Willard Van Orman Quine
Willard Van Orman Quine was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century." From 1930 until his death 70 years Quine was continually affiliated with Harvard University in one way or another, first as a student as a professor of philosophy and a teacher of logic and set theory, as a professor emeritus who published or revised several books in retirement. He filled the Edgar Pierce Chair of Philosophy at Harvard from 1956 to 1978. A 2009 poll conducted among analytic philosophers named Quine as the fifth most important philosopher of the past two centuries, he won the first Schock Prize in Logic and Philosophy in 1993 for "his systematical and penetrating discussions of how learning of language and communication are based on available evidence and of the consequences of this for theories on knowledge and linguistic meaning." In 1996 he was awarded the Kyoto Prize in Arts and Philosophy for his "outstanding contributions to the progress of philosophy in the 20th century by proposing numerous theories based on keen insights in logic, philosophy of science and philosophy of language."Quine falls squarely into the analytic philosophy tradition while being the main proponent of the view that philosophy is not conceptual analysis but the abstract branch of the empirical sciences.
His major writings include Two Dogmas of Empiricism, which attacked the distinction between analytic and synthetic propositions and advocated a form of semantic holism, Word and Object, which further developed these positions and introduced Quine's famous indeterminacy of translation thesis, advocating a behaviorist theory of meaning. He developed an influential naturalized epistemology that tried to provide "an improved scientific explanation of how we have developed elaborate scientific theories on the basis of meager sensory input." He is important in philosophy of science for his "systematic attempt to understand science from within the resources of science itself" and for his conception of philosophy as continuous with science. This led to his famous quip that "philosophy of science is philosophy enough." In philosophy of mathematics, he and his Harvard colleague Hilary Putnam developed the "Quine–Putnam indispensability thesis," an argument for the reality of mathematical entities. According to his autobiography, The Time of My Life, Quine grew up in Akron, where he lived with his parents and older brother Robert Cloyd.
His father, Cloyd Robert, was a manufacturing entrepreneur and his mother, Harriett E. was a schoolteacher and a housewife. He received his B. A. in mathematics from Oberlin College in 1930, his Ph. D. in philosophy from Harvard University in 1932. His thesis supervisor was Alfred North Whitehead, he was appointed a Harvard Junior Fellow, which excused him from having to teach for four years. During the academic year 1932–33, he travelled in Europe thanks to a Sheldon fellowship, meeting Polish logicians and members of the Vienna Circle, as well as the logical positivist A. J. Ayer, it was Quine who arranged for Tarski to be invited to the September 1939 Unity of Science Congress in Cambridge, for which Tarski sailed on the last ship to leave Danzig before the Third Reich invaded Poland. Tarski survived the war and worked another 44 years in the US. During World War II, Quine lectured on logic in Brazil, in Portuguese, served in the United States Navy in a military intelligence role, deciphering messages from German submarines, reaching the rank of lieutenant commander.
At Harvard, Quine helped supervise the Harvard graduate theses of, among others, David Lewis, Daniel Dennett, Gilbert Harman, Dagfinn Føllesdal, Hao Wang, Hugues LeBlanc, Henry Hiz and George Myro. For the academic year 1964–1965, Quine was a fellow on the faculty in the Center for Advanced Studies at Wesleyan University. In 1980 Quine received an honorary doctorate from the Faculty of Humanities at Uppsala University, Sweden. Quine was an atheist, he had four children by two marriages. Guitarist Robert Quine was his nephew. In the foreword to the new edition of Word and Object, Quine's student Dagfinn Føllesdal noted that Quine began to lose his memory toward the end of his life; the deterioration of his short-term memory was so severe that he struggled to continue following arguments. Quine had considerable difficulty in his project to make the desired revisions to Word and Object. Before passing away, Quine noted to Morton White, "I do not remember what my illness is called, Althusser or Alzheimer, but since I cannot remember it, it must be Alzheimer."
He passed away from the illness on Christmas Day in 2000. Quine was politically conservative, but the bulk of his writing was in technical areas of philosophy removed from direct political issues, he did, write in defense of several conservative positions: for example, in Quiddities: An Intermittently Philosophical Dictionary, he wrote a defense of moral censorship. Quine's Ph. D. thesis and early publications were on formal set theory. Only after World War II did he, by virtue of seminal papers on ontology and language, emerge as a major philosopher. By the 1960s, he had worked out his "naturalized epistemology" whose aim was to answer all substantive questions of knowledge and meaning using the methods and tools of the natural sciences. Quine roundly rejected the notion that there should be a "first philosophy"
William of Ockham
William of Ockham was an English Franciscan friar and scholastic philosopher and theologian, believed to have been born in Ockham, a small village in Surrey. He is considered to be one of the major figures of medieval thought and was at the centre of the major intellectual and political controversies of the 14th century, he is known for Occam's razor, the methodological principle that bears his name, produced significant works on logic and theology. In the Church of England, his day of commemoration is 10 April. William of Ockham was born in Ockham, Surrey in 1285 and joined the Franciscan order at an early age, it is believed that he studied theology at the University of Oxford from 1309 to 1321, but while he completed all the requirements for a master's degree in theology, he was never made a regent master. Because of this, he acquired the honorific title Venerabilis Inceptor, or "Venerable Beginner". During the Middle Ages, theologian Peter Lombard's Sentences had become a standard work of theology, many ambitious theological scholars wrote commentaries on it.
William of Ockham was among these scholarly commentators. However, William's commentary was not well received by the Church authorities. In 1324, his commentary was condemned as unorthodox by a synod of bishops, he was ordered to Avignon, France, to defend himself before a papal court. An alternative understanding proposed by George Knysh, suggests that he was appointed in Avignon as a professor of philosophy in the Franciscan school, that his disciplinary difficulties did not begin until 1327, it is believed that these charges were levied by Oxford chancellor John Lutterell. The Franciscan Minister General, Michael of Cesena, had been summoned to Avignon, to answer charges of heresy. A theological commission had been asked to review his Commentary on the Sentences, it was during this that William of Ockham found himself involved in a different debate. Michael of Cesena had asked William to review arguments surrounding Apostolic poverty; the Franciscans believed that Jesus and his apostles owned no property either individually or in common, the Rule of Saint Francis commanded members of the order to follow this practice.
This brought them into conflict with Pope John XXII. Because of the pope's attack on the Rule of Saint Francis, William of Ockham, Michael of Cesena and other leading Franciscans fled Avignon on 26 May 1328, took refuge in the court of the Holy Roman Emperor Louis IV of Bavaria, engaged in dispute with the papacy, became William's patron. After studying the works of John XXII and previous papal statements, William agreed with the Minister General. In return for protection and patronage William wrote treatises that argued for emperor Louis to have supreme control over church and state in the Holy Roman Empire. "On June 6, 1328, William was excommunicated for leaving Avignon without permission," and William argued that John XXII was a heretic for attacking the doctrine of Apostolic poverty and the Rule of Saint Francis, endorsed by previous popes. William of Ockham's philosophy was never condemned as heretical, he spent much of the remainder of his life writing about political issues, including the relative authority and rights of the spiritual and temporal powers.
After Michael of Cesena's death in 1342, William became the leader of the small band of Franciscan dissidents living in exile with Louis IV. William of Ockham died on 9 April 1347, he was rehabilitated by Innocent VI in 1359. William of Ockham espoused fideism; the ways of God are not open to reason, for God has chosen to create a world and establish a way of salvation within it apart from any necessary laws that human logic or rationality can uncover." He saw God as the only ontological necessity. His importance is as a theologian with a developed interest in logical method, whose approach was critical rather than system building. In scholasticism, William of Ockham advocated reform in both method and content, the aim of, simplification. William incorporated much of the work of some previous theologians Duns Scotus. From Duns Scotus, William of Ockham derived his view of divine omnipotence, his view of grace and justification, much of his epistemology and ethical convictions. However, he reacted to and against Scotus in the areas of predestination, his understanding of universals, his formal distinction ex parte rei, his view of parsimony which became known as Occam's Razor.
William of Ockham was a pioneer of nominalism, some consider him the father of modern epistemology, because of his argued position that only individuals exist, rather than supra-individual universals, essences, or forms, that universals are the products of abstraction from individuals by the human mind and have no extra-mental existence. He advocated the reduction of ontology. William of Ockham is sometimes considered an advocate of conceptualism rather than nominalism, for whereas nominalists held that universals were names, i.e. words rather than extant realities, conceptualists held that they were mental concepts, i.e. the names were names of concepts, which do exist, although only in the mind. Therefore, the universal concept has for its object, not a reality existing in the world outside us, but an
Saint Thomas Aquinas was an Italian Dominican friar, Catholic priest, Doctor of the Church. He is an immensely influential philosopher and jurist in the tradition of scholasticism, within which he is known as the Doctor Angelicus and the Doctor Communis; the name Aquinas identifies his ancestral origins in the county of Aquino in present-day Lazio, Italy. He was the father of Thomism, his influence on Western thought is considerable, much of modern philosophy developed or opposed his ideas in the areas of ethics, natural law and political theory. Unlike many currents in the Church of the time, Thomas embraced several ideas put forward by Aristotle—whom he called "the Philosopher"—and attempted to synthesize Aristotelian philosophy with the principles of Christianity, his best-known works are the Disputed Questions on Truth, the Summa contra Gentiles, the Summa Theologiae. His commentaries on Scripture and on Aristotle form an important part of his body of work. Furthermore, Thomas is distinguished for his eucharistic hymns, which form a part of the Church's liturgy.
The Catholic Church honors Thomas Aquinas as a saint and regards him as the model teacher for those studying for the priesthood, indeed the highest expression of both natural reason and speculative theology. In modern times, under papal directives, the study of his works was long used as a core of the required program of study for those seeking ordination as priests or deacons, as well as for those in religious formation and for other students of the sacred disciplines. Thomas Aquinas is considered philosophers. Pope Benedict XV declared: "This Order... acquired new luster when the Church declared the teaching of Thomas to be her own and that Doctor, honored with the special praises of the Pontiffs, the master and patron of Catholic schools." The English philosopher Anthony Kenny considers Thomas to be "one of the dozen greatest philosophers of the western world". Thomas was most born in the castle of Roccasecca, Aquino, in the Kingdom of Sicily, c. 1225, According to some authors, he was born in the castle of Landulf of Aquino.
Though he did not belong to the most powerful branch of the family, Landulf of Aquino was a man of means. As a knight in the service of King Roger II, he held the title miles. Thomas's mother, belonged to the Rossi branch of the Neapolitan Caracciolo family. Landulf's brother Sinibald was abbot of the first Benedictine monastery at Monte Cassino. While the rest of the family's sons pursued military careers, the family intended for Thomas to follow his uncle into the abbacy. At the age of five Thomas began his early education at Monte Cassino but after the military conflict between the Emperor Frederick II and Pope Gregory IX spilled into the abbey in early 1239, Landulf and Theodora had Thomas enrolled at the studium generale established by Frederick in Naples, it was here that Thomas was introduced to Aristotle and Maimonides, all of whom would influence his theological philosophy. It was during his study at Naples that Thomas came under the influence of John of St. Julian, a Dominican preacher in Naples, part of the active effort by the Dominican order to recruit devout followers.
There his teacher in arithmetic, geometry and music was Petrus de Ibernia. At the age of nineteen Thomas resolved to join the founded Dominican Order. Thomas's change of heart did not please his family. In an attempt to prevent Theodora's interference in Thomas's choice, the Dominicans arranged to move Thomas to Rome, from Rome, to Paris. However, while on his journey to Rome, per Theodora's instructions, his brothers seized him as he was drinking from a spring and took him back to his parents at the castle of Monte San Giovanni Campano. Thomas was held prisoner for one year in the family castles at Monte San Giovanni and Roccasecca in an attempt to prevent him from assuming the Dominican habit and to push him into renouncing his new aspiration. Political concerns prevented the Pope from ordering Thomas's release, which had the effect of extending Thomas's detention. Thomas passed this time of trial tutoring his sisters and communicating with members of the Dominican Order. Family members became desperate to dissuade Thomas.
At one point, two of his brothers resorted to the measure of hiring a prostitute to seduce him. According to legend, Thomas drove her away wielding a fire iron and two angels appeared to him as he slept and strengthened his determination to remain celibate. By 1244, seeing that all of her attempts to dissuade Thomas had failed, Theodora sought to save the family's dignity, arranging for Thomas to escape at night through his window. In her mind, a secret escape from detention was less damaging than an open surrender to the Dominicans. Thomas was sent first to Naples and to Rome to meet Johannes von Wildeshausen, the Master General of the Dominican Order. In 1245 Thomas was sent to study at the Faculty of the Arts at the University of Paris, where he most met Dominican scholar Albertus Magnus the holder of the Chair of Theology at the College of St. James in Paris; when Albertus was sent by his superiors to teach at the new studium generale at Cologne in 1248, Thomas followed him, declining Pope Innocent
René Descartes was a French philosopher and scientist. A native of the Kingdom of France, he spent about 20 years of his life in the Dutch Republic after serving for a while in the Dutch States Army of Maurice of Nassau, Prince of Orange and the Stadtholder of the United Provinces, he is considered one of the most notable intellectual figures of the Dutch Golden Age. Descartes' Meditations on First Philosophy continues to be a standard text at most university philosophy departments. Descartes' influence in mathematics is apparent, he is credited as the father of analytical geometry, the bridge between algebra and geometry, used in the discovery of infinitesimal calculus and analysis. Descartes was one of the key figures in the Scientific Revolution. Descartes refused to accept the authority of previous philosophers, he set his views apart from those of his predecessors. In the opening section of the Passions of the Soul, an early modern treatise on emotions, Descartes goes so far as to assert that he will write on this topic "as if no one had written on these matters before".
His best known philosophical statement is "I think, therefore I am", found in Discourse on the Method and Principles of Philosophy. Many elements of his philosophy have precedents in late Aristotelianism, the revived Stoicism of the 16th century, or in earlier philosophers like Augustine. In his natural philosophy, he differed from the schools on two major points: first, he rejected the splitting of corporeal substance into matter and form. In his theology, he insists on the absolute freedom of God's act of creation. Descartes laid the foundation for 17th-century continental rationalism advocated by Spinoza and Leibniz, was opposed by the empiricist school of thought consisting of Hobbes, Locke and Hume. Leibniz and Descartes were all well-versed in mathematics as well as philosophy, Descartes and Leibniz contributed to science as well. René Descartes was born in La Haye en Touraine, France, on 31 March 1596, his mother, Jeanne Brochard, died soon after giving birth to him, so he was not expected to survive.
Descartes' father, was a member of the Parlement of Brittany at Rennes. René lived with his great-uncle. Although the Descartes family was Roman Catholic, the Poitou region was controlled by the Protestant Huguenots. In 1607, late because of his fragile health, he entered the Jesuit Collège Royal Henry-Le-Grand at La Flèche, where he was introduced to mathematics and physics, including Galileo's work. After graduation in 1614, he studied for two years at the University of Poitiers, earning a Baccalauréat and Licence in canon and civil law in 1616, in accordance with his father's wishes that he should become a lawyer. From there he moved to Paris. In Discourse on the Method, Descartes recalls, I abandoned the study of letters. Resolving to seek no knowledge other than that of which could be found in myself or else in the great book of the world, I spent the rest of my youth traveling, visiting courts and armies, mixing with people of diverse temperaments and ranks, gathering various experiences, testing myself in the situations which fortune offered me, at all times reflecting upon whatever came my way so as to derive some profit from it.
Given his ambition to become a professional military officer, in 1618, Descartes joined, as a mercenary, the Protestant Dutch States Army in Breda under the command of Maurice of Nassau, undertook a formal study of military engineering, as established by Simon Stevin. Descartes, received much encouragement in Breda to advance his knowledge of mathematics. In this way, he became acquainted with Isaac Beeckman, the principal of a Dordrecht school, for whom he wrote the Compendium of Music. Together they worked on free fall, conic section, fluid statics. Both believed that it was necessary to create a method that linked mathematics and physics. While in the service of the Catholic Duke Maximilian of Bavaria since 1619, Descartes was present at the Battle of the White Mountain outside Prague, in November 1620. According to Adrien Baillet, on the night of 10–11 November 1619, while stationed in Neuburg an der Donau, Descartes shut himself in a room with an "oven" to escape the cold. While within, he had three dreams and believed that a divine spirit revealed to him a new philosophy.
However, it is that what Descartes considered to be his second dream was an episode of exploding head syndrome. Upon exiting, he had formulated analytical geometry and the idea of applying the mathematical method to philosophy, he concluded from these visions that the pursuit of science would prove to be, for him, the pursuit of true wisdom and a central part of his life's work. Descartes saw clearly that all truths were linked with one another so that finding a fundamental truth and proceeding with logic would open the way to all science. Descartes discovered this basic truth quite soon: his famous "I think, therefore I am". In 1620 Descartes left the army, he visited Basilica della Santa Casa in Loreto visited various countries before returning to France, during the next few years spent time in Paris. It was there that he compo
Aristotle was a philosopher during the Classical period in Ancient Greece, the founder of the Lyceum and the Peripatetic school of philosophy and Aristotelian tradition. Along with his teacher Plato, he is considered the "Father of Western Philosophy", his writings cover many subjects – including physics, zoology, logic, aesthetics, theatre, rhetoric, linguistics, economics and government. Aristotle provided a complex synthesis of the various philosophies existing prior to him, it was above all from his teachings that the West inherited its intellectual lexicon, as well as problems and methods of inquiry; as a result, his philosophy has exerted a unique influence on every form of knowledge in the West and it continues to be a subject of contemporary philosophical discussion. Little is known about his life. Aristotle was born in the city of Stagira in Northern Greece, his father, died when Aristotle was a child, he was brought up by a guardian. At seventeen or eighteen years of age, he joined Plato's Academy in Athens and remained there until the age of thirty-seven.
Shortly after Plato died, Aristotle left Athens and, at the request of Philip II of Macedon, tutored Alexander the Great beginning in 343 BC. He established a library in the Lyceum which helped him to produce many of his hundreds of books on papyrus scrolls. Though Aristotle wrote many elegant treatises and dialogues for publication, only around a third of his original output has survived, none of it intended for publication; the fact that Aristotle was a pupil of Plato contributed to his former views of Platonism, following Plato's death, Aristotle developed an increased interest in natural sciences and adopted the position of immanent realism. Aristotle's views on physical science profoundly shaped medieval scholarship, their influence extended from Late Antiquity and the Early Middle Ages into the Renaissance, were not replaced systematically until the Enlightenment and theories such as classical mechanics. Some of Aristotle's zoological observations found in his biology, such as on the hectocotyl arm of the octopus, were disbelieved until the 19th century.
His works contain the earliest known formal study of logic, studied by medieval scholars such as Peter Abelard and John Buridan. Aristotle's influence on logic continued well into the 19th century He influenced Islamic thought during the Middle Ages, as well as Christian theology the Neoplatonism of the Early Church and the scholastic tradition of the Catholic Church. Aristotle was revered among medieval Muslim scholars as "The First Teacher" and among medieval Christians like Thomas Aquinas as "The Philosopher", his ethics, though always influential, gained renewed interest with the modern advent of virtue ethics, such as in the thinking of Alasdair MacIntyre and Philippa Foot. In general, the details of Aristotle's life are not well-established; the biographies written in ancient times are speculative and historians only agree on a few salient points. Aristotle, whose name means "the best purpose" in Ancient Greek, was born in 384 BC in Stagira, about 55 km east of modern-day Thessaloniki.
His father Nicomachus was the personal physician to King Amyntas of Macedon. Both of Aristotle's parents died when he was about thirteen, Proxenus of Atarneus became his guardian. Although little information about Aristotle's childhood has survived, he spent some time within the Macedonian palace, making his first connections with the Macedonian monarchy. At the age of seventeen or eighteen, Aristotle moved to Athens to continue his education at Plato's Academy, he remained there for nearly twenty years before leaving Athens in 348/47 BC. The traditional story about his departure records that he was disappointed with the Academy's direction after control passed to Plato's nephew Speusippus, although it is possible that he feared the anti-Macedonian sentiments in Athens at that time and left before Plato died. Aristotle accompanied Xenocrates to the court of his friend Hermias of Atarneus in Asia Minor. After the death of Hermias, Aristotle travelled with his pupil Theophrastus to the island of Lesbos, where together they researched the botany and zoology of the island and its sheltered lagoon.
While in Lesbos, Aristotle married Hermias's adoptive daughter or niece. She bore him a daughter, whom they named Pythias. In 343 BC, Aristotle was invited by Philip II of Macedon to become the tutor to his son Alexander. Aristotle was appointed as the head of the royal academy of Macedon. During Aristotle's time in the Macedonian court, he gave lessons not only to Alexander, but to two other future kings: Ptolemy and Cassander. Aristotle encouraged Alexander toward eastern conquest and Aristotle's own attitude towards Persia was unabashedly ethnocentric. In one famous example, he counsels Alexander to be "a leader to the Greeks and a despot to the barbarians, to look after the former as after friends and relatives, to deal with the latter as with beasts or plants". By 335 BC, Aristotle had returned to Athens. Aristotle conducted courses at the school for the next twelve years. While in Athens, his wife Pythias died and Aristotle became involved with Herpyllis of Stagira, who bore him a son whom he named after his father, Nicomachus.
According to the Suda, he had an erômenos, Palaephatus of Abydus. This period in Athens, between 335 and 323 BC, is when Aristotle is believed to have composed many of his works, he wrote many dialogues. Those works that have survived are in treatise form and were not
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions and implications of mathematics, purports to provide a viewpoint of the nature and methodology of mathematics, to understand the place of mathematics in people's lives. The logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts. Recurrent themes include: What is the role of humankind in developing mathematics? What are the sources of mathematical subject matter? What is the ontological status of mathematical entities? What does it mean to refer to a mathematical object? What is the character of a mathematical proposition? What is the relation between logic and mathematics? What is the role of hermeneutics in mathematics? What kinds of inquiry play a role in mathematics? What are the objectives of mathematical inquiry? What gives mathematics its hold on experience? What are the human traits behind mathematics? What is mathematical beauty? What is the source and nature of mathematical truth?
What is the relationship between the abstract world of mathematics and the material universe? The origin of mathematics is subject to argument. Whether the birth of mathematics was a random happening or induced by necessity duly contingent upon other subjects, say for example physics, is still a matter of prolific debates. Many thinkers have contributed their ideas concerning the nature of mathematics. Today, some philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis. There are traditions of mathematical philosophy in Eastern philosophy. Western philosophies of mathematics go as far back as Pythagoras, who described the theory "everything is mathematics", who paraphrased Pythagoras, studied the ontological status of mathematical objects, Aristotle, who studied logic and issues related to infinity. Greek philosophy on mathematics was influenced by their study of geometry.
For example, at one time, the Greeks held the opinion that 1 was not a number, but rather a unit of arbitrary length. A number was defined as a multitude. Therefore, 3, for example, represented a certain multitude of units, was thus not "truly" a number. At another point, a similar argument was made that 2 was not a number but a fundamental notion of a pair; these views come from the geometric straight-edge-and-compass viewpoint of the Greeks: just as lines drawn in a geometric problem are measured in proportion to the first arbitrarily drawn line, so too are the numbers on a number line measured in proportion to the arbitrary first "number" or "one". These earlier Greek ideas of numbers were upended by the discovery of the irrationality of the square root of two. Hippasus, a disciple of Pythagoras, showed that the diagonal of a unit square was incommensurable with its edge: in other words he proved there was no existing number that depicts the proportion of the diagonal of the unit square to its edge.
This caused a significant re-evaluation of Greek philosophy of mathematics. According to legend, fellow Pythagoreans were so traumatized by this discovery that they murdered Hippasus to stop him from spreading his heretical idea. Simon Stevin was one of the first in Europe to challenge Greek ideas in the 16th century. Beginning with Leibniz, the focus shifted to the relationship between mathematics and logic; this perspective dominated the philosophy of mathematics through the time of Frege and of Russell, but was brought into question by developments in the late 19th and early 20th centuries. A perennial issue in the philosophy of mathematics concerns the relationship between logic and mathematics at their joint foundations. While 20th-century philosophers continued to ask the questions mentioned at the outset of this article, the philosophy of mathematics in the 20th century was characterized by a predominant interest in formal logic, set theory, foundational issues, it is a profound puzzle that on the one hand mathematical truths seem to have a compelling inevitability, but on the other hand the source of their "truthfulness" remains elusive.
Investigations into this issue are known as the foundations of mathematics program. At the start of the 20th century, philosophers of mathematics were beginning to divide into various schools of thought about all these questions, broadly distinguished by their pictures of mathematical epistemology and ontology. Three schools, formalism and logicism, emerged at this time in response to the widespread worry that mathematics as it stood, analysis in particular, did not live up to the standards of certainty and rigor, taken for granted; each school addressed the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge. Surprising and counter-intuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics; as the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted since the time of Euclid around 300 BCE as the natural basis for mathematics.
Notions of axiom and proof, as well as the notion of a proposition being true of a mathematical object, were formalized, allowing them to be treated mathematically. The Zermelo–Fraenkel axioms for set theory were formulated whi
Plato was an Athenian philosopher during the Classical period in Ancient Greece, founder of the Platonist school of thought, the Academy, the first institution of higher learning in the Western world. He is considered the pivotal figure in the history of Ancient Greek and Western philosophy, along with his teacher and his most famous student, Aristotle. Plato has often been cited as one of the founders of Western religion and spirituality; the so-called Neoplatonism of philosophers like Plotinus and Porphyry influenced Saint Augustine and thus Christianity. 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 was the innovator of the written dialogue and dialectic forms in philosophy. Plato appears to have been the founder of Western political philosophy, his most famous contribution bears his name, the doctrine of the Forms known by pure reason to provide a realist solution to the problem of universals.
He is the namesake of Platonic love and the Platonic solids. His own most decisive philosophical influences are thought to have been along with Socrates, the pre-Socratics Pythagoras and Parmenides, although few of his predecessors' works remain extant and much of what we know about these figures today derives from Plato himself. Unlike the work of nearly all of his contemporaries, Plato's entire oeuvre is believed to have survived intact for over 2,400 years. Although their popularity has fluctuated over the years, the works of Plato have never been without readers since the time they were written. Due to a lack of surviving accounts, little is known about education. Plato belonged to an influential family. According to a disputed tradition, reported by doxographer Diogenes Laërtius, Plato's father Ariston traced his descent from the king of Athens and the king of Messenia, Melanthus. Plato's mother was Perictione, whose family boasted of a relationship with the famous Athenian lawmaker and lyric poet Solon, one of the seven sages, who repealed the laws of Draco.
Perictione was sister of Charmides and niece of Critias, both prominent figures of the Thirty Tyrants, known as the Thirty, the brief oligarchic regime, which followed on the collapse of Athens at the end of the Peloponnesian War. According to some accounts, Ariston tried to force his attentions on Perictione, but failed in his purpose; the exact time and place of Plato's birth are unknown. Based on ancient sources, most modern scholars believe that he was born in Athens or Aegina between 429 and 423 BC, not long after the start of the Peloponnesian War; the traditional date of Plato's birth during the 87th or 88th Olympiad, 428 or 427 BC, is based on a dubious interpretation of Diogenes Laërtius, who says, "When was gone, joined Cratylus the Heracleitean and Hermogenes, who philosophized in the manner of Parmenides. At twenty-eight, Hermodorus says, went to Euclides in Megara." However, as Debra Nails argues, the text does not state that Plato left for Megara after joining Cratylus and Hermogenes.
In his Seventh Letter, Plato notes that his coming of age coincided with the taking of power by the Thirty, remarking, "But a youth under the age of twenty made himself a laughingstock if he attempted to enter the political arena." Thus, Nails dates Plato's birth to 424/423. According to Neanthes, Plato was six years younger than Isocrates, therefore was born the same year the prominent Athenian statesman Pericles died. Jonathan Barnes regards 428 BC as the year of Plato's birth; the grammarian Apollodorus of Athens in his Chronicles argues that Plato was born in the 88th Olympiad. Both the Suda and Sir Thomas Browne claimed he was born during the 88th Olympiad. Another legend related that, when Plato was an infant, bees settled on his lips while he was sleeping: an augury of the sweetness of style in which he would discourse about philosophy. Besides Plato himself and Perictione had three other children; the brothers Adeimantus and Glaucon are mentioned in the Republic as sons of Ariston, brothers of Plato, though some have argued they were uncles.
In a scenario in the Memorabilia, Xenophon confused the issue by presenting a Glaucon much younger than Plato. Ariston appears to have died in Plato's childhood, although the precise dating of his death is difficult. Perictione married Pyrilampes, her mother's brother, who had served many times as an ambassador to the Persian court and was a friend of Pericles, the leader of the democratic faction in Athens. Pyrilampes had a son from a previous marriage, famous for his beauty. Perictione gave birth to Pyrilampes' second son, the half-brother of Plato, who appears in Parmenides. In contrast to his reticence about himself, Plato introduced his distinguished relatives into his dialogues, or referred to them with some precision. In addition to Adeimantus and Glaucon in the Republic, Charmides has a dialogue named after him; these and other references suggest a considerable amount of family pride and enable us to reconstruct Plato's family tree. According to Burnet, "the opening scene of the Ch