described his literary style as "a river of gold"[7] – it is thought that only around a third of his original output has survived.[8] Life Aristotle was appointed as the head of the royal academy of Macedon.
Logic. Pythagoras. Greek philosopher (c. 570 – c. 495 BC) Pythagoras of Samos[a] (Ancient Greek: Πυθαγόρας ὁ Σάμιος, romanized: Pythagóras ho Sámios, lit. 'Pythagoras the Samian', or simply Πυθαγόρας; Πυθαγόρης in Ionian Greek; c. 570 – c. 495 BC)[b] was an ancient Ionian Greek philosopher, polymath and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, the West in general.
Knowledge of his life is clouded by legend, but he appears to have been the son of Mnesarchus, a gem-engraver on the island of Samos or the city of Tyre.[5] Modern scholars disagree regarding Pythagoras's education and influences, but they do agree that, around 530 BC, he travelled to Croton in southern Italy, where he founded a school in which initiates were sworn to secrecy and lived a communal, ascetic lifestyle. Biographical sources Life Early life Reputed travels Alleged Greek teachers In Croton Death . Square root of 2. "Pythagoras's constant" redirects here; not to be confused with Pythagoras number The square root of 2, often known as root 2, radical 2, or Pythagoras' constant, and written as Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. Its numerical value, truncated to 65 decimal places, is: 1.41421356237309504880168872420969807856967187537694807317667973799...
The square root of 2 (number line not to scale). The quick approximation 99/70 (≈ 1.41429) for the square root of two is frequently used. History[edit] Babylonian clay tablet YBC 7289 with annotations. The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of in four sexagesimal figures, 1 24 51 10, which is accurate to about six decimal digits:[1] . Computation algorithms[edit] There are a number of algorithms for approximating First, pick a guess, The value of . Pythagorean tuning. The syntonic tuning continuum, showing Pythagorean tuning at 702 cents.[1] Diatonic scale on C Play 12-tone equal tempered and Play just intonation. Pythagorean (tonic) major chord on C Play (compare Play equal tempered and Play just). Comparison of equal-tempered (black) and Pythagorean (green) intervals showing the relationship between frequency ratio and the intervals' values, in cents.
The system had been mainly attributed to Pythagoras (sixth century BC) by modern authors of music theory, while Ptolemy, and later Boethius, ascribed the division of the tetrachord by only two intervals, called "semitonium", "tonus", "tonus" in Latin (256:243 x 9:8 x 9:8), to Eratosthenes. Method[edit] Pythagorean tuning is based on a stack of intervals called perfect fifths, each tuned in the ratio 3:2, the next simplest ratio after 2:1. This succession of eleven 3:2 intervals spans across a wide range of frequency (on a piano keyboard, it encompasses 77 keys). Size of intervals[edit] (e.g. between E♭ and E) Pythagoreanism. Pythagoreanism was the system of esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics, music and astronomy.
Pythagoreanism originated in the 5th century BC and greatly influenced Platonism. Later revivals of Pythagorean doctrines led to what is now called Neopythagoreanism. Two schools[edit] According to tradition, Pythagoreanism developed at some point into two separate schools of thought: the mathēmatikoi (μαθηματικοί, Greek for "learners") andthe akousmatikoi (ἀκουσματικοί, Greek for "listeners"). The mathēmatikoi[edit] The mathēmatikoi were supposed to have extended and developed the more mathematical and scientific work begun by Pythagoras.
The akousmatikoi[edit] The akousmatikoi focused on the more religious and ritualistic aspects of his teachings: they claimed that the mathēmatikoi were not genuinely Pythagorean, but followers of the "renegade" Pythagorean Hippasus. Natural philosophy[edit] Hippasus. Hippasus of Metapontum Hippasus of Metapontum (/ˈhɪpəsəs/; Greek: Ἵππασος, Híppasos; fl. 5th century BC), was a Pythagorean philosopher. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers. The discovery of irrational numbers is said to have been shocking to the Pythagoreans, and Hippasus is supposed to have drowned at sea, apparently as a punishment from the gods, for divulging this. However, the few ancient sources which describe this story either do not mention Hippasus by name or alternatively tell that Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere. The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer.
Some modern scholars though have suggested that he discovered the irrationality of √2, which it is believed was discovered around the time that he lived. Life[edit] Little is known about the life of Hippasus. Doctrines[edit] Irrational number. The famous mathematical constant pi (π) is among the best known irrational numbers and is much represented in popular culture In mathematics, an irrational number is any real number that cannot be expressed as a ratio of integers. Informally, this means that an irrational number cannot be represented as a simple fraction. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals. As a consequence of Cantor's proof that the real numbers are uncountable (and the rationals countable) it follows that almost all real numbers are irrational.[1] History[edit] The number is irrational.
Ancient Greece[edit] Start with an isosceles right triangle with side lengths of integers a, b, and c. Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. The discovery of incommensurable ratios was indicative of another problem facing the Greeks: the relation of the discrete to the continuous. India[edit] Middle Ages[edit] Parmenides. Parmenides of Elea (/pɑrˈmɛnɨdiːz əv ˈɛliə/; Ancient Greek: Παρμενίδης ὁ Ἐλεάτης; fl. 5th century BCE) was an ancient Greek philosopher born in Elea, a Greek city on the southern coast of Magna Graecia. He was the founder of the Eleatic school of philosophy. The single known work of Parmenides is a poem, On Nature, which has survived only in fragmentary form.
In this poem, Parmenides describes two views of reality. In "the way of truth" (a part of the poem), he explains how reality (coined as "what-is") is one, change is impossible, and existence is timeless, uniform, necessary, and unchanging. Life[edit] On Nature[edit] Parmenides is one of the most significant of the pre-Socratic philosophers.[12] His only known work, conventionally titled On Nature, is a poem which has only survived in fragmentary form.
Proem[edit] The Way of Truth[edit] For never shall this prevail, that things that are not are. There are extremely delicate issues here. For to be aware and to be are the same. Monad (philosophy) Philosophical concept of a most basic substance, or supreme being The Euclidean symbolism of the centered sphere also concerns the secular debate on the existence of a center of the universe. The idea of the monad is also reflected in the demiurge, or the belief of one supreme being that brought about the creation of the universe. Empedocles. Empedocles (/ɛmˈpɛdəkliːz/; Ancient Greek: Ἐμπεδοκλῆς; Empedoklēs; Ancient Greek: [empedoklɛ̂ːs]; c. 490 – 430 BC) was a Greek pre-Socratic philosopher and a citizen of Agrigentum, a Greek city in Sicily. Empedocles' philosophy is best known for being the originator of the cosmogenic theory of the four Classical elements. He also proposed powers called Love and Strife which would act as forces to bring about the mixture and separation of the elements.
These physical speculations were part of a history of the universe which also dealt with the origin and development of life. Influenced by the Pythagoreans, he supported the doctrine of reincarnation. Empedocles is generally considered the last Greek philosopher to record his ideas in verse. Some of his work survives, more than in the case of any other Presocratic philosopher.
Empedocles' death was mythologized by ancient writers, and has been the subject of a number of literary treatments. Life[edit] Works[edit] Purifications[edit] Daimonic. Etymology[edit] The root of the word has a referent from daiomai, which is translated as something like to divide or to lacerate.[1] Marie-Louise von Franz delineated the term daiomai (see ref.), and indicates that its usage is specifically when someone perceived an occurrence which they attributed to the influence of a divine presence, amongst the examples provided by Franz, are from attributing to a daimon the occurrence of a horse becoming or being startled.[2] According to another source, the term is thought to have originated with the Greeks, by way of Latin—dæmon: "spirit", derived from Greek—daimon (gen. daimonos): "lesser god, guiding spirit, tutelary deity",[3] a meaning which is attested to elsewhere. History of usage[edit] For the Minoan (3000-1100 BC) and Mycenaean (1500-1100 BC), "daimons" were seen as attendants or servants to the deities, possessing spiritual power.
During the period in which Homer was alive, people believed ailments were both caused and cured by daimons.[2] Monoimus. According to Monoimus, the world is created from the Monad (or iota, or Yod meaning "one horn"), a tittle that brings forth the duad, triad, tetrad, pentad, hexad, heptad, ogdoad, ennead, up to ten, producing a decad. He thus possibly identifies the gnostic aeons with the first elements of the Pythagorean cosmology. He identifies these divisions of different entities with the description of creation in Genesis. This description from Hippolytus also corresponds to two versions of a text called Epistle of Eugnostos found in Nag Hammadi, where the same monad to decad relationship is described. (Eugnostos in turn, has apparent resemblances to the gnostic text The Sophia of Jesus Christ, where the word monad appears again.) Doctrine[edit] Monoimus is famous for his quote about the unity of God and man (from Hippolytus): This idea resembles the viewpoint of the much later Sufi Ibn Arabi, but no connection between the two is known.
Attribution[edit] External links[edit] Euclid.