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86-596: Fast's story is about how a group of scientists and educators, through a controlled environment, succeed in raising naturally gifted children into "man-plus"—people who possess comparatively super-human abilities. They possess unparalleled understanding of all technical subjects such as math , physics and unlock natural telepathy . They also excel at physical endeavors, such as sports and break numerous physical records . Their controlled environment, an isolated compound in California comprising 8,000 acres (32 km),

172-454: A White House operative, Eggerton, summons Harry and inquires him on his knowledge of the work in the compound. He truthfully tells them that he has no inside knowledge of the project, save what his sister told him in the project's beginnings, which he hadn't disclosed to anyone due to its secrecy. Harry is told that the compound was about to be visited for the first time since its beginning and, as they were about to enter, promptly vanished. It

258-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects

344-460: A controlled environment, tailored to allow them to reach their full potential. The children in question, all babies when arriving at the government-sponsored compound, are all to be naturally gifted. The compound the government grants them is a secluded 8,000 acres (32 km) compound in California . The researchers are given 15 years to raise the children. Next she sends Harry to find a Professor Hans Goldbaum who, before World War II , had written

430-614: A foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of

516-637: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)

602-724: A major protagonist in the Irish national epic the Táin Bo Cuailnge , ranks as a hero or as a demigod. He is the son of the Irish god Lugh and the mortal princess Deichtine . In the immediate pre-Roman period, the Celtic Gallaceian tribe in Portugal made powerful, large stone statues of deified local heroes , which stood on hill forts in the mountainous regions of - what is today - Northern Portugal and Spanish Galicia . In Hinduism ,

688-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were

774-422: A mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture . Through a series of rigorous arguments employing deductive reasoning , a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma . A proven instance that forms part of

860-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, " or " means "one, the other or both", while, in common language, it

946-620: A mortal scholar. Abe no Seimei , a famous onmyōji from the Heian period was supposed to be one. His father, Abe no Yasuna (安倍 保名), was human. Still, his mother Kuzunoha , was a Kitsune , a divine fox, being this the origin of Abe no Seimei's magical prowess. In the indigenous religions originating from the Philippines , collectively called Anitism , demigods abound in various ethnic stories. Many of these demigods equal major gods and goddesses in power and influence. Notable examples include Mayari,

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1032-417: A new race of super-intelligent demigods . Jean fears—correctly—that the government will react with fear and destroy the advanced super-race. The children, though incredibly advanced, are incapable of violence, even in self-defense. Jean is able to obtain a three-year extension, and then another of a few weeks when the three years expires. In that time, the children, now able to telepathically reach

1118-426: A paper about how he had discovered a set of characteristics in babies that would determine whether or not they would grow to be mentally gifted. Finding him, Goldbaum agrees to join their project and he and Harry set out to find a diverse set of gifted babies ( orphans and those they could buy) for the compound. After delivering the children, Harry has no more involvement with the project. About eighteen years later,

1204-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes

1290-603: A process that has been referred to as "heroization". Pindar also used the term frequently as a synonym for "hero". According to the Roman author Cassius Dio , the Roman Senate declared Julius Caesar a demigod after his 46 BCE victory at Thapsus . However, Dio was writing in the third century CE — centuries after the death of Caesar — and modern critics have cast doubt on whether the Senate really did this. The first Roman to employ

1376-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as

1462-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of

1548-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints. For example, designing a survey often involves minimizing the cost of estimating

1634-477: A wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory , type theory , computability theory and computational complexity theory . Although these aspects of mathematical logic were introduced before the rise of computers , their use in compiler design, formal verification , program analysis , proof assistants and other aspects of computer science , contributed in turn to

1720-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry

1806-403: Is flat " and "a field is always a ring ". Demigod A demigod is a half-god and half-human offspring of a god and a human , or a human or non-human creature that is accorded divine status after death, or someone who has attained the " divine spark " ( divine illumination ). An immortal demigod often has tutelary status and a religious cult following, while a mortal demigod

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1892-407: Is a government sponsored facility granted for the raising of the children. The scientists were given fifteen years, later extended by three more years and a few weeks, to experimentally raise the children. By a very early age, the children surpass their teachers' knowledge. The story begins as a series of communiqués between a sister, Jean Arbalaid, and her brother, Harry Felton, recently retired from

1978-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example

2064-509: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of

2150-407: Is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called " exclusive or "). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module

2236-487: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as

2322-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example

2408-401: Is necessary between them. When meeting with the researchers, who they love, but pity (due to their inferior intelligence and lack of telepathic abilities), only one child is necessary since all can hear and speak through the single representative's mind. Several children are born to the researchers while living at the facility. While these children are of normal intelligence, they flourish under

2494-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,

2580-1192: Is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations , unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts . Operation and relations are generally represented by specific symbols or glyphs , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of

2666-547: Is often held to be Archimedes ( c.  287  – c.  212 BC ) of Syracuse . He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and

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2752-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it

2838-493: Is one who has fallen or died, but is popular as a legendary hero in various polytheistic religions. Figuratively, it is used to describe a person whose talents or abilities are so superlative that they appear to approach being divine. The English term " demi- god" is a calque of the Latin word semideus , "half-god". The Roman poet Ovid probably coined semideus to refer to less important gods, such as dryads . Compare

2924-554: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after

3010-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,

3096-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of

3182-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with

3268-472: The Ashvins ). Queen Kunti had previously conceived another son, Karna , when she had tested the mantra out. Despite her protests, Surya the sun god was compelled by the mantra to impregnate her. Bhishma is another figures who fits the western definition of demigod, as he was the son of King Shantanu and Goddess Ganga . The Vaishnavites (who often translate deva as "demigod") cite various verses that speak of

3354-574: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It

3440-753: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during

3526-529: The Greek hemitheos . The term demigod first appeared in English in the late sixteenth or early seventeenth century, when it was used to render the Greek and Roman concepts of semideus and daemon. Since then, it has frequently been applied figuratively to people of extraordinary ability. In the ancient Greek and Roman world, the concept of a demigod did not have a consistent definition and associated terminology rarely appeared. The earliest recorded use of

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3612-505: The Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity

3698-524: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of

3784-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object

3870-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry

3956-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not

4042-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and

4128-764: The Devas are stated to be subordinate to Vishnu, or God. A. C. Bhaktivedanta Swami Prabhupada , the founder of the International Society for Krishna Consciousness (ISKCON) translates the Sanskrit word "deva" as "demigod" in his literature when the term referred to a God other than the Supreme Lord . This is because the Vaishnava tradition teaches that there is only one Supreme Lord and that all others are but His servants. In an effort to emphasize their subservience, Prabhupada uses

4214-620: The Middle Ages and made available in Europe. During the early modern period , mathematics began to develop at an accelerating pace in Western Europe , with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,

4300-558: The Tagalog moon goddess who governs the world every night, Tala, the Tagalog star goddess, Hanan, the Tagalog morning goddess, Apo Anno, a Kankanaey demigod hero, Oryol, a Bicolano half-snake demi-goddess who brought peace to the land after defeating all beasts in Ibalon, Laon, a Hiligaynon demigod who can talk to animals and defeated the mad dragon at Mount Kanlaon, Ovug, an Ifugao thunder and lightning demigod who has separate animations in both

4386-694: The Western definition of demigods though they are generally not referred to as such. Queen Kunti , the wife of King Pandu , was given a mantra that, when recited, meant that one of the gods would give her his child. When her husband was cursed to die if he ever engaged in sexual relations, Kunti used this mantra to provide her husband with children fathered by various deities. These children were Yudhishthira (child of Dharmaraj ), Bhima (child of Vayu ) and Arjuna (child of Indra ). She taught this mantra to Madri , King Pandu's other wife, and she immaculately conceived twin boys named Nakula and Sahadeva (children of

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4472-625: The basis of the shield, the " eggheads " will find a way in. And when they do, they'll eradicate "the disease." Mathematics Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as

4558-574: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During

4644-619: The children: no one child had one parent, no one teacher had one child. They then immersed the children in a knowledge-rich atmosphere. Since they were predetermined to be mentally gifted, they all progressed rapidly in knowledge and abilities. By the time they were five, the children were discovering their telepathic abilities. Jean relates that the children usually walk about nude, openly make love with one another, and possess unmatched knowledge in all academic and physical areas. They also share one mindset. With their advanced telepathic abilities, they constantly think as one, no verbal communication

4730-456: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,

4816-399: The current language, where expressions play the role of noun phrases and formulas play the role of clauses . Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is

4902-634: The demigods in Chinese mythology , Erlang Shen and Chen Xiang are most prominent. In the Journey to the West , the Jade Emperor 's younger sister Yaoji is mentioned to have descended to the mortal realm and given birth to a child named Yang Jian. He would eventually grow up to become a deity himself known as Erlang Shen. Chen Xiang is nephew of Erlang Shen, birth by his younger sister Huayue Sanniang who married with

4988-553: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely

5074-600: The devas' subordinate status. For example, the Rig Veda (1.22.20) reads, " oṃ tad viṣṇoḥ paramam padam sadā paśyanti sūrayaḥ ", which translates to, "All the suras [i.e., the devas] look always toward the feet of Lord Vishnu". Similarly, in the Vishnu Sahasranama, the concluding verses, read, "The Rishis [great sages], the ancestors, the devas, the great elements, in fact, all things moving and unmoving constituting this universe, have originated from Narayana," (i.e., Vishnu). Thus

5160-529: The entire Earth's population, build a defense mechanism which resulted in the gray barrier Eggerton described. Jean reveals that the barrier is based on time: the Earth outside the shield is a fraction of a millisecond in the future. They can pass outside without any difficulty. And though they can also get back in, Jean doesn't disclose how this is done. While cut off with the shield, the children are summoning other gifted children telepathically. Passing secretly outside

5246-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of

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5332-567: The first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant " astrology " (or sometimes " astronomy ") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematici , meaning "astrologers",

5418-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before

5504-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and

5590-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term

5676-449: The military. Jean and her husband, Mark, are renowned and well-connected psychologists. Since he is unemployed, Jean sends Harry on a series of trips to investigate some extreme cases of feral children . His reports back to her confirm her hypothesis that children are a result of their environment. Jean and Mark convince the US government to sponsor an experimental program to raise 40 children in

5762-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to

5848-536: The objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains

5934-514: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000  BC , when

6020-654: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been

6106-404: The shield, they are to bring them into the compound to be raised like the others. Hopefully, given time, they would be able to construct more secret compounds to raise more super-intelligent humans. By doing this, perhaps they could turn the tide of humanity. This all depends, of course, on the security provided by their shield. At the conclusion of the letter, Eggerton says that now that they know

6192-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become

6278-561: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and

6364-408: The term "demigod" may have been the poet Ovid (17 or 18 CE), who used the Latin semideus several times in reference to minor deities. The poet Lucan (39-65) also uses the term to speak of Pompey attaining divinity upon his death in 48 BCE. In later antiquity, the Roman writer Martianus Capella ( fl. 410-420) proposed a hierarchy of gods as follows: The Celtic warrior Cú Chulainn ,

6450-604: The term demigod is used to refer to deities who were once human and later became devas (gods). There are two notable demigods in Vedic Scriptures : Nandi (the divine vehicle of Shiva ), and Garuda (the divine vehicle of Vishnu ). Examples of demigods worshiped in South India are Madurai Veeran and Karuppu Sami . The heroes of the Hindu epic Mahabharata , the five Pandava brothers and their half brother Karna , fit

6536-423: The term occurs in texts attributed to the archaic Greek poets Homer and Hesiod . Both describe dead heroes as hemitheoi , or "half gods". In these cases, the word did not literally mean that these figures had one parent who was divine and one who was mortal. Instead, those who demonstrated "strength, power, good family, and good behavior" were termed heroes , and after death they could be called hemitheoi ,

6622-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,

6708-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in

6794-402: The tutelage of the older children and become almost as gifted as them. Near the end of the fifteenth year, realizing that their experiment is about to be investigated, Jean worries what may happen to the children, now young adults, when the government discovers them. The experiment was a success—too much of a success. Not only have they raised mentally gifted people, they have given rise to

6880-504: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in

6966-496: The word "demigod" as a translation of deva . However, there are at least three occurrences in the eleventh chapter of Bhagavad-Gita where the word deva, used in reference to Lord Krishna , is translated as "Lord". The word deva can be used to refer to the Supreme Lord, celestial beings, and saintly souls depending on the context. This is similar to the word Bhagavan , which is translated according to different contexts. Among

7052-457: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until

7138-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"

7224-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to

7310-406: Was replaced by a great, gray impenetrable barrier. In hearing this, Harry produces a letter he had received almost a year earlier from his sister. In it, they find an informal report on their progress in the experiment. Jean and Mark had recruited a group of educators, married couples only, to live and teach at the compound. All the educators, Jean and Mark included, acted as teacher/parents to all

7396-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In

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