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Can someone explain the joke that killed Chrysippus of Soli?

Can someone explain the joke that killed Chrysippus of Soli?


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There are a number of versions of how Chrysippus died, one of which says he drank some overproof wine while another says that he died of laughter*.

Apparently, he died laughing at his own joke. The story goes something like this*:

… one day on his way home Chrysippus came across an ass eating figs. He instructed the old woman who lived with him to give the ass some unmixed wine to drink afterwards, and with that he laughed so violently he died.

Or according to Wikipedia*:

In the second account, he was watching a donkey eat some figs and cried out: "Now give the donkey a drink of pure wine to wash down the figs", whereupon he died in a fit of laughter.

I don't know if it is the zeitgeist or some weird Stoic sense of humor but the joke is completely lost on me. I found multiple sources telling the story, but none provided an explanation.

Can someone explain why somebody in that time period (ca. 200 BC) would find that joke hilarious?


You asked if anyone could explain the joke that killed Chrysippus. I doubt that anyone can; humor is very difficult to translate across cultures. (My second language was German, and I spent far too long trying to understand German humor until I relaxed and accepted that it was just different).

I believe the joke relies on the incongruity raised by an animal eating figs. Figs are human food, and an animal eating human food should drink human beverages. The incongruity is heightened by giving the animal unwatered wine (Wine was normally drunk watered). I'm pretty sure that youtube includes multiple videos around the theme of drunken animals. If you took video of a cat eating a cheeseburger with a bottle of premium beer, people would laugh.

I think that there are two other factors at play (I have no evidence, but I think these are logical assumptions)

  1. Chrysippus apparently had a strong sense of humor.
  2. I suspect that Chrysippus had already indulged in some overproof wine.

Figs were extremely expensive and a status symbol at the time in Athens, the equivalent of good caviar now. To give someone the "sign of the fig" is still an obscene or rude gesture, as "fig" is slang for the female genital organs, and figs were the apples in some versions of the story of Adam and Eve, so again, it could be translated in several ways. However, I believe he's probably poking fun at his fellow attendees. He's at a gathering/party, and if you read accounts/look at the sculptures, he is a grumpy old man. The "donkey" is an ass. Drinking unwatered wine would make it a drunken ass. It's a drunken ass, eating the equivalent of caviar, being noisy and stupid at a party… which is evidently deadly hilarious for a grumpy old genius in 200 BC.


don't over-analyze the joke! The man was coming back from a feast, he clearly had a good time, he was probably a bit drunk too. He was in an excellent mood, and when you are in a good mood even plain jokes can make you laugh a bit too hard. Chrysippus probably laughed to the point where he couldn't breathe properly, probably even choking. The joke is not that complex to need an explanation, the meaning is all behind the context.


There is nothing you need to research about a joke. If you research upon the humour in a joke it's very likely you might end up thinking, "really? he died laughing at this joke?" No matter what is the humour in joke. Its a joke. Everybody knows that. He laughed, obviously we laugh after landing a good joke upon someone. Isn't it? Perhaps, it was merely the moment and situation when he cracked that joke made it more funny. So he died of cardiac arrest or asphyxiation while laughing. So nothing's special about that joke but it's very rare form of death and he was a good figure amongst people because he was a philosopher and all. Its been circling around since centuries. So is this question.


"Funny" is all relative and this is especially true given the times you live in. Have you ever been stoned, and then something really odd happens to you? Just something out of the ordinary, like the recent viral video of stoned Russian guys in a car following 2 vans carrying people who apparently worked for Disney on their way to work. They had a verbal altercation with one another and suddenly they are beating the crap out of one another in the street in full uniform; one of them Mickey, I think. And the Russians following behind just blew up with laughter, which itself was hilarious.

Back in his day, a donkey coming over out of the blue and eating one's precious figs as if he was a connoisseur of such things, then the idea of helping him finish it off properly with wine, was probably the equivalent of that… since, you know, they didn't have much by way of entertainment in the bad old days… so it may have been too much for him in his state of intoxication, and anyway, fate cannot be denied, according to him… so it was his time to go; ignominious death or not!


Sigurd Eysteinsson, also known as Sigurd the Mighty, reigned as the second Earl of Orkney from 875 until his death in 892. With the island becoming a popular refuge for exiled Vikings after the Battle of Hafrsfjord and the unification of Norway under Harald Fairhair, Orkney served as a base from which to conduct raids against their former homelands until King Harald pacified the inhabitants and granted an ally dominion over the territory. Seeking to expand his holdings, Sigurd repeatedly attempted to acquire a foothold on the northern Scottish mainlands, garnering a fearsome reputation as a warrior and raider during his lifetime.

Challenging a native ruler, Máel Brigte the Bucktoothed, to a 40 versus 40 man battle, Sigurd, in an act of great dishonor and deceit, secretly brought 80 men to the field. Easily besting his opponent and winning the unfair battle, he beheaded his defeated opponent. Strapping the head of Máel Brigte to his saddle as a trophy, at some point during his ride home the famed buck-tooth of his enemy scratched Sigurd&rsquos leg. The resultant wound became infected as a result of intimate contact with the necrotic tissue, with Sigurd dying soon after from the contracted illness in an apropos display of karmic vengeance.


5 most bizarre deaths in history

We all want to be remembered when we’re no longer around, and ideally, we’d like to be remembered for something good.

The people you’re about to meet certainly fill the first criteria, but sadly, rather than being remembered for their achievements (or their sins, for that matter), they’ve earned themselves a place in the history books for something else entirely.

Here, we’re taking a look at the stories behind some of history’s most unusual deaths.

1. Overeating

We’ve all overindulged once or twice, but we’ve got nothing on Adolf Frederick, King of Sweden, who died at the age of 60 after an unbelievably huge meal. His final feast included lobster, caviar, sauerkraut, kippers, champagne and 14 (yes, 14) servings of his favourite dessert – semla, a traditional Swedish sweet, almond cream-filled bun, served in a bowl of hot milk. We can think of worse ways to go!

2. Kicking a safe

Jack Daniel, the founder of Jack Daniel’s Tennessee whiskey distillery, died from blood poisoning at the age of 62. How it happened, however, is quite a story. You see, Daniel couldn’t remember the combination to his safe and kicked it out of frustration. The injury was so severe, it left him with a limp and considerable pain. The foot was eventually amputated, but the surrounding area became gangrenous.

People say they’re “dying of laughter” all the time, but who knew it could actually happen? Around 206 BC, ancient Greek philosopher Chrysippus of Soli died after a night of fun and drinking. While watching a donkey eat some figs, he cried, “Now give the donkey a drink of pure wine to wash down the figs”. Chrysippus found his joke so funny, he died of laughter.

4. Having a beard

Hans Steininger, the mayor of the small Austrian town of Braunau am Inn, took great pride in his impressive 1.4-metre-long beard, which reached to his feet. He usually kept it rolled up into a pouch, but one fateful day in 1567, he decided to let the majestic facial hair flow free. On September 28, when a large fire broke out in town, Steininger ran for his life, accidentally tripping on his beard and falling down a flight of stairs, breaking his neck.

5. Politeness

Sometimes, kindness really can kill. In 1601, Danish nobleman and astronomer Tycho Brahe attended a banquet in Prague. He ate, drank and eventually, as we all do, eventually needed to pee – desperately. Unfortunately, Brahe refused to leave the table as it would have been a breach of etiquette. When he returned home, he found he was unable to urinate except in tiny quantities and with excruciating pain. 11 days later, at the age of 54, he passed away.


Contents

Logic in India Edit

Hindu logic Edit

Origin Edit

The Nasadiya Sukta of the Rigveda (RV 10.129) contains ontological speculation in terms of various logical divisions that were later recast formally as the four circles of catuskoti: "A", "not A", "A and 'not A'", and "not A and not not A".

Who really knows?
Who will here proclaim it?
Whence was it produced? Whence is this creation?
The gods came afterwards, with the creation of this universe.
Who then knows whence it has arisen?

Logic began independently in ancient India and continued to develop to early modern times without any known influence from Greek logic. [8]

Before Gautama Edit

Though the origins in India of public debate (pariṣad), one form of rational inquiry, are not clear, we know that public debates were common in preclassical India, for they are frequently alluded to in various Upaniṣads and in the early Buddhist literature. Public debate is not the only form of public deliberations in preclassical India. Assemblies (pariṣad or sabhā) of various sorts, comprising relevant experts, were regularly convened to deliberate on a variety of matters, including administrative, legal and religious matters.

Dattatreya Edit

A philosopher named Dattatreya is stated in the Bhagavata purana to have taught Anvlksikl to Aiarka, Prahlada and others. It appears from the Markandeya purana that the Anvlksikl-vidya expounded by him consisted of a mere disquisition on soul in accordance with the yoga philosophy. Dattatreya expounded the philosophical side of Anvlksiki and not its logical aspect. [9] [10]

Medhatithi Gautama Edit

While the teachers mentioned before dealt with some particular topics of Anviksiki, the credit of founding the Anviksiki in its special sense of a science is to be attributed to Medhatithi Gautama (c. 6th century BC). Guatama founded the anviksiki school of logic. [11] The Mahabharata (12.173.45), around the 5th century BC, refers to the anviksiki and tarka schools of logic.

Panini Edit

Pāṇini (c. 5th century BC) developed a form of logic (to which Boolean logic has some similarities) for his formulation of Sanskrit grammar. Logic is described by Chanakya (c. 350-283 BC) in his Arthashastra as an independent field of inquiry. [12]

Nyaya-Vaisheshika Edit

Two of the six Indian schools of thought deal with logic: Nyaya and Vaisheshika. The Nyaya Sutras of Aksapada Gautama (c. 2nd century AD) constitute the core texts of the Nyaya school, one of the six orthodox schools of Hindu philosophy. This realist school developed a rigid five-member schema of inference involving an initial premise, a reason, an example, an application, and a conclusion. [13] The idealist Buddhist philosophy became the chief opponent to the Naiyayikas.

Jain Logic Edit

Jains made its own unique contribution to this mainstream development of logic by also occupying itself with the basic epistemological issues, namely, with those concerning the nature of knowledge, how knowledge is derived, and in what way knowledge can be said to be reliable.Jain logic

The Jains have doctrines of relativity used for logic and reasoning:

    – the theory of relative pluralism or manifoldness – the theory of conditioned predication and – The theory of partial standpoints.

These Jain philosophical concepts made most important contributions to the ancient Indian philosophy, especially in the areas of skepticism and relativity. [4] [14]

Buddhist logic Edit

Nagarjuna Edit

Nagarjuna (c. 150-250 AD), the founder of the Madhyamika ("Middle Way") developed an analysis known as the catuṣkoṭi (Sanskrit), a "four-cornered" system of argumentation that involves the systematic examination and rejection of each of the 4 possibilities of a proposition, P:

Dignaga Edit

However, Dignaga (c 480-540 AD) is sometimes said to have developed a formal syllogism, [15] and it was through him and his successor, Dharmakirti, that Buddhist logic reached its height it is contested whether their analysis actually constitutes a formal syllogistic system. In particular, their analysis centered on the definition of an inference-warranting relation, "vyapti", also known as invariable concomitance or pervasion. [16] To this end, a doctrine known as "apoha" or differentiation was developed. [17] This involved what might be called inclusion and exclusion of defining properties.

Dignāga's famous "wheel of reason" (Hetucakra) is a method of indicating when one thing (such as smoke) can be taken as an invariable sign of another thing (like fire), but the inference is often inductive and based on past observation. Matilal remarks that Dignāga's analysis is much like John Stuart Mill's Joint Method of Agreement and Difference, which is inductive. [18]

Syllogism and influence Edit

In addition, the traditional five-member Indian syllogism, though deductively valid, has repetitions that are unnecessary to its logical validity. As a result, some commentators see the traditional Indian syllogism as a rhetorical form that is entirely natural in many cultures of the world, and yet not as a logical form—not in the sense that all logically unnecessary elements have been omitted for the sake of analysis.

Though its departure from india, it continued to fascinates many great mathematician like Charles Babbage, George Boole, Augustus de Morgan, John Mill etc.

Logic in China Edit

In China, a contemporary of Confucius, Mozi, "Master Mo", is credited with founding the Mohist school, whose canons dealt with issues relating to valid inference and the conditions of correct conclusions. In particular, one of the schools that grew out of Mohism, the Logicians, are credited by some scholars for their early investigation of formal logic. Due to the harsh rule of Legalism in the subsequent Qin Dynasty, this line of investigation disappeared in China until the introduction of Indian philosophy by Buddhists.

Prehistory of logic Edit

Valid reasoning has been employed in all periods of human history. However, logic studies the principles of valid reasoning, inference and demonstration. It is probable that the idea of demonstrating a conclusion first arose in connection with geometry, which originally meant the same as "land measurement". [19] The ancient Egyptians discovered geometry, including the formula for the volume of a truncated pyramid. [20] Ancient Babylon was also skilled in mathematics. Esagil-kin-apli's medical Diagnostic Handbook in the 11th century BC was based on a logical set of axioms and assumptions, [21] while Babylonian astronomers in the 8th and 7th centuries BC employed an internal logic within their predictive planetary systems, an important contribution to the philosophy of science. [22]

Ancient Greece before Aristotle Edit

While the ancient Egyptians empirically discovered some truths of geometry, the great achievement of the ancient Greeks was to replace empirical methods by demonstrative proof. Both Thales and Pythagoras of the Pre-Socratic philosophers seem aware of geometry's methods.

Fragments of early proofs are preserved in the works of Plato and Aristotle, [23] and the idea of a deductive system was probably known in the Pythagorean school and the Platonic Academy. [20] The proofs of Euclid of Alexandria are a paradigm of Greek geometry. The three basic principles of geometry are as follows:

  • Certain propositions must be accepted as true without demonstration such a proposition is known as an axiom of geometry.
  • Every proposition that is not an axiom of geometry must be demonstrated as following from the axioms of geometry such a demonstration is known as a proof or a "derivation" of the proposition.
  • The proof must be formal that is, the derivation of the proposition must be independent of the particular subject matter in question. [20]

Further evidence that early Greek thinkers were concerned with the principles of reasoning is found in the fragment called dissoi logoi, probably written at the beginning of the fourth century BC. This is part of a protracted debate about truth and falsity. [24] In the case of the classical Greek city-states, interest in argumentation was also stimulated by the activities of the Rhetoricians or Orators and the Sophists, who used arguments to defend or attack a thesis, both in legal and political contexts. [25]

Thales Edit

It is said Thales, most widely regarded as the first philosopher in the Greek tradition, [26] [27] measured the height of the pyramids by their shadows at the moment when his own shadow was equal to his height. Thales was said to have had a sacrifice in celebration of discovering Thales' theorem just as Pythagoras had the Pythagorean theorem. [28]

Thales is the first known individual to use deductive reasoning applied to geometry, by deriving four corollaries to his theorem, and the first known individual to whom a mathematical discovery has been attributed. [29] Indian and Babylonian mathematicians knew his theorem for special cases before he proved it. [30] It is believed that Thales learned that an angle inscribed in a semicircle is a right angle during his travels to Babylon. [31]

Pythagoras Edit

Before 520 BC, on one of his visits to Egypt or Greece, Pythagoras might have met the c. 54 years older Thales. [32] The systematic study of proof seems to have begun with the school of Pythagoras (i. e. the Pythagoreans) in the late sixth century BC. [20] Indeed, the Pythagoreans, believing all was number, are the first philosophers to emphasize form rather than matter. [33]

Heraclitus and Parmenides Edit

The writing of Heraclitus (c. 535 – c. 475 BC) was the first place where the word logos was given special attention in ancient Greek philosophy, [34] Heraclitus held that everything changes and all was fire and conflicting opposites, seemingly unified only by this Logos. He is known for his obscure sayings.

This logos holds always but humans always prove unable to understand it, both before hearing it and when they have first heard it. For though all things come to be in accordance with this logos, humans are like the inexperienced when they experience such words and deeds as I set out, distinguishing each in accordance with its nature and saying how it is. But other people fail to notice what they do when awake, just as they forget what they do while asleep.

In contrast to Heraclitus, Parmenides held that all is one and nothing changes. He may have been a dissident Pythagorean, disagreeing that One (a number) produced the many. [35] "X is not" must always be false or meaningless. What exists can in no way not exist. Our sense perceptions with its noticing of generation and destruction are in grievous error. Instead of sense perception, Parmenides advocated logos as the means to Truth. He has been called the discoverer of logic, [36] [37]

For this view, that That Which Is Not exists, can never predominate. You must debar your thought from this way of search, nor let ordinary experience in its variety force you along this way, (namely, that of allowing) the eye, sightless as it is, and the ear, full of sound, and the tongue, to rule but (you must) judge by means of the Reason (Logos) the much-contested proof which is expounded by me. (B 7.1–8.2)

Zeno of Elea, a pupil of Parmenides, had the idea of a standard argument pattern found in the method of proof known as reductio ad absurdum. This is the technique of drawing an obviously false (that is, "absurd") conclusion from an assumption, thus demonstrating that the assumption is false. [38] Therefore, Zeno and his teacher are seen as the first to apply the art of logic. [39] Plato's dialogue Parmenides portrays Zeno as claiming to have written a book defending the monism of Parmenides by demonstrating the absurd consequence of assuming that there is plurality. Zeno famously used this method to develop his paradoxes in his arguments against motion. Such dialectic reasoning later became popular. The members of this school were called "dialecticians" (from a Greek word meaning "to discuss").

Plato Edit

Let no one ignorant of geometry enter here.

None of the surviving works of the great fourth-century philosopher Plato (428–347 BC) include any formal logic, [40] but they include important contributions to the field of philosophical logic. Plato raises three questions:

  • What is it that can properly be called true or false?
  • What is the nature of the connection between the assumptions of a valid argument and its conclusion?
  • What is the nature of definition?

The first question arises in the dialogue Theaetetus, where Plato identifies thought or opinion with talk or discourse (logos). [41] The second question is a result of Plato's theory of Forms. Forms are not things in the ordinary sense, nor strictly ideas in the mind, but they correspond to what philosophers later called universals, namely an abstract entity common to each set of things that have the same name. In both the Republic and the Sophist, Plato suggests that the necessary connection between the assumptions of a valid argument and its conclusion corresponds to a necessary connection between "forms". [42] The third question is about definition. Many of Plato's dialogues concern the search for a definition of some important concept (justice, truth, the Good), and it is likely that Plato was impressed by the importance of definition in mathematics. [43] What underlies every definition is a Platonic Form, the common nature present in different particular things. Thus, a definition reflects the ultimate object of understanding, and is the foundation of all valid inference. This had a great influence on Plato's student Aristotle, in particular Aristotle's notion of the essence of a thing. [44]

Aristotle Edit

The logic of Aristotle, and particularly his theory of the syllogism, has had an enormous influence in Western thought. [45] Aristotle was the first logician to attempt a systematic analysis of logical syntax, of noun (or term), and of verb. He was the first formal logician, in that he demonstrated the principles of reasoning by employing variables to show the underlying logical form of an argument. [46] He sought relations of dependence which characterize necessary inference, and distinguished the validity of these relations, from the truth of the premises. He was the first to deal with the principles of contradiction and excluded middle in a systematic way. [47]

The Organon Edit

His logical works, called the Organon, are the earliest formal study of logic that have come down to modern times. Though it is difficult to determine the dates, the probable order of writing of Aristotle's logical works is:

  • The Categories, a study of the ten kinds of primitive term.
  • The Topics (with an appendix called On Sophistical Refutations), a discussion of dialectics.
  • On Interpretation, an analysis of simple categorical propositions into simple terms, negation, and signs of quantity.
  • The Prior Analytics, a formal analysis of what makes a syllogism (a valid argument, according to Aristotle).
  • The Posterior Analytics, a study of scientific demonstration, containing Aristotle's mature views on logic.

These works are of outstanding importance in the history of logic. In the Categories, he attempts to discern all the possible things to which a term can refer this idea underpins his philosophical work Metaphysics, which itself had a profound influence on Western thought.

He also developed a theory of non-formal logic (i.e., the theory of fallacies), which is presented in Topics and Sophistical Refutations. [47]

On Interpretation contains a comprehensive treatment of the notions of opposition and conversion chapter 7 is at the origin of the square of opposition (or logical square) chapter 9 contains the beginning of modal logic.

The Prior Analytics contains his exposition of the "syllogism", where three important principles are applied for the first time in history: the use of variables, a purely formal treatment, and the use of an axiomatic system.

Stoics Edit

The other great school of Greek logic is that of the Stoics. [48] Stoic logic traces its roots back to the late 5th century BC philosopher Euclid of Megara, a pupil of Socrates and slightly older contemporary of Plato, probably following in the tradition of Parmenides and Zeno. His pupils and successors were called "Megarians", or "Eristics", and later the "Dialecticians". The two most important dialecticians of the Megarian school were Diodorus Cronus and Philo, who were active in the late 4th century BC.

The Stoics adopted the Megarian logic and systemized it. The most important member of the school was Chrysippus (c. 278–c. 206 BC), who was its third head, and who formalized much of Stoic doctrine. He is supposed to have written over 700 works, including at least 300 on logic, almost none of which survive. [49] [50] Unlike with Aristotle, we have no complete works by the Megarians or the early Stoics, and have to rely mostly on accounts (sometimes hostile) by later sources, including prominently Diogenes Laërtius, Sextus Empiricus, Galen, Aulus Gellius, Alexander of Aphrodisias, and Cicero. [51]

Three significant contributions of the Stoic school were (i) their account of modality, (ii) their theory of the Material conditional, and (iii) their account of meaning and truth. [52]

  • Modality. According to Aristotle, the Megarians of his day claimed there was no distinction between potentiality and actuality. [53] Diodorus Cronus defined the possible as that which either is or will be, the impossible as what will not be true, and the contingent as that which either is already, or will be false. [54] Diodorus is also famous for what is known as his Master argument, which states that each pair of the following 3 propositions contradicts the third proposition:
  • Everything that is past is true and necessary.
  • The impossible does not follow from the possible.
  • What neither is nor will be is possible.
  • Conditional statements. The first logicians to debate conditional statements were Diodorus and his pupil Philo of Megara. Sextus Empiricus refers three times to a debate between Diodorus and Philo. Philo regarded a conditional as true unless it has both a true antecedent and a false consequent. Precisely, let T0 and T1 be true statements, and let F0 and F1 be false statements then, according to Philo, each of the following conditionals is a true statement, because it is not the case that the consequent is false while the antecedent is true (it is not the case that a false statement is asserted to follow from a true statement):
  • If T0, then T1
  • If F0, then T0
  • If F0, then F1
  • If T0, then F0
  • Meaning and truth. The most important and striking difference between Megarian-Stoic logic and Aristotelian logic is that Megarian-Stoic logic concerns propositions, not terms, and is thus closer to modern propositional logic. [60] The Stoics distinguished between utterance (phone), which may be noise, speech (lexis), which is articulate but which may be meaningless, and discourse (logos), which is meaningful utterance. The most original part of their theory is the idea that what is expressed by a sentence, called a lekton, is something real this corresponds to what is now called a proposition. Sextus says that according to the Stoics, three things are linked together: that which signifies, that which is signified, and the object for example, that which signifies is the word Dion, and that which is signified is what Greeks understand but barbarians do not, and the object is Dion himself. [61]

Logic in the Middle East Edit

The works of Al-Kindi, Al-Farabi, Avicenna, Al-Ghazali, Averroes and other Muslim logicians were based on Aristotelian logic and were important in communicating the ideas of the ancient world to the medieval West. [62] Al-Farabi (Alfarabi) (873–950) was an Aristotelian logician who discussed the topics of future contingents, the number and relation of the categories, the relation between logic and grammar, and non-Aristotelian forms of inference. [63] Al-Farabi also considered the theories of conditional syllogisms and analogical inference, which were part of the Stoic tradition of logic rather than the Aristotelian. [64]

Ibn Sina (Avicenna) (980–1037) was the founder of Avicennian logic, which replaced Aristotelian logic as the dominant system of logic in the Islamic world, [65] and also had an important influence on Western medieval writers such as Albertus Magnus. [66] Avicenna wrote on the hypothetical syllogism [67] and on the propositional calculus, which were both part of the Stoic logical tradition. [68] He developed an original "temporally modalized" syllogistic theory, involving temporal logic and modal logic. [63] He also made use of inductive logic, such as the methods of agreement, difference, and concomitant variation which are critical to the scientific method. [67] One of Avicenna's ideas had a particularly important influence on Western logicians such as William of Ockham: Avicenna's word for a meaning or notion (ma'na), was translated by the scholastic logicians as the Latin intentio in medieval logic and epistemology, this is a sign in the mind that naturally represents a thing. [69] This was crucial to the development of Ockham's conceptualism: A universal term (e.g., "man") does not signify a thing existing in reality, but rather a sign in the mind (intentio in intellectu) which represents many things in reality Ockham cites Avicenna's commentary on Metaphysics V in support of this view. [70]

Fakhr al-Din al-Razi (b. 1149) criticised Aristotle's "first figure" and formulated an early system of inductive logic, foreshadowing the system of inductive logic developed by John Stuart Mill (1806–1873). [71] Al-Razi's work was seen by later Islamic scholars as marking a new direction for Islamic logic, towards a Post-Avicennian logic. This was further elaborated by his student Afdaladdîn al-Khûnajî (d. 1249), who developed a form of logic revolving around the subject matter of conceptions and assents. In response to this tradition, Nasir al-Din al-Tusi (1201–1274) began a tradition of Neo-Avicennian logic which remained faithful to Avicenna's work and existed as an alternative to the more dominant Post-Avicennian school over the following centuries. [72]

The Illuminationist school was founded by Shahab al-Din Suhrawardi (1155–1191), who developed the idea of "decisive necessity", which refers to the reduction of all modalities (necessity, possibility, contingency and impossibility) to the single mode of necessity. [73] Ibn al-Nafis (1213–1288) wrote a book on Avicennian logic, which was a commentary of Avicenna's Al-Isharat (The Signs) and Al-Hidayah (The Guidance). [74] Ibn Taymiyyah (1263–1328), wrote the Ar-Radd 'ala al-Mantiqiyyin, where he argued against the usefulness, though not the validity, of the syllogism [75] and in favour of inductive reasoning. [71] Ibn Taymiyyah also argued against the certainty of syllogistic arguments and in favour of analogy his argument is that concepts founded on induction are themselves not certain but only probable, and thus a syllogism based on such concepts is no more certain than an argument based on analogy. He further claimed that induction itself is founded on a process of analogy. His model of analogical reasoning was based on that of juridical arguments. [76] [77] This model of analogy has been used in the recent work of John F. Sowa. [77]

The Sharh al-takmil fi'l-mantiq written by Muhammad ibn Fayd Allah ibn Muhammad Amin al-Sharwani in the 15th century is the last major Arabic work on logic that has been studied. [78] However, "thousands upon thousands of pages" on logic were written between the 14th and 19th centuries, though only a fraction of the texts written during this period have been studied by historians, hence little is known about the original work on Islamic logic produced during this later period. [72]

Logic in medieval Europe Edit

"Medieval logic" (also known as "Scholastic logic") generally means the form of Aristotelian logic developed in medieval Europe throughout roughly the period 1200–1600. [1] For centuries after Stoic logic had been formulated, it was the dominant system of logic in the classical world. When the study of logic resumed after the Dark Ages, the main source was the work of the Christian philosopher Boethius, who was familiar with some of Aristotle's logic, but almost none of the work of the Stoics. [79] Until the twelfth century, the only works of Aristotle available in the West were the Categories, On Interpretation, and Boethius's translation of the Isagoge of Porphyry (a commentary on the Categories). These works were known as the "Old Logic" (Logica Vetus or Ars Vetus). An important work in this tradition was the Logica Ingredientibus of Peter Abelard (1079–1142). His direct influence was small, [80] but his influence through pupils such as John of Salisbury was great, and his method of applying rigorous logical analysis to theology shaped the way that theological criticism developed in the period that followed. [81]

By the early thirteenth century, the remaining works of Aristotle's Organon (including the Prior Analytics, Posterior Analytics, and the Sophistical Refutations) had been recovered in the West. [82] Logical work until then was mostly paraphrasis or commentary on the work of Aristotle. [83] The period from the middle of the thirteenth to the middle of the fourteenth century was one of significant developments in logic, particularly in three areas which were original, with little foundation in the Aristotelian tradition that came before. These were: [84]

  • The theory of supposition. Supposition theory deals with the way that predicates (e.g., 'man') range over a domain of individuals (e.g., all men). [85] In the proposition 'every man is an animal', does the term 'man' range over or 'supposit for' men existing just in the present, or does the range include past and future men? Can a term supposit for a non-existing individual? Some medievalists have argued that this idea is a precursor of modern first-order logic. [86] "The theory of supposition with the associated theories of copulatio (sign-capacity of adjectival terms), ampliatio (widening of referential domain), and distributio constitute one of the most original achievements of Western medieval logic". [87]
  • The theory of syncategoremata. Syncategoremata are terms which are necessary for logic, but which, unlike categorematic terms, do not signify on their own behalf, but 'co-signify' with other words. Examples of syncategoremata are 'and', 'not', 'every', 'if', and so on.
  • The theory of consequences. A consequence is a hypothetical, conditional proposition: two propositions joined by the terms 'if . then'. For example, 'if a man runs, then God exists' (Si homo currit, Deus est). [88] A fully developed theory of consequences is given in Book III of William of Ockham's work Summa Logicae. There, Ockham distinguishes between 'material' and 'formal' consequences, which are roughly equivalent to the modern material implication and logical implication respectively. Similar accounts are given by Jean Buridan and Albert of Saxony.

The last great works in this tradition are the Logic of John Poinsot (1589–1644, known as John of St Thomas), the Metaphysical Disputations of Francisco Suarez (1548–1617), and the Logica Demonstrativa of Giovanni Girolamo Saccheri (1667–1733).

The textbook tradition Edit

Traditional logic generally means the textbook tradition that begins with Antoine Arnauld's and Pierre Nicole's Logic, or the Art of Thinking, better known as the Port-Royal Logic. [89] Published in 1662, it was the most influential work on logic after Aristotle until the nineteenth century. [90] The book presents a loosely Cartesian doctrine (that the proposition is a combining of ideas rather than terms, for example) within a framework that is broadly derived from Aristotelian and medieval term logic. Between 1664 and 1700, there were eight editions, and the book had considerable influence after that. [90] The Port-Royal introduces the concepts of extension and intension. The account of propositions that Locke gives in the Essay is essentially that of the Port-Royal: "Verbal propositions, which are words, [are] the signs of our ideas, put together or separated in affirmative or negative sentences. So that proposition consists in the putting together or separating these signs, according as the things which they stand for agree or disagree." [91]

Dudley Fenner helped popularize Ramist logic, a reaction against Aristotle. Another influential work was the Novum Organum by Francis Bacon, published in 1620. The title translates as "new instrument". This is a reference to Aristotle's work known as the Organon. In this work, Bacon rejects the syllogistic method of Aristotle in favor of an alternative procedure "which by slow and faithful toil gathers information from things and brings it into understanding". [92] This method is known as inductive reasoning, a method which starts from empirical observation and proceeds to lower axioms or propositions from these lower axioms, more general ones can be induced. For example, in finding the cause of a phenomenal nature such as heat, 3 lists should be constructed:

  • The presence list: a list of every situation where heat is found.
  • The absence list: a list of every situation that is similar to at least one of those of the presence list, except for the lack of heat.
  • The variability list: a list of every situation where heat can vary.

Then, the form nature (or cause) of heat may be defined as that which is common to every situation of the presence list, and which is lacking from every situation of the absence list, and which varies by degree in every situation of the variability list.

Other works in the textbook tradition include Isaac Watts's Logick: Or, the Right Use of Reason (1725), Richard Whately's Logic (1826), and John Stuart Mill's A System of Logic (1843). Although the latter was one of the last great works in the tradition, Mill's view that the foundations of logic lie in introspection [93] influenced the view that logic is best understood as a branch of psychology, a view which dominated the next fifty years of its development, especially in Germany. [94]

Logic in Hegel's philosophy Edit

G.W.F. Hegel indicated the importance of logic to his philosophical system when he condensed his extensive Science of Logic into a shorter work published in 1817 as the first volume of his Encyclopaedia of the Philosophical Sciences. The "Shorter" or "Encyclopaedia" Logic, as it is often known, lays out a series of transitions which leads from the most empty and abstract of categories—Hegel begins with "Pure Being" and "Pure Nothing"—to the "Absolute", the category which contains and resolves all the categories which preceded it. Despite the title, Hegel's Logic is not really a contribution to the science of valid inference. Rather than deriving conclusions about concepts through valid inference from premises, Hegel seeks to show that thinking about one concept compels thinking about another concept (one cannot, he argues, possess the concept of "Quality" without the concept of "Quantity") this compulsion is, supposedly, not a matter of individual psychology, because it arises almost organically from the content of the concepts themselves. His purpose is to show the rational structure of the "Absolute"—indeed of rationality itself. The method by which thought is driven from one concept to its contrary, and then to further concepts, is known as the Hegelian dialectic.

Although Hegel's Logic has had little impact on mainstream logical studies, its influence can be seen elsewhere:

    's Geschichte der Logik im Abendland (1855–1867). [95]
  • The work of the British Idealists, such as F.H. Bradley's Principles of Logic (1883).
  • The economic, political, and philosophical studies of Karl Marx, and in the various schools of Marxism.

Logic and psychology Edit

Between the work of Mill and Frege stretched half a century during which logic was widely treated as a descriptive science, an empirical study of the structure of reasoning, and thus essentially as a branch of psychology. [96] The German psychologist Wilhelm Wundt, for example, discussed deriving "the logical from the psychological laws of thought", emphasizing that "psychological thinking is always the more comprehensive form of thinking." [97] This view was widespread among German philosophers of the period:

    described logic as "a specific discipline of psychology". [98] understood logical necessity as grounded in the individual's compulsion to think in a certain way. [99] argued that "logical laws only hold within the limits of our thinking". [100]

Such was the dominant view of logic in the years following Mill's work. [101] This psychological approach to logic was rejected by Gottlob Frege. It was also subjected to an extended and destructive critique by Edmund Husserl in the first volume of his Logical Investigations (1900), an assault which has been described as "overwhelming". [102] Husserl argued forcefully that grounding logic in psychological observations implied that all logical truths remained unproven, and that skepticism and relativism were unavoidable consequences.

Such criticisms did not immediately extirpate what is called "psychologism". For example, the American philosopher Josiah Royce, while acknowledging the force of Husserl's critique, remained "unable to doubt" that progress in psychology would be accompanied by progress in logic, and vice versa. [103]

The period between the fourteenth century and the beginning of the nineteenth century had been largely one of decline and neglect, and is generally regarded as barren by historians of logic. [2] The revival of logic occurred in the mid-nineteenth century, at the beginning of a revolutionary period where the subject developed into a rigorous and formalistic discipline whose exemplar was the exact method of proof used in mathematics. The development of the modern "symbolic" or "mathematical" logic during this period is the most significant in the 2000-year history of logic, and is arguably one of the most important and remarkable events in human intellectual history. [4]

A number of features distinguish modern logic from the old Aristotelian or traditional logic, the most important of which are as follows: [104] Modern logic is fundamentally a calculus whose rules of operation are determined only by the shape and not by the meaning of the symbols it employs, as in mathematics. Many logicians were impressed by the "success" of mathematics, in that there had been no prolonged dispute about any truly mathematical result. C.S. Peirce noted [105] that even though a mistake in the evaluation of a definite integral by Laplace led to an error concerning the moon's orbit that persisted for nearly 50 years, the mistake, once spotted, was corrected without any serious dispute. Peirce contrasted this with the disputation and uncertainty surrounding traditional logic, and especially reasoning in metaphysics. He argued that a truly "exact" logic would depend upon mathematical, i.e., "diagrammatic" or "iconic" thought. "Those who follow such methods will . escape all error except such as will be speedily corrected after it is once suspected". Modern logic is also "constructive" rather than "abstractive" i.e., rather than abstracting and formalising theorems derived from ordinary language (or from psychological intuitions about validity), it constructs theorems by formal methods, then looks for an interpretation in ordinary language. It is entirely symbolic, meaning that even the logical constants (which the medieval logicians called "syncategoremata") and the categoric terms are expressed in symbols.

The development of modern logic falls into roughly five periods: [106]

  • The embryonic period from Leibniz to 1847, when the notion of a logical calculus was discussed and developed, particularly by Leibniz, but no schools were formed, and isolated periodic attempts were abandoned or went unnoticed.
  • The algebraic period from Boole's Analysis to Schröder's Vorlesungen. In this period, there were more practitioners, and a greater continuity of development.
  • The logicist period from the Begriffsschrift of Frege to the Principia Mathematica of Russell and Whitehead. The aim of the "logicist school" was to incorporate the logic of all mathematical and scientific discourse in a single unified system which, taking as a fundamental principle that all mathematical truths are logical, did not accept any non-logical terminology. The major logicists were Frege, Russell, and the early Wittgenstein. [107] It culminates with the Principia, an important work which includes a thorough examination and attempted solution of the antinomies which had been an obstacle to earlier progress.
  • The metamathematical period from 1910 to the 1930s, which saw the development of metalogic, in the finitist system of Hilbert, and the non-finitist system of Löwenheim and Skolem, the combination of logic and metalogic in the work of Gödel and Tarski. Gödel's incompleteness theorem of 1931 was one of the greatest achievements in the history of logic. Later in the 1930s, Gödel developed the notion of set-theoretic constructibility.
  • The period after World War II, when mathematical logic branched into four inter-related but separate areas of research: model theory, proof theory, computability theory, and set theory, and its ideas and methods began to influence philosophy.

Embryonic period Edit

The idea that inference could be represented by a purely mechanical process is found as early as Raymond Llull, who proposed a (somewhat eccentric) method of drawing conclusions by a system of concentric rings. The work of logicians such as the Oxford Calculators [108] led to a method of using letters instead of writing out logical calculations (calculationes) in words, a method used, for instance, in the Logica magna by Paul of Venice. Three hundred years after Llull, the English philosopher and logician Thomas Hobbes suggested that all logic and reasoning could be reduced to the mathematical operations of addition and subtraction. [109] The same idea is found in the work of Leibniz, who had read both Llull and Hobbes, and who argued that logic can be represented through a combinatorial process or calculus. But, like Llull and Hobbes, he failed to develop a detailed or comprehensive system, and his work on this topic was not published until long after his death. Leibniz says that ordinary languages are subject to "countless ambiguities" and are unsuited for a calculus, whose task is to expose mistakes in inference arising from the forms and structures of words [110] hence, he proposed to identify an alphabet of human thought comprising fundamental concepts which could be composed to express complex ideas, [111] and create a calculus ratiocinator that would make all arguments "as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate." [112]

Gergonne (1816) said that reasoning does not have to be about objects about which one has perfectly clear ideas, because algebraic operations can be carried out without having any idea of the meaning of the symbols involved. [113] Bolzano anticipated a fundamental idea of modern proof theory when he defined logical consequence or "deducibility" in terms of variables: [114]

Algebraic period Edit

Modern logic begins with what is known as the "algebraic school", originating with Boole and including Peirce, Jevons, Schröder, and Venn. [115] Their objective was to develop a calculus to formalise reasoning in the area of classes, propositions, and probabilities. The school begins with Boole's seminal work Mathematical Analysis of Logic which appeared in 1847, although De Morgan (1847) is its immediate precursor. [116] The fundamental idea of Boole's system is that algebraic formulae can be used to express logical relations. This idea occurred to Boole in his teenage years, working as an usher in a private school in Lincoln, Lincolnshire. [117] For example, let x and y stand for classes let the symbol = signify that the classes have the same members, xy stand for the class containing all and only the members of x and y and so on. Boole calls these elective symbols, i.e. symbols which select certain objects for consideration. [118] An expression in which elective symbols are used is called an elective function, and an equation of which the members are elective functions, is an elective equation. [119] The theory of elective functions and their "development" is essentially the modern idea of truth-functions and their expression in disjunctive normal form. [118]

Boole's system admits of two interpretations, in class logic, and propositional logic. Boole distinguished between "primary propositions" which are the subject of syllogistic theory, and "secondary propositions", which are the subject of propositional logic, and showed how under different "interpretations" the same algebraic system could represent both. An example of a primary proposition is "All inhabitants are either Europeans or Asiatics." An example of a secondary proposition is "Either all inhabitants are Europeans or they are all Asiatics." [120] These are easily distinguished in modern predicate logic, where it is also possible to show that the first follows from the second, but it is a significant disadvantage that there is no way of representing this in the Boolean system. [121]

In his Symbolic Logic (1881), John Venn used diagrams of overlapping areas to express Boolean relations between classes or truth-conditions of propositions. In 1869 Jevons realised that Boole's methods could be mechanised, and constructed a "logical machine" which he showed to the Royal Society the following year. [118] In 1885 Allan Marquand proposed an electrical version of the machine that is still extant (picture at the Firestone Library).

The defects in Boole's system (such as the use of the letter v for existential propositions) were all remedied by his followers. Jevons published Pure Logic, or the Logic of Quality apart from Quantity in 1864, where he suggested a symbol to signify exclusive or, which allowed Boole's system to be greatly simplified. [122] This was usefully exploited by Schröder when he set out theorems in parallel columns in his Vorlesungen (1890–1905). Peirce (1880) showed how all the Boolean elective functions could be expressed by the use of a single primitive binary operation, "neither . nor . " and equally well "not both . and . ", [123] however, like many of Peirce's innovations, this remained unknown or unnoticed until Sheffer rediscovered it in 1913. [124] Boole's early work also lacks the idea of the logical sum which originates in Peirce (1867), Schröder (1877) and Jevons (1890), [125] and the concept of inclusion, first suggested by Gergonne (1816) and clearly articulated by Peirce (1870).

The success of Boole's algebraic system suggested that all logic must be capable of algebraic representation, and there were attempts to express a logic of relations in such form, of which the most ambitious was Schröder's monumental Vorlesungen über die Algebra der Logik ("Lectures on the Algebra of Logic", vol iii 1895), although the original idea was again anticipated by Peirce. [126]

Boole's unwavering acceptance of Aristotle's logic is emphasized by the historian of logic John Corcoran in an accessible introduction to Laws of Thought [127] Corcoran also wrote a point-by-point comparison of Prior Analytics and Laws of Thought. [128] According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were "to go under, over, and beyond" Aristotle's logic by 1) providing it with mathematical foundations involving equations, 2) extending the class of problems it could treat — from assessing validity to solving equations — and 3) expanding the range of applications it could handle — e.g. from propositions having only two terms to those having arbitrarily many.

More specifically, Boole agreed with what Aristotle said Boole's 'disagreements', if they might be called that, concern what Aristotle did not say. First, in the realm of foundations, Boole reduced the four propositional forms of Aristotelian logic to formulas in the form of equations — by itself a revolutionary idea. Second, in the realm of logic's problems, Boole's addition of equation solving to logic — another revolutionary idea — involved Boole's doctrine that Aristotle's rules of inference (the "perfect syllogisms") must be supplemented by rules for equation solving. Third, in the realm of applications, Boole's system could handle multi-term propositions and arguments whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, Aristotle's system could not deduce "No quadrangle that is a square is a rectangle that is a rhombus" from "No square that is a quadrangle is a rhombus that is a rectangle" or from "No rhombus that is a rectangle is a square that is a quadrangle".

Logicist period Edit

After Boole, the next great advances were made by the German mathematician Gottlob Frege. Frege's objective was the program of Logicism, i.e. demonstrating that arithmetic is identical with logic. [129] Frege went much further than any of his predecessors in his rigorous and formal approach to logic, and his calculus or Begriffsschrift is important. [129] Frege also tried to show that the concept of number can be defined by purely logical means, so that (if he was right) logic includes arithmetic and all branches of mathematics that are reducible to arithmetic. He was not the first writer to suggest this. In his pioneering work Die Grundlagen der Arithmetik (The Foundations of Arithmetic), sections 15–17, he acknowledges the efforts of Leibniz, J.S. Mill as well as Jevons, citing the latter's claim that "algebra is a highly developed logic, and number but logical discrimination." [130]

Frege's first work, the Begriffsschrift ("concept script") is a rigorously axiomatised system of propositional logic, relying on just two connectives (negational and conditional), two rules of inference (modus ponens and substitution), and six axioms. Frege referred to the "completeness" of this system, but was unable to prove this. [131] The most significant innovation, however, was his explanation of the quantifier in terms of mathematical functions. Traditional logic regards the sentence "Caesar is a man" as of fundamentally the same form as "all men are mortal." Sentences with a proper name subject were regarded as universal in character, interpretable as "every Caesar is a man". [132] At the outset Frege abandons the traditional "concepts subject and predicate", replacing them with argument and function respectively, which he believes "will stand the test of time. It is easy to see how regarding a content as a function of an argument leads to the formation of concepts. Furthermore, the demonstration of the connection between the meanings of the words if, and, not, or, there is, some, all, and so forth, deserves attention". [133] Frege argued that the quantifier expression "all men" does not have the same logical or semantic form as "all men", and that the universal proposition "every A is B" is a complex proposition involving two functions, namely ' – is A' and ' – is B' such that whatever satisfies the first, also satisfies the second. In modern notation, this would be expressed as

In English, "for all x, if Ax then Bx". Thus only singular propositions are of subject-predicate form, and they are irreducibly singular, i.e. not reducible to a general proposition. Universal and particular propositions, by contrast, are not of simple subject-predicate form at all. If "all mammals" were the logical subject of the sentence "all mammals are land-dwellers", then to negate the whole sentence we would have to negate the predicate to give "all mammals are not land-dwellers". But this is not the case. [134] This functional analysis of ordinary-language sentences later had a great impact on philosophy and linguistics.

This means that in Frege's calculus, Boole's "primary" propositions can be represented in a different way from "secondary" propositions. "All inhabitants are either men or women" is

whereas "All the inhabitants are men or all the inhabitants are women" is

As Frege remarked in a critique of Boole's calculus:

"The real difference is that I avoid [the Boolean] division into two parts . and give a homogeneous presentation of the lot. In Boole the two parts run alongside one another, so that one is like the mirror image of the other, but for that very reason stands in no organic relation to it' [135]

As well as providing a unified and comprehensive system of logic, Frege's calculus also resolved the ancient problem of multiple generality. The ambiguity of "every girl kissed a boy" is difficult to express in traditional logic, but Frege's logic resolves this through the different scope of the quantifiers. Thus

means that to every girl there corresponds some boy (any one will do) who the girl kissed. But

means that there is some particular boy whom every girl kissed. Without this device, the project of logicism would have been doubtful or impossible. Using it, Frege provided a definition of the ancestral relation, of the many-to-one relation, and of mathematical induction. [136]

This period overlaps with the work of what is known as the "mathematical school", which included Dedekind, Pasch, Peano, Hilbert, Zermelo, Huntington, Veblen and Heyting. Their objective was the axiomatisation of branches of mathematics like geometry, arithmetic, analysis and set theory. Most notable was Hilbert's Program, which sought to ground all of mathematics to a finite set of axioms, proving its consistency by "finitistic" means and providing a procedure which would decide the truth or falsity of any mathematical statement. The standard axiomatization of the natural numbers is named the Peano axioms eponymously. Peano maintained a clear distinction between mathematical and logical symbols. While unaware of Frege's work, he independently recreated his logical apparatus based on the work of Boole and Schröder. [137]

The logicist project received a near-fatal setback with the discovery of a paradox in 1901 by Bertrand Russell. This proved Frege's naive set theory led to a contradiction. Frege's theory contained the axiom that for any formal criterion, there is a set of all objects that meet the criterion. Russell showed that a set containing exactly the sets that are not members of themselves would contradict its own definition (if it is not a member of itself, it is a member of itself, and if it is a member of itself, it is not). [138] This contradiction is now known as Russell's paradox. One important method of resolving this paradox was proposed by Ernst Zermelo. [139] Zermelo set theory was the first axiomatic set theory. It was developed into the now-canonical Zermelo–Fraenkel set theory (ZF). Russell's paradox symbolically is as follows:

The monumental Principia Mathematica, a three-volume work on the foundations of mathematics, written by Russell and Alfred North Whitehead and published 1910–13 also included an attempt to resolve the paradox, by means of an elaborate system of types: a set of elements is of a different type than is each of its elements (set is not the element one element is not the set) and one cannot speak of the "set of all sets". The Principia was an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in symbolic logic.

Metamathematical period Edit

The names of Gödel and Tarski dominate the 1930s, [140] a crucial period in the development of metamathematics – the study of mathematics using mathematical methods to produce metatheories, or mathematical theories about other mathematical theories. Early investigations into metamathematics had been driven by Hilbert's program. Work on metamathematics culminated in the work of Gödel, who in 1929 showed that a given first-order sentence is deducible if and only if it is logically valid – i.e. it is true in every structure for its language. This is known as Gödel's completeness theorem. A year later, he proved two important theorems, which showed Hibert's program to be unattainable in its original form. The first is that no consistent system of axioms whose theorems can be listed by an effective procedure such as an algorithm or computer program is capable of proving all facts about the natural numbers. For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second is that if such a system is also capable of proving certain basic facts about the natural numbers, then the system cannot prove the consistency of the system itself. These two results are known as Gödel's incompleteness theorems, or simply Gödel's Theorem. Later in the decade, Gödel developed the concept of set-theoretic constructibility, as part of his proof that the axiom of choice and the continuum hypothesis are consistent with Zermelo–Fraenkel set theory. In proof theory, Gerhard Gentzen developed natural deduction and the sequent calculus. The former attempts to model logical reasoning as it 'naturally' occurs in practice and is most easily applied to intuitionistic logic, while the latter was devised to clarify the derivation of logical proofs in any formal system. Since Gentzen's work, natural deduction and sequent calculi have been widely applied in the fields of proof theory, mathematical logic and computer science. Gentzen also proved normalization and cut-elimination theorems for intuitionistic and classical logic which could be used to reduce logical proofs to a normal form. [141] [142]

Alfred Tarski, a pupil of Łukasiewicz, is best known for his definition of truth and logical consequence, and the semantic concept of logical satisfaction. In 1933, he published (in Polish) The concept of truth in formalized languages, in which he proposed his semantic theory of truth: a sentence such as "snow is white" is true if and only if snow is white. Tarski's theory separated the metalanguage, which makes the statement about truth, from the object language, which contains the sentence whose truth is being asserted, and gave a correspondence (the T-schema) between phrases in the object language and elements of an interpretation. Tarski's approach to the difficult idea of explaining truth has been enduringly influential in logic and philosophy, especially in the development of model theory. [143] Tarski also produced important work on the methodology of deductive systems, and on fundamental principles such as completeness, decidability, consistency and definability. According to Anita Feferman, Tarski "changed the face of logic in the twentieth century". [144]

Alonzo Church and Alan Turing proposed formal models of computability, giving independent negative solutions to Hilbert's Entscheidungsproblem in 1936 and 1937, respectively. The Entscheidungsproblem asked for a procedure that, given any formal mathematical statement, would algorithmically determine whether the statement is true. Church and Turing proved there is no such procedure Turing's paper introduced the halting problem as a key example of a mathematical problem without an algorithmic solution.

Church's system for computation developed into the modern λ-calculus, while the Turing machine became a standard model for a general-purpose computing device. It was soon shown that many other proposed models of computation were equivalent in power to those proposed by Church and Turing. These results led to the Church–Turing thesis that any deterministic algorithm that can be carried out by a human can be carried out by a Turing machine. Church proved additional undecidability results, showing that both Peano arithmetic and first-order logic are undecidable. Later work by Emil Post and Stephen Cole Kleene in the 1940s extended the scope of computability theory and introduced the concept of degrees of unsolvability.

The results of the first few decades of the twentieth century also had an impact upon analytic philosophy and philosophical logic, particularly from the 1950s onwards, in subjects such as modal logic, temporal logic, deontic logic, and relevance logic.

Logic after WWII Edit

After World War II, mathematical logic branched into four inter-related but separate areas of research: model theory, proof theory, computability theory, and set theory. [145]

In set theory, the method of forcing revolutionized the field by providing a robust method for constructing models and obtaining independence results. Paul Cohen introduced this method in 1963 to prove the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory. [146] His technique, which was simplified and extended soon after its introduction, has since been applied to many other problems in all areas of mathematical logic.

Computability theory had its roots in the work of Turing, Church, Kleene, and Post in the 1930s and 40s. It developed into a study of abstract computability, which became known as recursion theory. [147] The priority method, discovered independently by Albert Muchnik and Richard Friedberg in the 1950s, led to major advances in the understanding of the degrees of unsolvability and related structures. Research into higher-order computability theory demonstrated its connections to set theory. The fields of constructive analysis and computable analysis were developed to study the effective content of classical mathematical theorems these in turn inspired the program of reverse mathematics. A separate branch of computability theory, computational complexity theory, was also characterized in logical terms as a result of investigations into descriptive complexity.

Model theory applies the methods of mathematical logic to study models of particular mathematical theories. Alfred Tarski published much pioneering work in the field, which is named after a series of papers he published under the title Contributions to the theory of models. In the 1960s, Abraham Robinson used model-theoretic techniques to develop calculus and analysis based on infinitesimals, a problem that first had been proposed by Leibniz.

In proof theory, the relationship between classical mathematics and intuitionistic mathematics was clarified via tools such as the realizability method invented by Georg Kreisel and Gödel's Dialectica interpretation. This work inspired the contemporary area of proof mining. The Curry–Howard correspondence emerged as a deep analogy between logic and computation, including a correspondence between systems of natural deduction and typed lambda calculi used in computer science. As a result, research into this class of formal systems began to address both logical and computational aspects this area of research came to be known as modern type theory. Advances were also made in ordinal analysis and the study of independence results in arithmetic such as the Paris–Harrington theorem.

This was also a period, particularly in the 1950s and afterwards, when the ideas of mathematical logic begin to influence philosophical thinking. For example, tense logic is a formalised system for representing, and reasoning about, propositions qualified in terms of time. The philosopher Arthur Prior played a significant role in its development in the 1960s. Modal logics extend the scope of formal logic to include the elements of modality (for example, possibility and necessity). The ideas of Saul Kripke, particularly about possible worlds, and the formal system now called Kripke semantics have had a profound impact on analytic philosophy. [148] His best known and most influential work is Naming and Necessity (1980). [149] Deontic logics are closely related to modal logics: they attempt to capture the logical features of obligation, permission and related concepts. Although some basic novelties syncretizing mathematical and philosophical logic were shown by Bolzano in the early 1800s, it was Ernst Mally, a pupil of Alexius Meinong, who was to propose the first formal deontic system in his Grundgesetze des Sollens, based on the syntax of Whitehead's and Russell's propositional calculus.

Another logical system founded after World War II was fuzzy logic by Azerbaijani mathematician Lotfi Asker Zadeh in 1965.


Contents

No authentic writings of Pythagoras have survived, [5] [6] [7] and almost nothing is known for certain about his life. [8] [9] [10] The earliest sources on Pythagoras's life are brief, ambiguous, and often satirical. [7] [11] [12] The earliest source on Pythagoras's teachings is a satirical poem probably written after his death by Xenophanes of Colophon, who had been one of his contemporaries. [13] [14] In the poem, Xenophanes describes Pythagoras interceding on behalf of a dog that is being beaten, professing to recognize in its cries the voice of a departed friend. [12] [13] [15] [16] Alcmaeon of Croton, a doctor who lived in Croton at around the same time Pythagoras lived there, [13] incorporates many Pythagorean teachings into his writings [17] and alludes to having possibly known Pythagoras personally. [17] The poet Heraclitus of Ephesus, who was born across a few miles of sea away from Samos and may have lived within Pythagoras's lifetime, [18] mocked Pythagoras as a clever charlatan, [11] [18] remarking that "Pythagoras, son of Mnesarchus, practiced inquiry more than any other man, and selecting from these writings he manufactured a wisdom for himself—much learning, artful knavery." [11] [18]

The Greek poets Ion of Chios (c. 480 – c. 421 BC ) and Empedocles of Acragas (c. 493 – c. 432 BC ) both express admiration for Pythagoras in their poems. [19] The first concise description of Pythagoras comes from the historian Herodotus of Halicarnassus ( c. 484 – c. 420 BC ), [20] who describes him as "not the most insignificant" of Greek sages [21] and states that Pythagoras taught his followers how to attain immortality. [20] The accuracy of the works of Herodotus is controversial. [22] [23] [24] [25] [26] The writings attributed to the Pythagorean philosopher Philolaus of Croton, who lived in the late fifth century BC, are the earliest texts to describe the numerological and musical theories that were later ascribed to Pythagoras. [27] The Athenian rhetorician Isocrates (436–338 BC) was the first to describe Pythagoras as having visited Egypt. [20] Aristotle wrote a treatise On the Pythagoreans, which is no longer extant. [28] Some of it may be preserved in the Protrepticus. Aristotle's disciples Dicaearchus, Aristoxenus, and Heraclides Ponticus also wrote on the same subject. [29]

Most of the major sources on Pythagoras's life are from the Roman period, [30] by which point, according to the German classicist Walter Burkert, "the history of Pythagoreanism was already. the laborious reconstruction of something lost and gone." [29] Three lives of Pythagoras have survived from late antiquity, [10] [30] all of which are filled primarily with myths and legends. [10] [30] [31] The earliest and most respectable of these is the one from Diogenes Laërtius's Lives and Opinions of Eminent Philosophers. [30] [31] The two later lives were written by the Neoplatonist philosophers Porphyry and Iamblichus [30] [31] and were partially intended as polemics against the rise of Christianity. [31] The later sources are much lengthier than the earlier ones, [30] and even more fantastic in their descriptions of Pythagoras's achievements. [30] [31] Porphyry and Iamblichus used material from the lost writings of Aristotle's disciples [29] and material taken from these sources is generally considered to be the most reliable. [29]

Early life

There is not a single detail in the life of Pythagoras that stands uncontradicted. But it is possible, from a more or less critical selection of the data, to construct a plausible account.

Herodotus, Isocrates, and other early writers agree that Pythagoras was the son of Mnesarchus [20] [33] and that he was born on the Greek island of Samos in the eastern Aegean. [5] [33] [34] [35] His father is said to have been a gem-engraver or a wealthy merchant, [36] [37] but his ancestry is disputed and unclear. [38] [d] Pythagoras's name led him to be associated with Pythian Apollo (Pūthíā) Aristippus of Cyrene in the 4th century BC explained his name by saying, "He spoke [ ἀγορεύω , agoreúō] the truth no less than did the Pythian [ πυθικός puthikós]". [39] A late source gives Pythagoras's mother's name as Pythaïs. [40] [41] Iamblichus tells the story that the Pythia prophesied to her while she was pregnant with him that she would give birth to a man supremely beautiful, wise, and beneficial to humankind. [39] 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. [42]

During Pythagoras's formative years, Samos was a thriving cultural hub known for its feats of advanced architectural engineering, including the building of the Tunnel of Eupalinos, and for its riotous festival culture. [43] It was a major center of trade in the Aegean where traders brought goods from the Near East. [5] According to Christiane L. Joost-Gaugier, these traders almost certainly brought with them Near Eastern ideas and traditions. [5] Pythagoras's early life also coincided with the flowering of early Ionian natural philosophy. [33] [44] He was a contemporary of the philosophers Anaximander, Anaximenes, and the historian Hecataeus, all of whom lived in Miletus, across the sea from Samos. [44]

Reputed travels

Pythagoras is traditionally thought to have received most of his education in Ancient Egypt, the Neo-Babylonian Empire, the Achaemenid Empire, and Crete. [45] Modern scholarship has shown that the culture of Archaic Greece was heavily influenced by those of Levantine and Mesopotamian cultures. [45] Like many other important Greek thinkers, Pythagoras was said to have studied in Egypt. [20] [46] [47] By the time of Isocrates in the fourth century BC, Pythagoras's reputed studies in Egypt were already taken as fact. [20] [39] The writer Antiphon, who may have lived during the Hellenistic Era, claimed in his lost work On Men of Outstanding Merit, used as a source by Porphyry, that Pythagoras learned to speak Egyptian from the Pharaoh Amasis II himself, that he studied with the Egyptian priests at Diospolis (Thebes), and that he was the only foreigner ever to be granted the privilege of taking part in their worship. [45] [48] The Middle Platonist biographer Plutarch (c. 46 – c. 120 AD ) writes in his treatise On Isis and Osiris that, during his visit to Egypt, Pythagoras received instruction from the Egyptian priest Oenuphis of Heliopolis (meanwhile Solon received lectures from a Sonchis of Sais). [49] According to the Christian theologian Clement of Alexandria ( c. 150 – c. 215 AD ), "Pythagoras was a disciple of Soches, an Egyptian archprophet, as well as Plato of Sechnuphis of Heliopolis." [50] Some ancient writers claimed that Pythagoras learned geometry and the doctrine of metempsychosis from the Egyptians. [46] [51]

Other ancient writers, however, claimed that Pythagoras had learned these teachings from the Magi in Persia or even from Zoroaster himself. [52] [53] Diogenes Laërtius asserts that Pythagoras later visited Crete, where he went to the Cave of Ida with Epimenides. [52] The Phoenicians are reputed to have taught Pythagoras arithmetic and the Chaldeans to have taught him astronomy. [53] By the third century BC, Pythagoras was already reported to have studied under the Jews as well. [53] Contradicting all these reports, the novelist Antonius Diogenes, writing in the second century BC, reports that Pythagoras discovered all his doctrines himself by interpreting dreams. [53] The third-century AD Sophist Philostratus claims that, in addition to the Egyptians, Pythagoras also studied under Hindu sages or gymnosophists in India. [53] Iamblichus expands this list even further by claiming that Pythagoras also studied with the Celts and Iberians. [53]

Alleged Greek teachers

Ancient sources also record Pythagoras having studied under a variety of native Greek thinkers. [53] Some identify Hermodamas of Samos as a possible tutor. [53] [55] Hermodamas represented the indigenous Samian rhapsodic tradition and his father Creophylos was said to have been the host of his rival poet Homer. [53] Others credit Bias of Priene, Thales, [56] or Anaximander (a pupil of Thales). [53] [56] [57] Other traditions claim the mythic bard Orpheus as Pythagoras's teacher, thus representing the Orphic Mysteries. [53] The Neoplatonists wrote of a "sacred discourse" Pythagoras had written on the gods in the Doric Greek dialect, which they believed had been dictated to Pythagoras by the Orphic priest Aglaophamus upon his initiation to the orphic Mysteries at Leibethra. [53] Iamblichus credited Orpheus with having been the model for Pythagoras's manner of speech, his spiritual attitude, and his manner of worship. [58] Iamblichus describes Pythagoreanism as a synthesis of everything Pythagoras had learned from Orpheus, from the Egyptian priests, from the Eleusinian Mysteries, and from other religious and philosophical traditions. [58] Riedweg states that, although these stories are fanciful, Pythagoras's teachings were definitely influenced by Orphism to a noteworthy extent. [59]

Of the various Greek sages claimed to have taught Pythagoras, Pherecydes of Syros is mentioned most often. [59] [60] Similar miracle stories were told about both Pythagoras and Pherecydes, including one in which the hero predicts a shipwreck, one in which he predicts the conquest of Messina, and one in which he drinks from a well and predicts an earthquake. [59] Apollonius Paradoxographus, a paradoxographer who may have lived in the second century BC, identified Pythagoras's thaumaturgic ideas as a result of Pherecydes's influence. [59] Another story, which may be traced to the Neopythagorean philosopher Nicomachus, tells that, when Pherecydes was old and dying on the island of Delos, Pythagoras returned to care for him and pay his respects. [59] Duris, the historian and tyrant of Samos, is reported to have patriotically boasted of an epitaph supposedly penned by Pherecydes which declared that Pythagoras's wisdom exceeded his own. [59] On the grounds of all these references connecting Pythagoras with Pherecydes, Riedweg concludes that there may well be some historical foundation to the tradition that Pherecydes was Pythagoras's teacher. [59] Pythagoras and Pherecydes also appear to have shared similar views on the soul and the teaching of metempsychosis. [59]

Before 520 BC, on one of his visits to Egypt or Greece, Pythagoras might have met Thales of Miletus, who would have been around fifty-four years older than him. Thales was a philosopher, scientist, mathematician, and engineer, [61] also known for a special case of the inscribed angle theorem. Pythagoras's birthplace, the island of Samos, is situated in the Northeast Aegean Sea not far from Miletus. [62] Diogenes Laërtius cites a statement from Aristoxenus (fourth century BC) stating that Pythagoras learned most of his moral doctrines from the Delphic priestess Themistoclea. [63] [64] [65] Porphyry agrees with this assertion, [66] but calls the priestess Aristoclea (Aristokleia). [67] Ancient authorities furthermore note the similarities between the religious and ascetic peculiarities of Pythagoras with the Orphic or Cretan mysteries, [68] or the Delphic oracle. [69]

In Croton

Porphyry repeats an account from Antiphon, who reported that, while he was still on Samos, Pythagoras founded a school known as the "semicircle". [70] [71] Here, Samians debated matters of public concern. [70] [71] Supposedly, the school became so renowned that the brightest minds in all of Greece came to Samos to hear Pythagoras teach. [70] Pythagoras himself dwelled in a secret cave, where he studied in private and occasionally held discourses with a few of his close friends. [70] [71] Christoph Riedweg, a German scholar of early Pythagoreanism, states that it is entirely possible Pythagoras may have taught on Samos, [70] but cautions that Antiphon's account, which makes reference to a specific building that was still in use during his own time, appears to be motivated by Samian patriotic interest. [70]

Around 530 BC, when Pythagoras was around forty years old, he left Samos. [5] [33] [72] [73] [74] His later admirers claimed that he left because he disagreed with the tyranny of Polycrates in Samos, [61] [72] Riedweg notes that this explanation closely aligns with Nicomachus's emphasis on Pythagoras's purported love of freedom, but that Pythagoras's enemies portrayed him as having a proclivity towards tyranny. [72] Other accounts claim that Pythagoras left Samos because he was so overburdened with public duties in Samos, because of the high estimation in which he was held by his fellow-citizens. [75] He arrived in the Greek colony of Croton (today's Crotone, in Calabria) in what was then Magna Graecia. [33] [74] [76] [77] All sources agree that Pythagoras was charismatic and quickly acquired great political influence in his new environment. [33] [78] [79] He served as an advisor to the elites in Croton and gave them frequent advice. [80] Later biographers tell fantastical stories of the effects of his eloquent speeches 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. [81] [82]

Family and friends

Diogenes Laërtius states that Pythagoras "did not indulge in the pleasures of love" [86] and that he cautioned others to only have sex "whenever you are willing to be weaker than yourself". [87] According to Porphyry, Pythagoras married Theano, a lady of Crete and the daughter of Pythenax [87] and had several children with her. [87] Porphyry writes that Pythagoras had two sons named Telauges and Arignote, [87] and a daughter named Myia, [87] who "took precedence among the maidens in Croton and, when a wife, among married women." [87] Iamblichus mentions none of these children [87] and instead only mentions a son named Mnesarchus after his grandfather. [87] This son was raised by Pythagoras's appointed successor Aristaeus and eventually took over the school when Aristaeus was too old to continue running it. [87] Suda writes that Pythagoras had 4 children (Telauges, Mnesarchus, Myia and Arignote). [88]

The wrestler Milo of Croton was said to have been a close associate of Pythagoras [89] and was credited with having saved the philosopher's life when a roof was about to collapse. [89] This association may been the result of confusion with a different man named Pythagoras, who was an athletics trainer. [70] Diogenes Laërtius records Milo's wife's name as Myia. [87] Iamblichus mentions Theano as the wife of Brontinus of Croton. [87] Diogenes Laërtius states that the same Theano was Pythagoras's pupil [87] and that Pythagoras's wife Theano was her daughter. [87] Diogenes Laërtius also records that works supposedly written by Theano were still extant during his own lifetime [87] and quotes several opinions attributed to her. [87] These writings are now known to be pseudepigraphical. [87]

Death

Pythagoras's emphasis on dedication and asceticism are credited with aiding in Croton's decisive victory over the neighboring colony of Sybaris in 510 BC. [90] After the victory, some prominent citizens of Croton proposed a democratic constitution, which the Pythagoreans rejected. [90] The supporters of democracy, headed by Cylon and Ninon, the former of whom is said to have been irritated by his exclusion from Pythagoras's brotherhood, roused the populace against them. [91] Followers of Cylon and Ninon attacked the Pythagoreans during one of their meetings, either in the house of Milo or in some other meeting-place. [92] [93] Accounts of the attack are often contradictory and many probably confused it with later anti-Pythagorean rebellions. [91] The building was apparently set on fire, [92] and many of the assembled members perished [92] only the younger and more active members managed to escape. [94]

Sources disagree regarding whether Pythagoras was present when the attack occurred and, if he was, whether or not he managed to escape. [32] [93] In some accounts, Pythagoras was not at the meeting when the Pythagoreans were attacked because he was on Delos tending to the dying Pherecydes. [93] According to another account from Dicaearchus, Pythagoras was at the meeting and managed to escape, [95] leading a small group of followers to the nearby city of Locris, where they pleaded for sanctuary, but were denied. [95] They reached the city of Metapontum, where they took shelter in the temple of the Muses and died there of starvation after forty days without food. [32] [92] [95] [96] Another tale recorded by Porphyry claims that, as Pythagoras's enemies were burning the house, his devoted students laid down on the ground to make a path for him to escape by walking over their bodies across the flames like a bridge. [95] Pythagoras managed to escape, but was so despondent at the deaths of his beloved students that he committed suicide. [95] A different legend reported by both Diogenes Laërtius and Iamblichus states that Pythagoras almost managed to escape, but that he came to a fava bean field and refused to run through it, since doing so would violate his teachings, so he stopped instead and was killed. [95] [97] This story seems to have originated from the writer Neanthes, who told it about later Pythagoreans, not about Pythagoras himself. [95]

Metempsychosis

Although the exact details of Pythagoras's teachings are uncertain, [99] [100] it is possible to reconstruct a general outline of his main ideas. [99] [101] Aristotle writes at length about the teachings of the Pythagoreans, [16] [102] but without mentioning Pythagoras directly. [16] [102] One of Pythagoras's main doctrines appears to have been metempsychosis, [73] [103] [104] [105] [106] [107] the belief that all souls are immortal and that, after death, a soul is transferred into a new body. [103] [106] This teaching is referenced by Xenophanes, Ion of Chios, and Herodotus. [103] [108] Nothing whatsoever, however, is known about the nature or mechanism by which Pythagoras believed metempsychosis to occur. [109]

Empedocles alludes in one of his poems that Pythagoras may have claimed to possess the ability to recall his former incarnations. [110] Diogenes Laërtius reports an account from Heraclides Ponticus that Pythagoras told people that he had lived four previous lives that he could remember in detail. [111] [112] [113] The first of these lives was as Aethalides the son of Hermes, who granted him the ability to remember all his past incarnations. [114] Next, he was incarnated as Euphorbus, a minor hero from the Trojan War briefly mentioned in the Iliad. [115] He then became the philosopher Hermotimus, [116] who recognized the shield of Euphorbus in the temple of Apollo. [116] His final incarnation was as Pyrrhus, a fisherman from Delos. [116] One of his past lives, as reported by Dicaearchus, was as a beautiful courtesan. [104] [117]

Mysticism

Another belief attributed to Pythagoras was that of the "harmony of the spheres", [118] [119] which maintained that the planets and stars move according to mathematical equations, which correspond to musical notes and thus produce an inaudible symphony. [118] [119] According to Porphyry, Pythagoras taught that the seven Muses were actually the seven planets singing together. [120] In his philosophical dialogue Protrepticus, Aristotle has his literary double say:

When Pythagoras was asked [why humans exist], he said, "to observe the heavens," and he used to claim that he himself was an observer of nature, and it was for the sake of this that he had passed over into life. [121]

Pythagoras was said to have practiced divination and prophecy. [122] 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. [123]

Numerology

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.

According to Aristotle, the Pythagoreans used mathematics for solely mystical reasons, devoid of practical application. [128] They believed that all things were made of numbers. [129] [130] The number one (the monad) represented the origin of all things [131] and the number two (the dyad) represented matter. [131] The number three was an "ideal number" because it had a beginning, middle, and end [132] and was the smallest number of points that could be used to define a plane triangle, which they revered as a symbol of the god Apollo. [132] The number four signified the four seasons and the four elements. [133] The number seven was also sacred because it was the number of planets and the number of strings on a lyre, [133] and because Apollo's birthday was celebrated on the seventh day of each month. [133] They believed that odd numbers were masculine, [134] that even numbers were feminine, [134] and that the number five represented marriage, because it was the sum of two and three. [135] [136]

Ten was regarded as the "perfect number" [128] and the Pythagoreans honored it by never gathering in groups larger than ten. [137] Pythagoras was credited with devising the tetractys, the triangular figure of four rows which add up to the perfect number, ten. [124] [125] The Pythagoreans regarded the tetractys as a symbol of utmost mystical importance. [124] [125] [126] Iamblichus, in his Life of Pythagoras, states that the tetractys was "so admirable, and so divinised by those who understood [it]," that Pythagoras's students would swear oaths by it. [98] [125] [126] [138] Andrew Gregory concludes that the tradition linking Pythagoras to the tetractys is probably genuine. [139]

Modern scholars debate whether these numerological teachings were developed by Pythagoras himself or by the later Pythagorean philosopher Philolaus of Croton. [140] In his landmark study Lore and Science in Ancient Pythagoreanism, Walter Burkert argues that Pythagoras was a charismatic political and religious teacher, [141] but that the number philosophy attributed to him was really an innovation by Philolaus. [142] According to Burkert, Pythagoras never dealt with numbers at all, let alone made any noteworthy contribution to mathematics. [141] Burkert argues that the only mathematics the Pythagoreans ever actually engaged in was simple, proofless arithmetic, [143] but that these arithmetic discoveries did contribute significantly to the beginnings of mathematics. [144]

Communal lifestyle

Both Plato and Isocrates state that, above all else, Pythagoras was known as the founder of a new way of life. [145] [146] [147] The organization Pythagoras founded at Croton was called a "school", [148] [149] [61] but, in many ways, resembled a monastery. [150] 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. [151] The members of the sect shared all their possessions in common [152] and were devoted to each other to the exclusion of outsiders. [153] [154] Ancient sources record that the Pythagoreans ate meals in common after the manner of the Spartans. [155] [156] One Pythagorean maxim was "koinà tà phílōn" ("All things in common among friends"). [152] Both Iamblichus and Porphyry provide detailed accounts of the organization 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. [157] Iamblichus, in particular, presents the "Pythagorean Way of Life" as a pagan alternative to the Christian monastic communities of his own time. [150]

Two groups existed within early Pythagoreanism: the mathematikoi ("learners") and the akousmatikoi ("listeners"). [62] [158] The akousmatikoi are traditionally identified by scholars as "old believers" in mysticism, numerology, and religious teachings [158] whereas the mathematikoi are traditionally identified as a more intellectual, modernist faction who were more rationalist and scientific. [158] Gregory cautions that there was probably not a sharp distinction between them and that many Pythagoreans probably believed the two approaches were compatible. [158] The study of mathematics and music may have been connected to the worship of Apollo. [159] The Pythagoreans believed that music was a purification for the soul, just as medicine was a purification for the body. [120] One anecdote of Pythagoras reports that when he encountered some drunken youths trying to break into the home of a virtuous woman, he sang a solemn tune with long spondees and the boys' "raging willfulness" was quelled. [120] The Pythagoreans also placed particular emphasis on the importance of physical exercise [150] therapeutic dancing, daily morning walks along scenic routes, and athletics were major components of the Pythagorean lifestyle. [150] Moments of contemplation at the beginning and end of each day were also advised. [160]

Prohibitions and regulations

Pythagorean teachings were known as "symbols" (symbola) [83] and members took a vow of silence that they would not reveal these symbols to non-members. [83] [146] [161] Those who did not obey the laws of the community were expelled [162] and the remaining members would erect tombstones for them as though they had died. [162] A number of "oral sayings" (akoúsmata) attributed to Pythagoras have survived, [16] [163] dealing with how members of the Pythagorean community should perform sacrifices, how they should honor the gods, how they should "move from here", and how they should be buried. [164] Many of these sayings emphasize the importance of ritual purity and avoiding defilement. [165] [107] For instance, a saying which Leonid Zhmud concludes can probably be genuinely traced back to Pythagoras himself forbids his followers from wearing woolen garments. [166] Other extant oral sayings forbid Pythagoreans from breaking bread, poking fires with swords, or picking up crumbs [156] and teach that a person should always put the right sandal on before the left. [156] The exact meanings of these sayings, however, are frequently obscure. [167] Iamblichus preserves Aristotle's descriptions of the original, ritualistic intentions behind a few of these sayings, [168] but these apparently later fell out of fashion, because Porphyry provides markedly different ethical-philosophical interpretations of them: [169]

Pythagorean saying Original ritual purpose according to Aristotle/Iamblichus Porphyry's philosophical interpretation
"Do not take roads traveled by the public." [170] [16] "Fear of being defiled by the impure" [170] "with this he forbade following the opinions of the masses, yet to follow the ones of the few and the educated." [170]
"and [do] not wear images of the gods on rings" [170] "Fear of defiling them by wearing them." [170] "One should not have the teaching and knowledge of the gods quickly at hand and visible [for everyone], nor communicate them to the masses." [170]
"and pour libations for the gods from a drinking cup's handle [the 'ear']" [170] "Efforts to keep the divine and the human strictly separate" [170] "thereby he enigmatically hints that the gods should be honored and praised with music for it goes through the ears." [170]

New initiates were allegedly not permitted to meet Pythagoras until after they had completed a five-year initiation period, [71] during which they were required to remain silent. [71] Sources indicate that Pythagoras himself was unusually progressive in his attitudes towards women [85] and female members of Pythagoras's school appear to have played an active role in its operations. [83] [85] Iamblichus provides a list of 235 famous Pythagoreans, [84] seventeen of whom are women. [84] In later times, many prominent female philosophers contributed to the development of Neopythagoreanism. [171]

Pythagoreanism also entailed a number of dietary prohibitions. [107] [156] [172] It is more or less agreed that Pythagoras issued a prohibition against the consumption of fava beans [173] [156] and the meat of non-sacrificial animals such as fish and poultry. [166] [156] Both of these assumptions, however, have been contradicted. [174] [175] Pythagorean dietary restrictions may have been motivated by belief in the doctrine of metempsychosis. [146] [176] [177] [178] Some ancient writers present Pythagoras as enforcing a strictly vegetarian diet. [e] [146] [177] Eudoxus of Cnidus, a student of Archytas, writes, "Pythagoras was distinguished by such purity and so avoided killing and killers that he not only abstained from animal foods, but even kept his distance from cooks and hunters." [179] [180] Other authorities contradict this statement. [181] According to Aristoxenus, [182] Pythagoras allowed the use of all kinds of animal food except the flesh of oxen used for ploughing, and rams. [180] [183] According to Heraclides Ponticus, Pythagoras ate the meat from sacrifices [180] and established a diet for athletes dependent on meat. [180]

Within his own lifetime, Pythagoras was already the subject of elaborate hagiographic legends. [30] [184] Aristotle described Pythagoras as a wonder-worker and somewhat of a supernatural figure. [185] [186] In a fragment, Aristotle writes that Pythagoras had a golden thigh, [185] [187] [188] which he publicly exhibited at the Olympic Games [185] [189] and showed to Abaris the Hyperborean as proof of his identity as the "Hyperborean Apollo". [185] [190] Supposedly, the priest of Apollo gave Pythagoras a magic arrow, which he used to fly over long distances and perform ritual purifications. [191] He was supposedly once seen at both Metapontum and Croton at the same time. [192] [30] [189] [187] [188] When Pythagoras crossed the river Kosas (the modern-day Basento), "several witnesses" reported that they heard it greet him by name. [193] [189] [187] In Roman times, a legend claimed that Pythagoras was the son of Apollo. [194] [188] According to Muslim tradition, Pythagoras was said to have been initiated by Hermes (Egyptian Thoth). [195]

Pythagoras was said to have dressed all in white. [185] [196] He is also said to have borne a golden wreath atop his head [185] and to have worn trousers after the fashion of the Thracians. [185] Diogenes Laërtius presents Pythagoras as having exercised remarkable self-control [197] he was always cheerful, [197] but "abstained wholly from laughter, and from all such indulgences as jests and idle stories". [87] Pythagoras was said to have had extraordinary success in dealing with animals. [30] [198] [189] A fragment from Aristotle records that, when a deadly snake bit Pythagoras, he bit it back and killed it. [191] [189] [187] Both Porphyry and Iamblichus report that Pythagoras once persuaded a bull not to eat fava beans [30] [198] and that he once convinced a notoriously destructive bear to swear that it would never harm a living thing again, and that the bear kept its word. [30] [198]

Riedweg suggests that Pythagoras may have personally encouraged these legends, [184] but Gregory states that there is no direct evidence of this. [158] Anti-Pythagorean legends were also circulated. [199] Diogenes Laërtes retells a story told by Hermippus of Samos, which states that Pythagoras had once gone into an underground room, telling everyone that he was descending to the underworld. [200] He stayed in this room for months, while his mother secretly recorded everything that happened during his absence. [200] After he returned from this room, Pythagoras recounted everything that had happened while he was gone, [200] convincing everyone that he had really been in the underworld [200] and leading them to trust him with their wives. [200]

In mathematics

Although Pythagoras is most famous today for his alleged mathematical discoveries, [127] [201] classical historians dispute whether he himself ever actually made any significant contributions to the field. [143] [141] Many mathematical and scientific discoveries were attributed to Pythagoras, including his famous theorem, [202] as well as discoveries in the fields of music, [203] astronomy, [204] and medicine. [205] Since at least the first century BC, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, [206] [207] a theorem in geometry that states that "in a right-angled triangle the square of the hypotenuse is equal [to the sum of] the squares of the two other sides" [208] —that is, a 2 + b 2 = c 2 +b^<2>=c^<2>> . According to a popular legend, after he discovered this theorem, Pythagoras sacrificed an ox, or possibly even a whole hecatomb, to the gods. [208] [209] Cicero rejected this story as spurious [208] because of the much more widely held belief that Pythagoras forbade blood sacrifices. [208] Porphyry attempted to explain the story by asserting that the ox was actually made of dough. [208]

The Pythagorean theorem was known and used by the Babylonians and Indians centuries before Pythagoras, [210] [208] [211] [212] but he may have been the first to introduce it to the Greeks. [213] [211] Some historians of mathematics have even suggested that he—or his students—may have constructed the first proof. [214] Burkert rejects this suggestion as implausible, [213] noting that Pythagoras was never credited with having proved any theorem in antiquity. [213] Furthermore, the manner in which the Babylonians employed 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. [f] Pythagoras's biographers state that he also was the first to identify the five regular solids [127] and that he was the first to discover the Theory of Proportions. [127]

In music

According to legend, Pythagoras discovered that musical notes could be translated into mathematical equations when he passed blacksmiths at work one day and heard the sound of their hammers clanging against the anvils. [215] [216] Thinking that the sounds of the hammers were beautiful and harmonious, except for one, [217] he rushed into the blacksmith shop and began testing the hammers. [217] He then realized that the tune played when the hammer struck was directly proportional to the size of the hammer and therefore concluded that music was mathematical. [216] [217] However, this legend is demonstrably false, [126] [216] [218] as these ratios are only relevant to string length (such as the string of a monochord), and not to hammer weight. [218] [216]

In astronomy

In ancient times, Pythagoras and his contemporary Parmenides of Elea were both credited with having been the first to teach that the Earth was spherical, [219] the first to divide the globe into five climatic zones, [219] and the first to identify the morning star and the evening star as the same celestial object (now known as Venus). [220] Of the two philosophers, Parmenides has a much stronger claim to having been the first [221] and the attribution of these discoveries to Pythagoras seems to have possibly originated from a pseudepigraphal poem. [220] Empedocles, who lived in Magna Graecia shortly after Pythagoras and Parmenides, knew that the earth was spherical. [222] By the end of the fifth century BC, this fact was universally accepted among Greek intellectuals. [223] The identity of the morning star and evening star was known to the Babylonians over a thousand years earlier. [224]

On Greek philosophy

Sizeable Pythagorean communities existed in Magna Graecia, Phlius, and Thebes during the early fourth century BC. [226] Around the same time, the Pythagorean philosopher Archytas was highly influential on the politics of the city of Tarentum in Magna Graecia. [227] According to later tradition, Archytas was elected as strategos ("general") seven times, even though others were prohibited from serving more than a year. [227] Archytas was also a renowned mathematician and musician. [228] He was a close friend of Plato [229] and he is quoted in Plato's Republic. [230] [231] Aristotle states that the philosophy of Plato was heavily dependent on the teachings of the Pythagoreans. [232] [233] Cicero repeats this statement, remarking that Platonem ferunt didicisse Pythagorea omnia ("They say Plato learned all things Pythagorean"). [234] According to Charles H. Kahn, Plato's middle dialogues, including Meno, Phaedo, and The Republic, have a strong "Pythagorean coloring", [235] and his last few dialogues (particularly Philebus and Timaeus) [225] are extremely Pythagorean in character. [225]

According to R. M. Hare, Plato's Republic may be partially based on the "tightly organised community of like-minded thinkers" established by Pythagoras at Croton. [236] Additionally, Plato may have borrowed from Pythagoras the idea that mathematics and abstract thought are a secure basis for philosophy, science, and morality. [236] Plato and Pythagoras shared a "mystical approach to the soul and its place in the material world" [236] and it is probable that both were influenced by Orphism. [236] The historian of philosophy Frederick Copleston states that Plato probably borrowed his tripartite theory of the soul from the Pythagoreans. [237] Bertrand Russell, in his A History of Western Philosophy, contends that the influence of Pythagoras on Plato and others was so great that he should be considered the most influential philosopher of all time. [238] He concludes that "I do not know of any other man who has been as influential as he was in the school of thought." [239]

A revival of Pythagorean teachings occurred in the first century BC [240] when Middle Platonist philosophers such as Eudorus and Philo of Alexandria hailed the rise of a "new" Pythagoreanism in Alexandria. [241] At around the same time, Neopythagoreanism became prominent. [242] The first-century AD philosopher Apollonius of Tyana sought to emulate Pythagoras and live by Pythagorean teachings. [243] The later first-century Neopythagorean philosopher Moderatus of Gades expanded on Pythagorean number philosophy [243] and probably understood the soul as a "kind of mathematical harmony." [243] The Neopythagorean mathematician and musicologist Nicomachus likewise expanded on Pythagorean numerology and music theory. [242] Numenius of Apamea interpreted Plato's teachings in light of Pythagorean doctrines. [244]

On art and architecture

Greek sculpture sought to represent the permanent reality behind superficial appearances. [246] Early Archaic sculpture represents life in simple forms, and may have been influenced by the earliest Greek natural philosophies. [g] The Greeks generally believed that nature expressed itself in ideal forms and was represented by a type ( εἶδος ), which was mathematically calculated. [247] [248] When dimensions changed, architects sought to relay permanence through mathematics. [249] [250] Maurice Bowra believes that these ideas influenced the theory of Pythagoras and his students, who believed that "all things are numbers". [250]

During the sixth century BC, the number philosophy of the Pythagoreans triggered a revolution in Greek sculpture. [251] Greek sculptors and architects attempted to find the mathematical relation (canon) behind aesthetic perfection. [248] Possibly drawing on the ideas of Pythagoras, [248] the sculptor Polykleitos wrote in his Canon that beauty consists in the proportion, not of the elements (materials), but of the interrelation of parts with one another and with the whole. [248] [h] In the Greek architectural orders, every element was calculated and constructed by mathematical relations. Rhys Carpenter states that the ratio 2:1 was "the generative ratio of the Doric order, and in Hellenistic times an ordinary Doric colonnade, beats out a rhythm of notes." [248]

The oldest known building designed according to Pythagorean teachings is the Porta Maggiore Basilica, [252] a subterranean basilica which was built during the reign of the Roman emperor Nero as a secret place of worship for Pythagoreans. [253] The basilica was built underground because of the Pythagorean emphasis on secrecy [254] and also because of the legend that Pythagoras had sequestered himself in a cave on Samos. [255] The basilica's apse is in the east and its atrium in the west out of respect for the rising sun. [256] It has a narrow entrance leading to a small pool where the initiates could purify themselves. [257] The building is also designed according to Pythagorean numerology, [258] with each table in the sanctuary providing seats for seven people. [137] Three aisles lead to a single altar, symbolizing the three parts of the soul approaching the unity of Apollo. [137] The apse depicts a scene of the poet Sappho leaping off the Leucadian cliffs, clutching her lyre to her breast, while Apollo stands beneath her, extending his right hand in a gesture of protection, [259] symbolizing Pythagorean teachings about the immortality of the soul. [259] The interior of the sanctuary is almost entirely white because the color white was regarded by Pythagoreans as sacred. [260]

The emperor Hadrian's Pantheon in Rome was also built based on Pythagorean numerology. [245] The temple's circular plan, central axis, hemispherical dome, and alignment with the four cardinal directions symbolize Pythagorean views on the order of the universe. [261] The single oculus at the top of the dome symbolizes the monad and the sun-god Apollo. [262] The twenty-eight ribs extending from the oculus symbolize the moon, because twenty-eight was the same number of months on the Pythagorean lunar calendar. [263] The five coffered rings beneath the ribs represent the marriage of the sun and moon. [132]

In early Christianity

Many early Christians had a deep respect for Pythagoras. [264] Eusebius (c. 260 – c. 340 AD), bishop of Caesarea, praises Pythagoras in his Against Hierokles for his rule of silence, his frugality, his "extraordinary" morality, and his wise teachings. [265] In another work, Eusebius compares Pythagoras to Moses. [265] In one of his letter, the Church Father Jerome (c. 347 – 420 AD) praises Pythagoras for his wisdom [265] and, in another letter, he credits Pythagoras for his belief in the immortality of the soul, which he suggests Christians inherited from him. [266] Augustine of Hippo (354 – 430 AD) rejected Pythagoras's teaching of metempsychosis without explicitly naming him, but otherwise expressed admiration for him. [267] In On the Trinity, Augustine lauds the fact that Pythagoras was humble enough to call himself a philosophos or "lover of wisdom" rather than a "sage". [268] In another passage, Augustine defends Pythagoras's reputation, arguing that Pythagoras certainly never taught the doctrine of metempsychosis. [268]

In the Middle Ages

During the Middle Ages, Pythagoras was revered as the founder of mathematics and music, two of the Seven Liberal Arts. [269] He appears in numerous medieval depictions, in illuminated manuscripts and in the relief sculptures on the portal of the Cathedral of Chartres. [269] The Timaeus was the only dialogue of Plato to survive in Latin translation in western Europe, [269] which led William of Conches (c. 1080–1160) to declare that Plato was Pythagorean. [269] In the 1430s, the Camaldolese friar Ambrose Traversari translated Diogenes Laërtius's Lives and Opinions of Eminent Philosophers from Greek into Latin [269] and, in the 1460s, the philosopher Marsilio Ficino translated Porphyry and Iamblichus's Lives of Pythagoras into Latin as well, [269] thereby allowing them to be read and studied by western scholars. [269] In 1494, the Greek Neopythagorean scholar Constantine Lascaris published The Golden Verses of Pythagoras, translated into Latin, with a printed edition of his Grammatica, [270] thereby bringing them to a widespread audience. [270] In 1499, he published the first Renaissance biography of Pythagoras in his work Vitae illustrium philosophorum siculorum et calabrorum, issued in Messina. [270]

On modern science

In his preface to his book On the Revolution of the Heavenly Spheres (1543), Nicolaus Copernicus cites various Pythagoreans as the most important influences on the development of his heliocentric model of the universe, [269] [271] deliberately omitting mention of Aristarchus of Samos, a non-Pythagorean astronomer who had developed a fully heliocentric model in the fourth century BC, in effort to portray his model as fundamentally Pythagorean. [271] Johannes Kepler considered himself to be a Pythagorean. [269] [272] [273] He believed in the Pythagorean doctrine of musica universalis [274] and it was his search for the mathematical equations behind this doctrine that led to his discovery of the laws of planetary motion. [274] Kepler titled his book on the subject Harmonices Mundi (Harmonics of the World), after the Pythagorean teaching that had inspired him. [269] [275] Near the conclusion of the book, Kepler describes himself falling asleep to the sound of the heavenly music, "warmed by having drunk a generous draught. from the cup of Pythagoras." [276] He also called Pythagoras the "grandfather" of all Copernicans. [277]

Isaac Newton firmly believed in the Pythagorean teaching of the mathematical harmony and order of the universe. [278] Though Newton was notorious for rarely giving others credit for their discoveries, [279] he attributed the discovery of the Law of Universal Gravitation to Pythagoras. [279] Albert Einstein believed that a scientist may also be "a Platonist or a Pythagorean insofar as he considers the viewpoint of logical simplicity as an indispensable and effective tool of his research." [280] The English philosopher Alfred North Whitehead argued that "In a sense, Plato and Pythagoras stand nearer to modern physical science than does Aristotle. The two former were mathematicians, whereas Aristotle was the son of a doctor". [281] By this measure, Whitehead declared that Einstein and other modern scientists like him are "following the pure Pythagorean tradition." [280] [282]

On vegetarianism

A fictionalized portrayal of Pythagoras appears in Book XV of Ovid's Metamorphoses, [284] in which he delivers a speech imploring his followers to adhere to a strictly vegetarian diet. [285] It was through Arthur Golding's 1567 English translation of Ovid's Metamorphoses that Pythagoras was best known to English-speakers throughout the early modern period. [285] John Donne's Progress of the Soul discusses the implications of the doctrines expounded in the speech, [286] and Michel de Montaigne quoted the speech no less than three times in his treatise "Of Cruelty" to voice his moral objections against the mistreatment of animals. [286] William Shakespeare references the speech in his play The Merchant of Venice. [287] John Dryden included a translation of the scene with Pythagoras in his 1700 work Fables, Ancient and Modern, [286] and John Gay's 1726 fable "Pythagoras and the Countryman" reiterates its major themes, linking carnivorism with tyranny. [286] Lord Chesterfield records that his conversion to vegetarianism had been motivated by reading Pythagoras's speech in Ovid's Metamorphoses. [286] Until the word vegetarianism was coined in the 1840s, vegetarians were referred to in English as "Pythagoreans". [286] Percy Bysshe Shelley wrote an ode entitled "To the Pythagorean Diet", [288] and Leo Tolstoy adopted the Pythagorean diet himself. [288]

On Western esotericism

Early modern European esotericism drew heavily on the teachings of Pythagoras. [269] The German humanist scholar Johannes Reuchlin (1455–1522) synthesized Pythagoreanism with Christian theology and Jewish Kabbalah, [289] arguing that Kabbalah and Pythagoreanism were both inspired by Mosaic tradition [290] and that Pythagoras was therefore a kabbalist. [290] In his dialogue De verbo mirifico (1494), Reuchlin compared the Pythagorean tetractys to the ineffable divine name YHWH, [289] ascribing each of the four letters of the tetragrammaton a symbolic meaning according to Pythagorean mystical teachings. [290]

Heinrich Cornelius Agrippa's popular and influential three-volume treatise De Occulta Philosophia cites Pythagoras as a "religious magi" [291] and indicates that Pythagoras's mystical numerology operates on a supercelestial level. [291] The freemasons deliberately modeled their society on the community founded by Pythagoras at Croton. [292] Rosicrucianism used Pythagorean symbolism, [269] as did Robert Fludd (1574–1637), [269] who believed his own musical writings to have been inspired by Pythagoras. [269] John Dee was heavily influenced by Pythagorean ideology, [293] [291] particularly the teaching that all things are made of numbers. [293] [291] Adam Weishaupt, the founder of the Illuminati, was a strong admirer of Pythagoras [294] and, in his book Pythagoras (1787), he advocated that society should be reformed to be more like Pythagoras's commune at Croton. [295] Wolfgang Amadeus Mozart incorporated Masonic and Pythagorean symbolism into his opera The Magic Flute. [296] Sylvain Maréchal, in his six-volume 1799 biography The Voyages of Pythagoras, declared that all revolutionaries in all time periods are the "heirs of Pythagoras". [297]

On literature

Dante Alighieri was fascinated by Pythagorean numerology [298] and based his descriptions of Hell, Purgatory, and Heaven on Pythagorean numbers. [298] Dante wrote that Pythagoras saw Unity as Good and Plurality as Evil [299] and, in Paradiso XV, 56–57, he declares: "five and six, if understood, ray forth from unity." [300] The number eleven and its multiples are found throughout the Divine Comedy, each book of which has thirty-three cantos, except for the Inferno, which has thirty-four, the first of which serves as a general introduction. [301] Dante describes the ninth and tenth bolgias in the Eighth Circle of Hell as being twenty-two miles and eleven miles respectively, [301] which correspond to the fraction 22 / 7 , which was the Pythagorean approximation of pi. [301] Hell, Purgatory, and Heaven are all described as circular [301] and Dante compares the wonder of God's majesty to the mathematical puzzle of squaring the circle. [301] The number three also features prominently: [301] the Divine Comedy has three parts [302] and Beatrice is associated with the number nine, which is equal to three times three. [303]

The Transcendentalists read the ancient Lives of Pythagoras as guides on how to live a model life. [304] Henry David Thoreau was impacted by Thomas Taylor's translations of Iamblichus's Life of Pythagoras and Stobaeus's Pythagoric Sayings [304] and his views on nature may have been influenced by the Pythagorean idea of images corresponding to archetypes. [304] The Pythagorean teaching of musica universalis is a recurring theme throughout Thoreau's magnum opus, Walden. [304]

Footnotes

  1. ^US:/ p ɪ ˈ θ æ ɡ ər ə s / , [2]UK:/ p aɪ -/ [3]Ancient Greek: Πυθαγόρας ὁ Σάμιος , romanized:Pythagóras ho Sámios, lit.'Pythagoras the Samian', or simply Πυθαγόρας Πυθαγόρης in Ionian Greek
  2. ^ "The dates of his life cannot be fixed exactly, but assuming the approximate correctness of the statement of Aristoxenus (ap. Porph. V.P. 9) that he left Samos to escape the tyranny of Polycrates at the age of forty, we may put his birth round about 570 BC, or a few years earlier. The length of his life was variously estimated in antiquity, but it is agreed that he lived to a fairly ripe old age, and most probably he died at about seventy-five or eighty." [4]
  3. ^Cicero, Tusculan Disputations, 5.3.8–9 (citing Heraclides Ponticus fr. 88 Wehrli), Diogenes Laërtius 1.12, 8.8, IamblichusVP 58. Burkert attempted to discredit this ancient tradition, but it has been defended by C.J. De Vogel, Pythagoras and Early Pythagoreanism (1966), pp. 97–102, and C. Riedweg, Pythagoras: His Life, Teaching, And Influence (2005), p. 92.
  4. ^ Some writers call him a Tyrrhenian or Phliasian, and give Marmacus, or Demaratus, as the name of his father: Diogenes Laërtius, viii. 1 Porphyry, Vit. Pyth. 1, 2 Justin, xx. 4 Pausanias, ii. 13.
  5. ^ as Empedocles did afterwards, Aristotle, Rhet. i. 14. § 2 Sextus Empiricus, ix. 127. This was also one of the Orphic precepts, Aristoph. Ran. 1032
  6. ^ There are about 100,000 unpublished cuneiform sources in the British Museum alone. Babylonian knowledge of proof of the Pythagorean Theorem is discussed by J. Høyrup, 'The Pythagorean "Rule" and "Theorem" – Mirror of the Relation between Babylonian and Greek Mathematics,' in: J. Renger (red.): Babylon. Focus mesopotamischer Geschichte, Wiege früher Gelehrsamkeit, Mythos in der Moderne (1999).
  7. ^ "For Thales, the origin was water, and for Anaximander the infinite (apeiron), which must be considered a material form" [246]
  8. ^ "Each part (finger, palm, arm, etc) transmitted its individual existence to the next, and then to the whole": Canon of Polykleitos, also Plotinus, Ennead I.vi.i: Nigel Spivey, pp. 290–294.

Citations

  1. ^ abcJoost-Gaugier 2006, p. 143.
  2. ^American: Pythagoras, Collins Dictionary, n.d. , retrieved 25 September 2014
  3. ^
  4. British: Pythagoras, Collins Dictionary, n.d. , retrieved 25 September 2014
  5. ^William Keith Chambers Guthrie, (1978), A history of Greek philosophy, Volume 1: The earlier Presocratics and the Pythagoreans, p. 173. Cambridge University Press
  6. ^ abcdeJoost-Gaugier 2006, p. 11.
  7. ^Celenza 2010, p. 796.
  8. ^ abFerguson 2008, p. 4.
  9. ^Ferguson 2008, pp. 3–5.
  10. ^Gregory 2015, pp. 21–23.
  11. ^ abcCopleston 2003, p. 29.
  12. ^ abcKahn 2001, p. 2.
  13. ^ abBurkert 1985, p. 299.
  14. ^ abcJoost-Gaugier 2006, p. 12.
  15. ^Riedweg 2005, p. 62.
  16. ^ Diogenes Laërtius, viii. 36
  17. ^ abcdeCopleston 2003, p. 31.
  18. ^ abJoost-Gaugier 2006, pp. 12–13.
  19. ^ abcJoost-Gaugier 2006, p. 13.
  20. ^Joost-Gaugier 2006, pp. 14–15.
  21. ^ abcdefJoost-Gaugier 2006, p. 16.
  22. ^ 4. 95.
  23. ^Marincola (2001), p. 59
  24. ^Roberts (2011), p. 2
  25. ^Sparks (1998), p. 58
  26. ^Asheri, Lloyd & Corcella (2007)
  27. ^Cameron (2004), p. 156
  28. ^Joost-Gaugier 2006, p. 88.
  29. ^ He alludes to it himself, Met. i. 5. p. 986. 12, ed. Bekker.
  30. ^ abcdBurkert 1972, p. 109.
  31. ^ abcdefghijklKahn 2001, p. 5.
  32. ^ abcdeZhmud 2012, p. 9.
  33. ^ abcBurkert 1972, p. 106.
  34. ^ abcdefKahn 2001, p. 6.
  35. ^Ferguson 2008, p. 12.
  36. ^Kenny 2004, p. 9.
  37. ^ Clemens von Alexandria: Stromata I 62, 2–3, cit.
  38. Eugene V. Afonasin John M. Dillon John Finamore, eds. (2012), Iamblichus and the Foundations of Late Platonism, Leiden and Boston: Brill, p. 15, ISBN978-90-04-23011-8
  39. ^Joost-Gaugier 2006, p. 21.
  40. ^Ferguson 2008, pp. 11–12.
  41. ^ abcRiedweg 2005, p. 59.
  42. ^Taub 2017, p. 122.
  43. ^Apollonius of Tyana ap. Porphyry, Vit. Pyth. 2.
  44. ^ Porphyry, Vit. Pyth. 9
  45. ^Riedweg 2005, pp. 45–47.
  46. ^ abRiedweg 2005, pp. 44–45.
  47. ^ abcRiedweg 2005, p. 7.
  48. ^ abRiedweg 2005, pp. 7–8.
  49. ^Gregory 2015, pp. 22–23.
  50. ^ Porphyry, Vit. Pyth. 6.
  51. ^ Plutarch, On Isis And Osiris, ch. 10.
  52. ^Press 2003, p. 83.
  53. ^ cf. Antiphon. ap. Porphyry, Vit. Pyth. 7 Isocrates, Busiris, 28–9 Cicero, de Finibus, v. 29 Strabo, 14.1.16.
  54. ^ ab Diogenes Laërtius, viii. 1, 3.
  55. ^ abcdefghijklRiedweg 2005, p. 8.
  56. ^Dillon 2005, p. 163.
  57. ^ Porphyry, Vit. Pyth. 2, Diogenes Laërtius, viii. 2.
  58. ^ ab Iamblichus, Vit. Pyth. 9.
  59. ^ Porphyry, Vit. Pyth. 2.
  60. ^ abRiedweg 2005, pp. 8–9.
  61. ^ abcdefghRiedweg 2005, p. 9.
  62. ^ Aristoxenus and others in Diogenes Laërtius, i. 118, 119 Cicero, de Div. i. 49
  63. ^ abc
  64. Boyer, Carl B. (1968), A History of Mathematics
  65. ^ abZhmud 2012, pp. 2, 16.
  66. ^ Diogenes Laërtius, Lives of Eminent Philosophers, viii. 1, 8.
  67. ^
  68. Waithe, M. E. (April 30, 1987), Ancient Women Philosophers: 600 B.C.-500 A. D., Springer Science & Business Media, ISBN9789024733682 – via Google Books
  69. ^
  70. Malone, John C. (30 June 2009), Psychology: Pythagoras to present, MIT Press, p. 22, ISBN978-0-262-01296-6 , retrieved 25 October 2010
  71. ^ Porphyry, Life of Pythagoras, 41.
  72. ^Gilles Ménage: The history of women philosophers. Translated from the Latin with an introduction by Beatrice H. Zedler. University Press of America, Lanham 1984, p. 47. "The person who is referred to as Themistoclea in Laërtius and Theoclea in Suidas, Porphyry calls Aristoclea."
  73. ^ Iamblichus, Vit. Pyth. 25 Porphyry, Vit. Pyth. 17 Diogenes Laërtius, viii. 3.
  74. ^ Ariston. ap. Diogenes Laërtius, viii. 8, 21 Porphyry, Vit. Pyth. 41.
  75. ^ abcdefgRiedweg 2005, p. 10.
  76. ^ abcdeCornelli & McKirahan 2013, p. 64.
  77. ^ abcRiedweg 2005, p. 11.
  78. ^ abFerguson 2008, p. 5.
  79. ^ abGregory 2015, p. 22.
  80. ^ Iamblichus, Vit. Pyth. 28 Porphyry, Vit. Pyth. 9
  81. ^ Cornelia J. de Vogel: Pythagoras and Early Pythagoreanism. Assen 1966, pp. 21ff. Cfr. Cicero, De re publica 2, 28–30.
  82. ^Riedweg 2005, pp. 11–12.
  83. ^ Cornelia J. de Vogel: Pythagoras and Early Pythagoreanism, Assen 1966, S. 148–150.
  84. ^Riedweg 2005, pp. 12–13.
  85. ^Riedweg 2005, pp. 12–18.
  86. ^ Porphyry, Vit. Pyth. 18 Iamblichus, Vit. Pyth. 37, etc.
  87. ^Riedweg 2005, pp. 13–18.
  88. ^ abcdKahn 2001, p. 8.
  89. ^ abcPomeroy 2013, p. 1.
  90. ^ abcPomeroy 2013, p. xvi.
  91. ^Ferguson 2008, p. 58.
  92. ^ abcdefghijklmnopqFerguson 2008, p. 59.
  93. ^Suda Encyclopedia, th.84
  94. ^ abRiedweg 2005, pp. 5–6, 59, 73.
  95. ^ abKahn 2001, pp. 6–7.
  96. ^ abRiedweg 2005, p. 19.
  97. ^ abcdKahn 2001, p. 7.
  98. ^ abcRiedweg 2005, pp. 19–20.
  99. ^ Iamblichus, Vit. Pyth. 255–259 Porphyry, Vit. Pyth. 54–57 Diogenes Laërtius, viii. 39 comp. Plutarch, de Gen. Socr. p. 583
  100. ^ abcdefgRiedweg 2005, p. 20.
  101. ^Grant 1989, p. 278.
  102. ^Simoons 1998, pp. 225–228.
  103. ^ abBruhn 2005, p. 66.
  104. ^ abBurkert 1972, pp. 106–109.
  105. ^Kahn 2001, pp. 5–6.
  106. ^Kahn 2001, pp. 9–11.
  107. ^ abBurkert 1972, pp. 29–30.
  108. ^ abcKahn 2001, p. 11.
  109. ^ abZhmud 2012, p. 232.
  110. ^Burkert 1985, pp. 300–301.
  111. ^ abGregory 2015, pp. 24–25.
  112. ^ abcCopleston 2003, pp. 30–31.
  113. ^ Diogenes Laërtius, viii. 36, comp. Aristotle, de Anima, i. 3 Herodotus, ii. 123.
  114. ^Gregory 2015, p. 25.
  115. ^Kahn 2001, p. 12.
  116. ^ Diogenes Laërtius, viii. 3–4
  117. ^Cornelli & McKirahan 2013, pp. 164–167.
  118. ^ Porphyry, Vit. Pyth. 26 Pausanias, ii. 17 Diogenes Laërtius, viii. 5 Horace, Od. i. 28,1. 10
  119. ^Cornelli & McKirahan 2013, pp. 164–165.
  120. ^Cornelli & McKirahan 2013, pp. 165–166.
  121. ^ abcCornelli & McKirahan 2013, p. 167.
  122. ^ Aulus Gellius, iv. 11
  123. ^ abRiedweg 2005, pp. 29–30.
  124. ^ abGregory 2015, pp. 38–39.
  125. ^ abcRiedweg 2005, p. 30.
  126. ^
  127. D. S. Hutchinson Monte Ransome Johnson (25 January 2015), New Reconstruction, includes Greek text, p. 48
  128. ^ Cicero, de Divin. i. 3, 46 Porphyry, Vit. Pyth. 29.
  129. ^ Iamblichus, Vit. Pyth. 25 Porphyry, Vit. Pyth. 17 Diogenes Laërtius, viii. 3, 13 Cicero, Tusc. Qu. v. 3.
  130. ^ abcBruhn 2005, pp. 65–66.
  131. ^ abcdGregory 2015, pp. 28–29.
  132. ^ abcdRiedweg 2005, p. 29.
  133. ^ abcdKahn 2001, pp. 1–2.
  134. ^ abBurkert 1972, pp. 467–468.
  135. ^Burkert 1972, p. 265.
  136. ^Kahn 2001, p. 27.
  137. ^ abRiedweg 2005, p. 23.
  138. ^ abcJoost-Gaugier 2006, pp. 170–172.
  139. ^ abcJoost-Gaugier 2006, p. 172.
  140. ^ abBurkert 1972, p. 433.
  141. ^Burkert 1972, p. 467.
  142. ^Joost-Gaugier 2006, p. 170.
  143. ^ abcJoost-Gaugier 2006, p. 161.
  144. ^ Iamblichus, Vit. Pyth., 29
  145. ^ abGregory 2015, p. 28.
  146. ^Joost-Gaugier 2006, pp. 87–88.
  147. ^ abcKahn 2001, pp. 2–3.
  148. ^Kahn 2001, p. 3.
  149. ^ abBurkert 1972, pp. 428–433.
  150. ^Burkert 1972, p. 465.
  151. ^ Plato, Republic, 600a, Isocrates, Busiris, 28
  152. ^ abcdCornelli & McKirahan 2013, p. 168.
  153. ^Grant 1989, p. 277.
  154. ^ Porphyry, Vit. Pyth. 19
  155. ^ Thirlwall, Hist. of Greece, vol. ii. p. 148
  156. ^ abcdRiedweg 2005, p. 31.
  157. ^ comp. Cicero, de Leg. i. 12, de Off. i. 7 Diogenes Laërtius, viii. 10
  158. ^ abCornelli & McKirahan 2013, p. 65.
  159. ^ Aristonexus ap. Iamblichus, Vit. Pyth. 94, 101, etc., 229, etc. comp. the story of Damon and Phintias Porphyry, Vit. Pyth. 60 Iamblichus, Vit. Pyth. 233, etc.
  160. ^Cornelli & McKirahan 2013, pp. 68–69.
  161. ^ Iamblichus, Vit. Pyth. 98 Strabo, vi.
  162. ^ abcdefKenny 2004, p. 10.
  163. ^ John Dillon and Jackson Hershbell, (1991), Iamblichus, On the Pythagorean Way of Life, page 14. Scholars Press. D. J. O'Meara, (1989), Pythagoras Revived. Mathematics and Philosophy in Late Antiquity, pages 35–40. Clarendon Press.
  164. ^ abcdeGregory 2015, p. 31.
  165. ^ Aelian, Varia Historia, ii. 26 Diogenes Laërtius, viii. 13 Iamblichus, Vit. Pyth. 8, 91, 141
  166. ^Riedweg 2005, pp. 33–34.
  167. ^ Scholion ad Aristophanes, Nub. 611 Iamblichus, Vit. Pyth. 237, 238
  168. ^ abCornelli & McKirahan 2013, p. 69.
  169. ^Riedweg 2005, pp. 64–67.
  170. ^Riedweg 2005, p. 64.
  171. ^Riedweg 2005, p. 65.
  172. ^ abZhmud 2012, p. 200.
  173. ^Riedweg 2005, pp. 65–67.
  174. ^Riedweg 2005, pp. 65–66.
  175. ^Riedweg 2005, pp. 66–67.
  176. ^ abcdefghiRiedweg 2005, p. 66.
  177. ^Pomeroy 2013, pp. xvi–xvii.
  178. ^ comp. Porphyry, Vit. Pyth. 32 Iamblichus, Vit. Pyth. 96, etc.
  179. ^Zhmud 2012, pp. 137, 200.
  180. ^Copleston 2003, p. 30.
  181. ^ Diogenes Laërtius, viii. 19, 34 Aulus Gellius, iv. 11 Porphyry, Vit. Pyth. 34, de Abst. i. 26 Iamblichus, Vit. Pyth. 98
  182. ^ Plutarch, de Esu Carn. pp. 993, 996, 997
  183. ^ abKahn 2001, p. 9.
  184. ^Kenny 2004, pp. 10–11.
  185. ^ Eudoxus, frg. 325
  186. ^ abcdZhmud 2012, p. 235.
  187. ^ Aristo ap. Diogenes Laërtius, viii. 20 comp. Porphyry, Vit. Pyth. 7 Iamblichus, Vit. Pyth. 85, 108
  188. ^ Aristoxenus ap. Diogenes Laërtius, viii. 20
  189. ^ comp. Porphyry, Vit. Pyth. 7 Iamblichus, Vit. Pyth. 85, 108
  190. ^ abRiedweg 2005, p. 1.
  191. ^ abcdefgRiedweg 2005, p. 2.
  192. ^Gregory 2015, pp. 30–31.
  193. ^ abcdGregory 2015, p. 30.
  194. ^ abcKenny 2004, p. 11.
  195. ^ abcdeFerguson 2008, p. 60.
  196. ^ Porphyry, Vit. Pyth. 20 Iamblichus, Vit. Pyth. 31, 140 Aelian, Varia Historia, ii. 26 Diogenes Laërtius, viii. 36.
  197. ^ abMcKeown 2013, p. 155.
  198. ^ Comp. Herodian, iv. 94, etc.
  199. ^Burkert 1972, p. 144.
  200. ^Ferguson 2008, p. 10.
  201. ^ See Antoine Faivre, in The Eternal Hermes (1995)
  202. ^Joost-Gaugier 2006, p. 47.
  203. ^ abFerguson 2008, pp. 58–59.
  204. ^ abcCornelli & McKirahan 2013, p. 160.
  205. ^Ferguson 2008, pp. 60–61.
  206. ^ abcdeFerguson 2008, p. 61.
  207. ^Gregory 2015, pp. 21–22.
  208. ^ Diogenes Laërtius, viii. 12 Plutarch, Non posse suav. vivi sec. Ep. p. 1094
  209. ^ Porphyry, in Ptol. Harm. p. 213 Diogenes Laërtius, viii. 12.
  210. ^ Diogenes Laërtius, viii. 14 Pliny, Hist. Nat. ii. 8.
  211. ^ Diogenes Laërtius, viii. 12, 14, 32.
  212. ^Kahn 2001, pp. 32–33.
  213. ^Riedweg 2005, pp. 26–27.
  214. ^ abcdefRiedweg 2005, p. 27.
  215. ^Burkert 1972, p. 428.
  216. ^Burkert 1972, pp. 429, 462.
  217. ^ abKahn 2001, p. 32.
  218. ^Ferguson 2008, pp. 6–7.
  219. ^ abcBurkert 1972, p. 429.
  220. ^Kahn 2001, p. 33.
  221. ^Riedweg 2005, pp. 27–28.
  222. ^ abcdGregory 2015, p. 27.
  223. ^ abcRiedweg 2005, p. 28.
  224. ^ abChristensen 2002, p. 143.
  225. ^ abBurkert 1972, p. 306.
  226. ^ abBurkert 1972, pp. 307–308.
  227. ^Burkert 1972, pp. 306–308.
  228. ^Kahn 2001, p. 53.
  229. ^Dicks 1970, p. 68.
  230. ^Langdon & Fotheringham 1928.
  231. ^ abcKahn 2001, pp. 55–62.
  232. ^Kahn 2001, pp. 48–49.
  233. ^ abKahn 2001, p. 39.
  234. ^Kahn 2001, pp. 39–43.
  235. ^Kahn 2001, pp. 39–40.
  236. ^Kahn 2001, pp. 40, 44–45.
  237. ^ Plato, Republic VII, 530d
  238. ^ Metaphysics, 1.6.1 (987a)
  239. ^Kahn 2001, p. 1.
  240. ^ Tusc. Disput. 1.17.39.
  241. ^Kahn 2001, p. 55.
  242. ^ abcdHare 1999, pp. 117–119.
  243. ^Copleston 2003, p. 37.
  244. ^Russell 2008, pp. 33–37.
  245. ^Russell 2008, p. 37.
  246. ^Riedweg 2005, pp. 123–124.
  247. ^Riedweg 2005, p. 124.
  248. ^ abRiedweg 2005, pp. 125–126.
  249. ^ abcRiedweg 2005, p. 125.
  250. ^Riedweg 2005, pp. 126–127.
  251. ^ abJoost-Gaugier 2006, pp. 166–181.
  252. ^ abHomann-Wedeking 1968, p. 63.
  253. ^Homann-Wedeking 1968, p. 62.
  254. ^ abcdeCarpenter 1921, pp. 107, 122, 128.
  255. ^Homann-Wedeking 1968, pp. 62–63.
  256. ^ abBowra 1994, p. 166.
  257. ^Homann-Wedeking 1968, pp. 62–65.
  258. ^Joost-Gaugier 2006, p. 154.
  259. ^Joost-Gaugier 2006, pp. 154–156.
  260. ^Joost-Gaugier 2006, pp. 157–158.
  261. ^Joost-Gaugier 2006, p. 158.
  262. ^Joost-Gaugier 2006, pp. 158–159.
  263. ^Joost-Gaugier 2006, p. 159.
  264. ^Joost-Gaugier 2006, pp. 159–161.
  265. ^ abJoost-Gaugier 2006, p. 162.
  266. ^Joost-Gaugier 2006, pp. 162–164.
  267. ^Joost-Gaugier 2006, pp. 167–168.
  268. ^Joost-Gaugier 2006, p. 168.
  269. ^Joost-Gaugier 2006, pp. 169–170.
  270. ^Joost-Gaugier 2006, pp. 57–65.
  271. ^ abcJoost-Gaugier 2006, p. 57.
  272. ^Joost-Gaugier 2006, pp. 57–58.
  273. ^Joost-Gaugier 2006, pp. 58–59.
  274. ^ abJoost-Gaugier 2006, p. 59.
  275. ^ abcdefghijklmnoCelenza 2010, p. 798.
  276. ^ abcRusso 2004, pp. 5–87, especially 51–53.
  277. ^ abKahn 2001, p. 160.
  278. ^Kahn 2001, pp. 161–171.
  279. ^Ferguson 2008, p. 265.
  280. ^ abFerguson 2008, pp. 264–274.
  281. ^Kahn 2001, p. 162.
  282. ^Ferguson 2008, p. 274.
  283. ^ Jamie James, The Music of the Spheres: Music, Science, and the Natural Order of the Universe, p 142.
  284. ^Ferguson 2008, p. 279.
  285. ^ abFerguson 2008, pp. 279–280.
  286. ^ abKahn 2001, p. 172.
  287. ^Whitehead 1953, pp. 36–37.
  288. ^Whitehead 1953, p. 36.
  289. ^ abBorlik 2011, p. 192.
  290. ^Borlik 2011, p. 189.
  291. ^ abBorlik 2011, pp. 189–190.
  292. ^ abcdefBorlik 2011, p. 190.
  293. ^Ferguson 2008, p. 282.
  294. ^ abFerguson 2008, p. 294.
  295. ^ abRiedweg 2005, pp. 127–128.
  296. ^ abcRiedweg 2005, p. 128.
  297. ^ abcdFrench 2002, p. 30.
  298. ^Riedweg 2005, p. 133.
  299. ^ abSherman 1995, p. 15.
  300. ^Ferguson 2008, pp. 284–288.
  301. ^Ferguson 2008, pp. 287–288.
  302. ^Ferguson 2008, pp. 286–287.
  303. ^Ferguson 2008, p. 288.
  304. ^ abcHaag 2013, p. 89.
  305. ^Haag 2013, p. 90.
  306. ^Haag 2013, pp. 90–91.
  307. ^ abcdefHaag 2013, p. 91.
  308. ^Haag 2013, pp. 91–92.
  309. ^Haag 2013, p. 92.
  310. ^ abcdBregman 2002, p. 186.

Works cited

Only a few relevant source texts deal with Pythagoras and the Pythagoreans most are available in different translations. Later texts usually build solely upon information in these works.


Time's Orphans Have Names

A journey into the furthest depths of ancient history

The ancient Egyptians, as you might expect, immortalized their sense of humor in stone. In the Temple of Hatshepsut — Egypt’s first female pharaoh — at Deir el Bahari on the west bank of the Nile, artists engraved an image of the Queen of Punt, who clearly suffers from some kind of disease. Close behind her comes a donkey — with an inscription that says, “this poor donkey had to carry the queen.”

I do find it kind of ironic, though, that the royals of the Eighteenth Dynasty think it’s amusing to mock the disabled, given that their own beloved King Tutankhamun was “feminine-hipped and inbred” —National Geographic’s words, not mine.

If you’ve read my article, “Fear and Loathing in World History” —which, if you want to read about Adolf Hitler high on methamphetamine, I highly recommend you do — you’ll remember our friend Herodotus, the inventor of “finding stuff out,” a.k.a. historia, a.k.a. history. That’s the guy who heard this next little tidbit from one of his Egyptian tabloid sources.


Can someone explain the joke that killed Chrysippus of Soli? - History

    • Laius, who ruled Thebes at the time, was told the prophecy that his son would kill him and sleep with his wife.
    • He and his wife gave their baby son to one of their slaves, who was to bring the baby to Mt. Cithaeron, which was haunted by wild beasts.
    • However, the slave felt pity for the baby, so he gave him to another shepherd from the city of Corinth located on the other side of the mountain.
    • King Polybus of Corinth was presented with the baby and decided to bring him up on his own.
    • When Oedipus was older, someone calls him a bastard.
    • He decided to leave Corinth for Delphi, so he could learn of his parentage at the oracle of Apollo.
    • There he was given the news that he would kill his father and sleep with his mother.
    • To prevent the oracle from coming true, Oedipus went to Thebes.
    • On the way he ran into an old man driving a wagon at a place where three roads cross. The man ordered Oedipus to move off the road, but he refused.
    • He became aggressive and killed the man and what he thought to be all the guards.
    • Before Oedipus could enter Thebes, he had to solve the riddle the Sphinx, who guarded the entrance to the city, asked him. No one had ever solved the riddle before and as a consequence, they were killed by the Sphinx.
    • The riddle is, "Which animal has one voice, but two, three, or four feet being slowest on three?" Oedipus answered correctly with the answer, "Man."
    • The city welcomed Oedipus and offered him the vacant job of king and the marriage to Laius' widow, Jocasta.
    • Years passed while Oedipus was king of Thebes. He had four children by Jocasta.
    • Eventually the city was infected by a plague. Oedipus promised to save his city, so he ordered his brother-in-law Creon to consult the oracle at Delphi.
    • He returned with news that the plague was caused by the unpunished murderer who killed Laius. Oedipus cursed the killer, but Tiresias said that Oedipus was the killer.
    • Oedipus was furious and blamed Tiresias and Creon for creating such a story to dethrone him so that they could have power.
    • Jocasta explained to Oedipus that robbers killed Laius at a place where three roads crossed. Oedipus remembered that he killed a man at such a place.
    • He contemplated the possibility of him being the killer, but Jocasta reassured him that a witness saw several robbers kill Laius.
    • Oedipus sent for the witness, so the issue could be resolved.
    • While he waited, a Corinthian messenger arrived with news that Polybus had died, so Oedipus would be King of Corinth.
    • Oedipus told the messenger that he could not go back while his mother was alive.
    • Surprise overwhelmed Oedipus, for the messenger told him that she was not his mother. He explained that he was given the baby many years ago by a Theban shepherd.
    • Jocasta then realized that Oedipus was her son.
    • The witness finally arrived and revealed that he was given the baby by Jocasta and passed it to the messenger because he did not want to kill him.
    • Oedipus realizes the truth and went to tell Jocasta, but she had already killed herself.
    • He blinds himself and was ordered to leave Thebes by Creon, the new king.
    • It is the female analog to the Oedipus Complex.
    • It is the female child's erotic desire for the father and simultaneous fear of the mother.
    • Electa and Orestes meet unexpectedly at the tomb of their father.
    • Electa brings gifts to the tomb.
    • It is named after Electra who, in the Greek myth, sought to avenge the death of her father Agamemnon.
    • She was away from home when her mother, Clytemnestra, and her lover, Aegisthus, killed Agamemnon.
    • Electra and her brother, Orestes, came home years later to the tomb of their father.
    • There they plotted how Orestes would kill Clytemnestra, which he successfully accomplished.
    • Freud created the term Oedipus complex during his self-analysis.
    • He claimed that this psychological condition is universal.
    • Oedipus Tyrannus effected both the ancient Greeks and today's audiences everyone has oedipal feelings.
    • When Freud saw the play's incredible success in Germany and France in the 1880's, it supported his belief that the play moves modern audiences with as much force as it did originally in Greece.
    • He characterized it as loving and hostile wishes that children experienced toward their parents at the height of the phallic phase. The phallic phase is the stage of psychosexual development occurring between the ages of three and seven.
    • In its positive form, the child's rival is the parent of the same sex and the child desires sexual union with the parent of opposite sex.
    • In its negative form, the child's rival is the parent of the opposite sex and the child desires the parent of the same sex.
    • The decline of the Oedipus complex and entry into the latency period are related to threat of castration for boys and the desire for a baby for girls.
    • After puberty, the resolution of the complex happens through the choice of a suitable substitute for the object of love.
    • However, the Oedipus complex continues to be an unconscious organizer throughout life. It also forms an indissoluble link between wish and law.
    • The Picture
      • This is a picture of Sigmund Freud.
      • The song, "The End" by the Doors has a verse where the vocalist claims to want to kill his father and have sex with his mother.
        • The killer awoke before dawn, he put his boots on
          He took a face from the ancient gallery
          And he walked on down the hall
          He went into the room where his sister lived, and. then he
          Paid a visit to his brother, and then he
          He walked on down the hall, and
          And he came to a door. and he looked inside
          Father, yes son, I want to kill you
          Mother. i want to. fuck you
        • Will Wuu, the antagonist from Gun X Sword, suffered very heavily from the Oedipus complex.
          • Gun X Sword is an anime production that premiered in July 2005. The protagonist traveled the world to find the man that killed his bride.
          • Will Wuu was a villain who was an aloof and philosophical individual who suffered from Sigmund Freud's Oedipus complex.
          • He despised his father for receiving his mother's complete attention.
          • He tried to kill his father, but instead his mother took the blow and died instead.
          • Will Wuu's guilt tormented him years later.
          • Fei, the main character of the video game Xenogears, seemed to suffer from the complex.
            • In the game, Fei Fong Wong, was an eighteen-year-old male wno had no clear memory of his childhood.
            • However, eventually he discovered the truth of his childhood through a flashback.
              • In his youth, he hated his mother for painfully testing him. She though that he was God's "chosen one."
              • He begged his father to stop his mother, which was opposite the normal hatred of the father at his age.
              • In the Star Wars film, Return of the Jedi, Yoda explained that Luke Skywalker had to confront his father, Darth Vader, in order to realize Jedi.
                • Because of his mother's death, Luke did not focus his sexual desires during the phallic stage of his development.
                • Therefore, he had a deviant form of the Oedipus complex. His sexual desires were focused on the other female in his family - Leia.
                • They tried to hide their forbidden love because Luke feared that his father would punish him by "castrating" him. This was a normal feeling for a boy suffering from the Oedipus complex.
                • Vader did "castrate" him by cutting his saber-phallus from his hand.
                • It is believed that Vader had a sick and terrible sexual love for his son. The "dark side" was a metaphor for the sexual abuse Vader inflicted on Luke. Luke resisted Vaders seductions.
                • Vader then tried to get Leia to join him, but Luke rebelled. He cut Vader's saber-phallus and hand to protect his sister.
                • It is ironic that Luke, the son, castrated Vader, his father, to protect Leia, his lover. It is a reversal of the Oedipus complex.
                • The picture is of Luke Skywalker and Darth Vader fighting with their sabers.
                • The character Norman Bates, from the novel and movie Psycho , showed many signs of having an Oedipus complex when he murdered his mother and her partner.
                  • The movie was released in 1960 when Freud was heavily followed and praised for his work.
                  • The movie was remade in 1998.
                  • Norman Bates suffered abuse as a child by his mother. His father died when he was young.
                  • Bates' mother took a new lover, which made him jealous.
                  • Norman killed his mother, Norma, and her lover. He preserved his mother's corpse.
                  • He developed multiple personality disorder by assuming his mother's personality.
                  • As Norma, he dressed in her clothes and killed people who came between her and her son. He believed that she was still alive.
                  • As Norman, he was a barely functional adult who ran a hotel.
                  • The picture is of Norma Bates in the shower. She about to murder a girl.

                  I won’t wait! I’d said it’s great
                  to be, punctually tardy
                  You think? Shrinking my head, by swelling my brain,
                  with vice instead you’d best
                  analyze this bliss oh Oedipus Rex!
                  my complexion it’s the best!
                  whilst stroking my molten scales,
                  oh Oedipus Rex!

                  Bending, crawling, bleeding, backwards
                  Sailing, smiling, singing backwards
                  Floating, dying, waiting, backwards
                  Kicking, crying, screaming, backwards

                  Why stand in line waiting for bread?
                  Nether instead I’ll lay
                  in Vodkaville euphoria
                  Is dancing in my head
                  true to Magazine I’ll masturbate
                  Catholic-guilt it seems to cramp my wrists
                  Whilst stroking my molten scales,
                  oh Oedipus Wrecks!

                  Bending, falling, dying backwards
                  Kicking, screaming, calling backwards
                  Floating, flying, falling backwards
                  Bending, breathing, dying backwards

                  American-Pop witness a quest!
                  popular and universal
                  Impetus, must I digress?
                  I’m ‘Bastard-Son’ my home
                  so long ago I left ‘twas best!
                  color me unimpressed
                  It sat next to me on the U.F.O.
                  the Easter Bunny and a lot of dope(s)!

                  Bending, falling, dying backwards
                  Kicking, screaming, calling backwards
                  Floating, flying, falling backwards
                  Bending, breathing, dying backwards


                  A study reveals the Lebanese DNA of the Phoenicians of Ibiza

                  "The Ibizans were more Lebanese than the Lebanese themselves"

                  A study reveals the Lebanese DNA of the Phoenicians of Ibiza and Phoenician distribution in the Mediterranean, including analysis of Ibizan Punic DNA
                  By José Miguel L. Romero, Ibiza

                  Reproduced without permission from
                  the original Spanish in DIARIO de Ibiza
                  Translated to English by Salim G. Khalaf, Phoenicia.org

                  "The Ibizans were more Lebanese than the Lebanese themselves"

                  The Ibizan Phoenicians were more Lebanese than the Canaanites themselves, while the citizens of Tyre or Sidon possessed in their genes more European traces than those of the Balearic islands themselves. It is one of the most surprising results obtained from the analysis of the DNA of the inhabitants of that civilization. This is what the Lebanese biologist, Pierre Zalloua said in a talk in the Museum of the Necroplis of Puig des Molins, in Ibiza, Spain.

                  Pierre Zalloua during the conference
                  at the Museum of the Necropolis.

                  "No, it is not a joke," although it seems so. The end of the conference 'Mitochondrial genomes of the ancient Phoenicians,' given on Friday in Eivissa by biologist Pierre Zalloua of the American University of Lebanon. What was incredible was that in the DNA sequences of the Phoenicians people of Ibiza have found to have more eastern traces than expected. Traces identified were more than those found in the Lebanese Phoenicians themselves. But, in addition, in the genome of the Lebanese they have detected more European traces of what was expected in principle. They are even more than those found in the old Balearic island people. It seems that the world was changed upside down. Is not it a heavy joke, as if someone moved the graves or maybe there was an error in the analysis? "No, it's not a joke," Zalloua said at the end of his talk. He smiled for having accomplished his purpose while his audience were disconcerted. They filled the meeting room of the Museum of the Necroplis of Puig des Molins.

                  The conclusion of the study, which was carried out after analyzing Punic skeletal remains of the Eivissa tombs, is difficult to assimilate, even for the researchers themselves. The samples from Ibiza have little to do with those of Bronze Age Europe, "but they are more close to those of the Neolithic Levant. "The Ibizan Phoenicians look like Lebanese!" Exclaimed Zalloua. On the contrary, "the analyzed samples of the Phoenician tombs of Beirut are very similar to those of the ancient European populations of the late Neolithic." In the genetic map, the Ibizian DNA appears mapped where the Lebanese should be, and vice versa. It is as if the map of the Mediterranean had folded, so that Lebanon appeared on its western coast, and Eivissa between Syria and Israel.

                  Underestimated Migratory Flow

                  Now, in addition to digesting this information, finding an explanation is not easy. "It is difficult to explain," says the Lebanese biologist: "Perhaps," he continues, "we have underestimated the frequency of people's movement at that time." That could be the key: that in those times there would be more migratory flows than we imagined and that the borders would fluid than the current ones. The world of such genetic analyzes resembles that described in 'Sinuhé the Egyptian' or 'the African Lion,' rather than that of modern civilization, whose borders were increasingly hard to cross. In his view, "it is difficult to understand why there was then so much movement of population."

                  "We are at the beginning of this research," says the Lebanese researcher, who specializes in genetic links between populations in the Middle East and the Mediterranean. Research on this subject could be decisive and is being developed with the Phoenician DNA of Cadiz and Malta.

                  For the study, they chose to analyze mitochondrial DNA (in which only the genetic information of the mother appears is studied). It is more abundant and easier to obtain than the nucleus (in this case, father and mother). In the case of Eivissa (also investigated in Sardinia and Beirut) 11 samples were collected from four Punic sites. This is so because obtaining eight sequences with three cases DNA "was not of sufficient quality." Only in one of the samples, in the extracted from the deposit of Can Portes of Jurat, they obtained mitochondrial as well as nucleus DNA. Zalloua clarified that the remains analyzed belonged to Punic and not the first Phoenicians who arrived in Eivissa — as the latter burned their dead, so getting their DNA was mission impossible.

                  According to Elisa A. Matisoo-Smith, Professor of Biological Anthropology at the University of Otago (New Zealand), who is part of the research team of this project and was present at the talk, concluded. The results of mitochondrial DNA studied "imply that men from the East — from the narrow fringe of present-day Lebanon in which the Phoenicians lived — intermarried with the local Ibizan women, just as in Sardinia. " Further, she believes the same thing happened in the opposite direction when Europeans intermarried with the Lebanese.

                  This, according to Zalloua, shows a society more concerned with integration than with conquest and slavery. This means to say that "The Phoenicians had an inclusive, multicultural nature."

                  He did not want to give many clues, but Pierre Zalloua left in the air the suspicion that there may be more surprises coming soon. For example, the result from the study of the haplogroup (set of haplotypes, which are sets of DNA variations) T2b, found in one of the 18 samples taken from Ibizans today. The 18 were different, but one of them contained that T2b. It is something surprising because, according to the biologist, it was also found in the genetic sample extracted from a Punic in the deposit of Can Portes of Jurat. Similarly that same haplogroup has been detected in the Phoenicians in Lebanon: "The existence of the T2b haplogroup calls our attention," said Zalloua with a hoarse smile, as if he knew that he would soon bewilder us.

                  By José Miguel L. Romero, DIARIO de Ibiza, Ibiza
                  Translated to English by Salim G. Khalaf, Phoenicia.Org

                  DISCLAIMER: Opinions expressed in this site do not necessarily represent Phoenicia.org nor do they necessarily reflect those of the various authors, editors, and owner of this site. Consequently, parties mentioned or implied cannot be held liable or responsible for such opinions.

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                  Contact: Salim George Khalaf, Byzantine Phoenician Descendent
                  Salim is from Shalim, Phoenician god of dusk, whose place was Urushalim/Jerusalem
                  "A Bequest Unearthed, Phoenicia" &mdash Encyclopedia Phoeniciana

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                  5 most bizarre deaths in history

                  We all want to be remembered when we’re no longer around, and ideally, we’d like to be remembered for something good.

                  The people you’re about to meet certainly fill the first criteria, but sadly, rather than being remembered for their achievements (or their sins, for that matter), they’ve earned themselves a place in the history books for something else entirely.

                  Here, we’re taking a look at the stories behind some of history’s most unusual deaths.

                  1. Overeating

                  We’ve all overindulged once or twice, but we’ve got nothing on Adolf Frederick, King of Sweden, who died at the age of 60 after an unbelievably huge meal. His final feast included lobster, caviar, sauerkraut, kippers, champagne and 14 (yes, 14) servings of his favourite dessert – semla, a traditional Swedish sweet, almond cream-filled bun, served in a bowl of hot milk. We can think of worse ways to go!

                  2. Kicking a safe

                  Jack Daniel, the founder of Jack Daniel’s Tennessee whiskey distillery, died from blood poisoning at the age of 62. How it happened, however, is quite a story. You see, Daniel couldn’t remember the combination to his safe and kicked it out of frustration. The injury was so severe, it left him with a limp and considerable pain. The foot was eventually amputated, but the surrounding area became gangrenous.

                  People say they’re “dying of laughter” all the time, but who knew it could actually happen? Around 206 BC, ancient Greek philosopher Chrysippus of Soli died after a night of fun and drinking. While watching a donkey eat some figs, he cried, “Now give the donkey a drink of pure wine to wash down the figs”. Chrysippus found his joke so funny, he died of laughter.

                  4. Having a beard

                  Hans Steininger, the mayor of the small Austrian town of Braunau am Inn, took great pride in his impressive 1.4-metre-long beard, which reached to his feet. He usually kept it rolled up into a pouch, but one fateful day in 1567, he decided to let the majestic facial hair flow free. On September 28, when a large fire broke out in town, Steininger ran for his life, accidentally tripping on his beard and falling down a flight of stairs, breaking his neck.

                  5. Politeness

                  Sometimes, kindness really can kill. In 1601, Danish nobleman and astronomer Tycho Brahe attended a banquet in Prague. He ate, drank and eventually, as we all do, eventually needed to pee – desperately. Unfortunately, Brahe refused to leave the table as it would have been a breach of etiquette. When he returned home, he found he was unable to urinate except in tiny quantities and with excruciating pain. 11 days later, at the age of 54, he passed away.


                  Watch the video: How People Literally Laughed to Death (May 2022).


Comments:

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  3. Markey

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  4. Euryalus

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