1.
Euclid
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Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost
2.
Calipers
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A caliper is a device used to measure the distance between two opposite sides of an object. A caliper can be as simple as a compass with inward or outward-facing points. The tips of the caliper are adjusted to fit across the points to be measured, the caliper is then removed and it is used in many fields such as mechanical engineering, metalworking, forestry, woodworking, science and medicine. A plurale tantum sense of the word calipers coexists in natural usage with the regular noun sense of caliper, also existing colloquially but not in formal usage is referring to a vernier caliper as a vernier or a pair of verniers. In imprecise colloquial usage, some extend this even to dial calipers. In machine-shop usage, the caliper is often used in contradistinction to micrometer. In this usage, caliper implies only the factor of the vernier or dial caliper. The earliest caliper has been found in the Greek Giglio wreck near the Italian coast, the ship find dates to the 6th century BC. The wooden piece already featured a fixed and a movable jaw, although rare finds, caliper remained in use by the Greeks and Romans. A bronze caliper, dating from 9 AD, was used for minute measurements during the Chinese Xin dynasty, the caliper had an inscription stating that it was made on a gui-you day at new moon of the first month of the first year of the Shijian guo period. The calipers included a slot and pin and graduated in inches, the modern vernier caliper, reading to thousandths of an inch, was invented by American Joseph R. Brown in 1851. It was the first practical tool for exact measurements that could be sold at a price within the reach of ordinary machinists, the inside calipers are used to measure the internal size of an object. The upper caliper in the image requires manual adjustment prior to fitting, fine setting of this caliper type is performed by tapping the caliper legs lightly on a handy surface until they will almost pass over the object. A light push against the resistance of the pivot screw then spreads the legs to the correct dimension and provides the required. The lower caliper in the image has a screw that permits it to be carefully adjusted without removal of the tool from the workpiece. Outside calipers are used to measure the size of an object. The same observations and technique apply to this type of caliper, with some understanding of their limitations and usage, these instruments can provide a high degree of accuracy and repeatability. They are especially useful when measuring over very large distances, consider if the calipers are used to measure a large diameter pipe, a vernier caliper does not have the depth capacity to straddle this large diameter while at the same time reach the outermost points of the pipes diameter
3.
Raphael
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Raffaello Sanzio da Urbino, known as Raphael, was an Italian painter and architect of the High Renaissance. His work is admired for its clarity of form, ease of composition, together with Michelangelo and Leonardo da Vinci, he forms the traditional trinity of great masters of that period. Raphael was enormously productive, running a large workshop and, despite his death at 37. Many of his works are found in the Vatican Palace, where the frescoed Raphael Rooms were the central, the best known work is The School of Athens in the Vatican Stanza della Segnatura. After his early years in Rome much of his work was executed by his workshop from his drawings and he was extremely influential in his lifetime, though outside Rome his work was mostly known from his collaborative printmaking. Raphael was born in the small but artistically significant central Italian city of Urbino in the Marche region and his poem to Federico shows him as keen to show awareness of the most advanced North Italian painters, and Early Netherlandish artists as well. In the very court of Urbino he was probably more integrated into the central circle of the ruling family than most court painters. Under them, the court continued as a centre for literary culture, growing up in the circle of this small court gave Raphael the excellent manners and social skills stressed by Vasari. Castiglione moved to Urbino in 1504, when Raphael was no longer based there but frequently visited, Raphael mixed easily in the highest circles throughout his life, one of the factors that tended to give a misleading impression of effortlessness to his career. He did not receive a humanistic education however, it is unclear how easily he read Latin. His mother Màgia died in 1491 when Raphael was eight, followed on August 1,1494 by his father, Raphael was thus orphaned at eleven, his formal guardian became his only paternal uncle Bartolomeo, a priest, who subsequently engaged in litigation with his stepmother. He probably continued to live with his stepmother when not staying as an apprentice with a master and he had already shown talent, according to Vasari, who says that Raphael had been a great help to his father. A self-portrait drawing from his teenage years shows his precocity and his fathers workshop continued and, probably together with his stepmother, Raphael evidently played a part in managing it from a very early age. In Urbino, he came into contact with the works of Paolo Uccello, previously the court painter, and Luca Signorelli, according to Vasari, his father placed him in the workshop of the Umbrian master Pietro Perugino as an apprentice despite the tears of his mother. The evidence of an apprenticeship comes only from Vasari and another source, an alternative theory is that he received at least some training from Timoteo Viti, who acted as court painter in Urbino from 1495. An excess of resin in the varnish often causes cracking of areas of paint in the works of both masters, the Perugino workshop was active in both Perugia and Florence, perhaps maintaining two permanent branches. Raphael is described as a master, that is to say fully trained and his first documented work was the Baronci altarpiece for the church of Saint Nicholas of Tolentino in Città di Castello, a town halfway between Perugia and Urbino. Evangelista da Pian di Meleto, who had worked for his father, was named in the commission
4.
The School of Athens
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The School of Athens is one of the most famous frescoes by the Italian Renaissance artist Raphael. It was painted between 1509 and 1511 as a part of Raphaels commission to decorate the rooms now known as the Stanze di Raffaello, the picture has long been seen as Raphaels masterpiece and the perfect embodiment of the classical spirit of the Renaissance. The School of Athens is one of a group of four main frescoes on the walls of the Stanza that depict distinct branches of knowledge, accordingly, the figures on the walls below exemplify Philosophy, Poetry, Theology, and Law. The traditional title is not Raphaels, indeed, Plato and Aristotle appear to be the central figures in the scene. However, all the philosophers depicted sought knowledge of first causes, many lived before Plato and Aristotle, and hardly a third were Athenians. The architecture contains Roman elements, but the general semi-circular setting having Plato, compounding the problem, Raphael had to invent a system of iconography to allude to various figures for whom there were no traditional visual types. For example, while the Socrates figure is immediately recognizable from Classical busts, aside from the identities of the figures depicted, many aspects of the fresco have been variously interpreted, but few such interpretations are unanimously accepted among scholars. The popular idea that the gestures of Plato and Aristotle are kinds of pointing is very likely. Aristotle, with his four-elements theory, held that all change on Earth was owing to motions of the heavens, in the painting Aristotle carries his Ethics, which he denied could be reduced to a mathematical science. Finally, according to Vasari, the scene includes Raphael himself, however, as Heinrich Wölfflin observed, it is quite wrong to attempt interpretations of the School of Athens as an esoteric treatise. The all-important thing was the motive which expressed a physical or spiritual state. An interpretation of the fresco relating to hidden symmetries of the figures, the identities of some of the philosophers in the picture, such as Plato or Aristotle, are certain. Beyond that, identifications of Raphaels figures have always been hypothetical, to complicate matters, beginning from Vasaris efforts, some have received multiple identifications, not only as ancients but also as figures contemporary with Raphael. Luitpold Dussler counts among those who can be identified with certainty, Plato, Aristotle, Socrates, Pythagoras, Euclid, Ptolemy, Zoroaster, Raphael, Sodoma. Other identifications he holds to be more or less speculative, both figures hold modern, bound copies of their books in their left hands, while gesturing with their right. Plato holds Timaeus, Aristotle his Nicomachean Ethics, Plato is depicted as old, grey, wise-looking, and bare-foot. By contrast Aristotle, slightly ahead of him, is in manhood, handsome, well-shod and dressed with gold. Many interpret the painting to show a divergence of the two philosophical schools, Plato argues a sense of timelessness whilst Aristotle looks into the physicality of life and the present realm
5.
Ancient Greek
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Ancient Greek includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD. It is often divided into the Archaic period, Classical period. It is antedated in the second millennium BC by Mycenaean Greek, the language of the Hellenistic phase is known as Koine. Koine is regarded as a historical stage of its own, although in its earliest form it closely resembled Attic Greek. Prior to the Koine period, Greek of the classic and earlier periods included several regional dialects, Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, and philosophers. It has contributed many words to English vocabulary and has been a subject of study in educational institutions of the Western world since the Renaissance. This article primarily contains information about the Epic and Classical phases of the language, Ancient Greek was a pluricentric language, divided into many dialects. The main dialect groups are Attic and Ionic, Aeolic, Arcadocypriot, some dialects are found in standardized literary forms used in literature, while others are attested only in inscriptions. There are also several historical forms, homeric Greek is a literary form of Archaic Greek used in the epic poems, the Iliad and Odyssey, and in later poems by other authors. Homeric Greek had significant differences in grammar and pronunciation from Classical Attic, the origins, early form and development of the Hellenic language family are not well understood because of a lack of contemporaneous evidence. Several theories exist about what Hellenic dialect groups may have existed between the divergence of early Greek-like speech from the common Proto-Indo-European language and the Classical period and they have the same general outline, but differ in some of the detail. The invasion would not be Dorian unless the invaders had some relationship to the historical Dorians. The invasion is known to have displaced population to the later Attic-Ionic regions, the Greeks of this period believed there were three major divisions of all Greek people—Dorians, Aeolians, and Ionians, each with their own defining and distinctive dialects. Often non-west is called East Greek, Arcadocypriot apparently descended more closely from the Mycenaean Greek of the Bronze Age. Boeotian had come under a strong Northwest Greek influence, and can in some respects be considered a transitional dialect, thessalian likewise had come under Northwest Greek influence, though to a lesser degree. Most of the dialect sub-groups listed above had further subdivisions, generally equivalent to a city-state and its surrounding territory, Doric notably had several intermediate divisions as well, into Island Doric, Southern Peloponnesus Doric, and Northern Peloponnesus Doric. The Lesbian dialect was Aeolic Greek and this dialect slowly replaced most of the older dialects, although Doric dialect has survived in the Tsakonian language, which is spoken in the region of modern Sparta. Doric has also passed down its aorist terminations into most verbs of Demotic Greek, by about the 6th century AD, the Koine had slowly metamorphosized into Medieval Greek
6.
Space
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Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. Many of these classical philosophical questions were discussed in the Renaissance and then reformulated in the 17th century, in Isaac Newtons view, space was absolute—in the sense that it existed permanently and independently of whether there was any matter in the space. Other natural philosophers, notably Gottfried Leibniz, thought instead that space was in fact a collection of relations between objects, given by their distance and direction from one another. In the 18th century, the philosopher and theologian George Berkeley attempted to refute the visibility of spatial depth in his Essay Towards a New Theory of Vision. Kant referred to the experience of space in his Critique of Pure Reason as being a pure a priori form of intuition. In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean, in space is conceived as curved. According to Albert Einsteins theory of relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a model for the shape of space. In the seventeenth century, the philosophy of space and time emerged as an issue in epistemology. At its heart, Gottfried Leibniz, the German philosopher-mathematician, and Isaac Newton, unoccupied regions are those that could have objects in them, and thus spatial relations with other places. For Leibniz, then, space was an abstraction from the relations between individual entities or their possible locations and therefore could not be continuous but must be discrete. Space could be thought of in a way to the relations between family members. Although people in the family are related to one another, the relations do not exist independently of the people, but since there would be no observational way of telling these universes apart then, according to the identity of indiscernibles, there would be no real difference between them. According to the principle of sufficient reason, any theory of space that implied that there could be two possible universes must therefore be wrong. Newton took space to be more than relations between objects and based his position on observation and experimentation
7.
Calculus
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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, in other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today
8.
Patterns
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A pattern, apart from the terms use to mean Template, is a discernible regularity in the world or in a manmade design. As such, the elements of a repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes, any of the senses may directly observe patterns. Conversely, abstract patterns in science, mathematics, or language may be only by analysis. Direct observation in practice means seeing visual patterns, which are widespread in nature, visual patterns in nature are often chaotic, never exactly repeating, and often involve fractals. Natural patterns include spirals, meanders, waves, foams, tilings, cracks, Patterns have an underlying mathematical structure, indeed, mathematics can be seen as the search for regularities, and the output of any function is a mathematical pattern. Similarly in the sciences, theories explain and predict regularities in the world, in art and architecture, decorations or visual motifs may be combined and repeated to form patterns designed to have a chosen effect on the viewer. In computer science, a design pattern is a known solution to a class of problems in programming. In fashion, the pattern is a used to create any number of similar garments. Nature provides examples of many kinds of pattern, including symmetries, trees and other structures with a dimension, spirals, meanders, waves, foams, tilings, cracks. Symmetry is widespread in living things, animals that move usually have bilateral or mirror symmetry as this favours movement. Plants often have radial or rotational symmetry, as do many flowers, as well as animals which are largely static as adults, fivefold symmetry is found in the echinoderms, including starfish, sea urchins, and sea lilies. Among non-living things, snowflakes have striking sixfold symmetry, each flake is unique, crystals have a highly specific set of possible crystal symmetries, they can be cubic or octahedral, but cannot have fivefold symmetry. Many natural patterns are shaped by this apparent randomness, including vortex streets, waves are disturbances that carry energy as they move. Mechanical waves propagate through a medium – air or water, making it oscillate as they pass by, wind waves are surface waves that create the chaotic patterns of the sea. As they pass over sand, such waves create patterns of ripples, similarly, as the passes over sand. Foams obey Plateaus laws, which films to be smooth and continuous. Foam and bubble patterns occur widely in nature, for example in radiolarians, sponge spicules, cracks form in materials to relieve stress, with 120 degree joints in elastic materials, but at 90 degrees in inelastic materials
9.
Conjecture
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In mathematics, a conjecture is a conclusion or proposition based on incomplete information, for which no proof has been found. Conjectures such as the Riemann hypothesis or Fermats Last Theorem have shaped much of history as new areas of mathematics are developed in order to prove them. In number theory, Fermats Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any value of n greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, the unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics, two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. For example, in the map of the United States of America, Utah and Arizona are adjacent, but Utah and New Mexico, möbius mentioned the problem in his lectures as early as 1840. The conjecture was first proposed on October 23,1852 when Francis Guthrie, while trying to color the map of counties of England, a number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852. The four color theorem was proven in 1976 by Kenneth Appel and it was the first major theorem to be proved using a computer. Appel and Hakens approach started by showing that there is a set of 1,936 maps. Appel and Haken used a computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps, showing this required hundreds of pages of hand analysis. Appel and Haken concluded that no smallest counterexamples exists because any must contain, yet do not contain and this contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was not accepted by all mathematicians because the proof was infeasible for a human to check by hand. Since then the proof has gained acceptance, although doubts remain. The Hauptvermutung of geometric topology is the conjecture that any two triangulations of a triangulable space have a refinement, a single triangulation that is a subdivision of both of them. It was originally formulated in 1908, by Steinitz and Tietze and this conjecture is now known to be false. The non-manifold version was disproved by John Milnor in 1961 using Reidemeister torsion, the manifold version is true in dimensions m ≤3
10.
Mathematical proof
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In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, in principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies, Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is true, rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture, Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to examination of current and historical mathematical practice, quasi-empiricism in mathematics. The philosophy of mathematics is concerned with the role of language and logic in proofs, the word proof comes from the Latin probare meaning to test. Related modern words are the English probe, probation, and probability, the Spanish probar, Italian provare, the early use of probity was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. It is likely that the idea of demonstrating a conclusion first arose in connection with geometry, the development of mathematical proof is primarily the product of ancient Greek mathematics, and one of the greatest achievements thereof. Thales proved some theorems in geometry, eudoxus and Theaetetus formulated theorems but did not prove them. Aristotle said definitions should describe the concept being defined in terms of other concepts already known and his book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. Further advances took place in medieval Islamic mathematics, while earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, division and he used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures, there is no longer an assumption that axioms are true in any sense, this allows for parallel mathematical theories built on alternate sets of axioms
11.
Logic
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Logic, originally meaning the word or what is spoken, is generally held to consist of the systematic study of the form of arguments. A valid argument is one where there is a relation of logical support between the assumptions of the argument and its conclusion. Historically, logic has been studied in philosophy and mathematics, and recently logic has been studied in science, linguistics, psychology. The concept of form is central to logic. The validity of an argument is determined by its logical form, traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic. Informal logic is the study of natural language arguments, the study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as an application of a wholly abstract rule, that is. The works of Aristotle contain the earliest known study of logic. Modern formal logic follows and expands on Aristotle, in many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language, Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is divided into two main branches, propositional logic and predicate logic. Mathematical logic is an extension of logic into other areas, in particular to the study of model theory, proof theory, set theory. Logic is generally considered formal when it analyzes and represents the form of any valid argument type, the form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. Simply put, formalising simply means translating English sentences into the language of logic and this is called showing the logical form of the argument. It is necessary because indicative sentences of ordinary language show a variety of form. Second, certain parts of the sentence must be replaced with schematic letters, thus, for example, the expression all Ps are Qs shows the logical form common to the sentences all men are mortals, all cats are carnivores, all Greeks are philosophers, and so on. The schema can further be condensed into the formula A, where the letter A indicates the judgement all - are -, the importance of form was recognised from ancient times
12.
Counting
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Counting is the action of finding the number of elements of a finite set of objects. The related term refers to uniquely identifying the elements of a finite set or infinite set by assigning a number to each element. Counting sometimes involves numbers other than one, for example, when counting money, counting out change, counting by twos, there is archeological evidence suggesting that humans have been counting for at least 50,000 years. Counting was primarily used by ancient cultures to keep track of social and economic data such as number of members, prey animals, property. The development of counting led to the development of mathematical notation, numeral systems, counting can occur in a variety of forms. Counting can be verbal, that is, speaking every number out loud to keep track of progress and this is often used to count objects that are present already, instead of counting a variety of things over time. Counting can also be in the form of tally marks, making a mark for each number and this is useful when counting objects over time, such as the number of times something occurs during the course of a day. Tallying is base 1 counting, normal counting is done in base 10, counting can also be in the form of finger counting, especially when counting small numbers. This is often used by children to facilitate counting and simple mathematical operations, finger-counting uses unary notation, and is thus limited to counting 10. Older finger counting used the four fingers and the three bones in each finger to count to the number twelve, other hand-gesture systems are also in use, for example the Chinese system by which one can count 10 using only gestures of one hand. By using finger binary, it is possible to keep a count up to 1023 =210 −1. Various devices can also be used to facilitate counting, such as tally counters. Inclusive counting is usually encountered when dealing with time in the Romance languages, in exclusive counting languages such as English, when counting 8 days from Sunday, Monday will be day 1, Tuesday day 2, and the following Monday will be the eighth day. When counting inclusively, the Sunday will be day 1 and therefore the following Sunday will be the eighth day, for example, the French phrase for fortnight is quinzaine, and similar words are present in Greek, Spanish and Portuguese. In contrast, the English word fortnight itself derives from a fourteen-night, as the archaic sennight does from a seven-night, learning to count is an important educational/developmental milestone in most cultures of the world. Learning to count is a very first step into mathematics. However, some cultures in Amazonia and the Australian Outback do not count, many children at just 2 years of age have some skill in reciting the count list. They can also answer questions of ordinality for small numbers, e. g and they can even be skilled at pointing to each object in a set and reciting the words one after another
13.
Measurement
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Measurement is the assignment of a number to a characteristic of an object or event, which can be compared with other objects or events. The scope and application of a measurement is dependent on the context, however, in other fields such as statistics as well as the social and behavioral sciences, measurements can have multiple levels, which would include nominal, ordinal, interval, and ratio scales. Measurement is a cornerstone of trade, science, technology, historically, many measurement systems existed for the varied fields of human existence to facilitate comparisons in these fields. Often these were achieved by local agreements between trading partners or collaborators, since the 18th century, developments progressed towards unifying, widely accepted standards that resulted in the modern International System of Units. This system reduces all physical measurements to a combination of seven base units. The science of measurement is pursued in the field of metrology, the measurement of a property may be categorized by the following criteria, type, magnitude, unit, and uncertainty. They enable unambiguous comparisons between measurements, the type or level of measurement is a taxonomy for the methodological character of a comparison. For example, two states of a property may be compared by ratio, difference, or ordinal preference, the type is commonly not explicitly expressed, but implicit in the definition of a measurement procedure. The magnitude is the value of the characterization, usually obtained with a suitably chosen measuring instrument. A unit assigns a mathematical weighting factor to the magnitude that is derived as a ratio to the property of a used as standard or a natural physical quantity. An uncertainty represents the random and systemic errors of the measurement procedure, errors are evaluated by methodically repeating measurements and considering the accuracy and precision of the measuring instrument. Measurements most commonly use the International System of Units as a comparison framework, the system defines seven fundamental units, kilogram, metre, candela, second, ampere, kelvin, and mole. Instead, the measurement unit can only ever change through increased accuracy in determining the value of the constant it is tied to and this directly influenced the Michelson–Morley experiment, Michelson and Morley cite Peirce, and improve on his method. With the exception of a few fundamental quantum constants, units of measurement are derived from historical agreements, nothing inherent in nature dictates that an inch has to be a certain length, nor that a mile is a better measure of distance than a kilometre. Over the course of history, however, first for convenience and then for necessity. Laws regulating measurement were originally developed to prevent fraud in commerce.9144 metres, in the United States, the National Institute of Standards and Technology, a division of the United States Department of Commerce, regulates commercial measurements. Before SI units were adopted around the world, the British systems of English units and later imperial units were used in Britain, the Commonwealth. The system came to be known as U. S. customary units in the United States and is still in use there and in a few Caribbean countries. S
14.
Shape
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A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material composition. Psychologists have theorized that humans mentally break down images into simple geometric shapes called geons, examples of geons include cones and spheres. Some simple shapes can be put into broad categories, for instance, polygons are classified according to their number of edges as triangles, quadrilaterals, pentagons, etc. Each of these is divided into categories, triangles can be equilateral, isosceles, obtuse, acute, scalene, etc. while quadrilaterals can be rectangles, rhombi, trapezoids, squares. Other common shapes are points, lines, planes, and conic sections such as ellipses, circles, among the most common 3-dimensional shapes are polyhedra, which are shapes with flat faces, ellipsoids, which are egg-shaped or sphere-shaped objects, cylinders, and cones. If an object falls into one of these categories exactly or even approximately, thus, we say that the shape of a manhole cover is a disk, because it is approximately the same geometric object as an actual geometric disk. Similarity, Two objects are similar if one can be transformed into the other by a scaling, together with a sequence of rotations, translations. Isotopy, Two objects are isotopic if one can be transformed into the other by a sequence of deformations that do not tear the object or put holes in it. Sometimes, two similar or congruent objects may be regarded as having a different shape if a reflection is required to transform one into the other. For instance, the b and d are a reflection of each other, and hence they are congruent and similar. Sometimes, only the outline or external boundary of the object is considered to determine its shape, for instance, an hollow sphere may be considered to have the same shape as a solid sphere. Procrustes analysis is used in many sciences to determine whether or not two objects have the shape, or to measure the difference between two shapes. In advanced mathematics, quasi-isometry can be used as a criterion to state that two shapes are approximately the same. Simple shapes can often be classified into basic objects such as a point, a line, a curve, a plane. However, most shapes occurring in the world are complex. Some, such as plant structures and coastlines, may be so complicated as to defy traditional mathematical description – in which case they may be analyzed by differential geometry, or as fractals. In geometry, two subsets of a Euclidean space have the shape if one can be transformed to the other by a combination of translations, rotations. In other words, the shape of a set of points is all the information that is invariant to translations, rotations
15.
Motion (physics)
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In physics, motion is a change in position of an object over time. Motion is described in terms of displacement, distance, velocity, acceleration, time, motion of a body is observed by attaching a frame of reference to an observer and measuring the change in position of the body relative to that frame. If the position of a body is not changing with respect to a frame of reference. An objects motion cannot change unless it is acted upon by a force, momentum is a quantity which is used for measuring motion of an object. As there is no frame of reference, absolute motion cannot be determined. Thus, everything in the universe can be considered to be moving, more generally, motion is a concept that applies to objects, bodies, and matter particles, to radiation, radiation fields and radiation particles, and to space, its curvature and space-time. One can also speak of motion of shapes and boundaries, so, the term motion in general signifies a continuous change in the configuration of a physical system. For example, one can talk about motion of a wave or about motion of a quantum particle, in physics, motion is described through two sets of apparently contradictory laws of mechanics. Motions of all large scale and familiar objects in the universe are described by classical mechanics, whereas the motion of very small atomic and sub-atomic objects is described by quantum mechanics. It produces very accurate results within these domains, and is one of the oldest and largest in science, engineering, classical mechanics is fundamentally based on Newtons laws of motion. These laws describe the relationship between the acting on a body and the motion of that body. They were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica and his three laws are, A body either is at rest or moves with constant velocity, until and unless an outer force is applied to it. An object will travel in one direction only until an outer force changes its direction, whenever one body exerts a force F onto a second body, the second body exerts the force −F on the first body. F and −F are equal in magnitude and opposite in sense, so, the body which exerts F will go backwards. Newtons three laws of motion, along with his Newtons law of motion, which were the first to provide a mathematical model for understanding orbiting bodies in outer space. This explanation unified the motion of bodies and motion of objects on earth. Classical mechanics was later enhanced by Albert Einsteins special relativity. Motion of objects with a velocity, approaching the speed of light
16.
History of Mathematics
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Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322, the Rhind Mathematical Papyrus, All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. Greek mathematics greatly refined the methods and expanded the subject matter of mathematics, Chinese mathematics made early contributions, including a place value system. Islamic mathematics, in turn, developed and expanded the known to these civilizations. Many Greek and Arabic texts on mathematics were then translated into Latin, from ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, the origins of mathematical thought lie in the concepts of number, magnitude, and form. Modern studies of cognition have shown that these concepts are not unique to humans. Such concepts would have part of everyday life in hunter-gatherer societies. The idea of the number concept evolving gradually over time is supported by the existence of languages which preserve the distinction between one, two, and many, but not of numbers larger than two. Prehistoric artifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time. The Ishango bone, found near the headwaters of the Nile river, may be more than 20,000 years old, common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of prime numbers or a six-month lunar calendar. He also writes that no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10, predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian, Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity. The majority of Babylonian mathematical work comes from two widely separated periods, The first few hundred years of the second millennium BC, and it is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic mathematics, in contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed whilst the clay was moist, Some of these appear to be graded homework. The earliest evidence of written mathematics dates back to the ancient Sumerians and they developed a complex system of metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises, the earliest traces of the Babylonian numerals also date back to this period
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Greek mathematics
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Greek mathematics, as the term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture, Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word mathematics itself derives from the ancient Greek μάθημα, meaning subject of instruction, the study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations. The origin of Greek mathematics is not well documented, the earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilization, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. The two earliest mathematical theorems, Thales theorem and Intercept theorem are attributed to Thales. The former, which states that an angle inscribed in a semicircle is a right angle and it is for this reason that Thales is often hailed as the father of the deductive organization of mathematics and as the first true mathematician. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed, another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar, Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order. And since in antiquity it was customary to give all credit to the master, aristotle for one refused to attribute anything specifically to Pythagoras as an individual and only discussed the work of the Pythagoreans as a group. One of the most important characteristics of the Pythagorean order was that it maintained that the pursuit of philosophical and mathematical studies was a basis for the conduct of life. Indeed, the philosophy and mathematics are said to have been coined by Pythagoras. From this love of knowledge came many achievements and it has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclids Elements. The reason it is not clear exactly what either Thales or Pythagoras actually did is that almost no documentation has survived. The only evidence comes from traditions recorded in such as Proclus’ commentary on Euclid written centuries later. Some of these works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments
18.
Euclid's Elements
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Euclids Elements is a mathematical and geometric treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt circa 300 BC. It is a collection of definitions, postulates, propositions, the books cover Euclidean geometry and the ancient Greek version of elementary number theory. Elements is the second-oldest extant Greek mathematical treatise after Autolycus On the Moving Sphere and it has proven instrumental in the development of logic and modern science. According to Proclus, the element was used to describe a theorem that is all-pervading. The word element in the Greek language is the same as letter and this suggests that theorems in the Elements should be seen as standing in the same relation to geometry as letters to language. Euclids Elements has been referred to as the most successful and influential textbook ever written, for centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclids Elements was required of all students. Not until the 20th century, by which time its content was taught through other school textbooks. Scholars believe that the Elements is largely a collection of theorems proven by other mathematicians, the Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures. This manuscript, the Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions, papyrus Oxyrhynchus 29 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. Although known to, for instance, Cicero, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. The Arabs received the Elements from the Byzantines around 760, this version was translated into Arabic under Harun al Rashid circa 800, the Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, the first printed edition appeared in 1482, and since then it has been translated into many languages and published in about a thousand different editions. Theons Greek edition was recovered in 1533, in 1570, John Dee provided a widely respected Mathematical Preface, along with copious notes and supplementary material, to the first English edition by Henry Billingsley. Copies of the Greek text still exist, some of which can be found in the Vatican Library, the manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text, ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of the text, also of importance are the scholia, or annotations to the text. These additions, which distinguished themselves from the main text. The Elements is still considered a masterpiece in the application of logic to mathematics, in historical context, it has proven enormously influential in many areas of science
19.
Giuseppe Peano
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Giuseppe Peano was an Italian mathematician. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, the standard axiomatization of the natural numbers is named the Peano axioms in his honor. As part of effort, he made key contributions to the modern rigorous. He spent most of his career teaching mathematics at the University of Turin, Peano was born and raised on a farm at Spinetta, a hamlet now belonging to Cuneo, Piedmont, Italy. Due to Genocchis poor health, Peano took over the teaching of calculus course within two years and his first major work, a textbook on calculus, was published in 1884 and was credited to Genocchi. A few years later, Peano published his first book dealing with mathematical logic, here the modern symbols for the union and intersection of sets appeared for the first time. In 1887, Peano married Carola Crosio, the daughter of the Turin-based painter Luigi Crosio, in 1886, he began teaching concurrently at the Royal Military Academy, and was promoted to Professor First Class in 1889. The next year, the University of Turin also granted him his full professorship, Peanos famous space-filling curve appeared in 1890 as a counterexample. He used it to show that a continuous curve cannot always be enclosed in a small region. This was an example of what came to be known as a fractal. In 1890 Peano founded the journal Rivista di Matematica, which published its first issue in January 1891, in 1891 Peano started the Formulario Project. It was to be an Encyclopedia of Mathematics, containing all known formulae, in 1897, the first International Congress of Mathematicians was held in Zürich. Peano was a key participant, presenting a paper on mathematical logic and he also started to become increasingly occupied with Formulario to the detriment of his other work. In 1898 he presented a note to the Academy about binary numeration and he also became so frustrated with publishing delays that he purchased a printing press. Paris was the venue for the Second International Congress of Mathematicians in 1900, the conference was preceded by the First International Conference of Philosophy where Peano was a member of the patronage committee. He presented a paper which posed the question of correctly formed definitions in mathematics and this became one of Peanos main philosophical interests for the rest of his life. At the conference Peano met Bertrand Russell and gave him a copy of Formulario, Russell was so struck by Peanos innovative logical symbols that he left the conference and returned home to study Peanos text. Peanos students Mario Pieri and Alessandro Padoa had papers presented at the congress also
20.
David Hilbert
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David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th, Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis, Hilbert adopted and warmly defended Georg Cantors set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in mathematical physics. Hilbert is known as one of the founders of theory and mathematical logic. In late 1872, Hilbert entered the Friedrichskolleg Gymnasium, but, after a period, he transferred to. Upon graduation, in autumn 1880, Hilbert enrolled at the University of Königsberg, in early 1882, Hermann Minkowski, returned to Königsberg and entered the university. Hilbert knew his luck when he saw it, in spite of his fathers disapproval, he soon became friends with the shy, gifted Minkowski. In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius, Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen. Hilbert remained at the University of Königsberg as a Privatdozent from 1886 to 1895, in 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world and he remained there for the rest of his life. Among Hilberts students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, john von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a circle of some of the most important mathematicians of the 20th century, such as Emmy Noether. Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, good, he did not have enough imagination to become a mathematician. Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen in 1933 and those forced out included Hermann Weyl, Emmy Noether and Edmund Landau. One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic and this was a sequel to the Hilbert-Ackermann book Principles of Mathematical Logic from 1928. Hermann Weyls successor was Helmut Hasse, about a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust
21.
Truth
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Truth is most often used to mean being in accord with fact or reality, or fidelity to an original or standard. Truth may also often be used in modern contexts to refer to an idea of truth to self, the commonly understood opposite of truth is falsehood, which, correspondingly, can also take on a logical, factual, or ethical meaning. The concept of truth is discussed and debated in several contexts, including philosophy, art, Some philosophers view the concept of truth as basic, and unable to be explained in any terms that are more easily understood than the concept of truth itself. Commonly, truth is viewed as the correspondence of language or thought to an independent reality, other philosophers take this common meaning to be secondary and derivative. On this view, the conception of truth as correctness is a derivation from the concepts original essence. Various theories and views of truth continue to be debated among scholars, philosophers, language and words are a means by which humans convey information to one another and the method used to determine what is a truth is termed a criterion of truth. The English word truth is derived from Old English tríewþ, tréowþ, trýwþ, Middle English trewþe, cognate to Old High German triuwida, like troth, it is a -th nominalisation of the adjective true. Old Norse trú, faith, word of honour, religious faith, thus, truth involves both the quality of faithfulness, fidelity, loyalty, sincerity, veracity, and that of agreement with fact or reality, in Anglo-Saxon expressed by sōþ. All Germanic languages besides English have introduced a distinction between truth fidelity and truth factuality. To express factuality, North Germanic opted for nouns derived from sanna to assert, affirm, while continental West Germanic opted for continuations of wâra faith, trust, pact. Romance languages use terms following the Latin veritas, while the Greek aletheia, Russian pravda, each presents perspectives that are widely shared by published scholars. However, the theories are not universally accepted. More recently developed deflationary or minimalist theories of truth have emerged as competitors to the substantive theories. Minimalist reasoning centres around the notion that the application of a term like true to a statement does not assert anything significant about it, for instance, anything about its nature. Minimalist reasoning realises truth as a label utilised in general discourse to express agreement, to stress claims, correspondence theories emphasise that true beliefs and true statements correspond to the actual state of affairs. This type of theory stresses a relationship between thoughts or statements on one hand, and things or objects on the other and it is a traditional model tracing its origins to ancient Greek philosophers such as Socrates, Plato, and Aristotle. This class of theories holds that the truth or the falsity of a representation is determined in principle entirely by how it relates to things, Aquinas also restated the theory as, A judgment is said to be true when it conforms to the external reality. Many modern theorists have stated that this ideal cannot be achieved without analysing additional factors, for example, language plays a role in that all languages have words to represent concepts that are virtually undefined in other languages
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Definition
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A definition is a statement of the meaning of a term. Definitions can be classified into two categories, intensional definitions and extensional definitions. Another important category of definitions is the class of ostensive definitions, a term may have many different senses and multiple meanings, and thus require multiple definitions. In mathematics, a definition is used to give a meaning to a new term. Definitions and axioms are the basis on all of mathematics is constructed. In modern usage, a definition is something, typically expressed in words, the word or group of words that is to be defined is called the definiendum, and the word, group of words, or action that defines it is called the definiens. In the definition An elephant is a large gray animal native to Asia and Africa, the elephant is the definiendum. Note that the definiens is not the meaning of the word defined, there are many sub-types of definitions, often specific to a given field of knowledge or study. An intensional definition, also called a connotative definition, specifies the necessary, any definition that attempts to set out the essence of something, such as that by genus and differentia, is an intensional definition. An extensional definition, also called a denotative definition, of a concept or term specifies its extension and it is a list naming every object that is a member of a specific set. An extensional definition would be the list of wrath, greed, sloth, pride, lust, envy, a genus–differentia definition is a type of intensional definition that takes a large category and narrows it down to a smaller category by a distinguishing characteristic. The differentia, The portion of the new definition that is not provided by the genus, for example, consider the following genus-differentia definitions, a triangle, A plane figure that has three straight bounding sides. A quadrilateral, A plane figure that has four straight bounding sides and those definitions can be expressed as a genus and two differentiae. It is possible to have two different genus-differentia definitions that describe the same term, especially when the term describes the overlap of two large categories, for instance, both of these genus-differentia definitions of square are equally acceptable, a square, a rectangle that is a rhombus. A square, a rhombus that is a rectangle, thus, a square is a member of both the genus rectangle and the genus rhombus. One important form of the definition is ostensive definition. This gives the meaning of a term by pointing, in the case of an individual, to the thing itself, or in the case of a class, to examples of the right kind. So one can explain who Alice is by pointing her out to another, or what a rabbit is by pointing at several, the process of ostensive definition itself was critically appraised by Ludwig Wittgenstein
23.
Renaissance
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The Renaissance was a period in European history, from the 14th to the 17th century, regarded as the cultural bridge between the Middle Ages and modern history. It started as a movement in Italy in the Late Medieval period and later spread to the rest of Europe. This new thinking became manifest in art, architecture, politics, science, Early examples were the development of perspective in oil painting and the recycled knowledge of how to make concrete. Although the invention of movable type sped the dissemination of ideas from the later 15th century. In politics, the Renaissance contributed to the development of the customs and conventions of diplomacy, the Renaissance began in Florence, in the 14th century. Other major centres were northern Italian city-states such as Venice, Genoa, Milan, Bologna, the word Renaissance, literally meaning Rebirth in French, first appeared in English in the 1830s. The word also occurs in Jules Michelets 1855 work, Histoire de France, the word Renaissance has also been extended to other historical and cultural movements, such as the Carolingian Renaissance and the Renaissance of the 12th century. The Renaissance was a movement that profoundly affected European intellectual life in the early modern period. Renaissance scholars employed the humanist method in study, and searched for realism, however, a subtle shift took place in the way that intellectuals approached religion that was reflected in many other areas of cultural life. In addition, many Greek Christian works, including the Greek New Testament, were back from Byzantium to Western Europe. Political philosophers, most famously Niccolò Machiavelli, sought to describe life as it really was. Others see more competition between artists and polymaths such as Brunelleschi, Ghiberti, Donatello, and Masaccio for artistic commissions as sparking the creativity of the Renaissance. Yet it remains much debated why the Renaissance began in Italy, accordingly, several theories have been put forward to explain its origins. During the Renaissance, money and art went hand in hand, Artists depended entirely on patrons while the patrons needed money to foster artistic talent. Wealth was brought to Italy in the 14th, 15th, and 16th centuries by expanding trade into Asia, silver mining in Tyrol increased the flow of money. Luxuries from the Eastern world, brought home during the Crusades, increased the prosperity of Genoa, unlike with Latin texts, which had been preserved and studied in Western Europe since late antiquity, the study of ancient Greek texts was very limited in medieval Western Europe. One of the greatest achievements of Renaissance scholars was to bring this entire class of Greek cultural works back into Western Europe for the first time since late antiquity, Arab logicians had inherited Greek ideas after they had invaded and conquered Egypt and the Levant. Their translations and commentaries on these ideas worked their way through the Arab West into Spain and Sicily and this work of translation from Islamic culture, though largely unplanned and disorganized, constituted one of the greatest transmissions of ideas in history
24.
Timeline of scientific discoveries
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The timeline below shows the date of publication of possible major scientific theories and discoveries, along with the discoverer. In many cases, the discoveries spanned several years, 4th century BCE - Mandragora was described by Theophrastus in the fourth century B. C. E. for treatment of wounds, gout, and sleeplessness, and as a love potion. By the first century C. E. Dioscorides recognized wine of mandrake as an anaesthetic for treatment of pain or sleeplessness, first use of controlled experiments and reproducibility of its results. 1020s – Avicennas The Canon of Medicine 1054 – Various early astronomers observe supernova, shen Kuo, Discovers the concepts of true north and magnetic declination. In addition, he develops the first theory of Geomorphology,1821 – Thomas Johann Seebeck is the first to observe a property of semiconductors. 1873 – Frederick Guthrie discovers thermionic emission,1873 - Willoughby Smith discovers photoconductivity. 1887 – Albert A. Michelson and Edward W. Morley,1996 – Roslin Institute, Dolly the sheep was cloned. 1997 – CDF and DØ experiments at Fermilab, Top quark,1998 – Supernova Cosmology Project and the High-Z Supernova Search Team, discovery of the accelerated expansion of the Universe / Dark Energy. 2000 – The Tau neutrino is discovered by the DONUT collaboration 2001 – The first draft of the Human Genome Project is published,2003 - Grigori Perelman presents proof of the Poincaré Conjecture. 2005 – Grid cells in the brain are discovered by Edvard Moser and May-Britt Moser
25.
Galileo Galilei
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Galileo Galilei was an Italian polymath, astronomer, physicist, engineer, philosopher, and mathematician. He played a role in the scientific revolution of the seventeenth century. Galileo also worked in applied science and technology, inventing an improved military compass, Galileos championing of heliocentrism and Copernicanism was controversial during his lifetime, when most subscribed to either geocentrism or the Tychonic system. He met with opposition from astronomers, who doubted heliocentrism because of the absence of a stellar parallax. He was tried by the Inquisition, found vehemently suspect of heresy and he spent the rest of his life under house arrest. He has been called the father of observational astronomy, the father of modern physics, the father of scientific method, and the father of science. Galileo was born in Pisa, Italy, on 15 February 1564, the first of six children of Vincenzo Galilei, a famous lutenist, composer, and music theorist, and Giulia, three of Galileos five siblings survived infancy. The youngest, Michelangelo, also became a noted lutenist and composer although he contributed to financial burdens during Galileos young adulthood, Michelangelo was unable to contribute his fair share of their fathers promised dowries to their brothers-in-law, who would later attempt to seek legal remedies for payments due. Michelangelo would also occasionally have to borrow funds from Galileo to support his musical endeavours and these financial burdens may have contributed to Galileos early fire to develop inventions that would bring him additional income. When Galileo Galilei was eight, his family moved to Florence and he then was educated in the Vallombrosa Abbey, about 30 km southeast of Florence. Galileo Bonaiuti was buried in the church, the Basilica of Santa Croce in Florence. It was common for mid-sixteenth century Tuscan families to name the eldest son after the parents surname, hence, Galileo Galilei was not necessarily named after his ancestor Galileo Bonaiuti. The Italian male given name Galileo derives from the Latin Galilaeus, meaning of Galilee, the biblical roots of Galileos name and surname were to become the subject of a famous pun. In 1614, during the Galileo affair, one of Galileos opponents, in it he made a point of quoting Acts 1,11, Ye men of Galilee, why stand ye gazing up into heaven. Despite being a genuinely pious Roman Catholic, Galileo fathered three children out of wedlock with Marina Gamba and they had two daughters, Virginia and Livia, and a son, Vincenzo. Their only worthy alternative was the religious life, both girls were accepted by the convent of San Matteo in Arcetri and remained there for the rest of their lives. Virginia took the name Maria Celeste upon entering the convent and she died on 2 April 1634, and is buried with Galileo at the Basilica of Santa Croce, Florence. Livia took the name Sister Arcangela and was ill for most of her life, Vincenzo was later legitimised as the legal heir of Galileo and married Sestilia Bocchineri
26.
Carl Friedrich Gauss
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Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, in the Duchy of Brunswick-Wolfenbüttel, as the son of poor working-class parents. Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter and he was christened and confirmed in a church near the school he attended as a child. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100, there are many other anecdotes about his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completed Disquisitiones Arithmeticae, his opus, in 1798 at the age of 21. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day, while at university, Gauss independently rediscovered several important theorems. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone, the stonemason declined, stating that the difficult construction would essentially look like a circle. The year 1796 was most productive for both Gauss and number theory and he discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in number theory, on 8 April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic, the prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers. Gauss also discovered that every integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note, ΕΥΡΗΚΑ. On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, in 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism and the discovery of Kirchhoffs circuit laws in electricity. It was during this time that he formulated his namesake law and they constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation. Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula. In 1845, he became associated member of the Royal Institute of the Netherlands, in 1854, Gauss selected the topic for Bernhard Riemanns Habilitationvortrag, Über die Hypothesen, welche der Geometrie zu Grunde liegen. On the way home from Riemanns lecture, Weber reported that Gauss was full of praise, Gauss died in Göttingen, on 23 February 1855 and is interred in the Albani Cemetery there. Two individuals gave eulogies at his funeral, Gausss son-in-law Heinrich Ewald and Wolfgang Sartorius von Waltershausen and his brain was preserved and was studied by Rudolf Wagner who found its mass to be 1,492 grams and the cerebral area equal to 219,588 square millimeters. Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation of his genius, Gauss was a Lutheran Protestant, a member of the St. Albans Evangelical Lutheran church in Göttingen
27.
Benjamin Peirce
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Benjamin Peirce was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and he was the son of Benjamin Peirce, later librarian of Harvard, and Lydia Ropes Nichols Peirce. After graduating from Harvard, he remained as a tutor, and was appointed professor of mathematics in 1831. He added astronomy to his portfolio in 1842, and remained as Harvard professor until his death. In addition, he was instrumental in the development of Harvards science curriculum, served as the college librarian, Benjamin Peirce is often regarded as the earliest American scientist whose research was recognized as world class. He was an apologist for slavery, opining that it should be condoned if it was used to allow an elite to pursue scientific enquiry, in number theory, he proved there is no odd perfect number with fewer than four prime factors. In algebra, he was notable for the study of associative algebras and he first introduced the terms idempotent and nilpotent in 1870 to describe elements of these algebras, and he also introduced the Peirce decomposition. In the philosophy of mathematics, he known for the statement that Mathematics is the science that draws necessary conclusions. Peirces definition of mathematics was credited by his son, Charles Sanders Peirce, like George Boole, Peirce believed that mathematics could be used to study logic. These ideas were developed by Charles Sanders Peirce, who noted that logic also includes the study of faulty reasoning. In contrast, the later logicist program of Gottlob Frege and Bertrand Russell attempted to base mathematics on logic, Peirce proposed what came to be known as Peirces Criterion for the statistical treatment of outliers, that is, of apparently extreme observations. His ideas were developed by Charles Sanders Peirce. Peirce was a witness in the Howland will forgery trial. Their analysis of the questioned signature showed that it resembled another particular handwriting example so closely that the chances of such a match were statistically extremely remote and he was devoutly religious, though he seldom published his theological thoughts. Peirce credited God as shaping nature in ways that account for the efficacy of pure mathematics in describing empirical phenomena, Peirce viewed mathematics as study of Gods work by Gods creatures, according to an encyclopedia. He married Sarah Hunt Mills, the daughter of U. S, the lunar crater Peirce is named for Peirce. Post-doctoral positions in Harvard Universitys mathematics department are named in his honor as Benjamin Peirce Fellows, the United States Coast Survey ship USCS Benjamin Peirce, in commission from 1855 to 1868, was named for him. An Elementary Treatise on Plane and Spherical Trigonometry, Boston, James Munroe, google Eprints of successive editions 1840–1862
28.
Albert Einstein
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Albert Einstein was a German-born theoretical physicist. He developed the theory of relativity, one of the two pillars of modern physics, Einsteins work is also known for its influence on the philosophy of science. Einstein is best known in popular culture for his mass–energy equivalence formula E = mc2, near the beginning of his career, Einstein thought that Newtonian mechanics was no longer enough to reconcile the laws of classical mechanics with the laws of the electromagnetic field. This led him to develop his theory of relativity during his time at the Swiss Patent Office in Bern. Briefly before, he aquired the Swiss citizenship in 1901, which he kept for his whole life and he continued to deal with problems of statistical mechanics and quantum theory, which led to his explanations of particle theory and the motion of molecules. He also investigated the properties of light which laid the foundation of the photon theory of light. In 1917, Einstein applied the theory of relativity to model the large-scale structure of the universe. He was visiting the United States when Adolf Hitler came to power in 1933 and, being Jewish, did not go back to Germany and he settled in the United States, becoming an American citizen in 1940. This eventually led to what would become the Manhattan Project, Einstein supported defending the Allied forces, but generally denounced the idea of using the newly discovered nuclear fission as a weapon. Later, with the British philosopher Bertrand Russell, Einstein signed the Russell–Einstein Manifesto, Einstein was affiliated with the Institute for Advanced Study in Princeton, New Jersey, until his death in 1955. Einstein published more than 300 scientific papers along with over 150 non-scientific works, on 5 December 2014, universities and archives announced the release of Einsteins papers, comprising more than 30,000 unique documents. Einsteins intellectual achievements and originality have made the word Einstein synonymous with genius, Albert Einstein was born in Ulm, in the Kingdom of Württemberg in the German Empire, on 14 March 1879. His parents were Hermann Einstein, a salesman and engineer, the Einsteins were non-observant Ashkenazi Jews, and Albert attended a Catholic elementary school in Munich from the age of 5 for three years. At the age of 8, he was transferred to the Luitpold Gymnasium, the loss forced the sale of the Munich factory. In search of business, the Einstein family moved to Italy, first to Milan, when the family moved to Pavia, Einstein stayed in Munich to finish his studies at the Luitpold Gymnasium. His father intended for him to electrical engineering, but Einstein clashed with authorities and resented the schools regimen. He later wrote that the spirit of learning and creative thought was lost in strict rote learning, at the end of December 1894, he travelled to Italy to join his family in Pavia, convincing the school to let him go by using a doctors note. During his time in Italy he wrote an essay with the title On the Investigation of the State of the Ether in a Magnetic Field
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Natural science
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Natural science is a branch of science concerned with the description, prediction, and understanding of natural phenomena, based on observational and empirical evidence. Mechanisms such as review and repeatability of findings are used to try to ensure the validity of scientific advances. Natural science can be divided into two branches, life science and physical science. Physical science is subdivided into branches, including physics, space science, chemistry and these branches of natural science may be further divided into more specialized branches. Modern natural science succeeded more classical approaches to natural philosophy, usually traced to ancient Greece, galileo, Descartes, Francis Bacon, and Newton debated the benefits of using approaches which were more mathematical and more experimental in a methodical way. Still, philosophical perspectives, conjectures, and presuppositions, often overlooked, systematic data collection, including discovery science, succeeded natural history, which emerged in the 16th century by describing and classifying plants, animals, minerals, and so on. Today, natural history suggests observational descriptions aimed at popular audiences, philosophers of science have suggested a number of criteria, including Karl Poppers controversial falsifiability criterion, to help them differentiate scientific endeavors from non-scientific ones. Validity, accuracy, and quality control, such as peer review and this field encompasses a set of disciplines that examines phenomena related to living organisms. The scale of study can range from sub-component biophysics up to complex ecologies, biology is concerned with the characteristics, classification and behaviors of organisms, as well as how species were formed and their interactions with each other and the environment. The biological fields of botany, zoology, and medicine date back to periods of civilization. However, it was not until the 19th century that became a unified science. Once scientists discovered commonalities between all living things, it was decided they were best studied as a whole, modern biology is divided into subdisciplines by the type of organism and by the scale being studied. Molecular biology is the study of the chemistry of life, while cellular biology is the examination of the cell. At a higher level, anatomy and physiology looks at the internal structures, constituting the scientific study of matter at the atomic and molecular scale, chemistry deals primarily with collections of atoms, such as gases, molecules, crystals, and metals. The composition, statistical properties, transformations and reactions of these materials are studied, chemistry also involves understanding the properties and interactions of individual atoms and molecules for use in larger-scale applications. Most chemical processes can be studied directly in a laboratory, using a series of techniques for manipulating materials, chemistry is often called the central science because of its role in connecting the other natural sciences. Early experiments in chemistry had their roots in the system of Alchemy, the science of chemistry began to develop with the work of Robert Boyle, the discoverer of gas, and Antoine Lavoisier, who developed the theory of the Conservation of mass. The success of science led to a complementary chemical industry that now plays a significant role in the world economy
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Social sciences
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Social science is a major category of academic disciplines, concerned with society and the relationships among individuals within a society. It in turn has many branches, each of which is considered a social science, the social sciences include economics, political science, human geography, demography, psychology, sociology, anthropology, archaeology, jurisprudence, history, and linguistics. The term is sometimes used to refer specifically to the field of sociology. A more detailed list of sub-disciplines within the sciences can be found at Outline of social science. Positivist social scientists use methods resembling those of the sciences as tools for understanding society. In modern academic practice, researchers are often eclectic, using multiple methodologies, the term social research has also acquired a degree of autonomy as practitioners from various disciplines share in its aims and methods. Social sciences came forth from the philosophy of the time and were influenced by the Age of Revolutions, such as the Industrial Revolution. The social sciences developed from the sciences, or the systematic knowledge-bases or prescriptive practices, the beginnings of the social sciences in the 18th century are reflected in the grand encyclopedia of Diderot, with articles from Jean-Jacques Rousseau and other pioneers. The growth of the sciences is also reflected in other specialized encyclopedias. The modern period saw social science first used as a distinct conceptual field, Social science was influenced by positivism, focusing on knowledge based on actual positive sense experience and avoiding the negative, metaphysical speculation was avoided. Auguste Comte used the term science sociale to describe the field, taken from the ideas of Charles Fourier, following this period, there were five paths of development that sprang forth in the social sciences, influenced by Comte on other fields. One route that was taken was the rise of social research, large statistical surveys were undertaken in various parts of the United States and Europe. Another route undertaken was initiated by Émile Durkheim, studying social facts, a third means developed, arising from the methodological dichotomy present, in which social phenomena were identified with and understood, this was championed by figures such as Max Weber. The fourth route taken, based in economics, was developed and furthered economic knowledge as a hard science, the last path was the correlation of knowledge and social values, the antipositivism and verstehen sociology of Max Weber firmly demanded this distinction. In this route, theory and prescription were non-overlapping formal discussions of a subject, around the start of the 20th century, Enlightenment philosophy was challenged in various quarters. The development of social science subfields became very quantitative in methodology, examples of boundary blurring include emerging disciplines like social research of medicine, sociobiology, neuropsychology, bioeconomics and the history and sociology of science. Increasingly, quantitative research and qualitative methods are being integrated in the study of action and its implications. In the first half of the 20th century, statistics became a discipline of applied mathematics
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Applied mathematics
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Applied mathematics is a branch of mathematics that deals with mathematical methods that find use in science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of science and specialized knowledge. The term applied mathematics also describes the professional specialty in which work on practical problems by formulating and studying mathematical models. The activity of applied mathematics is thus connected with research in pure mathematics. Historically, applied mathematics consisted principally of applied analysis, most notably differential equations, approximation theory, quantitative finance is now taught in mathematics departments across universities and mathematical finance is considered a full branch of applied mathematics. Engineering and computer science departments have made use of applied mathematics. Today, the applied mathematics is used in a broader sense. It includes the areas noted above as well as other areas that have become increasingly important in applications. Even fields such as number theory that are part of mathematics are now important in applications. There is no consensus as to what the various branches of applied mathematics are, such categorizations are made difficult by the way mathematics and science change over time, and also by the way universities organize departments, courses, and degrees. Many mathematicians distinguish between applied mathematics, which is concerned with methods, and the applications of mathematics within science. Mathematicians such as Poincaré and Arnold deny the existence of applied mathematics, similarly, non-mathematicians blend applied mathematics and applications of mathematics. The use and development of mathematics to industrial problems is also called industrial mathematics. Historically, mathematics was most important in the sciences and engineering. Academic institutions are not consistent in the way they group and label courses, programs, at some schools, there is a single mathematics department, whereas others have separate departments for Applied Mathematics and Mathematics. It is very common for Statistics departments to be separated at schools with graduate programs, many applied mathematics programs consist of primarily cross-listed courses and jointly appointed faculty in departments representing applications. Some Ph. D. programs in applied mathematics require little or no coursework outside of mathematics, in some respects this difference reflects the distinction between application of mathematics and applied mathematics. Research universities dividing their mathematics department into pure and applied sections include MIT, brigham Young University also has an Applied and Computational Emphasis, a program that allows student to graduate with a Mathematics degree, with an emphasis in Applied Math
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Game theory
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Game theory is the study of mathematical models of conflict and cooperation between intelligent rational decision-makers. Game theory is used in economics, political science, and psychology, as well as logic, computer science. Originally, it addressed zero-sum games, in one persons gains result in losses for the other participants. Today, game theory applies to a range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals. Modern game theory began with the idea regarding the existence of equilibria in two-person zero-sum games. Von Neumanns original proof used Brouwer fixed-point theorem on continuous mappings into compact convex sets and his paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this provided an axiomatic theory of expected utility. This theory was developed extensively in the 1950s by many scholars, Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been recognized as an important tool in many fields. With the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole in 2014, John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology. Early discussions of examples of two-person games occurred long before the rise of modern, the first known discussion of game theory occurred in a letter written by Charles Waldegrave, an active Jacobite, and uncle to James Waldegrave, a British diplomat, in 1713. In this letter, Waldegrave provides a mixed strategy solution to a two-person version of the card game le Her. James Madison made what we now recognize as an analysis of the ways states can be expected to behave under different systems of taxation. In 1913 Ernst Zermelo published Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels and it proved that the optimal chess strategy is strictly determined. This paved the way for more general theorems, the Danish mathematician Zeuthen proved that the mathematical model had a winning strategy by using Brouwers fixed point theorem. In his 1938 book Applications aux Jeux de Hasard and earlier notes, Borel conjectured that non-existence of mixed-strategy equilibria in two-person zero-sum games would occur, a conjecture that was proved false. Game theory did not really exist as a field until John von Neumann published a paper in 1928. Von Neumanns original proof used Brouwers fixed-point theorem on continuous mappings into compact convex sets and his paper was followed by his 1944 book Theory of Games and Economic Behavior co-authored with Oskar Morgenstern
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Pure mathematics
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Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts. Even though the pure and applied viewpoints are distinct philosophical positions, in there is much overlap in the activity of pure. To develop accurate models for describing the world, many applied mathematicians draw on tools. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research, ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between arithmetic, now called number theory, and logistic, now called arithmetic. Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, the term itself is enshrined in the full title of the Sadleirian Chair, founded in the mid-nineteenth century. The idea of a discipline of pure mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, in the following years, specialisation and professionalisation started to make a rift more apparent. At the start of the twentieth century mathematicians took up the axiomatic method, in fact in an axiomatic setting rigorous adds nothing to the idea of proof. Pure mathematics, according to a view that can be ascribed to the Bourbaki group, is what is proved, Pure mathematician became a recognized vocation, achievable through training. One central concept in mathematics is the idea of generality. One can use generality to avoid duplication of effort, proving a general instead of having to prove separate cases independently. Generality can facilitate connections between different branches of mathematics, category theory is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math. Generalitys impact on intuition is both dependent on the subject and a matter of preference or learning style. Often generality is seen as a hindrance to intuition, although it can function as an aid to it. Each of these branches of abstract mathematics have many sub-specialties. A steep rise in abstraction was seen mid 20th century, in practice, however, these developments led to a sharp divergence from physics, particularly from 1950 to 1983. Later this was criticised, for example by Vladimir Arnold, as too much Hilbert, the point does not yet seem to be settled, in that string theory pulls one way, while discrete mathematics pulls back towards proof as central
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History of mathematics
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Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322, the Rhind Mathematical Papyrus, All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. Greek mathematics greatly refined the methods and expanded the subject matter of mathematics, Chinese mathematics made early contributions, including a place value system. Islamic mathematics, in turn, developed and expanded the known to these civilizations. Many Greek and Arabic texts on mathematics were then translated into Latin, from ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, the origins of mathematical thought lie in the concepts of number, magnitude, and form. Modern studies of cognition have shown that these concepts are not unique to humans. Such concepts would have part of everyday life in hunter-gatherer societies. The idea of the number concept evolving gradually over time is supported by the existence of languages which preserve the distinction between one, two, and many, but not of numbers larger than two. Prehistoric artifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time. The Ishango bone, found near the headwaters of the Nile river, may be more than 20,000 years old, common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of prime numbers or a six-month lunar calendar. He also writes that no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10, predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian, Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity. The majority of Babylonian mathematical work comes from two widely separated periods, The first few hundred years of the second millennium BC, and it is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic mathematics, in contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed whilst the clay was moist, Some of these appear to be graded homework. The earliest evidence of written mathematics dates back to the ancient Sumerians and they developed a complex system of metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises, the earliest traces of the Babylonian numerals also date back to this period
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Pythagoras
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Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and the putative founder of the movement called Pythagoreanism. Most of the information about Pythagoras was written centuries after he lived. He was born on the island of Samos, and travelled, visiting Egypt and Greece, around 530 BC, he moved to Croton, in Magna Graecia, and there established some kind of school or guild. In 520 BC, he returned to Samos, Pythagoras made influential contributions to philosophy and religion in the late 6th century BC. He is often revered as a mathematician and scientist and is best known for the Pythagorean theorem which bears his name. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues, some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important. It was said that he was the first man to himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Plato. Burkert states that Aristoxenus and Dicaearchus are the most important accounts, Aristotle had written a separate work On the Pythagoreans, which is no longer extant. However, the Protrepticus possibly contains parts of On the Pythagoreans and his disciples Dicaearchus, Aristoxenus, and Heraclides Ponticus had written on the same subject. These writers, late as they are, were among the best sources from whom Porphyry and Iamblichus drew, while adding some legendary accounts. Herodotus, Isocrates, and other writers agree that Pythagoras was the son of Mnesarchus and born on the Greek island of Samos. His father is said to have been a gem-engraver or a wealthy merchant, a late source gives his mothers name as Pythais. As to the date of his birth, Aristoxenus stated that Pythagoras left Samos in the reign of Polycrates, at the age of 40, around 530 BC he arrived in the Greek colony of Croton in what was then Magna Graecia. There he founded his own school the members of which he engaged to a disciplined. He furthermore aquired some political influence, on Greeks and non-Greeks of the region, following a conflict with the neighbouring colony of Sybaris, internal discord drove most of the Pythagoreans out of Croton. Pythagoras left the city before the outbreak of civil unrest and moved to Metapontum, after his death, his house was transformed into a sanctuary of Demeter, out of veneration for the philosopher, by the local population. In ancient sources there was disagreement and inconsistency about the late life of Pythagoras. His tomb was shown at Metapontum in the time of Cicero, according to Walter Burkert, Most obvious is the contradiction between Aristoxenus and Dicaearchus, regarding the catastrophe that overwhelmed the Pythagorean society
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Pythagorean theorem
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In mathematics, the Pythagorean theorem, also known as Pythagorass theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the two sides. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework, Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases. The theorem has been given numerous proofs – possibly the most for any mathematical theorem and they are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The Pythagorean theorem was known long before Pythagoras, but he may well have been the first to prove it, in any event, the proof attributed to him is very simple, and is called a proof by rearrangement. The two large squares shown in the figure each contain four triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem and that Pythagoras originated this very simple proof is sometimes inferred from the writings of the later Greek philosopher and mathematician Proclus. Several other proofs of this theorem are described below, but this is known as the Pythagorean one, If the length of both a and b are known, then c can be calculated as c = a 2 + b 2. If the length of the c and of one side are known. The Pythagorean equation relates the sides of a triangle in a simple way. Another corollary of the theorem is that in any triangle, the hypotenuse is greater than any one of the other sides. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. This theorem may have more known proofs than any other, the book The Pythagorean Proposition contains 370 proofs, Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB, point H divides the length of the hypotenuse c into parts d and e. By a similar reasoning, the triangle CBH is also similar to ABC, the proof of similarity of the triangles requires the triangle postulate, the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the leads to the equality of ratios of corresponding sides. The first result equates the cosines of the angles θ, whereas the second result equates their sines, the role of this proof in history is the subject of much speculation
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Mayan numerals
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The Maya numeral system is a vigesimal positional notation used in the Maya civilization to represent numbers. The numerals are made up of three symbols, zero, one and five, for example, thirteen is written as three dots in a horizontal row above two horizontal lines stacked above each other. Numbers after 19 were written vertically in powers of twenty, for example, thirty-three would be written as one dot above three dots, which are in turn atop two lines. The first dot represents one twenty or 1×20, which is added to three dots and two bars, or thirteen, upon reaching 202 or 400, another row is started. The number 429 would be written as one dot above one dot above four dots, the powers of twenty are numerals, just as the Hindu-Arabic numeral system uses powers of tens. Other than the bar and dot notation, Maya numerals can be illustrated by face type glyphs or pictures, the face glyph for a number represents the deity associated with the number. These face number glyphs were used, and are mostly seen on some of the most elaborate monumental carving. Addition and subtraction, Adding and subtracting numbers below 20 using Maya numerals is very simple, addition is performed by combining the numeric symbols at each level, If five or more dots result from the combination, five dots are removed and replaced by a bar. If four or more bars result, four bars are removed, similarly with subtraction, remove the elements of the subtrahend symbol from the minuend symbol, If there are not enough dots in a minuend position, a bar is replaced by five dots. If there are not enough bars, a dot is removed from the next higher minuend symbol in the column, the Maya/Mesoamerican Long Count calendar required the use of zero as a place-holder within its vigesimal positional numeral system. A shell glyph – – was used as a symbol for these Long Count dates. However, since the eight earliest Long Count dates appear outside the Maya homeland, it is assumed that the use of zero predated the Maya, indeed, many of the earliest Long Count dates were found within the Olmec heartland. However, the Olmec civilization had come to an end by the 4th century BC, in the Long Count portion of the Maya calendar, a variation on the strictly vigesimal numbering is used. The Long Count changes in the place value, it is not 20×20 =400, as would otherwise be expected. This is supposed to be because 360 is roughly the number of days in a year, subsequent place values return to base-twenty. In fact, every known example of large numbers uses this modified vigesimal system and it is reasonable to assume, but not proven by any evidence, that the normal system in use was a pure base-20 system. Maya Mathematics - online converter from decimal numeration to Maya numeral notation, anthropomorphic Maya numbers - online story of number representations
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Tally sticks
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A tally stick was an ancient memory aid device used to record and document numbers, quantities, or even messages. Tally sticks first appear as animal bones carved with notches during the Upper Paleolithic, historical reference is made by Pliny the Elder about the best wood to use for tallies, and by Marco Polo who mentions the use of the tally in China. Tallies have been used for purposes such as messaging and scheduling. Principally, there are two different kinds of sticks, the single tally and the split tally. A common form of the kind of primitive counting device is seen in various kinds of prayer beads. A number of artefacts have been conjectured to be tally sticks. It is a dark brown length of bone, the fibula of a baboon and it has a series of tally marks carved in three columns running the length of the tool. It was found in 1960 in Belgian Congo, the Lebombo Bone is a baboons fibula with 29 distinct notches, discovered within the Border Cave in the Lebombo Mountains of Swaziland. The so-called Wolf bone is a prehistoric artefact discovered in 1937 in Czechoslovakia during excavations at Vestonice, Moravia, dated to the Aurignacian, approximately 30,000 years ago, the bone is marked with 55 marks which some believe to be tally marks. The head of an ivory Venus figurine was excavated close to the bone, the single tally stick was an elongated piece of bone, ivory, wood, or stone which is marked with a system of notches. The single tally stick serves predominantly mnemonic purposes, related to the single tally concept are messenger sticks, the knotted cords, khipus or quipus, as used by the Inca. Herodotus reported the use of a knotted cord by Darius I of Persia, the split tally was a technique which became common in medieval Europe, which was constantly short of money and predominantly illiterate, in order to record bilateral exchange and debts. A stick was marked with a system of notches and then split lengthwise and this way the two halves both record the same notches and each party to the transaction received one half of the marked stick as proof. Later this technique was refined in ways and became virtually tamper proof. One of the refinements was to make the two halves of the stick of different lengths, the longer part was called stock and was given to the party which had advanced money to the receiver. The shorter portion of the stick was called foil and was given to the party which had received the funds or goods, using this technique each of the parties had an identifiable record of the transaction. If one party tried to change the value of his half of the tally stick by adding more notches. The split tally was accepted as proof in medieval courts
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Prehistoric
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Prehistory means literally before history, from the Latin word for before, præ, and Greek ιστορία. Neighbouring civilisations were the first to follow, most other civilisations reached the end of prehistory during the Iron Age. The period when a culture is written about by others, but has not developed its own writing is known as the protohistory of the culture. By definition, there are no records from human prehistory. Clear techniques for dating were not well-developed until the 19th century and this article is concerned with human prehistory as defined here above. There are separate articles for the history of the Earth. However, for the race as a whole, prehistory ends when recorded history begins with the accounts of the ancient world around the 4th millennium BC. For example, in Egypt it is accepted that prehistory ended around 3200 BC, whereas in New Guinea the end of the prehistoric era is set much more recently. The three-age system is the periodization of prehistory into three consecutive time periods, named for their respective predominant tool-making technologies, Stone Age Bronze Age Iron Age. The notion of prehistory began to surface during the Enlightenment in the work of antiquarians who used the word primitive to describe societies that existed before written records, the first use of the word prehistory in English, however, occurred in the Foreign Quarterly Review in 1836. The main source for prehistory is archaeology, but some scholars are beginning to more use of evidence from the natural and social sciences. This view has been articulated by advocates of deep history, human population geneticists and historical linguists are also providing valuable insight for these questions. Human prehistory differs from history not only in terms of its chronology, restricted to material processes, remains and artifacts rather than written records, prehistory is anonymous. Because of this, reference terms that use, such as Neanderthal or Iron Age are modern labels with definitions sometimes subject to debate. Palaeolithic means Old Stone Age, and begins with the first use of stone tools, the Paleolithic is the earliest period of the Stone Age. The early part of the Palaeolithic is called the Lower Palaeolithic, evidence of control of fire by early humans during the Lower Palaeolithic Era is uncertain and has at best limited scholarly support. The most widely accepted claim is that H. erectus or H. ergaster made fires between 790,000 and 690,000 BP in a site at Bnot Yaakov Bridge, Israel. The use of fire enabled early humans to cook food, provide warmth, Early Homo sapiens originated some 200,000 years ago, ushering in the Middle Palaeolithic