Socrates

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Socrates
Socrates
Born	470/469 BC[1] Deme Alopece, Athens
Died	399 BC (age approx. 71) Athens
Nationality	Greek
Era	Ancient philosophy
Region	Western philosophy
School	Classical Greek
Main interests	Epistemology, ethics
Notable ideas	Socratic method, Socratic irony
Influenced	[show]

Socrates (/ˈsɒkrətiːz/;^[2] Greek: Σωκράτης Sōkrátēs [sɔːkrátɛːs]; 470/469 BC – 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. The latter remains a commonly used tool in a wide range of discussions, and is a type of pedagogy in which a series of questions is asked not only to draw individual answers, but also to encourage fundamental insight into the issue at hand. Plato's Socrates also made important and lasting contributions to the field of epistemology, and the influence of his ideas and approach remains a strong foundation for much western philosophy that followed.

Biography

Socrates and Alcibiades, by Christoffer Wilhelm Eckersberg

The Socratic problem

An accurate picture of the historical Socrates and his philosophical viewpoints is problematic: an issue known as the Socratic problem.

As Socrates did not write philosophical texts, the knowledge of the man, his life, and his philosophy is entirely based on writings by his students and contemporaries. Foremost among them is Plato. Works by Xenophon, Aristotle, and Aristophanes also provide important insights.^[4] The difficulty of finding the “real” Socrates arises because these works are mostly philosophical or dramatic texts rather than straightforward histories. Aside from Thucydides (who makes no mention of Socrates or philosophers in general) and Xenophon, there are in fact no straightforward histories contemporary with Socrates that dealt with his own time and place. A corollary of this is that sources that do mention Socrates do not necessarily claim to be historically accurate, and are often partisan (those who prosecuted and convicted Socrates have left no testament). Historians therefore face the challenge of reconciling the various texts that come from these men to create an accurate and consistent account of Socrates' life and work. The result of such an effort is not necessarily realistic, merely consistent.

Plato is frequently viewed as the most informative source about Socrates' life and philosophy.^[5] At the same time, however, many scholars believe that in some works Plato, being a literary artist, pushed his avowedly brightened-up version of "Socrates" far beyond anything the historical Socrates was likely to have done or said; and that Xenophon, being an historian, is a more reliable witness to the historical Socrates. It is a matter of much debate over just which Socrates it is whom Plato is describing at any given point—the historical figure, or Plato's fictionalization. As Martin Cohen has put it, Plato, the idealist, offers "an idol, a master figure, for philosophy. A Saint, a prophet of the 'Sun-God', a teacher condemned for his teachings as a heretic."^[6]

It is also clear from other writings and historical artifacts, however, that Socrates was not simply a character, or an invention, of Plato. The testimony of Xenophon and Aristotle, alongside some of Aristophanes' work (especially The Clouds), is useful in fleshing out a perception of Socrates beyond Plato's work.

Life

Carnelian gem imprint representing Socrates, Rome, 1st century BC-1st century AD.

Details about Socrates can be derived from three contemporary sources: the dialogues of Plato and Xenophon (both devotees of Socrates), and the plays of Aristophanes. He has been depicted by some scholars, including Eric Havelock and Walter Ong, as a champion of oral modes of communication, standing up at the dawn of writing against its haphazard diffusion.^[7]

Aristophanes' play The Clouds portrays Socrates as a clown who teaches his students how to bamboozle their way out of debt. Most of Aristophanes' works, however, function as parodies. Thus, it is presumed this characterization was also not literal.

Socrates' father was Sophroniscus,^[8] a sculptor,^[9] and his mother Phaenarete,^[10] a midwife. Socrates married Xanthippe,^[11] who was much younger than he and was characterized as undesirable in temperament. She bore for him three sons,^[12] Lamprocles, Sophroniscus and Menexenus. His friend Crito of Alopece criticized him for abandoning his sons when he refused to try to escape before his execution.^[13]

Socrates initially earned his living as a master stonecutter. According to Timon of Phlius and later sources, Socrates took over the profession of stonemasonry from his father who cut stone for the Parthenon. There was a tradition in antiquity, not credited by modern scholarship, that Socrates crafted the statues of the Three Graces, which stood near the Acropolis until the 2nd century AD.^[14] Ancient texts seem to indicate that Socrates did not work at stonecutting after retiring. In Xenophon's Symposium, Socrates is reported as saying he devotes himself only to what he regards as the most important art or occupation: discussing philosophy. In The Clouds Aristophanes portrays Socrates as accepting payment for teaching and running a sophist school with Chaerephon, while in Plato's Apology and Symposium and in Xenophon's accounts, Socrates explicitly denies accepting payment for teaching. More specifically, in the Apology Socrates cites his poverty as proof he is not a teacher.

Socrates Tears Alcibiades from the Embrace of Sensual Pleasure

Several of Plato's dialogues refer to Socrates' military service. Socrates says he served in the Athenian army during three campaigns: at Potidaea, Amphipolis, and Delium. In the Symposium Alcibiades describes Socrates' valour in the battles of Potidaea and Delium, recounting how Socrates saved his life in the former battle (219e-221b). Socrates' exceptional service at Delium is also mentioned in the Laches by the General after whom the dialogue is named (181b). In the Apology, Socrates compares his military service to his courtroom troubles, and says anyone on the jury who thinks he ought to retreat from philosophy must also think soldiers should retreat when it seems likely that they will be killed in battle.^[15]

In 406, he was a member of the Boule, and his tribe the Antiochis held the Prytany on the day the Generals of the Battle of Arginusae, who abandoned the slain and the survivors of foundered ships to pursue the defeated Spartan navy, were discussed. Socrates was the Epistates and resisted the unconstitutional demand for a collective trial to establish the guilt of all eight Generals, proposed by Callixeinus. Eventually, Socrates refused to be cowed by threats of impeachment and imprisonment and blocked the vote until his Prytany ended the next day, whereupon the six Generals who had returned to Athens were condemned to death.

In 404, the Thirty Tyrants sought to ensure the loyalty of those opposed to them by making them complicit in their activities. Socrates and four others were ordered to bring a certain Leon of Salamis from his home for unjust execution. Socrates quietly refused, his death averted only by the overthrow of the Tyrants soon afterwards.^[16]

Trial and death

The Death of Socrates, by Jacques-Louis David (1787)

Socrates lived during the time of the transition from the height of the Athenian hegemony to its decline with the defeat by Sparta and its allies in the Peloponnesian War. At a time when Athens sought to stabilize and recover from its humiliating defeat, the Athenian public may have been entertaining doubts about democracy as an efficient form of government. Socrates appears to have been a critic of democracy,^[17] and some scholars interpret his trial as an expression of political infighting.^[18] Claiming loyalty to his city, Socrates clashed with the current course of Athenian politics and society.^[19] He praises Sparta, archrival to Athens, directly and indirectly in various dialogues. One of Socrates' purported offenses to the city was his position as a social and moral critic. Rather than upholding a status quo and accepting the development of what he perceived as immorality within his region, Socrates questioned the collective notion of "might makes right" that he felt was common in Greece during this period. Plato refers to Socrates as the "gadfly" of the state (as the gadfly stings the horse into action, so Socrates stung various Athenians), insofar as he irritated some people with considerations of justice and the pursuit of goodness.^[20] His attempts to improve the Athenians' sense of justice may have been the cause of his execution.

According to Plato's Apology, Socrates' life as the "gadfly" of Athens began when his friend Chaerephon asked the oracle at Delphi if anyone were wiser than Socrates; the Oracle responded that no-one was wiser. Socrates believed the Oracle's response was a paradox, because he believed he possessed no wisdom whatsoever. He proceeded to test the riddle by approaching men considered wise by the people of Athens—statesmen, poets, and artisans—in order to refute the Oracle's pronouncement. Questioning them, however, Socrates concluded: while each man thought he knew a great deal and was wise, in fact they knew very little and were not wise at all. Socrates realized the Oracle was correct; while so-called wise men thought themselves wise and yet were not, he himself knew he was not wise at all, which, paradoxically, made him the wiser one since he was the only person aware of his own ignorance. Socrates' paradoxical wisdom made the prominent Athenians he publicly questioned look foolish, turning them against him and leading to accusations of wrongdoing. Socrates defended his role as a gadfly until the end: at his trial, when Socrates was asked to propose his own punishment, he suggested a wage paid by the government and free dinners for the rest of his life instead, to finance the time he spent as Athens' benefactor.^[21] He was, nevertheless, found guilty of both corrupting the minds of the youth of Athens and of impiety ("not believing in the gods of the state"),^[22] and subsequently sentenced to death by drinking a mixture containing poison hemlock.^{[23][24][25][26]}

Bust of Socrates in the Vatican Museum

According to Xenophon's story, Socrates purposefully gave a defiant defense to the jury because "he believed he would be better off dead". Xenophon goes on to describe a defense by Socrates that explains the rigors of old age, and how Socrates would be glad to circumvent them by being sentenced to death. It is also understood that Socrates also wished to die because he "actually believed the right time had come for him to die."

Xenophon and Plato agree that Socrates had an opportunity to escape, as his followers were able to bribe the prison guards. There have been several suggestions offered as reasons why he chose to stay:

1. He believed such a flight would indicate a fear of death, which he believed no true philosopher has.
2. If he fled Athens his teaching would fare no better in another country, as he would continue questioning all he met and undoubtedly incur their displeasure.
3. Having knowingly agreed to live under the city's laws, he implicitly subjected himself to the possibility of being accused of crimes by its citizens and judged guilty by its jury. To do otherwise would have caused him to break his "social contract" with the state, and so harm the state, an unprincipled act.
4. If he escaped at the instigation of his friends, then his friends would become liable in law.^[27]

The full reasoning behind his refusal to flee is the main subject of the Crito.^[28]

Socrates' death is described at the end of Plato's Phaedo. Socrates turned down Crito's pleas to attempt an escape from prison. After drinking the poison, he was instructed to walk around until his legs felt numb. After he lay down, the man who administered the poison pinched his foot; Socrates could no longer feel his legs. The numbness slowly crept up his body until it reached his heart. Shortly before his death, Socrates speaks his last words to Crito: "Crito, we owe a rooster to Asclepius. Please, don't forget to pay the debt."

Asclepius was the Greek god for curing illness, and it is likely Socrates' last words meant that death is the cure—and freedom, of the soul from the body. Additionally, in Why Socrates Died: Dispelling the Myths, Robin Waterfield adds another interpretation of Socrates' last words. He suggests that Socrates was a voluntary scapegoat; his death was the purifying remedy for Athens’ misfortunes. In this view, the token of appreciation for Asclepius would represent a cure for Athens' ailments.^[20]

Philosophy

Socratic method

Perhaps his most important contribution to Western thought is his dialectic method of inquiry, known as the Socratic method or method of "elenchus", which he largely applied to the examination of key moral concepts such as the Good and Justice. It was first described by Plato in the Socratic Dialogues. To solve a problem, it would be broken down into a series of questions, the answers to which gradually distill the answer a person would seek. The influence of this approach is most strongly felt today in the use of the scientific method, in which hypothesis is the first stage. The development and practice of this method is one of Socrates' most enduring contributions, and is a key factor in earning his mantle as the father of political philosophy, ethics or moral philosophy, and as a figurehead of all the central themes in Western philosophy.
To illustrate the use of the Socratic method; a series of questions are posed to help a person or group to determine their underlying beliefs and the extent of their knowledge. The Socratic method is a negative method of hypothesis elimination, in that better hypotheses are found by steadily identifying and eliminating those that lead to contradictions. It was designed to force one to examine one's own beliefs and the validity of such beliefs.

An alternative interpretation of the dialectic is that it is a method for direct perception of the Form of the Good. Philosopher Karl Popper describes the dialectic as "the art of intellectual intuition, of visualising the divine originals, the Forms or Ideas, of unveiling the Great Mystery behind the common man's everyday world of appearances."^[29] In a similar vein, French philosopher Pierre Hadot suggests that the dialogues are a type of spiritual exercise. "Furthermore," writes Hadot, "in Plato's view, every dialectical exercise, precisely because it is an exercise of pure thought, subject to the demands of the Logos, turns the soul away from the sensible world, and allows it to convert itself towards the Good."^[30]

Philosophical beliefs

The beliefs of Socrates, as distinct from those of Plato, are difficult to discern. Little in the way of concrete evidence exists to demarcate the two. The lengthy presentation of ideas given in most of the dialogues may be deformed by Plato, and some scholars think Plato so adapted the Socratic style as to make the literary character and the philosopher himself impossible to distinguish. Others argue that he did have his own theories and beliefs, but there is much controversy over what these might have been, owing to the difficulty of separating Socrates from Plato and the difficulty of interpreting even the dramatic writings concerning Socrates. Consequently, distinguishing the philosophical beliefs of Socrates from those of Plato and Xenophon is not easy and it must be remembered that what is attributed to Socrates might more closely reflect the specific concerns of these thinkers.

The matter is complicated because the historical Socrates seems to have been notorious for asking questions but not answering, claiming to lack wisdom concerning the subjects about which he questioned others.^[31]

If anything in general can be said about the philosophical beliefs of Socrates, it is that he was morally, intellectually, and politically at odds with many of his fellow Athenians. When he is on trial for heresy and corrupting the minds of the youth of Athens, he uses his method of elenchos to demonstrate to the jurors that their moral values are wrong-headed. He tells them they are concerned with their families, careers, and political responsibilities when they ought to be worried about the "welfare of their souls". Socrates' assertion that the gods had singled him out as a divine emissary seemed to provoke irritation, if not outright ridicule. Socrates also questioned the Sophistic doctrine that arete (virtue) can be taught. He liked to observe that successful fathers (such as the prominent military general Pericles) did not produce sons of their own quality. Socrates argued that moral excellence was more a matter of divine bequest than parental nurture. This belief may have contributed to his lack of anxiety about the future of his own sons.

Also, according to A. A. Long, "There should be no doubt that, despite his claim to know only that he knew nothing, Socrates had strong beliefs about the divine", and, citing Xenophon's Memorabilia, 1.4, 4.3,:

According to Xenophon, he was a teleologist who held that god arranges everything for the best.^[32]

Socrates frequently says his ideas are not his own, but his teachers'. He mentions several influences: Prodicus the rhetor and Anaxagoras the philosopher. Perhaps surprisingly, Socrates claims to have been deeply influenced by two women besides his mother: he says that Diotima, a witch and priestess from Mantinea, taught him all he knows about eros, or love; and that Aspasia, the mistress of Pericles, taught him the art of rhetoric.^[33] John Burnet argued that his principal teacher was the Anaxagorean Archelaus but his ideas were as Plato described them; Eric A. Havelock, on the other hand, considered Socrates' association with the Anaxagoreans to be evidence of Plato's philosophical separation from Socrates.

Socratic paradoxes

Many of the beliefs traditionally attributed to the historical Socrates have been characterized as "paradoxical" because they seem to conflict with common sense. The following are among the so-called Socratic Paradoxes:^[34]

• No one desires evil.
• No one errs or does wrong willingly or knowingly.
• Virtue—all virtue—is knowledge.
• Virtue is sufficient for happiness.

The term, "Socratic paradox" can also refer to a self-referential paradox, originating in Socrates' utterance, "what I do not know I do not think I know",^[35] often paraphrased as "I know that I know nothing."

Knowledge

One of the best known sayings of Socrates is "what I do not know I do not think I know". The conventional interpretation of this remark is that Socrates' wisdom was limited to an awareness of his own ignorance. Socrates believed wrongdoing was a consequence of ignorance and those who did wrong knew no better. The one thing Socrates consistently claimed to have knowledge of was "the art of love", which he connected with the concept of "the love of wisdom", i.e., philosophy. He never actually claimed to be wise, only to understand the path a lover of wisdom must take in pursuing it. It is debatable whether Socrates believed humans (as opposed to gods like Apollo) could actually become wise. On the one hand, he drew a clear line between human ignorance and ideal knowledge; on the other, Plato's Symposium (Diotima's Speech) and Republic (Allegory of the Cave) describe a method for ascending to wisdom.

In Plato's Theaetetus (150a), Socrates compares himself to a true matchmaker (προμνηστικός promnestikós), as distinguished from a panderer (προᾰγωγός proagogos). This distinction is echoed in Xenophon's Symposium (3.20), when Socrates jokes about his certainty of being able to make a fortune, if he chose to practice the art of pandering. For his part as a philosophical interlocutor, he leads his respondent to a clearer conception of wisdom, although he claims he is not himself a teacher (Apology). His role, he claims, is more properly to be understood as analogous to a midwife (μαῖα maia). Socrates explains that he is himself barren of theories, but knows how to bring the theories of others to birth and determine whether they are worthy or mere "wind eggs" (ἀνεμιαῖον anemiaion). Perhaps significantly, he points out that midwives are barren due to age, and women who have never given birth are unable to become midwives; they would have no experience or knowledge of birth and would be unable to separate the worthy infants from those that should be left on the hillside to be exposed. To judge this, the midwife must have experience and knowledge of what she is judging.

Virtue

Bust of Socrates in the Palermo Archaeological Museum.

Socrates believed the best way for people to live was to focus on the pursuit of virtue rather than the pursuit, for instance, of material wealth.^[36] He always invited others to try to concentrate more on friendships and a sense of true community, for Socrates felt this was the best way for people to grow together as a populace.^[37] His actions lived up to this: in the end, Socrates accepted his death sentence when most thought he would simply leave Athens, as he felt he could not run away from or go against the will of his community; as mentioned above, his reputation for valor on the battlefield was without reproach.

The idea that there are certain virtues formed a common thread in Socrates' teachings. These virtues represented the most important qualities for a person to have, foremost of which were the philosophical or intellectual virtues. Socrates stressed that "the unexamined life is not worth living [and] ethical virtue is the only thing that matters."^[38]

Politics

It is argued that Socrates believed "ideals belong in a world only the wise man can understand",^[39] making the philosopher the only type of person suitable to govern others. In Plato's dialogue the Republic, Socrates openly objected to the democracy that ran Athens during his adult life. It was not only Athenian democracy: Socrates found short of ideal any government that did not conform to his presentation of a perfect regime led by philosophers, and Athenian government was far from that. It is, however, possible that the Socrates of Plato's Republic is colored by Plato's own views. During the last years of Socrates' life, Athens was in continual flux due to political upheaval. Democracy was at last overthrown by a junta known as the Thirty Tyrants, led by Plato's relative, Critias, who had once been a student and friend of Socrates. The Tyrants ruled for about a year before the Athenian democracy was reinstated, at which point it declared an amnesty for all recent events.

Socrates' opposition to democracy is often denied, and the question is one of the biggest philosophical debates when trying to determine exactly what Socrates believed. The strongest argument of those who claim Socrates did not actually believe in the idea of philosopher kings is that the view is expressed no earlier than Plato's Republic, which is widely considered one of Plato's "Middle" dialogues and not representative of the historical Socrates' views. Furthermore, according to Plato's Apology of Socrates, an "early" dialogue, Socrates refused to pursue conventional politics; he often stated he could not look into other's matters or tell people how to live their lives when he did not yet understand how to live his own. He believed he was a philosopher engaged in the pursuit of Truth, and did not claim to know it fully. Socrates' acceptance of his death sentence after his conviction can also be seen to support this view. It is often claimed much of the anti-democratic leanings are from Plato, who was never able to overcome his disgust at what was done to his teacher.
In any case, it is clear Socrates thought the rule of the Thirty Tyrants was also objectionable; when called before them to assist in the arrest of a fellow Athenian, Socrates refused and narrowly escaped death before the Tyrants were overthrown. He did, however, fulfill his duty to serve as Prytanis when a trial of a group of Generals who presided over a disastrous naval campaign were judged; even then he maintained an uncompromising attitude, being one of those who refused to proceed in a manner not supported by the laws, despite intense pressure.^[40] Judging by his actions, he considered the rule of the Thirty Tyrants less legitimate than the Democratic Senate that sentenced him to death.

Socrates' apparent respect for democracy is one of the themes emphasized in the 2008 play Socrates on Trial by Andrew Irvine. Irvine argues that it was because of his loyalty to Athenian democracy that Socrates was willing to accept the verdict of his fellow citizens. As Irvine puts it, “During a time of war and great social and intellectual upheaval, Socrates felt compelled to express his views openly, regardless of the consequences. As a result, he is remembered today, not only for his sharp wit and high ethical standards, but also for his loyalty to the view that in a democracy the best way for a man to serve himself, his friends, and his city – even during times of war – is by being loyal to, and by speaking publicly about, the truth.”^[41]

 Covertness

In the Dialogues of Plato, though Socrates sometimes seems to support a mystical side, discussing reincarnation and the mystery religions, this is generally attributed to Plato.^[42] Regardless, this view of Socrates cannot be dismissed out of hand, as we cannot be sure of the differences between the views of Plato and Socrates; in addition, there seem to be some corollaries in the works of Xenophon.
In the culmination of the philosophic path as discussed in Plato's Symposium, one comes to the Sea of Beauty or to the sight of "the beautiful itself" (211C); only then can one become wise. (In the Symposium, Socrates credits his speech on the philosophic path to his teacher, the priestess Diotima, who is not even sure if Socrates is capable of reaching the highest mysteries.) In the Meno, he refers to the Eleusinian Mysteries, telling Meno he would understand Socrates' answers better if only he could stay for the initiations next week. Further confusions result from the nature of these sources, insofar as the Platonic Dialogues are arguably the work of an artist-philosopher, whose meaning does not volunteer itself to the passive reader nor again the lifelong scholar. According to Olympiodorus the Younger in his Life of Plato,^[43] Plato himself "received instruction from the writers of tragedy" before taking up the study of philosophy. His works are, indeed, dialogues; Plato's choice of this, the medium of Sophocles, Euripides, and the fictions of theatre, may reflect the ever-interpretable nature of his writings, as he has been called a "dramatist of reason". What is more, the first word of nearly all Plato's works is a significant term for that respective dialogue, and is used with its many connotations in mind. Finally, the Phaedrus and the Symposium each allude to Socrates' coy delivery of philosophic truths in conversation; the Socrates of the Phaedrus goes so far as to demand such dissembling and mystery in all writing. The covertness we often find in Plato, appearing here and there couched in some enigmatic use of symbol and/or irony, may be at odds with the mysticism Plato's Socrates expounds in some other dialogues. These indirect methods may fail to satisfy some readers.

Perhaps the most interesting facet of this is Socrates' reliance on what the Greeks called his "daemonic sign", an averting (ἀποτρεπτικός apotreptikos) inner voice Socrates heard only when he was about to make a mistake. It was this sign that prevented Socrates from entering into politics. In the Phaedrus, we are told Socrates considered this to be a form of "divine madness", the sort of insanity that is a gift from the gods and gives us poetry, mysticism, love, and even philosophy itself. Alternately, the sign is often taken to be what we would call "intuition"; however, Socrates' characterization of the phenomenon as "daemonic" may suggest that its origin is divine, mysterious, and independent of his own thoughts. Today, such a voice would be classified under the Diagnostic and Statistical Manual of Mental Disorders as a command hallucination.^[44]

Satirical playwrights

He was prominently lampooned in Aristophanes' comedy The Clouds, produced when Socrates was in his mid-forties; he said at his trial (according to Plato) that the laughter of the theater was a harder task to answer than the arguments of his accusers. Søren Kierkegaard believed this play was a more accurate representation of Socrates than those of his students. In the play, Socrates is ridiculed for his dirtiness, which is associated with the Laconizing fad; also in plays by Callias, Eupolis, and Telecleides. Other comic poets who lampooned Socrates include Mnesimachus and Ameipsias. In all of these, Socrates and the Sophists were criticised for "the moral dangers inherent in contemporary thought and literature".

Prose sources

Plato, Xenophon, and Aristotle are the main sources for the historical Socrates; however, Xenophon and Plato were students of Socrates, and they may idealize him; however, they wrote the only continuous descriptions of Socrates that have come down to us in their complete form. Aristotle refers frequently, but in passing, to Socrates in his writings. Almost all of Plato's works center on Socrates. However, Plato's later works appear to be more his own philosophy put into the mouth of his mentor.

The Socratic dialogues

The Socratic Dialogues are a series of dialogues written by Plato and Xenophon in the form of discussions between Socrates and other persons of his time, or as discussions between Socrates' followers over his concepts. Plato's Phaedo is an example of this latter category. Although his Apology is a monologue delivered by Socrates, it is usually grouped with the Dialogues. The Apology professes to be a record of the actual speech Socrates delivered in his own defense at the trial. In the Athenian jury system, an "apology" is composed of three parts: a speech, followed by a counter-assessment, then some final words. "Apology" is a transliteration, not a translation, of the Greek apologia, meaning "defense"; in this sense it is not apologetic according to our contemporary use of the term.

Plato generally does not place his own ideas in the mouth of a specific speaker; he lets ideas emerge via the Socratic Method, under the guidance of Socrates. Most of the dialogues present Socrates applying this method to some extent, but nowhere as completely as in the Euthyphro. In this dialogue, Socrates and Euthyphro go through several iterations of refining the answer to Socrates' question, "...What is the pious, and what the impious?"

In Plato's Dialogues, learning appears as a process of remembering. The soul, before its incarnation in the body, was in the realm of Ideas (very similar to the Platonic "Forms"). There, it saw things the way they truly are, rather than the pale shadows or copies we experience on earth. By a process of questioning, the soul can be brought to remember the ideas in their pure form, thus bringing wisdom.^[45]

Especially for Plato's writings referring to Socrates, it is not always clear which ideas brought forward by Socrates (or his friends) actually belonged to Socrates and which of these may have been new additions or elaborations by Plato – this is known as the Socratic Problem. Generally, the early works of Plato are considered to be close to the spirit of Socrates, whereas the later works – including Phaedo and Republic – are considered to be possibly products of Plato's elaborations.^[46]

Legacy

Immediate influence

Statue of Socrates in front of the Academy of Athens (modern)

Immediately, the students of Socrates set to work both on exercising their perceptions of his teachings in politics and also on developing many new philosophical schools of thought. Some of Athens' controversial and anti-democratic tyrants were contemporary or posthumous students of Socrates including Alcibiades and Critias. Critias' cousin, Plato would go on to found the Academy in 385 BC, which gained so much renown that 'Academy' became the standard word for educational institutions in later European languages such as English, French, and Italian.^[47] Plato's protege, another important figure of the Classical era, Aristotle went on to tutor Alexander the Great and also to found his own school in 335 BC — the Lyceum — whose name also now means an educational institution.^[48]

While "Socrates dealt with moral matters and took no notice at all of nature in general",^[49] in his Dialogues, Plato would emphasize mathematics with metaphysical overtones mirroring that of Pythagoras – the former who would dominate Western thought well into the Renaissance. Aristotle himself was as much of a philosopher as he was a scientist with extensive work in the fields of biology and physics.

Socratic thought which challenged conventions, especially in stressing a simplistic way of living, became divorced from Plato's more detached and philosophical pursuits. This idea was inherited by one of Socrates' older students, Antisthenes, who became the originator of another philosophy in the years after Socrates' death: Cynicism. The idea of asceticism being hand in hand with an ethical life or one with piety, ignored by Plato and Aristotle and somewhat dealt with by the Cynics, formed the core of another philosophy in 281 BC – Stoicism when Zeno of Citium would discover Socrates' works and then learn from Crates, a Cynic philosopher.^[50]

Later historical influence

While some of the later contributions of Socrates to Hellenistic Era culture and philosophy as well as the Roman Era have been lost to time, his teachings began a resurgence in both medieval Europe and the Islamic Middle East alongside those of Aristotle and Stoicism. Socrates is mentioned in the dialogue Kuzari by Jewish philosopher and rabbi Yehuda Halevi in which a Jew instructs the Khazar king about Judaism.^[51] Al-Kindi, a well-known Arabic philosopher, introduced and tried to reconcile Socrates and Hellenistic philosophy to an Islamic audience,^[52] referring to him by the name 'Suqrat'.
Socrates' stature in Western philosophy returned in full force with the Renaissance and the Age of Reason in Europe when political theory began to resurface under those like Locke and Hobbes.^[53] Voltaire even went so far as to write a satirical play about the Trial of Socrates. There were a number of paintings about his life including Socrates Tears Alcibiades from the Embrace of Sensual Pleasure by Jean-Baptiste Regnault and The Death of Socrates by Jacques-Louis David in the later 18th century.

To this day, the Socratic Method is still used in classroom and law school discourse to expose underlying issues in both subject and the speaker. He has been recognized with accolades ranging from frequent mentions in pop culture (such as the movie Bill & Ted's Excellent Adventure and a Greek rock band called Socrates Drank the Conium) to numerous busts in academic institutions in recognition of his contribution to education.

Over the past century, numerous plays about Socrates have also focused on Socratess life and influence. One of the most recent has been Socrates on Trial, a play based on Aristophanes' Clouds and Plato's Apology, Crito, and Phaedo, all adapted for modern performance.

Criticism

Evaluation of and reaction to Socrates has been undertaken by both historians and philosophers from the time of his death to the present day with a multitude of conclusions and perspectives. Although he was not directly prosecuted for his connection to Critias, leader of the Spartan-backed Thirty Tyrants, and "showed considerable personal courage in refusing to submit to [them]", he was seen by some as a figure who mentored oligarchs who became abusive tyrants, and undermined Athenian democracy. The Sophistic movement that he railed at in life survived him, but by the 3rd century BC, was rapidly overtaken by the many philosophical schools of thought that Socrates influenced.^[54]

Socrates' death is considered iconic and his status as a martyr of philosophy overshadows most contemporary and posthumous criticism. However, Xenophon mentions Socrates' "arrogance" and that he was "an expert in the art of pimping" or "self-presentation". ^[55] Direct criticism of Socrates the man almost disappears after this time, but there is a noticeable preference for Plato or Aristotle over the elements of Socratic philosophy distinct from those of his students, even into the Middle Ages.

Some modern scholarship holds that, with so much of his own thought obscured and possibly altered by Plato, it is impossible to gain a clear picture of Socrates amidst all the contradictory evidence.
That both Cynicism and Stoicism, which carried heavy influence from Socratic thought, were unlike or even contrary to Platonism further illustrates this. The ambiguity and lack of reliability serves as the modern basis of criticism—that it is nearly impossible to know the real Socrates. Some controversy also exists about Socrates' attitude towards homosexuality^[56] and as to whether or not he believed in the Olympian gods, was monotheistic, or held some other religious viewpoint.^[57] However, it is still commonly taught and held with little exception that Socrates is the progenitor of subsequent Western philosophy, to the point that philosophers before him are referred to as pre-Socratic.

Pythagoras

From Wikipedia, the free encyclopedia

Pythagoras (Πυθαγόρας)
Bust of Pythagoras of Samos in the Capitoline Museums, Rome
Born	c. 570 BC Samos
Died	c. 495 BC (aged around 75) Metapontum
Era	Ancient philosophy
Region	Western philosophy
School	Pythagoreanism
Main interests	Metaphysics, Music, Mathematics, Ethics, Politics
Notable ideas	Musica universalis, Pythagorean tuning, Pythagorean theorem
Influences	[show]
Influenced	[show]

Pythagoras of Samos (/pɪˈθæɡərəs/; Ancient Greek: Πυθαγόρας ὁ Σάμιος Pythagóras ho Sámios “Pythagoras the Samian”, or simply Πυθαγόρας; Πυθαγόρης in Ionian Greek; c. 570 BC – c. 495 BC)^[1][2] was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him. He was born on the island of Samos, and might have travelled widely in his youth, visiting Egypt and other places seeking knowledge. Around 530 BC, he moved to Croton, in Magna Graecia, and there set up a religious sect. His followers pursued the religious rites and practices developed by Pythagoras and studied his philosophical theories. The society took an active role in the politics of Croton but this eventually led to their downfall. Pythagorean meeting-places were burned and Pythagoras was forced to flee the city. He is said to have died in Metapontum.

Pythagoras made influential contributions to philosophy and religion in the late 6th century BC. He is often revered as a great mathematician, mystic, and scientist and is best known for the Pythagorean theorem which bears his name. However, because legend and obfuscation cloud his work even more than that of the other pre-Socratic philosophers, one can give only a tentative account of his teachings, and some have questioned whether he contributed much to mathematics or natural philosophy. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors. Whether or not his disciples believed that everything was related to mathematics and that numbers were the ultimate reality is unknown. It was said that he was the first man to call himself a philosopher, or lover of wisdom,^[3] and Pythagorean ideas exercised a marked influence on Plato, and through him, all of Western philosophy.

Biographical sources

Accurate facts about the life of Pythagoras are so few that most information concerning him is untrustworthy, making it nearly impossible to provide more than a vague outline of his life. The lack of information by contemporary writers, together with the secrecy which surrounded the Pythagorean brotherhood, meant that invention took the place of facts. The stories which were created were eagerly sought by the Neoplatonist writers who provide most of the details about Pythagoras, but who were uncritical concerning anything which related to the gods or which was considered divine.^[4]
Thus many myths were created – such as that Apollo was his father; that Pythagoras gleamed with a supernatural brightness; that he had a golden thigh; that Abaris came flying to him on a golden arrow; that he was seen in different places at the same time.^[5] With the exception of a few remarks by Xenophanes, Heraclitus, Herodotus, Plato, Aristotle, and Isocrates, we are mainly dependent on Diogenes Laërtius, Porphyry, and Iamblichus for biographical details. Aristotle had written a separate work on the Pythagoreans, which unfortunately has not survived.^[6] His disciples Dicaearchus, Aristoxenus, and Heraclides Ponticus had written on the same subject. These writers, late as they are, were among the best sources from whom Porphyry and Iamblichus drew, while still adding some legendary accounts and their own inventions to the mix. Hence, historians are often reduced to considering the statements based on their inherent probability, but even then, if all the credible stories concerning Pythagoras were supposed true, his range of activity would be impossibly vast.^[7]

Life

Bust of Pythagoras, Vatican

Herodotus, Isocrates, and other early writers all agree that Pythagoras was born on Samos, the Greek island in the eastern Aegean, and we also learn that Pythagoras was the son of Mnesarchus.^[8] His father was a gem-engraver or a merchant. His name led him to be associated with Pythian Apollo; Aristippus explained his name by saying, "He spoke (agor-) the truth no less than did the Pythian (Pyth-)," and Iamblichus tells the story that the Pythia prophesied that his pregnant mother would give birth to a man supremely beautiful, wise, and beneficial to humankind.^[9] A late source gives his mother's name as Pythais.^[10] As to the date of his birth, Aristoxenus stated that Pythagoras left Samos in the reign of Polycrates, at the age of 40, which would give a date of birth around 570 BC.^[11]

It was natural for the ancient biographers to inquire as to the origins of Pythagoras' remarkable system. In the absence of reliable information, however, a huge range of teachers were assigned to Pythagoras. Some made his training almost entirely Greek, others exclusively Egyptian and Oriental. We find mentioned as his instructors Creophylus,^[12] Hermodamas of Samos,^[13] Bias,^[12] Thales,^[12] Anaximander,^[14] and Pherecydes of Syros.^[15] He is said too, to have been taught by a Delphic priestess named Themistoclea, who introduced him to the principles of ethics.^[16][17] The Egyptians are said to have taught him geometry, the Phoenicians arithmetic, the Chaldeans astronomy, the Magians the principles of religion and practical maxims for the conduct of life.^[18] Of the various claims regarding his Greek teachers, Pherecydes is mentioned most often.

Diogenes Laërtius reported that Pythagoras had undertaken extensive travels, having not only visited Egypt but also "journeyed among the Chaldaeans and Magi", for the purpose of collecting all available knowledge and especially to learn information concerning the secret or mystic cults of the gods.^[19] Plutarch asserted in his book On Isis and Osiris that during his visit to Egypt, Pythagoras received instruction from the Egyptian priest Oenuphis of Heliopolis.^[20] Other ancient writers asserted his visit to Egypt.^[21] Enough of Egypt was known to attract the curiosity of an inquiring Greek, and contact between Samos and other parts of Greece with Egypt is mentioned.^[22]

It is not easy to say how much Pythagoras learned from the Egyptian priests, or indeed, whether he learned anything at all from them. There was nothing in the symbolism which the Pythagoreans adopted which showed the distinct traces of Egypt. The secret religious rites of the Pythagoreans exhibited nothing but what might have been adopted in the spirit of Greek religion, by those who knew nothing of Egyptian mysteries. The philosophy and the institutions of Pythagoras might easily have been developed by a Greek mind exposed to the ordinary influences of the age. Even the ancient authorities note the similarities between the religious and ascetic peculiarities of Pythagoras with the Orphic or Cretan mysteries,^[23] or the Delphic oracle.^[24]

There is little direct evidence as to the kind and amount of knowledge which Pythagoras acquired, or as to his definite philosophical views. Everything of the kind mentioned by Plato and Aristotle is attributed not to Pythagoras, but to the Pythagoreans. Heraclitus stated that he was a man of extensive learning;^[25] and Xenophanes claimed that he believed in the transmigration of souls.^[26] Xenophanes mentions the story of his interceding on behalf of a dog that was being beaten, professing to recognise in its cries the voice of a departed friend. Pythagoras is supposed to have claimed that he had been Euphorbus, the son of Panthus, in the Trojan war, as well as various other characters, a tradesman, a courtesan, etc.^[27] In his book The Life of Apollonius of Tyana, Philostratus wrote that Pythagoras knew not only who he was himself, but also who he had been.^[28]

Many mathematical and scientific discoveries were attributed to Pythagoras, including his famous theorem,^[29] as well as discoveries in the field of music,^[30] astronomy,^[31] and medicine.^[32] But it was the religious element which made the profoundest impression upon his contemporaries. Thus the people of Croton were supposed to have identified him with the Hyperborean Apollo,^[33] and he was said to have practised divination and prophecy.^[34] In the visits to various places in Greece – Delos, Sparta, Phlius, Crete, etc. which are ascribed to him, he usually appears either in his religious or priestly guise, or else as a lawgiver.^[35]

Croton on the southern coast of Magna Graecia (Southern Italy), to which Pythagoras ventured after feeling overburdened in Samos.

After his travels, Pythagoras moved (around 530 BC) to Croton, in Italy (Magna Graecia). Possibly the tyranny of Polycrates in Samos made it difficult for him to achieve his schemes there. His later admirers claimed that Pythagoras was so overburdened with public duties in Samos, because of the high estimation in which he was held by his fellow-citizens, that he moved to Croton.^[36] On his arrival in Croton, he quickly attained extensive influence, and many people began to follow him. Later biographers tell fantastical stories of the effects of his eloquent speech in leading the people of Croton to abandon their luxurious and corrupt way of life and devote themselves to the purer system which he came to introduce.^[37]

His followers established a select brotherhood or club for the purpose of pursuing the religious and ascetic practices developed by their master. The accounts agree that what was done and taught among the members was kept a profound secret. The esoteric teachings may have concerned science and mathematics, or the secret religious doctrines and usages which were undoubtedly prominent in the Pythagorean system, and may have been connected with the worship of Apollo.^[38] Temperance of all kinds seems to have been strictly urged. There is disagreement among the biographers as to whether Pythagoras forbade all animal food,^[39] or only certain types.^[40] The club was in practice at once "a philosophical school, a religious brotherhood, and a political association."^[41]

Pythagoras, depicted on a 3rd-century coin

Such an aristocratic and exclusive club could easily have made many people in Croton jealous and hostile, and this seems to have led to its destruction. The circumstances, however, are uncertain. Conflict seems to have broken out between the towns of Sybaris and Croton. The forces of Croton were headed by the Pythagorean Milo, and it is likely that the members of the brotherhood took a prominent part. After the decisive victory by Croton, a proposal for establishing a more democratic constitution, was unsuccessfully resisted by the Pythagoreans. Their enemies, headed by Cylon and Ninon, the former of whom is said to have been irritated by his exclusion from the brotherhood, roused the populace against them. An attack was made upon them while assembled either in the house of Milo, or in some other meeting-place. The building was set on fire, and many of the assembled members perished; only the younger and more active escaping.^[42] Similar commotions ensued in the other cities of Magna Graecia in which Pythagorean clubs had been formed.

As an active and organised brotherhood the Pythagorean order was everywhere suppressed, and did not again revive. Still the Pythagoreans continued to exist as a sect, the members of which kept up among themselves their religious observances and scientific pursuits, while individuals, as in the case of Archytas, acquired now and then great political influence. Concerning the fate of Pythagoras himself, the accounts varied. Some say that he perished in the temple with his disciples,^[43] others that he fled first to Tarentum, and that, being driven from there, he escaped to Metapontum, and there starved himself to death.^[44] His tomb was shown at Metapontum in the time of Cicero.^[45]

According to some accounts Pythagoras married Theano, a lady of Croton. Their children are variously stated to have included a son, Telauges, and three daughters, Damo, Arignote, and Myia.

Writings

No texts by Pythagoras are known to have survived, although forgeries under his name — a few of which remain extant — did circulate in antiquity. Critical ancient sources like Aristotle and Aristoxenus cast doubt on these writings. Ancient Pythagoreans usually quoted their master's doctrines with the phrase autos ephe ("he himself said") — emphasizing the essentially oral nature of his teaching.

Mathematics

The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).

The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things.

—Aristotle, Metaphysics 1–5 , cc. 350 BC

There are good reasons to believe that Pythagoras never dealt with Mathematics at all.^[46] Therefore, although the mathematical ideas exposed below circulated among the Pythagoreans, they may well not be due to Pythagoras himself.

Pythagorean theorem

A visual proof of the Pythagorean theorem

Since the fourth century AD, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides—that is, .

While the theorem that now bears his name was known and previously utilized by the Babylonians and Indians, he, or his students, are often said to have constructed the first proof. It must, however, be stressed that the way in which the Babylonians handled Pythagorean numbers implies that they knew that the principle was generally applicable, and knew some kind of proof, which has not yet been found in the (still largely unpublished) cuneiform sources.^[47] Because of the secretive nature of his school and the custom of its students to attribute everything to their teacher, there is no evidence that Pythagoras himself worked on or proved this theorem. For that matter, there is no evidence that he worked on any mathematical or meta-mathematical problems. Some attribute it as a carefully constructed myth by followers of Plato over two centuries after the death of Pythagoras, mainly to bolster the case for Platonic meta-physics, which resonate well with the ideas they attributed to Pythagoras. This attribution has stuck down the centuries up to modern times.^[48] The earliest known mention of Pythagoras's name in connection with the theorem occurred five centuries after his death, in the writings of Cicero and Plutarch.

Musical theories and investigations

Medieval woodcut showing Pythagoras with bells and other instruments in Pythagorean tuning

According to legend, the way Pythagoras discovered that musical notes could be translated into mathematical equations was when he passed blacksmiths at work one day and thought that the sounds emanating from their anvils were beautiful and harmonious and decided that whatever scientific law caused this to happen must be mathematical and could be applied to music. He went to the blacksmiths to learn how the sounds were produced by looking at their tools. He discovered that it was because the hammers were "simple ratios of each other, one was half the size of the first, another was 2/3 the size, and so on."

This legend has since proven to be false by virtue of the fact that these ratios are only relevant to string length (such as the string of a monochord), and not to hammer weight.^[49][50] However, it may be that Pythagoras was indeed responsible for discovering the properties of string length.

Pythagoreans elaborated on a theory of numbers, the exact meaning of which is still debated among scholars. Another belief attributed to Pythagoras was that of the "harmony of the spheres". Thus the planets and stars moved according to mathematical equations, which corresponded to musical notes and thus produced a symphony.^[51]

Tetractys

Pythagoras was also credited with devising the tetractys, the triangular figure of four rows which add up to the perfect number, ten. As a mystical symbol, it was very important to the worship of the Pythagoreans who would swear oaths by it:

And the inventions were so admirable, and so divinised by those who understood them, that the members used them as forms of oath: "By him who handed to our generation the tetractys, source of the roots of ever-flowing nature."

—Iamblichus, Vit. Pyth., 29

Religion and science

Pythagoras' religious and scientific views were, in his opinion, inseparably interconnected.
Religiously, Pythagoras was a believer of metempsychosis. He believed in transmigration, or the reincarnation of the soul again and again into the bodies of humans, animals, or vegetables until it became immortal. His ideas of reincarnation were influenced by ancient Greek religion. Heraclides Ponticus reports the story that Pythagoras claimed that he had lived four previous lives that he could remember in detail.^[52] One of his past lives, as reported by Aulus Gellius, was as a beautiful courtesan.^[53] According to Xenophanes, Pythagoras heard the cry of his dead friend in the bark of a dog.^[54]

Lore

Pythagoras became the subject of elaborate legends surrounding his historic persona. Aristotle described Pythagoras as a wonder-worker and somewhat of a supernatural figure, attributing to him such aspects as a golden thigh, which was a sign of divinity. According to Muslim tradition,
Pythagoras was said to have been initiated by Hermes (Egyptian Thoth).^[55] According to accounts of Aristotle and others, some ancients believed that he had the ability to travel through space and time and to communicate with animals and plants.^[56] An extract from Brewer's Dictionary of Phrase and Fable's entry entitled "Golden Thigh":

Pythagoras is said to have had a golden thigh, which he showed to Abaris, the Hyperborean priest, and exhibited in the Olympic games.^[57]

Another legend describes his writing on the moon:

Pythagoras asserted he could write on the moon. His plan of operation was to write on a looking-glass in blood, and place it opposite the moon, when the inscription would appear photographed or reflected on the moon's disc.^[58]

Pythagoreans

Pythagoras, the man in the center with the book, teaching music, in The School of Athens by Raphael

Both Plato and Isocrates affirm that, above all else, Pythagoras was famous for leaving behind him a way of life.^[59] Both Iamblichus and Porphyry give detailed accounts of the organisation of the school, although the primary interest of both writers is not historical accuracy, but rather to present Pythagoras as a divine figure, sent by the gods to benefit humankind.^[60]

Pythagoras set up an organization which was in some ways a school, in some ways a brotherhood (and here it should be noted that sources indicate that as well as men there were many women among the adherents of Pythagoras),^[61] and in some ways a monastery. It was based upon the religious teachings of Pythagoras and was very secretive. The adherents were bound by a vow to Pythagoras and each other, for the purpose of pursuing the religious and ascetic observances, and of studying his religious and philosophical theories. The claim that they put all their property into a common stock is perhaps only a later inference from certain Pythagorean maxims and practices.^[62]

As to the internal arrangements of the sect, we are informed that what was done and taught among the members was kept a profound secret towards all. Porphyry stated that this silence was "of no ordinary kind." Candidates had to pass through a period of probation, in which their powers of maintaining silence (echemythia) were especially tested, as well as their general temper, disposition, and mental capacity.^[63] There were also gradations among the members themselves. It was an old Pythagorean maxim, that every thing was not to be told to every body.^[64] Thus the Pythagoreans were divided into an inner circle called the mathematikoi ("learners") and an outer circle called the akousmatikoi ("listeners").^[65] Iamblichus describes them in terms of esoterikoi and exoterikoi (or alternatively Pythagoreioi and Pythagoristai),^[66] according to the degree of intimacy which they enjoyed with Pythagoras. Porphyry wrote "the mathematikoi learned the more detailed and exactly elaborated version of this knowledge, the akousmatikoi (were) those who had heard only the summary headings of his (Pythagoras's) writings, without the more exact exposition."

Pythagoreans celebrate sunrise, painting by Fyodor Bronnikov (1827–1902)

There were ascetic practices (many of which had, perhaps, a symbolic meaning) in the way of life of the sect.^[67] Some represent Pythagoras as forbidding all animal food, advocating a plant-based diet, and prohibiting consumption of beans. This may have been due to the doctrine of metempsychosis.^[68] Other authorities contradict the statement. According to Aristoxenus,^[69] he allowed the use of all kinds of animal food except the flesh of oxen used for ploughing, and rams.^[70] There is a similar discrepancy as to the prohibition of fish and beans.^[71] But temperance of all kinds seems to have been urged. It is also stated that they had common meals, resembling the Spartan system, at which they met in companies of ten.^[72]

Considerable importance seems to have been attached to music and gymnastics in the daily exercises of the disciples. Their whole discipline is represented as encouraging a lofty serenity and self-possession, of which, there were various anecdotes in antiquity.^[73] Iamblichus (apparently on the authority of Aristoxenus)^[74] gives a long description of the daily routine of the members, which suggests many similarities with Sparta. The members of the sect showed a devoted attachment to each other, to the exclusion of those who did not belong to their ranks.^[75] There were even stories of secret symbols, by which members of the sect could recognise each other, even if they had never met before.^[76]

Influence

Influence on Plato

Pythagoras, depicted as a medieval scholar in the Nuremberg Chronicle

Pythagoras, or in a broader sense, the Pythagoreans, allegedly exercised an important influence on the work of Plato. According to R. M. Hare, this influence consists of three points: (1) The platonic Republic might be related to the idea of "a tightly organized community of like-minded thinkers", like the one established by Pythagoras in Croton. (2) There is evidence that Plato possibly took from Pythagoras the idea that mathematics and, generally speaking, abstract thinking is a secure basis for philosophical thinking as well as "for substantial theses in science and morals". (3) Plato and Pythagoras shared a "mystical approach to the soul and its place in the material world". It is probable that both were influenced by Orphism.^[77]

Aristotle claimed that the philosophy of Plato closely followed the teachings of the Pythagoreans,^[78] and Cicero repeats this claim: Platonem ferunt didicisse Pythagorea omnia ("They say Plato learned all things Pythagorean").^[79] Bertrand Russell, in his A History of Western Philosophy, contended that the influence of Pythagoras on Plato and others was so great that he should be considered the most influential of all Western philosophers.

Politics and science

Pythagoras was the first person known to have taught the earth was spherical, with antipodes and that it revolved around the sun. Pythagoras was also said to have spread the seeds of political liberty to Crotona, Sybaris, Meapontum, Rhegium, Sicily, Tauromenium, Catana, Agrigentum and Himera.^[80]

Influence on esoteric groups

Pythagoras started a secret society called the Pythagorean Brotherhood devoted to the study of mathematics. This had a great effect on future esoteric traditions, such as Freemasonry and Rosicrucianism, both of which were scientific/mystical groups dedicated to the study of mathematics/geometry and logical reasoning as opposed to religious dogma. Both Freemasonry and Rosicrucianism have claimed to have evolved out of the Pythagorean Brotherhood. The mystical and occult qualities of Pythagorean mathematics are discussed in a chapter of Manly P. Hall's The Secret Teachings of All Ages entitled "Pythagorean Mathematics".^[81]

Euclid

From Wikipedia, the free encyclopedia

Euclid
Euclid (central figure, bald-pated, marking a slate with a pair of compasses) as featured in Raphael's School of Athens.
Born	unknown  (mid-4th century BC)
Died	unknown  (mid-3rd century BC)
Residence	Alexandria, Egypt
Fields	Mathematics
Known for	Euclidean geometry Euclid's Elements

Euclid (/ˈjuːklɪd/; Greek: Εὐκλείδης Eukleidēs; fl. 300 BC), also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I (323–283 BC). His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century.^[1][2][3] In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor.

"Euclid" is the anglicized version of the Greek name Εὐκλείδης, meaning "Good Glory".^[4]

Life

Very few original references to Euclid survive, so little is known about his life. The date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other figures mentioned alongside him. He is rarely, if ever, referred to by name by other Greek mathematicians from Archimedes onward, who instead call him "ό στοιχειώτης" ("the author of Elements").^[5] The few historical references to Euclid were written centuries after he lived, by Proclus c. 450 AD and Pappus of Alexandria c. 320 AD.^[6]

Proclus introduces Euclid only briefly in his Commentary on the Elements. According to Proclus, Euclid belonged to Plato's "persuasion" and brought together the Elements, drawing on prior work by several pupils of Plato (particularly Eudoxus of Cnidus, Theaetetus and Philip of Opus.) Proclus believes that Euclid is not much younger than these, and that he must have lived during the time of Ptolemy I because he was mentioned by Archimedes (287-212 BC). Although the purported citation of Euclid by Archimedes has been judged to be an interpolation by later editors of his works, it is still believed that Euclid wrote his works before those of Archimedes.^[7][8][9]

Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclid's Elements, "Euclid replied there is no royal road to geometry."^[10] This anecdote is questionable since it is similar to a story told about Menaechmus and Alexander the Great.^[11]

In the only other key reference to Euclid, Pappus briefly mentioned in the fourth century that Apollonius "spent a very long time with the pupils of Euclid at Alexandria, and it was thus that he acquired such a scientific habit of thought" circa 247-222 BC.^[12][13]

A detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious.^[8]

Partly due to the lack of biographical information that is somewhat atypical for the period (extensive biographies are available for most significant Greek mathematicians for several centuries before and after Euclid), it has been proposed that Euclid was not, in fact, a historical character, and his works were written by a team of mathematicians (not unlike Bourbaki) who took the name Euclid from the historical character Euclid of Megara. However, this hypothesis is not well accepted by scholars and there is little evidence in its favor.^[8][14]

Elements

One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 (P. Oxy. 29). The diagram accompanies Book II, Proposition 5.^[15]

Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.^[16]

There is no mention of Euclid in the earliest remaining copies of the Elements, and most of the copies say they are "from the edition of Theon" or the "lectures of Theon",^[17] while the text considered to be primary, held by the Vatican, mentions no author. The only reference that historians rely on of Euclid having written the Elements was from Proclus, who briefly in his Commentary on the Elements ascribes Euclid as its author.

Although best known for its geometric results, the Elements also includes number theory. It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

The geometrical system described in the Elements was long known simply as geometry, and was considered to be the only geometry possible. Today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century.

Other works

Euclid's construction of a regular dodecahedron.

Construction of a dodecahedron by placing faces on the edges of a cube.

In addition to the Elements, at least five works of Euclid have survived to the present day. They follow the same logical structure as Elements, with definitions and proved propositions.

• Data deals with the nature and implications of "given" information in geometrical problems; the subject matter is closely related to the first four books of the Elements.
• On Divisions of Figures, which survives only partially in Arabic translation, concerns the division of geometrical figures into two or more equal parts or into parts in given ratios. It is similar to a third-century AD work by Heron of Alexandria.
• Catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J O'Connor and E F Robertson who name Theon of Alexandria as a more likely author.^[18]
• Phaenomena, a treatise on spherical astronomy, survives in Greek; it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC.

Statue of Euclid in the Oxford University Museum of Natural History.

• Optics is the earliest surviving Greek treatise on perspective. In its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth: "Things seen under a greater angle appear greater, and those under a lesser angle less, while those under equal angles appear equal." In the 36 propositions that follow, Euclid relates the apparent size of an object to its distance from the eye and investigates the apparent shapes of cylinders and cones when viewed from different angles. Proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Pappus believed these results to be important in astronomy and included Euclid's Optics, along with his Phaenomena, in the Little Astronomy, a compendium of smaller works to be studied before the Syntaxis (Almagest) of Claudius Ptolemy.

Other works are credibly attributed to Euclid, but have been lost.

• Conics was a work on conic sections that was later extended by Apollonius of Perga into his famous work on the subject. It is likely that the first four books of Apollonius's work come directly from Euclid. According to Pappus, "Apollonius, having completed Euclid's four books of conics and added four others, handed down eight volumes of conics." The Conics of Apollonius quickly supplanted the former work, and by the time of Pappus, Euclid's work was already lost.
• Porisms might have been an outgrowth of Euclid's work with conic sections, but the exact meaning of the title is controversial.
• Pseudaria, or Book of Fallacies, was an elementary text about errors in reasoning.
• Surface Loci concerned either loci (sets of points) on surfaces or loci which were themselves surfaces; under the latter interpretation, it has been hypothesized that the work might have dealt with quadric surfaces.
• Several works on mechanics are attributed to Euclid by Arabic sources. On the Heavy and the Light contains, in nine definitions and five propositions, Aristotelian notions of moving bodies and the concept of specific gravity. On the Balance treats the theory of the lever in a similarly Euclidean manner, containing one definition, two axioms, and four propositions. A third fragment, on the circles described by the ends of a moving lever, contains four propositions. These three works complement each other in such a way that it has been suggested that they are remnants of a single treatise on mechanics written by Euclid.