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109-403: In mathematics and formal logic , first principles are referred to as axioms or postulates. In physics and other sciences, theoretical work is said to be from first principles, or ab initio , if it starts directly at the level of established science and does not make assumptions such as empirical model and parameter fitting. "First principles thinking" consists of decomposing things down to

218-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects

327-463: A few similarities. Others argue that the two traditions share a common origin and can even be considered a single entity, termed " Orphico-Pythagoreanism ." The belief that Pythagoreanism was a subset or direct descendant of Orphic religion existed by late antiquity, when Neoplatonist philosophers took the Orphic origin of Pythagorean teachings at face value. Proclus wrote: In the fifteenth century,

436-414: A final state. This source of entity is always preserved. Although their theories were primitive, these philosophers were the first to give an explanation of the physical world without referencing the supernatural; this opened the way for much of modern science (and philosophy), which has the same goal of explaining the world without dependence on the supernatural. Thales of Miletus (7th to 6th century BC),

545-614: A foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of

654-669: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)

763-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were

872-422: A mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture . Through a series of rigorous arguments employing deductive reasoning , a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma . A proven instance that forms part of

981-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, " or " means "one, the other or both", while, in common language, it

1090-461: A mythological figure. Despite this, even these authors of the 5th and 4th centuries BC noted a strong similarity between the two doctrines. In fact, some claimed that rather than being an initiate of Orphism, Pythagoras was actually the original author of the first Orphic texts. Specifically, Ion of Chios claimed that Pythagoras authored poetry which he attributed to the mythical Orpheus, and Epigenes, in his On Works Attributed to Orpheus , attributed

1199-502: A number of beliefs about the afterlife similar to those in the "Orphic" mythology about Dionysus ' death and resurrection. Bone tablets found in Olbia (5th century BC) carry short and enigmatic inscriptions like: "Life. Death. Life. Truth. Dio(nysus). Orphics." The function of these bone tablets is unknown. Gold-leaf tablets found in graves from Thurii , Hipponium , Thessaly and Crete (4th century BC and after) give instructions to

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1308-510: A philosophical treatise that is an allegorical commentary on an Orphic poem in hexameters, a theogony concerning the birth of the gods, produced in the circle of the philosopher Anaxagoras , written in the second half of the fifth century BC. Fragments of the poem are quoted making it "the most important new piece of evidence about Greek philosophy and religion to come to light since the Renaissance". The papyrus dates to around 340 BC, during

1417-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes

1526-528: A priori truths. His most famous proposition is "Je pense, donc je suis" ( I think, therefore I am , or Cogito ergo sum ), which he indicated in his Discourse on the Method was "the first principle of the philosophy of which I was in search." Descartes describes the concept of a first principle in the following excerpt from the preface to the Principles of Philosophy (1644): I should have desired, in

1635-520: A re-ordering of Hesiod 's Theogony , based in part on pre-Socratic philosophy . The suffering and death of the god Dionysus at the hands of the Titans has been considered the central myth of Orphism. According to this myth, the infant Dionysus is killed, torn apart, and consumed by the Titans. In retribution, Zeus strikes the Titans with a thunderbolt, turning them to ash. From these ashes, humanity

1744-402: A ritual purification and reliving of the suffering and death of the god. Orphics believed that they would, after death, spend eternity alongside Orpheus and other heroes. The uninitiated ( Ancient Greek : ἀμύητος , romanized :  amúētos ), they believed, would be reincarnated indefinitely. Orphism is named after the legendary poet-hero Orpheus , who was said to have originated

1853-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as

1962-423: A set of definitions, postulates, and common notions: all three types constitute first principles. In philosophy, "first principles" are from first cause attitudes commonly referred to as a priori terms and arguments, which are contrasted to a posteriori terms, reasoning, or arguments, in that the former are simply assumed and exist prior to the reasoning process, and the latter are deduced or inferred after

2071-522: A shorter length composed in the Roman Imperial age. The Orphic Argonautica ( ‹See Tfd› Greek : Ὀρφέως Ἀργοναυτικά ) is a Greek epic poem dating from the 4th century CE of unknown authorship. It is narrated in the first person in the name of Orpheus and tells the story of Jason and the Argonauts . The narrative is basically similar to that in other versions of the story, such as

2180-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of

2289-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints. For example, designing a survey often involves minimizing the cost of estimating

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2398-477: A wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory , type theory , computability theory and computational complexity theory . Although these aspects of mathematical logic were introduced before the rise of computers , their use in compiler design, formal verification , program analysis , proof assistants and other aspects of computer science , contributed in turn to

2507-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry

2616-486: Is flat " and "a field is always a ring ". Orphic Orphism (more rarely Orphicism ; Ancient Greek : Ὀρφικά , romanized :  Orphiká ) is the name given to a set of religious beliefs and practices originating in the ancient Greek and Hellenistic world, associated with literature ascribed to the mythical poet Orpheus , who descended into the Greek underworld and returned. This type of journey

2725-429: Is possible that some of the statements can be deduced from other statements. For example, in the syllogism , "All men are mortal; Socrates is a man; Socrates is mortal" the last claim can be deduced from the first two. A first principle is an axiom that cannot be deduced from any other within that system. The classic example is that of Euclid's Elements ; its hundreds of geometric propositions can be deduced from

2834-544: Is born. In Orphic belief, this myth describes humanity as having a dual nature: body ( Ancient Greek : σῶμα , romanized :  sôma ), inherited from the Titans, and a divine spark or soul ( Ancient Greek : ψυχή , romanized :  psukhḗ ), inherited from Dionysus. In order to achieve salvation from the Titanic, material existence, one had to be initiated into the Dionysian mysteries and undergo teletē ,

2943-461: Is called a katabasis and is the basis of several hero worships and journeys. Orphics revered Dionysus (who once descended into the Underworld and returned) and Persephone (who annually descended into the Underworld for a season and then returned). Orphism has been described as a reform of the earlier Dionysian religion , involving a re-interpretation or re-reading of the myth of Dionysus and

3052-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example

3161-513: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of

3270-407: Is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called " exclusive or "). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module

3379-493: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as

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3488-481: Is looking for "first principles" (or "origins"; archai ): In every systematic inquiry (methodos) where there are first principles, or causes, or elements, knowledge and science result from acquiring knowledge of these; for we think we know something just in case we acquire knowledge of the primary causes, the primary first principles, all the way to the elements. It is clear, then, that in the science of nature as elsewhere, we should try first to determine questions about

3597-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example

3706-601: Is not axiomatic as expressed in Aristotle's account of a first principle (in one sense) as "the first basis from which a thing is known" (Met. 1013a14–15). For Aristotle , the arche is the condition necessary for the existence of something, the basis for what he calls "first philosophy" or metaphysics. The search for first principles is not peculiar to philosophy; philosophy shares this aim with biological, meteorological, and historical inquiries, among others. But Aristotle's references to first principles in this opening passage of

3815-562: Is not certain. Orphic views and practices have parallels to elements of Pythagoreanism , and various traditions hold that the Pythagoreans or Pythagoras himself authored early Orphic works; alternately, later philosophers believed that Pythagoras was an initiate of Orphism. The extent to which one movement may have influenced the other remains controversial. Some scholars maintain that Orphism and Pythagoreanism began as separate traditions which later became confused and conflated due to

3924-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,

4033-1192: Is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations , unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts . Operation and relations are generally represented by specific symbols or glyphs , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of

4142-547: Is often held to be Archimedes ( c.  287  – c.  212 BC ) of Syracuse . He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and

4251-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it

4360-441: Is something completely indefinite; and Anaximander was probably influenced by the original chaos of Hesiod (yawning abyss). Anaximander was the first philosopher that used arche for that which writers from Aristotle onwards called "the substratum" ( Simplicius Phys. 150, 22). He probably intended it to mean primarily "indefinite in kind" but assumed it also to be "of unlimited extent and duration". The notion of temporal infinity

4469-567: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after

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4578-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,

4687-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of

4796-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with

4905-706: The Argonautica of Apollonius Rhodius , on which it is probably based. The main differences are the emphasis on the role of Orpheus and a more mythological, less realistic technique of narration. In the Argonautica Orphica , unlike in Apollonius Rhodius, it is claimed that the Argo was the first ship ever built. The Derveni papyrus, found in Derveni , Macedonia (Greece) , in 1962, contains

5014-586: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It

5123-768: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during

5232-505: The Mysteries of Dionysus . However, Orpheus was more closely associated with Apollo than to Dionysus in the earliest sources and iconography. According to some versions of his mythos, he was the son of Apollo, and during his last days, he shunned the worship of other gods and devoted himself to Apollo alone. Poetry containing distinctly Orphic beliefs has been traced back to the 6th century BC or at least 5th century BC, and graffiti of

5341-511: The Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity

5450-476: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of

5559-564: The Sibyl . Of this vast literature, only two works survived whole: the Orphic Hymns , a set of 87 poems, possibly composed at some point in the second or third century, and the epic Orphic Argonautica , composed somewhere between the fourth and sixth centuries. Earlier Orphic literature, which may date back as far as the sixth century BC, survives only in papyrus fragments or in quotations. The Orphic Hymns are 87 hexametric poems of

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5668-467: The Titans to murder the child. Zagreus is then tricked with a mirror and children's toys by the Titans, who shred him to pieces and consume him. Athena saves the heart and tells Zeus of the crime, who in turn hurls a thunderbolt on the Titans . The resulting soot, from which sinful mankind is born, contains the bodies of the Titans and Zagreus. The soul of man (the Dionysus part) is therefore divine, but

5777-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object

5886-414: The "ultimate underlying substance" and "ultimate indemonstrable principle". The heritage of Greek mythology already embodied the desire to articulate reality as a whole and this universalizing impulse was fundamental for the first projects of speculative theorizing. It appears that the order of "being" was first imaginatively visualized before it was abstractly thought. In the mythological cosmogonies of

5995-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry

6104-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not

6213-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and

6322-414: The 5th century BC apparently refers to "Orphics". The Derveni papyrus allows Orphic mythology to be dated to the end of the 5th century BC, and it is probably even older. Orphic views and practices are attested as by Herodotus , Euripides , and Plato . Plato refers to "Orpheus-initiators" ( Ὀρφεοτελεσταί ), and associated rites, although how far "Orphic" literature in general related to these rites

6431-637: The Middle Ages and made available in Europe. During the early modern period , mathematics began to develop at an accelerating pace in Western Europe , with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,

6540-547: The Near East, the universe is formless and empty and the only existing thing prior to creation was the water abyss. In the Babylonian creation story, Enuma Elish , the primordial world is described as a "watery chaos" from which everything else appeared. This watery chaos has similarities in the cosmogony of the Greek mythographer Pherecydes of Syros . In the mythical Greek cosmogony of Hesiod (8th to 7th century BC),

6649-492: The Neoplatonic Greek scholar Constantine Lascaris (who found the poem Argonautica Orphica ) considered a Pythagorean Orpheus. Bertrand Russell (1947) noted: Study of early Orphic and Pythagorean sources, however, is more ambiguous concerning their relationship, and authors writing closer to Pythagoras' own lifetime never mentioned his supposed initiation into Orphism, and in general regarded Orpheus himself as

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6758-636: The Physics and at the start of other philosophical inquiries imply that it is a primary task of philosophy. Profoundly influenced by Euclid , Descartes was a rationalist who invented the foundationalist system of philosophy. He used the method of doubt , now called Cartesian doubt , to systematically doubt everything he could possibly doubt until he was left with what he saw as purely indubitable truths. Using these self-evident propositions as his axioms, or foundations, he went on to deduce his entire body of knowledge from them. The foundations are also called

6867-477: The arts, and that knowledge to subserve these ends must necessarily be deduced from first causes; so that in order to study the acquisition of it (which is properly called [284] philosophizing), we must commence with the investigation of those first causes which are called Principles. Now, these principles must possess two conditions: in the first place, they must be so clear and evident that the human mind, when it attentively considers them, cannot doubt their truth; in

6976-678: The authorship of several influential Orphic poems to notable early Pythagoreans, including Cercops. According to Cicero , Aristotle also claimed that Orpheus never existed, and that the Pythagoreans ascribed some Orphic poems to Cercon (see Cercops ). Belief in metempsychosis was common to both currents, although it also seems to contain differences. Where the Orphics taught about a cycle of grievous embodiments that could be escaped through their rites, Pythagoras seemed to teach about an eternal, neutral metempsychosis against which personal actions would be irrelevant. The Neoplatonists regarded

7085-583: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During

7194-648: The body (the Titan part) holds the soul in bondage. Thus, it was declared that the soul returns to a host ten times, bound to the wheel of rebirth . Following the punishment, the dismembered limbs of Zagreus were cautiously collected by Apollo who buried them in his sacred land Delphi . In Orphic theogonies, the Orphic Egg is a cosmic egg from which hatched the primordial hermaphroditic deity Phanes/Protogonus (variously equated also with Zeus , Pan , Metis , Eros , Erikepaios and Bromius ), who in turn created

7303-460: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,

7412-399: The current language, where expressions play the role of noun phrases and formulas play the role of clauses . Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is

7521-409: The dead . Although these thin tablets are often highly fragmentary, collectively they present a shared scenario of the passage into the afterlife. When the deceased arrives in the underworld, he is expected to confront obstacles. He must take care not to drink of Lethe ("Forgetfulness"), but of the pool of Mnemosyne ("Memory"). He is provided with formulaic expressions with which to present himself to

7630-569: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely

7739-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of

7848-575: The father of philosophy, claimed that the first principle of all things is water, and considered it as a substance that contains in it motion and change. His theory was supported by the observation of moisture throughout the world and coincided with his theory that the Earth floated on water. His ideas were influenced by the Near-Eastern mythological cosmogony and probably by the Homeric statement that

7957-408: The first place, to explain in it what philosophy is, by commencing with the most common matters, as, for example, that the word philosophy signifies the study of wisdom, and that by wisdom is to be understood not merely prudence in the management of affairs, but a perfect knowledge of all that man can know, as well for the conduct of his life as for the preservation of his health and the discovery of all

8066-532: The first principles. The naturally proper direction of our road is from things better known and clearer to us, to things that are clearer and better known by nature; for the things that are known to us are not the same as the things known unconditionally (haplôs). Hence it is necessary for us to progress, following this procedure, from the things that are less clear by nature, but clearer to us, towards things that are clearer and better known by nature. (Phys. 184a10–21) The connection between knowledge and first principles

8175-567: The first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant " astrology " (or sometimes " astronomy ") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematici , meaning "astrologers",

8284-419: The fundamental axioms in the given arena, before reasoning up by asking which ones are relevant to the question at hand, then cross referencing conclusions based on chosen axioms and making sure conclusions do not violate any fundamental laws. Physicists include counterintuitive concepts with reiteration . In a formal logical system , that is, a set of propositions that are consistent with one another, it

8393-511: The guardians of the afterlife. As said in the Petelia tablet : I am a son of Earth and starry sky. I am parched with thirst and am dying; but quickly grant me cold water from the Lake of Memory to drink. Other gold leaves offer instructions for addressing the rulers of the underworld: Now you have died and now you have come into being, O thrice happy one, on this same day. Tell Persephone that

8502-581: The initial reasoning process. First principles are generally treated in the realm of philosophy known as epistemology but are an important factor in any metaphysical speculation. In philosophy, "first principles" are often somewhat synonymous with a priori , datum, and axiomatic reasoning . In Ancient Greek philosophy , a first principle from which other principles are derived is called an arche and later "first principle" or "element". By extension, it may mean "first place", "method of government", "empire, realm", "authorities" The concept of an arche

8611-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before

8720-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and

8829-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term

8938-403: The material monists were the three Milesian philosophers: Thales , who believed that everything was composed of water; Anaximander , who believed it was apeiron ; and Anaximenes , who believed it was air. This is considered as a permanent substance or either one or more which is conserved in the generation of rest of it. From this all things first come to be and into this they are resolved in

9047-573: The model to experimental data is an ab initio approach . Mathematics Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as

9156-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to

9265-536: The objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains

9374-434: The origin of the world is Chaos , considered as a divine primordial condition, from which everything else appeared. In the creation "chaos" is a gaping-void, but later the word is used to describe the space between the Earth and the sky, after their separation. "Chaos" may mean infinite space, or a formless matter which can be differentiated. The notion of temporal infinity was familiar to the Greek mind from remote antiquity in

9483-563: The other gods. The egg is often depicted with the serpent-like creature, Ananke , wound about it. Phanes is the golden winged primordial being who was hatched from the shining cosmic egg that was the source of the universe. Called Protogonos (First-Born) and Eros (Love) an ancient Orphic hymn addresses him thus: Ineffable, hidden, brilliant scion, whose motion is whirring, you scattered the dark mist that lay before your eyes and, flapping your wings, you whirled about, and through this world you brought pure light. There are two Orphic stories of

9592-521: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000  BC , when

9701-491: The primordial succession: But there are other differences, notably in the treatment of Dionysos: In later centuries, these versions underwent a development where Apollo's act of burying became responsible for the reincarnation of Dionysus, thus giving Apollo the title Dionysiodotes (bestower of Dionysus). Apollo plays an important part in the dismemberment myth because he represents the reverting of Encosmic Soul back towards unification. Surviving written fragments show

9810-658: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been

9919-400: The rebirth of Dionysus : in one it is the heart of Dionysus that is implanted into the thigh of Zeus ; in the other Zeus has impregnated the mortal woman Semele , resulting in Dionysus's literal rebirth. Many of these details differ from accounts in the classical authors. Damascius says that Apollo "gathers him (Dionysus) together and brings him back up". The main difference seems to be in

10028-523: The reign of Philip II of Macedon , making it Europe's oldest surviving manuscript. The Orphic theogonies are works which present accounts of the origin of the gods, much like the Theogony of Hesiod . These theogonies are symbolically similar to Near Eastern models. The main story has it that Zagreus , Dionysus' previous incarnation, is the son of Zeus and Persephone . Zeus names the child as his successor, which angers his wife Hera . She instigates

10137-539: The religious conception of immortality. The conception of the "divine" as an origin influenced the first Greek philosophers. In the Orphic cosmogony, the unaging Chronos produced Aether and Chaos and made in divine Aether a silvery egg, from which everything else appeared. The earliest Pre-Socratic philosophers, the Ionian material monists, sought to explain all of nature ( physis ) in terms of one unifying arche. Among

10246-405: The second place, the knowledge of other things must be so dependent on them as that though the principles themselves may indeed be known apart from what depends on them, the latter cannot nevertheless be known apart from the former. It will accordingly be necessary thereafter to endeavor so to deduce from those principles the knowledge of the things that depend on them, as that there may be nothing in

10355-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become

10464-568: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and

10573-480: The surrounding Oceanus (ocean) is the source of all springs and rivers. Anaximander argued that water could not be the arche, because it could not give rise to its opposite, fire. Anaximander claimed that none of the elements ( earth , fire , air , water ) could be arche for the same reason. Instead, he proposed the existence of the apeiron , an indefinite substance from which all things are born and to which all things will return. Apeiron (endless or boundless)

10682-467: The theology of Orpheus, carried forward through Pythagoreanism, as the core of the original Greek religious tradition. Proclus , an influential neoplatonic philosopher, one of the last major classical philosophers of late antiquity, says (trans. Thomas Taylor, 1816) A number of Greek religious poems in hexameters were attributed to Orpheus, as they were to similar miracle-working figures, like Bakis , Musaeus , Abaris , Aristeas , Epimenides , and

10791-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,

10900-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in

11009-508: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in

11118-441: The whole series of deductions which is not perfectly manifest. In physics , a calculation is said to be from first principles , or ab initio , if it starts directly at the level of established laws of physics and does not make assumptions such as empirical model and fitting parameters. For example, calculation of electronic structure using Schrödinger's equation within a set of approximations that do not include fitting

11227-448: Was adapted from the earliest cosmogonies of Hesiod and Orphism , through the physical theories of Pre-Socratic philosophy and Plato before being formalized as a part of metaphysics by Aristotle . Arche sometimes also transcribed as arkhé ) is an Ancient Greek word with primary senses "beginning", "origin" or "source of action": from the beginning, οr the original argument, "command". The first principle or element corresponds to

11336-441: Was familiar to the Greek mind from remote antiquity in the religious conception of immortality and Anaximander's description was in terms appropriate to this conception. This arche is called "eternal and ageless". (Hippolitus I,6, I;DK B2) Anaximenes, Anaximander's pupil, advanced yet another theory. He returns to the elemental theory, but this time posits air, rather than water, as the arche and ascribes to it divine attributes. He

11445-462: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until

11554-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"

11663-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to

11772-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In

11881-548: Was the first recorded philosopher who provided a theory of change and supported it with observation. Using two contrary processes of rarefaction and condensation (thinning or thickening), he explains how air is part of a series of changes. Rarefied air becomes fire, condensed it becomes first wind, then cloud, water, earth, and stone in order. The arche is technically what underlies all of reality/appearances. Terence Irwin writes: When Aristotle explains in general terms what he tries to do in his philosophical works, he says he

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