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145-484: Though there remains dissent among scientific historians concerning organicism's pregeneration, most scholars agree on Ancient Athens as its birthplace. Surfacing in Athenian writing in the 4th-century BC, Plato was among the first philosophers to consider the universe an intelligent living (almost sentient) being, which he posits in his Philebus and Timaeus . At the turn of the 18th-century, Immanuel Kant championed

290-522: A Second Athenian League . Finally Thebes defeated Sparta in 371 BC in the Battle of Leuctra . But then the Greek cities (including Athens and Sparta) turned against Thebes, whose dominance was stopped at the Battle of Mantinea (362 BC) with the death of its military-genius leader Epaminondas . By the mid-4th century BC, however, the northern Greek kingdom of Macedon was becoming dominant in Athenian affairs. In

435-415: A cumulative hierarchy of sets. New Foundations takes a different approach; it allows objects such as the set of all sets at the cost of restrictions on its set-existence axioms. The system of Kripke–Platek set theory is closely related to generalized recursion theory. Two famous statements in set theory are the axiom of choice and the continuum hypothesis . The axiom of choice, first stated by Zermelo,

580-460: A battle against Eumenes in 320 BC, leaving Antipater alone to rule for a year, until his death in 319 BC. Athens had a central role in the struggle for his succession, when Antipater's son, Cassander , secured the Piraeus leaving Athens without a source of supplies, to contest Antipater's successor, Polyperchon . To consolidate power against Cassander, Polyperchon restored Athens's democracy, as it

725-565: A center of paganism. Christian items do not appear in the archaeological record until the early 5th century. The sack of the city by the Herules in 267 and by the Visigoths under their king Alaric I ( r.  395–410 ) in 396, however, dealt a heavy blow to the city's fabric and fortunes, and Athens was henceforth confined to a small fortified area that embraced a fraction of the ancient city. The emperor Justinian I ( r.  527–565 ) banned

870-449: A comprehensive mathematical and category-theoretic treatment of irreducible causal relations he believed to be responsible for life. The early biologists of the organicist movement have influenced the organism-centered perspective of the extended evolutionary synthesis . In the early 1930s Joseph Henry Woodger and Joseph Needham , together with Conrad Hal Waddington , John Desmond Bernal , Dorothy Needham, and Dorothy Wrinch , formed

1015-434: A correspondence between syntax and semantics in first-order logic. Gödel used the completeness theorem to prove the compactness theorem , demonstrating the finitary nature of first-order logical consequence . These results helped establish first-order logic as the dominant logic used by mathematicians. In 1931, Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems , which proved

1160-430: A definition of the real numbers in terms of Dedekind cuts of rational numbers, a definition still employed in contemporary texts. Georg Cantor developed the fundamental concepts of infinite set theory. His early results developed the theory of cardinality and proved that the reals and the natural numbers have different cardinalities. Over the next twenty years, Cantor developed a theory of transfinite numbers in

1305-400: A finitistic system together with a principle of transfinite induction . Gentzen's result introduced the ideas of cut elimination and proof-theoretic ordinals , which became key tools in proof theory. Gödel gave a different consistency proof, which reduces the consistency of classical arithmetic to that of intuitionistic arithmetic in higher types. The first textbook on symbolic logic for

1450-415: A formalized mathematical statement, whether the statement is true or false. Ernst Zermelo gave a proof that every set could be well-ordered , a result Georg Cantor had been unable to obtain. To achieve the proof, Zermelo introduced the axiom of choice , which drew heated debate and research among mathematicians and the pioneers of set theory. The immediate criticism of the method led Zermelo to publish

1595-521: A foundational theory for mathematics. Fraenkel proved that the axiom of choice cannot be proved from the axioms of Zermelo's set theory with urelements . Later work by Paul Cohen showed that the addition of urelements is not needed, and the axiom of choice is unprovable in ZF. Cohen's proof developed the method of forcing , which is now an important tool for establishing independence results in set theory. Leopold Löwenheim and Thoralf Skolem obtained

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1740-494: A function as a rule for computation, or a smooth graph, were no longer adequate. Weierstrass began to advocate the arithmetization of analysis , which sought to axiomatize analysis using properties of the natural numbers. The modern (ε, δ)-definition of limit and continuous functions was already developed by Bolzano in 1817, but remained relatively unknown. Cauchy in 1821 defined continuity in terms of infinitesimals (see Cours d'Analyse, page 34). In 1858, Dedekind proposed

1885-417: A mechanical collection of separate components. Ancient Athens Athens is one of the oldest named cities in the world, having been continuously inhabited for perhaps 5,000 years. Situated in southern Europe, Athens became the leading city of ancient Greece in the first millennium BC, and its cultural achievements during the 5th century BC laid the foundations of Western civilization . During

2030-421: A milestone in recursion theory and proof theory, but has also led to Löb's theorem in modal logic. The method of forcing is employed in set theory, model theory, and recursion theory, as well as in the study of intuitionistic mathematics. The mathematical field of category theory uses many formal axiomatic methods, and includes the study of categorical logic , but category theory is not ordinarily considered

2175-561: A model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory , and they are a key reason for the prominence of first-order logic in mathematics. Gödel's incompleteness theorems establish additional limits on first-order axiomatizations. The first incompleteness theorem states that for any consistent, effectively given (defined below) logical system that

2320-453: A natural advantage over inland rivals such as Thebes and Sparta . According to legend, Athens was formerly ruled by kings , a situation which may have continued up until the 9th century BC. From later accounts, it is believed that these kings stood at the head of a land-owning aristocracy known as the Eupatridae (the 'well-born'), whose instrument of government was a Council which met on

2465-403: A new concept – the computable function – had been discovered, and that this definition was robust enough to admit numerous independent characterizations. In his work on the incompleteness theorems in 1931, Gödel lacked a rigorous concept of an effective formal system; he immediately realized that the new definitions of computability could be used for this purpose, allowing him to state

2610-485: A particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has

2755-422: A portion of set theory directly in their semantics. The most well studied infinitary logic is L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} . In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it

2900-543: A result of the reciprocal play of all the components on each other. Examples of 20th-century biologists who were organicists are Ross Harrison , Paul Weiss , and Joseph Needham . Donna Haraway discusses them in her first book Crystals, Fabrics, and Fields . John Scott Haldane (father of J. B. S. Haldane ), William Emerson Ritter , Edward Stuart Russell , Joseph Henry Woodger , Ludwig von Bertalanffy , and Ralph Stayner Lillie are other early 20th-century organicists. Robert Rosen , founder of "relational biology", provided

3045-460: A revival of organicisitic thought by stressing, in his written works, "the inter-relatedness of the organism and its parts[,] and the circular causality" inherent to the inextricable entanglement of the greater whole. Organicism flourished for a period during the German romanticism intellectual movement and was a position considered by Friedrich Wilhelm Joseph Schelling to be an important principle in

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3190-478: A rough division of contemporary mathematical logic into four areas: Additionally, sometimes the field of computational complexity theory is also included as part of mathematical logic. Each area has a distinct focus, although many techniques and results are shared among multiple areas. The borderlines amongst these fields, and the lines separating mathematical logic and other fields of mathematics, are not always sharp. Gödel's incompleteness theorem marks not only

3335-485: A second exposition of his result, directly addressing criticisms of his proof. This paper led to the general acceptance of the axiom of choice in the mathematics community. Skepticism about the axiom of choice was reinforced by recently discovered paradoxes in naive set theory . Cesare Burali-Forti was the first to state a paradox: the Burali-Forti paradox shows that the collection of all ordinal numbers cannot form

3480-423: A separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of

3625-404: A series of publications. In 1891, he published a new proof of the uncountability of the real numbers that introduced the diagonal argument , and used this method to prove Cantor's theorem that no set can have the same cardinality as its powerset . Cantor believed that every set could be well-ordered , but was unable to produce a proof for this result, leaving it as an open problem in 1895. In

3770-457: A set. Very soon thereafter, Bertrand Russell discovered Russell's paradox in 1901, and Jules Richard discovered Richard's paradox . Zermelo provided the first set of axioms for set theory. These axioms, together with the additional axiom of replacement proposed by Abraham Fraenkel , are now called Zermelo–Fraenkel set theory (ZF). Zermelo's axioms incorporated the principle of limitation of size to avoid Russell's paradox. In 1910,

3915-571: A stronger limitation than the one established by the Löwenheim–Skolem theorem. The second incompleteness theorem states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be reached. Many logics besides first-order logic are studied. These include infinitary logics , which allow for formulas to provide an infinite amount of information, and higher-order logics , which include

4060-402: A subfield of mathematical logic. Because of its applicability in diverse fields of mathematics, mathematicians including Saunders Mac Lane have proposed category theory as a foundational system for mathematics, independent of set theory. These foundations use toposes , which resemble generalized models of set theory that may employ classical or nonclassical logic. Mathematical logic emerged in

4205-508: A symbol that Athena still had her mark there on the city. Plato , in his dialogue Cratylus , offers an etymology of Athena's name connecting it to the phrase ἁ θεονόα or hē theoû nóēsis (ἡ θεοῦ νόησις, 'the mind of god'). There is evidence that the site on which the Acropolis ('high city') stands was first inhabited in the Neolithic period, perhaps as a defensible settlement, around

4350-500: A variation of the logical system of Boole and Schröder but adding quantifiers. Peano was unaware of Frege's work at the time. Around the same time Richard Dedekind showed that the natural numbers are uniquely characterized by their induction properties. Dedekind proposed a different characterization, which lacked the formal logical character of Peano's axioms. Dedekind's work, however, proved theorems inaccessible in Peano's system, including

4495-432: A very small area compared to the sprawling metropolis of modern Greece. The ancient walled city encompassed an area measuring about two kilometres (1.5 mi) from east to west and slightly less than that from north to south, although at its peak the ancient city had suburbs extending well beyond these walls. The Acropolis was situated just south of the centre of this walled area. The Agora , the commercial and social centre of

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4640-581: Is computable ; this is not true in classical theories of arithmetic such as Peano arithmetic . Algebraic logic uses the methods of abstract algebra to study the semantics of formal logics. A fundamental example is the use of Boolean algebras to represent truth values in classical propositional logic, and the use of Heyting algebras to represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as cylindric algebras . Set theory

4785-517: Is a particular formal system of logic . Its syntax involves only finite expressions as well-formed formulas , while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse . Early results from formal logic established limitations of first-order logic. The Löwenheim–Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it

4930-449: Is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system). For example, in every logical system capable of expressing the Peano axioms , the Gödel sentence holds for

5075-468: Is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to isomorphism . As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark. Gödel's completeness theorem established the equivalence between semantic and syntactic definitions of logical consequence in first-order logic. It shows that if

5220-646: Is known as Hellenistic Greece . The period from the end of the Persian Wars to the Macedonian conquest marked the zenith of Athens as a center of literature, philosophy, and the arts. In Athens at this time, the political satire of the Comic poets at the theatres had a remarkable influence on public opinion . Some of the most important figures of Western cultural and intellectual history lived in Athens during this period:

5365-416: Is known to have existed during the 10th millennium BC . In addition, no evidence exists of any possible cultural or other ties between Egypt and any part of present-day Greece at such early a date. During the 1st millennium BC , Athens succeeded in bringing the other towns of Attica under its rule. This process of synoikismos  – the bringing together into one home – created

5510-414: Is nothing but the result of independent phenomena. Scott F. Gilbert and Sahotra Sarkar distinguish organicism from holism to avoid what they see as the vitalistic or spiritualistic connotations of holism. Val Dusek notes that holism contains a continuum of degrees of the top-down control of organization, ranging from monism (the doctrine that the only complete object is the whole universe, or that there

5655-399: Is only one entity, the universe) to organicism, which allows relatively more independence of the parts from the whole, despite the whole being more than the sum of the parts, and/or the whole exerting some control on the behavior of the parts. Still more independence is present in relational holism. This doctrine does not assert top-down control of the whole over its parts, but does claim that

5800-446: Is possible to say that an object is a whole number using a formula of L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} such as Higher-order logics allow for quantification not only of elements of the domain of discourse , but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having

5945-642: Is the study of sets , which are abstract collections of objects. Many of the basic notions, such as ordinal and cardinal numbers, were developed informally by Cantor before formal axiomatizations of set theory were developed. The first such axiomatization , due to Zermelo, was extended slightly to become Zermelo–Fraenkel set theory (ZF), which is now the most widely used foundational theory for mathematics. Other formalizations of set theory have been proposed, including von Neumann–Bernays–Gödel set theory (NBG), Morse–Kelley set theory (MK), and New Foundations (NF). Of these, ZF, NBG, and MK are similar in describing

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6090-401: Is the theorem that a line contains at least two points, or that circles of the same radius whose centers are separated by that radius must intersect. Hilbert developed a complete set of axioms for geometry , building on previous work by Pasch. The success in axiomatizing geometry motivated Hilbert to seek complete axiomatizations of other areas of mathematics, such as the natural numbers and

6235-529: The Organon , found wide application and acceptance in Western science and mathematics for millennia. The Stoics , especially Chrysippus , began the development of predicate logic . In 18th-century Europe, attempts to treat the operations of formal logic in a symbolic or algebraic way had been made by philosophical mathematicians including Leibniz and Lambert , but their labors remained isolated and little known. In

6380-438: The Battle of Aegospotami which ended Athenian naval supremacy. Due to its poor handling of the war, the democracy in Athens was briefly overthrown by a coup in 411 BC; however, it was quickly restored. The Peloponnesian War ended in 404 BC with the complete defeat of Athens. Since the loss of the war was largely blamed on democratic politicians such as Cleon and Cleophon , there was a brief reaction against democracy, aided by

6525-525: The Battle of Chaeronea (338 BC) , Philip II 's armies defeated an alliance of some of the Greek city-states including Athens and Thebes, forcing them into a confederation and effectively limiting Athenian independence. Philippides of Paiania , one of the wealthiest Athenian aristocratic oligarchs, campaigned for Philip II during the Battle of Chaeronea and proposed in the Assembly decrees honoring Alexander

6670-699: The Battle of Salamis and routed the Persians, a great turning point in the war. In 479 BC, the Athenians and Spartans, with their allies, defeated the Persian army conclusively at the Battle of Plataea . Athens then took the war to Asia Minor. These victories enabled it to bring most of the Aegean and many other parts of Greece together in the Delian League , an Athenian-dominated alliance. The resentment felt by other cities at

6815-613: The Burgundian family called De la Roche , it replaced Athens as the capital and seat of government, although Athens remained the most influential ecclesiastical centre in the duchy and site of a prime fortress. Under the Burgundian dukes, a bell tower was added to the Parthenon, known as the Frankish Tower . The Burgundians brought chivalry and tournaments to Athens; they also fortified

6960-410: The Eupatridae was reduced by forbidding the enslavement of Athenian citizens as a punishment for debt ( debt bondage ), by breaking up large landed estates and freeing up trade and commerce, which allowed the emergence of a prosperous urban trading class. Politically, Solon divided the Athenians into four classes, based on their wealth and their ability to perform military service. The poorest class,

7105-655: The First Mithridatic War , Athens was ruled by Aristion , a tyrant installed by Mithridates the Great . In 88–85 BC, most Athenian fortifications and homes were leveled by the Roman general Sulla after the Siege of Athens and Piraeus , although many civic buildings and monuments were left intact. The Macedonian astronomer Andronicus of Cyrrhus subsequently designed the Tower of

7250-604: The Kerameikos and other locations, are often richly provided for and demonstrate that from 900 BC onwards Athens was one of the leading centres of trade and prosperity in the region; as were Lefkandi in Euboea and Knossos in Crete. This position may well have resulted from its central location in the Greek world, its secure stronghold on the Acropolis and its access to the sea, which gave it

7395-500: The Löwenheim–Skolem theorem , which says that first-order logic cannot control the cardinalities of infinite structures. Skolem realized that this theorem would apply to first-order formalizations of set theory, and that it implies any such formalization has a countable model . This counterintuitive fact became known as Skolem's paradox . In his doctoral thesis, Kurt Gödel proved the completeness theorem , which establishes

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7540-780: The Peace of Apamea , and the Third Macedonian War (171–168), after which Macedonian territory was divided into four client republics and Macedonia was formally annexed to the Roman Republic after the Fourth Macedonian War (150–148). The Achaean League was defeated and dissolved by the Romans in the Achaean War in 146. Greece was divided into the Roman provinces of Macedonia and Achaea ; thus, Athens came under Roman rule. During

7685-873: The Pyrrhic War (280–275 BC) Rome asserted its hegemony over Magna Graecia and became increasingly involved in Greece and the Balkans peninsula. The First Macedonian War (214–205 BC) between the Roman Republic and the Kingdom of Macedon ended with the Treaty of Phoenice . During the Second Macedonian War (200–197), the Romans declared "the freedom of Greece" from the Macedonian Kings. The Roman–Seleucid War (192–188) ended with

7830-473: The Sack of Athens , the city to the north of the Acropolis was hastily refortified on a smaller scale, with the agora left outside the walls. Athens remained a centre of learning and philosophy during its 500 years of Roman rule, patronized by emperors such as Nero and Hadrian. In the early 4th century AD, the eastern Roman empire began to be governed from Constantinople , and with the construction and expansion of

7975-519: The Temple of Hephaestus (which is still largely intact) and the Temple of Olympian Zeus or Olympeion (once the largest temple in mainland Greece but now in ruins) also lay within the city walls. Athens has been inhabited from Neolithic times, possibly from the end of the fourth millennium BC , or over 5,000 years. By 1412 BC, the settlement had become an important center of the Mycenaean civilization and

8120-570: The Thetai , (Ancient Greek Θήται ) who formed the majority of the population, received political rights for the first time and were able to vote in the Ecclesia (Assembly). But only the upper classes could hold political office. The Areopagus continued to exist but its powers were reduced. The new system laid the foundations for what eventually became Athenian democracy , but in the short-term it failed to quell class conflict and after twenty years of unrest

8265-450: The cells were first observed in 17th-century Europe following the invention of the microscope . Before that period, individual organisms were studied as a whole in a field known as "organismic biology"; that area of research remains an important component of the biological sciences. In biology , organicism considers that the observable structures of life, its overall form and the properties and characteristics of its component parts, are

8410-468: The extended evolutionary synthesis . Organicism as a doctrine rejects mechanism and reductionism (doctrines that claim that the smallest parts by themselves explain the behavior of larger organized systems of which they are a part). However, organicism also rejects vitalism , the doctrine that there is a vital force different from physical forces that accounts for living things. As Fritjof Capra puts it, both schools, organicism and vitalism, were born from

8555-511: The iconophile position, chiefly due to the role played by Empress Irene of Athens in the ending of the first period of Iconoclasm at the Second Council of Nicaea in 787. A few years later, another Athenian, Theophano , became empress as the wife of Staurakios (r. 811–812). Invasion of the empire by the Turks after the Battle of Manzikert in 1071, and the ensuing civil wars, largely passed

8700-421: The real line . This would prove to be a major area of research in the first half of the 20th century. The 19th century saw great advances in the theory of real analysis , including theories of convergence of functions and Fourier series . Mathematicians such as Karl Weierstrass began to construct functions that stretched intuition, such as nowhere-differentiable continuous functions . Previous conceptions of

8845-554: The " Tory Radicalism" of Thomas Carlyle , John Ruskin , Samuel Taylor Coleridge , and Benjamin Disraeli . Thomas Hobbes arguably put forward a form of organicism. In the Leviathan , he argued that the state is like a secular God whose constituents (individual people) make up a larger organism. However, the body of the Leviathan is composed of many human faces (all looking outwards from

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8990-460: The 12th century. The agora or marketplace, which had been deserted since late antiquity, began to be built over, and soon the town became an important centre for the production of soaps and dyes. The growth of the town attracted the Venetians , and various other traders who frequented the ports of the Aegean, to Athens. This interest in trade appears to have further increased the economic prosperity of

9135-403: The 19th century. Concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systems for fundamental areas of mathematics such as arithmetic, analysis, and geometry. In logic, the term arithmetic refers to the theory of the natural numbers . Giuseppe Peano published a set of axioms for arithmetic that came to bear his name ( Peano axioms ), using

9280-452: The Acropolis was the site of a major Mycenaean fortress whose remains can be recognised from sections of the characteristic Cyclopean walls. On the summit of the Acropolis, below the later Erechtheion , cuttings in the rock have been identified as the location of a Mycenaean palace. Between 1250 and 1200 BC, to feed the needs of the Mycenaean settlement, a staircase was built down a cleft in

9425-514: The Acropolis. They were themselves influenced by Byzantine Greek culture. In 1311, Athens was conquered by the Catalan Company , a band of mercenaries called Almogavars . It was held by the Catalans until 1388. After 1379, when Thebes was lost, Athens became the capital of the duchy again. The history of Aragonese Athens, called Cetines (rarely Athenes ) by the conquerors, is obscure. Athens

9570-461: The Acropolis—according to Benizelos, if the pasha did not treat them well and heed their opinion, he was liable to be removed before his annual term of office was out—particularly through the influence at Constantinople of the two Athenian-born patriarchs of Jerusalem , Parthenius (1737–1766) and Ephram II (1766–1770). Taxation was also light, with only the haraç tax payable to

9715-462: The Areopagus. Most public offices were filled by lot, although the ten strategoi (generals) were elected. This system remained remarkably stable and, with a few brief interruptions, it remained in place for 170 years, until Philip II of Macedon defeated Athens and Thebes at the Battle of Chaeronea in 338 BC. Prior to the rise of Athens, Sparta considered itself to be the leader (or 2 ) of

9860-467: The Great for the Macedonian victory. Philippides was prosecuted in trial by Hypereides , who detested his pro-Macedonian sympathies. Subsequently, the conquests of Alexander the Great widened Greek horizons and made the traditional Greek city state obsolete. Athens remained a wealthy city with a brilliant cultural life, but ceased to be a leading power. The period following the death of Alexander in 323 BC

10005-730: The Greeks . In 499 BC, Athens sent troops to aid the Ionian Greeks of Asia Minor , who were rebelling against the Persian Empire (the Ionian Revolt ). This provoked two Persian invasions of Greece by the Achaemenid Empire . In 490 BC, the Athenians, led by the soldier-statesman Miltiades , defeated the first invasion of the Persians under Darius I at the Battle of Marathon . In 480 BC,

10150-706: The Hill of Ares , called the Areopagus and appointed the chief city officials, the archons and the polemarch (commander-in-chief). The most famous king of Athens was Theseus , a prominent figure in Greek Mythology who killed the Minotaur . A slightly different mythical version of Athens' past is given in Plato's dialogue Timaeus . In this dialogue, a story is told about information given to Athenian leader Solon from Egyptian priests of

10295-508: The Hymettos aqueduct; many sections had round, oval or square access holes on top of about 10 cm × 10 cm (4  in × 4 in). Pipe segments of this system are displayed at the Evangelismos and Syntagma Metro stations. Peisistratos died in 527 BC and was succeeded by his sons Hippias and Hipparchus . They proved to be much less adept rulers and in 514 BC, Hipparchus

10440-756: The Ottoman government, as well as the salt tax and a water-tax for the olive yards and gardens. Symbolic logic Mathematical logic is the study of formal logic within mathematics . Major subareas include model theory , proof theory , set theory , and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics . Since its inception, mathematical logic has both contributed to and been motivated by

10585-634: The Parthenon and Propylaea . In 1640, a lightning bolt struck the Propylaea, causing its destruction. In 1687, during the Morean War , the Acropolis was besieged by the Venetians under Francesco Morosini , and the temple of Athena Nike was dismantled by the Ottomans to fortify the Parthenon. A shot fired during the bombardment of the Acropolis caused a powder magazine in the Parthenon to explode (26 September), and

10730-400: The Persians returned under Darius's son Xerxes . When a small Greek force holding the pass of Thermopylae was defeated , the Persians proceeded to capture an evacuated Athens. The city of Athens was twice captured and sacked by the Persians within one year after Thermopylae. Subsequently, the Athenians (led by Themistocles ), with their allies, engaged the much larger Persian navy at sea in

10875-723: The Roman Empire (" Rhomaioi "). The conversion of the empire from paganism to Christianity greatly affected Athens, resulting in reduced reverence for the city. Ancient monuments such as the Parthenon, Erechtheion and the Hephaisteion (Theseion) were converted into churches. As the empire became increasingly anti-pagan, Athens became a provincial town and experienced fluctuating fortunes. The city remained an important center of learning, especially of Neoplatonism —with notable pupils including Gregory of Nazianzus , Basil of Caesarea and emperor Julian ( r.  355–363 )—and consequently

11020-559: The Spartan army (the rule of the Thirty Tyrants ). In 403 BC, however, democracy was restored by Thrasybulus and an amnesty was declared. Sparta's former allies soon turned against her, due to her imperialist policy, and soon Athens' former enemies Thebes and Corinth had become her allies; they fought with Athens and Argos against Sparta in the indecisive Corinthian War (395 – 387 BC). Opposition to Sparta enabled Athens to establish

11165-691: The Theoretical Biology Club, to promote the organicist approach to biology . The club was in opposition to mechanistic philosophy , reductionism , and the gene-centric view of evolution . Most of the members were influenced by the philosophy of Alfred North Whitehead . The club disbanded as the Rockefeller Foundation refused to fund their investigations. In ecology , "organicism" and "organicistic" (or "organismic") are used to designate theories which conceptualize populations, particularly ecological communities or ecosystems, according to

11310-447: The Venetians abandoned Athens again to the Ottomans. In the 18th century, the city recovered much of its prosperity. During Michel Fourmont 's visit in the city in the 1720s, he witnessed much construction going on, and by the time the Athenian teacher Ioannis Benizelos wrote an account of the city's affairs in the 1770s, Athens was once again enjoying some prosperity, so that, according to Benizelos, it "could be cited as an example to

11455-556: The Winds for the Roman forum , which mostly survives to the present day. Under Roman rule, Athens was given the status of a free city because of its widely admired schools. The Roman emperor Hadrian ( r.  117–138 AD ), constructed the Library of Hadrian , a gymnasium , an aqueduct which is still in use, several temples and sanctuaries, a bridge, and finally completed the Temple of Olympian Zeus . The Arch of Hadrian commemorates

11600-404: The beauty of its ancient monuments and issued a firman (imperial edict) forbidding their looting or destruction, on pain of death. The Parthenon was converted into the main mosque of the city. Under Ottoman rule, Athens was denuded of any importance and its population severely declined, leaving it as a "small country town" ( Franz Babinger ). From the early 17th century, Athens came under

11745-470: The body), and these faces do not symbolize different organs of a complex organism but the individual people who themselves have consented to the social contract, and thereby ceded their power to the Leviathan. That the Leviathan is more like a constructed machine than like a literal organism is perfectly in line with Hobbes' elementaristic individualism and mechanical materialism. According to scholars Jean-Yves Camus and Nicolas Lebourg , organicism stands at

11890-496: The building was severely damaged, giving it largely the appearance it has today. The Venetian occupation of Athens lasted for six months, and both the Venetians and the Ottomans participated in the looting of the Parthenon. One of its western pediments was removed, causing even more damage to the structure. During the Venetian occupation, the two mosques of the city were converted into Catholic and Protestant churches, but on 9 April 1688

12035-640: The burgeoning field of biological studies . Within contemporary biology, organicism stresses the organization (particularly the self-organizing properties) rather than the composition (the reduction into biological components) of organisms . John Scott Haldane was the first modern biologist to use the term to expand his philosophical stance in 1917; other 20th-century academics and professionals, such as Theodor Adorno and Albert Dalcq  [ fr ] , have followed in Haldane's wake. Properly scientific interest in organicist biology has recently been revived with

12180-408: The cells of an organism. This sort of organicist sociology was articulated by Alfred Espinas, Paul von Lilienfeld , Jacques Novicow , Albert Schäffle , Herbert Spencer , and René Worms , among others. Prominent conservative political thinkers who have developed an organic view of society are Edmund Burke , G.W.F. Hegel , Adam Müller , and Julius Evola . Organicism has also been identified with

12325-509: The city (as their capital) until the Turkish conquest of 1458. The first Ottoman attack on Athens, which involved a short-lived occupation of the town, came in 1397, under the Ottoman generals Yaqub Pasha and Timurtash. Finally, in 1458, Athens was captured by the Ottomans under the personal leadership of Sultan Mehmed II . As the Ottoman Sultan rode into the city, he was greatly struck by

12470-463: The city and to give their name to it, so they competed with offering the city one gift each. Poseidon produced a spring by striking the ground with his trident, symbolizing naval power. Athena created the olive tree , symbolizing peace and prosperity. The Athenians, under their ruler Cecrops , accepted the olive tree and named the city after Athena. (Later the Southern Italian city of Paestum

12615-686: The city, lay about 400 m (1,300 ft) north of the Acropolis, in what is now the Monastiraki district. The hill of the Pnyx , where the Athenian Assembly met, lay at the western end of the city. The Eridanus (Ηριδανός) river flowed through the city. One of the most important religious sites in ancient Athens was the Temple of Athena, known today as the Parthenon, which stood on top of the Acropolis, where its evocative ruins still stand. Two other major religious sites,

12760-469: The collection is nonempty, the lack of a general, concrete rule by which the choice can be made renders the axiom nonconstructive. Stefan Banach and Alfred Tarski showed that the axiom of choice can be used to decompose a solid ball into a finite number of pieces which can then be rearranged, with no scaling, to make two solid balls of the original size. This theorem, known as the Banach–Tarski paradox ,

12905-608: The completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis. Another type of logics are fixed-point logic s that allow inductive definitions , like one writes for primitive recursive functions . One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic . Lindström's theorem implies that

13050-412: The consistency of elementary arithmetic, respectively; the tenth was to produce a method that could decide whether a multivariate polynomial equation over the integers has a solution. Subsequent work to resolve these problems shaped the direction of mathematical logic, as did the effort to resolve Hilbert's Entscheidungsproblem , posed in 1928. This problem asked for a procedure that would decide, given

13195-523: The constitution of complex systems and theorization of organic-racial societies. In particular, one of the most significant ideological influences on the Nazis was the 19th-century German nationalist philosopher Johann Gottlieb Fichte , whose works had served as an inspiration to Hitler and other Nazi Party members, and whose ideas were implemented among the philosophical and ideological foundations of Nazi-oriented Völkisch nationalism . In breathing organisms,

13340-679: The context of proof theory. At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems . These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language . The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties. Stronger classical logics such as second-order logic or infinitary logic are also studied, along with Non-classical logics such as intuitionistic logic . First-order logic

13485-631: The core of the historical far-right's worldview . Adolf Hitler himself along with other members of the National Socialist German Workers' Party (German: Nationalsozialistische Deutsche Arbeiterpartei , NSDAP) in the Weimar Republic (1918–1933) were greatly influenced by several 19th- and early 20th-century thinkers and proponents of philosophical, onto-epistemic, and theoretical perspectives on ecological anthropology , scientific racism , holistic science , and organicism regarding

13630-402: The dramatists Aeschylus , Sophocles , Euripides and Aristophanes , the physician Hippocrates , the philosophers Socrates , Plato and Aristotle , the historians Herodotus, Thucydides and Xenophon , the poet Simonides , the orators Antiphon , Isocrates , Aeschines , and Demosthenes , and the sculptor Phidias . The leading statesman of the mid-fifth century BC was Pericles , who used

13775-557: The early Middle Ages , the city experienced a decline, then recovered under the later Byzantine Empire and was relatively prosperous during the period of the Crusades (12th and 13th centuries), benefiting from Italian trade. Following a period of sharp decline under the rule of the Ottoman Empire , Athens re-emerged in the 19th century as the capital of the independent and self-governing Greek state . The name of Athens , connected to

13920-407: The early decades of the 20th century, the main areas of study were set theory and formal logic. The discovery of paradoxes in informal set theory caused some to wonder whether mathematics itself is inconsistent, and to look for proofs of consistency. In 1900, Hilbert posed a famous list of 23 problems for the next century. The first two of these were to resolve the continuum hypothesis and prove

14065-522: The end of the fourth millennium BC or a little later. The site is a natural defensive position which commands the surrounding plains. It is located about 20 km (12 mi) inland from the Saronic Gulf , in the centre of the Cephisian Plain , a fertile valley surrounded by rivers. To the east lies Mount Hymettus , to the north Mount Pentelicus . Ancient Athens, in the first millennium BC, occupied

14210-450: The first volume of Principia Mathematica by Russell and Alfred North Whitehead was published. This seminal work developed the theory of functions and cardinality in a completely formal framework of type theory , which Russell and Whitehead developed in an effort to avoid the paradoxes. Principia Mathematica is considered one of the most influential works of the 20th century, although the framework of type theory did not prove popular as

14355-411: The form of the body, the properties and characters of its diverse parts, as resulting from the reciprocal play or struggle of all its elements, cells, fibres, tissues, organs, which act the one on the other, modify one the other, allot among them each its place and part, and lead all together to the final result, giving thus the appearance of a consensus, or a pre-established harmony, where in reality there

14500-499: The foundation of the city by Hadrian, with the "city of Theseus" referred to on its inscription on one side of the arch, and the new quarter erected by Hadrian around the Temple of Zeus called the "city of Hadrian". The city was sacked by the Heruli in 267 AD, resulting in the burning of all the public buildings, the plundering of the lower city and the damaging of the Agora and Acropolis. After

14645-400: The goddess Neith while he visited Egypt, according to which a well advanced Athenian state was established 9,000 years prior to his time that preceded Egypt's oldest kingdom by a thousand years. The laws of that state were the most just and largely inspired the various kings of Egypt when making laws for their kingdom. This story is not supported by any scholarly evidence, as no Athenian state

14790-488: The hegemony of Athens led to the Peloponnesian War , which began in 431 BC and pitted Athens and its increasingly rebellious overseas empire against a coalition of land-based states led by Sparta. The conflict was a drawn out one that saw Sparta control the land while Athens was dominant at sea, however the disastrous Sicilian Expedition severely weakened Athens and the war eventually ended in an Athenian defeat following

14935-400: The imperial city, many of Athens's works of art were taken by the emperors to adorn it. The Empire became Christianized, and the use of Latin declined in favour of exclusive use of Greek ; in the Roman imperial period , both languages had been used. In the later Roman period, Athens was ruled by the emperors continuing until the 13th century, its citizens identifying themselves as citizens of

15080-423: The importance of the incompleteness theorem for some time. Gödel's theorem shows that a consistency proof of any sufficiently strong, effective axiom system cannot be obtained in the system itself, if the system is consistent, nor in any weaker system. This leaves open the possibility of consistency proofs that cannot be formalized within the system they consider. Gentzen proved the consistency of arithmetic using

15225-443: The incompleteness (in a different meaning of the word) of all sufficiently strong, effective first-order theories. This result, known as Gödel's incompleteness theorem , establishes severe limitations on axiomatic foundations for mathematics, striking a strong blow to Hilbert's program. It showed the impossibility of providing a consistency proof of arithmetic within any formal theory of arithmetic. Hilbert, however, did not acknowledge

15370-491: The incompleteness theorems in generality that could only be implied in the original paper. Numerous results in recursion theory were obtained in the 1940s by Stephen Cole Kleene and Emil Leon Post . Kleene introduced the concepts of relative computability, foreshadowed by Turing, and the arithmetical hierarchy . Kleene later generalized recursion theory to higher-order functionals. Kleene and Georg Kreisel studied formal versions of intuitionistic mathematics, particularly in

15515-408: The introduction of and emphasis on the importance of relations, whether in symbolic logic , in phenomenology , or in metaphysics. William Wimsatt has suggested that the number of terms in the relations considered distinguishes reductionism from holism. Reductionistic explanations claim that two or at most three term relations are sufficient to account for the system's behavior. At the other extreme

15660-559: The issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics ) rather than trying to find theories in which all of mathematics can be developed. The Handbook of Mathematical Logic in 1977 makes

15805-603: The jurisdiction of the Kizlar Agha , the chief black eunuch of the Sultan's harem . The city had originally been granted by Sultan Ahmed I ( r.  1603–1617 ) to Basilica, one of his favourite concubines, who hailed from the city, in response of complaints of maladministration by the local governors. After her death, Athens came under the purview of the Kizlar Agha. The Turks began a practice of storing gunpowder and explosives in

15950-509: The largest and wealthiest state on the Greek mainland, but it also created a larger class of people excluded from political life by the nobility. By the 7th century BC, social unrest had become widespread, and the Areopagus appointed Draco to draft a strict new code of law (hence the word 'draconian'). When this failed, they appointed Solon , with a mandate to create a new constitution (in 594 BC). The reforms that Solon initiated dealt with both political and economic issues. The economic power of

16095-569: The layman was written by Lewis Carroll , author of Alice's Adventures in Wonderland , in 1896. Alfred Tarski developed the basics of model theory . Beginning in 1935, a group of prominent mathematicians collaborated under the pseudonym Nicolas Bourbaki to publish Éléments de mathématique , a series of encyclopedic mathematics texts. These texts, written in an austere and axiomatic style, emphasized rigorous presentation and set-theoretic foundations. Terminology coined by these texts, such as

16240-457: The mid-19th century as a subfield of mathematics, reflecting the confluence of two traditions: formal philosophical logic and mathematics. Mathematical logic, also called 'logistic', 'symbolic logic', the ' algebra of logic ', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the nineteenth century with the aid of an artificial notation and a rigorously deductive method. Before this emergence, logic

16385-487: The middle of the nineteenth century, George Boole and then Augustus De Morgan presented systematic mathematical treatments of logic. Their work, building on work by algebraists such as George Peacock , extended the traditional Aristotelian doctrine of logic into a sufficient framework for the study of foundations of mathematics . In 1847, Vatroslav Bertić made substantial work on algebraization of logic, independently from Boole. Charles Sanders Peirce later built upon

16530-417: The model of the individual organism . As such, the term "organicism" is sometimes used interchangeably with " holism ", although there are versions of holism that are not organicistic/organismic but individualistic. Early iterations of Gaia theory took an explicitly organicist approach by conceptualizing the entire Earth as an integrated, self-regulating organic whole akin to a living being, rather than just

16675-489: The name of its patron goddess Athena , originates from an earlier Pre-Greek language. The origin myth explaining how Athens acquired this name through the legendary contest between Poseidon and Athena was described by Herodotus , Apollodorus , Ovid , Plutarch , Pausanias and others. It even became the theme of the sculpture on the west pediment of the Parthenon . Both Athena and Poseidon requested to be patrons of

16820-482: The natural numbers but cannot be proved. Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent ,

16965-480: The offices of state. Peisistratus built the first aqueduct tunnel at Athens, which most likely had its sources on the slopes of Mount Hymettos and along the Ilissos river . It supplied, among other structures, the fountain house in the southeast corner of the Agora, but it had a number of branches. In the 4th century BC it was replaced by a system of terracotta pipes in a stone-built underground channel, sometimes called

17110-486: The only extension of first-order logic satisfying both the compactness theorem and the downward Löwenheim–Skolem theorem is first-order logic. Modal logics include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability and set-theoretic forcing. Intuitionistic logic

17255-406: The other cities of Greece". Its Greek population possessed a considerable degree of self-government, under a council of primates composed of the leading aristocratic families, along with the city's metropolitan bishop. The community was quite influential with the Ottoman authorities, the pasha (governor), the kadi (judge), the mufti (Islamic prelate), and the garrison commander of

17400-464: The popular party, led by Peisistratos , seized power. Peisistratos is usually called a tyrant , but the Greek word tyrannos does not mean a cruel and despotic ruler, merely one who took power by force. Peisistratos was in fact a very popular ruler, who made Athens wealthy, powerful, and a centre of culture. He preserved the Solonian Constitution , but made sure that he and his family held all

17545-704: The quest for getting rid of the Cartesian picture of reality, a view that has been claimed to be the most destructive paradigm nowadays, from science to politics. A number of biologists in the early to mid-twentieth century embraced organicism. They wished to reject earlier vitalisms but also to stress that whole organism biology was not fully explainable by atomic mechanism. The larger organization of an organic system has features that must be taken into account to explain its behavior. The French zoologist Yves Delage , in his seminal text L'Hérédité Et Les Grands Problèmes de la Biologie Générale, described organicism thus: [L]ife,

17690-468: The region by and Athens continued its provincial existence unharmed. When the Byzantine Empire was rescued by the resolute leadership of the three Komnenos emperors Alexios , John and Manuel , Attica and the rest of Greece prospered. Archaeological evidence tells us that the medieval town experienced a period of rapid and sustained growth, starting in the 11th century and continuing until the end of

17835-406: The relations of the parts are essential to explanation of behavior of the system. Aristotle and early modern philosophers and scientists tended to describe reality as made of substances and their qualities, and to neglect relations. Gottfried Wilhelm Leibniz showed the bizarre conclusions to which a doctrine of the non-existence of relations led. Twentieth century philosophy has been characterized by

17980-705: The rock to reach a water supply that was protected from enemy incursions, comparable to similar works carried out at Mycenae. Unlike other Mycenaean centers, such as Mycenae and Pylos , it is unclear whether Athens suffered destruction in about 1200 BC, an event traditionally attributed to a Dorian invasion (though now commonly attributed to a systems collapse, part of the Late Bronze Age collapse ). The Athenians always maintained that they were 'pure' Ionians with no Dorian element. However, Athens, like many other Bronze Age settlements, went into economic decline for around 150 years following this. Iron Age burials, in

18125-715: The series of feudal fiefs , similar to the Crusader states established in Syria and on Cyprus after the First Crusade . This period is known as the Frankokratia . Athens was initially the capital of the eponymous Duchy of Athens , a fief of the Latin Empire which replaced the Byzantine Empire, ruling from Constantinople. After Thebes became a possession of the Latin dukes, which were of

18270-410: The study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry , arithmetic , and analysis . In the early 20th century it was shaped by David Hilbert 's program to prove the consistency of foundational theories. Results of Kurt Gödel , Gerhard Gentzen , and others provided partial resolution to the program, and clarified

18415-466: The system could be considered as a single ten to the twenty-sixth term relation, for instance. Organicism as a structuring or worldview motif finds implementation in the dogmatic theology of the nineteenth century. It was implemented prominently by Dutch Reformed theologian Herman Bavinck , and has been variously debated and appraised. Nonetheless, Bavinck's consideration of reality as both being and becoming, with an organic unity-and-diversity rooted in

18560-494: The teaching of philosophy by pagans in 529, an event whose impact on the city is much debated, but is generally taken to mark the end of the ancient history of Athens. Athens was sacked by the Slavs in 582, but remained in imperial hands thereafter, as highlighted by the visit of the emperor Constans II ( r.  641–668 ) in 662/3 and its inclusion in the Theme of Hellas . The city

18705-575: The town. The 11th and 12th centuries were the Golden Age of Byzantine art in Athens. Almost all of the most important Middle Byzantine churches in and around Athens were built during these two centuries, and this reflects the growth of the town in general. However, this medieval prosperity was not to last. In 1204, the Fourth Crusade conquered Athens and the city was not recovered from the Latins before it

18850-584: The tribute paid by the members of the Delian League to build the Parthenon and other great monuments of classical Athens. The city became, in Pericles's words, "the school of Hellas [Greece]." Shortly after the death of Alexander the Great, Antipater and Craterus became joint generals of Greece and Macedonia. Athens joined Aetolia and Thessaly in facing their power, known as the Lamian War . Craterus fell in

18995-402: The turn of the century. The two-dimensional notation Frege developed was never widely adopted and is unused in contemporary texts. From 1890 to 1905, Ernst Schröder published Vorlesungen über die Algebra der Logik in three volumes. This work summarized and extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference to symbolic logic as it was understood at the end of

19140-553: The uniqueness of the set of natural numbers (up to isomorphism) and the recursive definitions of addition and multiplication from the successor function and mathematical induction. In the mid-19th century, flaws in Euclid's axioms for geometry became known. In addition to the independence of the parallel postulate , established by Nikolai Lobachevsky in 1826, mathematicians discovered that certain theorems taken for granted by Euclid were not in fact provable from his axioms. Among these

19285-675: The united essence of the Trinity, stands as a prominent example of a theological implementation of organicism. Likewise, organicist language can be found in relation to faith and the church by Norwegian Lutheran theologian Gisle Johnson . Johnson will construe the relationship between faith, church, and historical development as "an organic development," likening it to a sprout. Defunct Defunct Organicism has also been used to characterize notions put forth by various late 19th-century social scientists who considered human society to be analogous to an organism, and individual humans to be analogous to

19430-463: The words bijection , injection , and surjection , and the set-theoretic foundations the texts employed, were widely adopted throughout mathematics. The study of computability came to be known as recursion theory or computability theory , because early formalizations by Gödel and Kleene relied on recursive definitions of functions. When these definitions were shown equivalent to Turing's formalization involving Turing machines , it became clear that

19575-430: The work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885. Gottlob Frege presented an independent development of logic with quantifiers in his Begriffsschrift , published in 1879, a work generally considered as marking a turning point in the history of logic. Frege's work remained obscure, however, until Bertrand Russell began to promote it near

19720-665: Was a veguería with its own castellan , captain, and veguer . At some point during the Aragonese period, the Acropolis was further fortified and the Athenian archdiocese received an extra two suffragan sees. In 1388, the Florentine Nerio I Acciajuoli took the city and made himself duke. The Florentines had to dispute the city with the Republic of Venice , but they ultimately emerged victorious after seven years of Venetian rule (1395–1402). The descendants of Nerio I Acciajuoli ruled

19865-539: Was assassinated in a private dispute over a young man (see Harmodius and Aristogeiton ). This led Hippias to establish a real dictatorship, which proved very unpopular. He was overthrown in 510 BC. A radical politician with an aristocratic background named Cleisthenes then took charge, and it was he who established democracy in Athens. The reforms of Cleisthenes replaced the traditional four phyle ('tribes') with ten new ones, named after legendary heroes and having no class basis; they were in fact electorates. Each phyle

20010-494: Was before the Lamian War. However, after losing the fleet one year prior, Polyperchon had to flee Macedon when in 316 BC Cassander secured control of Athens. Cassander appointed Demetrius of Phalerum as head of the administration of Athens. Demetrius remained in power until 307 BC when Cassander's enemy, Demetrius Poliorcetes captured Athens, and Macedon, ending the short-lived Antipatrid dynasty and installing his own. After

20155-486: Was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the law of the excluded middle , which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic

20300-402: Was founded under the name of Poseidonia at about 600 BC.) A sacred olive tree said to be the one created by the goddess was still kept on the Acropolis at the time of Pausanias (2nd century AD). It was located by the temple of Pandrosus , next to the Parthenon. According to Herodotus, the tree had been burnt down during the Persian Wars , but a shoot sprung from the stump. The Greeks saw this as

20445-465: Was in turn divided into three trittyes and each trittys had one or more demes , which became the basis of local government. The phyle each elected fifty members to the Boule , a council which governed Athens on a day-to-day basis. The Assembly was open to all citizens and was both a legislature and a supreme court, except in murder cases and religious matters, which became the only remaining functions of

20590-407: Was proved independent of ZF by Fraenkel, but has come to be widely accepted by mathematicians. It states that given a collection of nonempty sets there is a single set C that contains exactly one element from each set in the collection. The set C is said to "choose" one element from each set in the collection. While the ability to make such a choice is considered obvious by some, since each set in

20735-496: Was studied with rhetoric , with calculationes , through the syllogism , and with philosophy . The first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over the foundations of mathematics. Theories of logic were developed in many cultures in history, including China , India , Greece and the Islamic world . Greek methods, particularly Aristotelian logic (or term logic) as found in

20880-632: Was taken by the Ottoman Turks . It did not become Greek in government again until the 19th century. From 1204 until 1458, Athens was ruled by Latins in three separate periods, following the Crusades . The "Latins", or " Franks ", were western Europeans and followers of the Latin Church brought to the Eastern Mediterranean during the Crusades. Along with rest of Byzantine Greece, Athens was part of

21025-404: Was threatened by Saracen raids in the 8th–9th centuries—in 896, Athens was raided and possibly occupied for a short period, an event which left some archaeological remains and elements of Arabic ornamentation in contemporary buildings —but there is also evidence of a mosque existing in the city at the time. In the great dispute over Byzantine Iconoclasm , Athens is commonly held to have supported

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