160-396: Probability is the branch of mathematics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable;
320-591: 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
480-424: A σ-algebra of such events (such as those arising from a continuous random variable ). For example, in a bag of 2 red balls and 2 blue balls (4 balls in total), the probability of taking a red ball is 1 / 2 ; {\displaystyle 1/2;} however, when taking a second ball, the probability of it being either a red ball or a blue ball depends on the ball previously taken. For example, if
640-587: A bachelor. During visits to his sister at Port-Royal in 1654, he displayed contempt for affairs of the world but was not drawn to God. On the 23 of November, 1654, between 10:30 and 12:30 at night, Pascal had an intense religious experience and immediately wrote a brief note to himself which began: "Fire. God of Abraham, God of Isaac, God of Jacob, not of the philosophers and the scholars..." and concluded by quoting Psalm 119:16: "I will not forget thy word. Amen." He seems to have carefully sewn this document into his coat and always transferred it when he changed clothes;
800-405: A barometer tube. This work was followed by Récit de la grande expérience de l'équilibre des liqueurs ("Account of the great experiment on equilibrium in liquids") published in 1648. The Torricellian vacuum found that air pressure is equal to the weight of 30 inches of mercury. If air has a finite weight, Earth's atmosphere must have a maximum height. Pascal reasoned that if true, air pressure on
960-421: A card from a deck of cards, the chance of getting a heart or a face card (J, Q, K) (or both) is 13 52 + 12 52 − 3 52 = 11 26 , {\displaystyle {\tfrac {13}{52}}+{\tfrac {12}{52}}-{\tfrac {3}{52}}={\tfrac {11}{26}},} since among the 52 cards of a deck, 13 are hearts, 12 are face cards, and 3 are both: here
1120-515: A duplicitous world that shapes us into duplicitous subjects and so we find it easy to reject God continually and deceive ourselves about our own sinfulness". Pascal's major contribution to the philosophy of mathematics came with his De l'Esprit géométrique ("Of the Geometrical Spirit"), originally written as a preface to a geometry textbook for one of the famous Petites écoles de Port-Royal ("Little Schools of Port-Royal"). The work
1280-669: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)
1440-422: A game early and, given the current circumstances of the game, want to divide the stakes fairly , based on the chance each has of winning the game from that point. From this discussion, the notion of expected value was introduced. John Ross writes, "Probability theory and the discoveries following it changed the way we regard uncertainty, risk, decision-making, and an individual's and society's ability to influence
1600-556: A high mountain must be less than at a lower altitude. He lived near the Puy de Dôme mountain, 4,790 feet (1,460 m) tall, but his health was poor so could not climb it. On 19 September 1648, after many months of Pascal's friendly but insistent prodding, Florin Périer , husband of Pascal's elder sister Gilberte, was finally able to carry out the fact-finding mission vital to Pascal's theory. The account, written by Périer, reads: The weather
1760-479: A long letter, as he had not had time to write a shorter one. From Letter XVI, as translated by Thomas M'Crie: 'Reverend fathers, my letters were not wont either to be so prolix, or to follow so closely on one another. Want of time must plead my excuse for both of these faults. The present letter is a very long one, simply because I had no leisure to make it shorter.' Charles Perrault wrote of the Letters : "Everything
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#17327727333841920-467: A maid who eventually became a key member of the family. Étienne, who never remarried, decided that he alone would educate his children. The young Pascal showed an extraordinary intellectual ability, with an amazing aptitude for mathematics and science. Etienne had tried to keep his son from learning mathematics; but by the age of 12, Pascal had rediscovered, on his own, using charcoal on a tile floor, Euclid ’s first thirty-two geometric propositions, and
2080-477: A manufacturer's decisions on a product's warranty . The cache language model and other statistical language models that are used in natural language processing are also examples of applications of probability theory. Consider an experiment that can produce a number of results. The collection of all possible results is called the sample space of the experiment, sometimes denoted as Ω {\displaystyle \Omega } . The power set of
2240-404: 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
2400-422: 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
2560-733: A mechanical calculator capable of addition and subtraction, called Pascal's calculator or the Pascaline . Of the eight Pascalines known to have survived, four are held by the Musée des Arts et Métiers in Paris and one more by the Zwinger museum in Dresden , Germany, exhibit two of his original mechanical calculators. Although these machines are pioneering forerunners to a further 400 years of development of mechanical methods of calculation, and in
2720-402: 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
2880-458: A popular ethical method used by Catholic thinkers in the early modern period (especially the Jesuits , and in particular Antonio Escobar ). Pascal denounced casuistry as the mere use of complex reasoning to justify moral laxity and all sorts of sins . The 18-letter series was published between 1656 and 1657 under the pseudonym Louis de Montalte and incensed Louis XIV . The king ordered that
3040-535: 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
3200-426: A red ball was taken, then the probability of picking a red ball again would be 1 / 3 , {\displaystyle 1/3,} since only 1 red and 2 blue balls would have been remaining. And if a blue ball was taken previously, the probability of taking a red ball will be 2 / 3. {\displaystyle 2/3.} In probability theory and applications, Bayes' rule relates
3360-588: A sense to the later field of computer engineering , the calculator failed to be a great commercial success. Partly because it was still quite cumbersome to use in practice, but probably primarily because it was extraordinarily expensive, the Pascaline became little more than a toy, and a status symbol , for the very rich both in France and elsewhere in Europe. Pascal continued to make improvements to his design through
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#17327727333843520-411: 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
3680-560: A servant discovered it only by chance after his death. This piece is now known as the Memorial . The story of a carriage accident as having led to the experience described in the Memorial is disputed by some scholars. His belief and religious commitment revitalized, Pascal visited the older of two convents at Port-Royal for a two-week retreat in January 1655. For the next four years, he regularly travelled between Port-Royal and Paris. It
3840-478: A significant treatise on the subject of conic sections at the age of 16. He later corresponded with Pierre de Fermat on probability theory , strongly influencing the development of modern economics and social science . In 1642, he started some pioneering work on calculating machines (called Pascal's calculators and later Pascalines), establishing him as one of the first two inventors of the mechanical calculator . Like his contemporary René Descartes , Pascal
4000-424: 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
4160-463: A single one of the phenomena, that suffices to establish its falsity." Blaise Pascal Chairs are given to outstanding international scientists to conduct their research in the Ile de France region. In the winter of 1646, Pascal's 58-year-old father broke his hip when he slipped and fell on an icy street of Rouen; given the man's age and the state of medicine in the 17th century, a broken hip could be
4320-491: A sniff: "I do not find it strange that he has offered demonstrations about conics more appropriate than those of the ancients," adding, "but other matters related to this subject can be proposed that would scarcely occur to a 16-year-old child." In France at that time offices and positions could be—and were—bought and sold. In 1631, Étienne sold his position as second president of the Cour des Aides for 65,665 livres . The money
4480-521: A splinter group from Catholic teaching known as Jansenism . This still fairly small sect was making surprising inroads into the French Catholic community at that time. It espoused rigorous Augustinism . Blaise spoke with the doctors frequently, and after their successful treatment of his father, borrowed from them works by Jansenist authors. In this period, Pascal experienced a sort of "first conversion" and began to write on theological subjects in
4640-418: 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
4800-403: A substance such as aether rather than vacuum filled the space. Following more experimentation in this vein, in 1647 Pascal produced Experiences nouvelles touchant le vide ("New experiments with the vacuum"), which detailed basic rules describing to what degree various liquids could be supported by air pressure . It also provided reasons why it was indeed a vacuum above the column of liquid in
4960-638: A vacuum . In 1646, he and his sister Jacqueline identified with the religious movement within Catholicism known by its detractors as Jansenism . Following a religious experience in late 1654, he began writing influential works on philosophy and theology. His two most famous works date from this period: the Lettres provinciales and the Pensées , the former set in the conflict between Jansenists and Jesuits . The latter contains Pascal's wager , known in
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5120-480: A very serious condition, perhaps even fatal. Rouen was home to two of the finest doctors in France, Deslandes and de la Bouteillerie. The elder Pascal "would not let anyone other than these men attend him...It was a good choice, for the old man survived and was able to walk again..." However treatment and rehabilitation took three months, during which time La Bouteillerie and Deslandes had become regular visitors. Both men were followers of Jean Guillebert , proponent of
5280-423: A vessel...then took several glass tubes...each four feet long and hermetically sealed at one end and opened at the other...then placed them in the vessel [of quicksilver]...I found the quick silver stood at 26" and 3 + 1 ⁄ 2 lines above the quicksilver in the vessel...I repeated the experiment two more times while standing in the same spot...[they] produced the same result each time... I attached one of
5440-477: 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
5600-703: 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
5760-453: Is flat " and "a field is always a ring ". Blaise Pascal Blaise Pascal (19 June 1623 – 19 August 1662) was a French mathematician , physicist , inventor, philosopher , and Catholic writer. Pascal was a child prodigy who was educated by his father, a tax collector in Rouen . His earliest mathematical work was on projective geometry ; he wrote
5920-421: Is arrived at from inductive reasoning and statistical inference . The scientific study of probability is a modern development of mathematics. Gambling shows that there has been an interest in quantifying the ideas of probability throughout history, but exact mathematical descriptions arose much later. There are reasons for the slow development of the mathematics of probability. Whereas games of chance provided
6080-403: 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
6240-513: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of
6400-544: Is denoted as P ( A ∪ B ) {\displaystyle P(A\cup B)} and P ( A or B ) = P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) = P ( A ) + P ( B ) − 0 = P ( A ) + P ( B ) {\displaystyle P(A{\mbox{ or }}B)=P(A\cup B)=P(A)+P(B)-P(A\cap B)=P(A)+P(B)-0=P(A)+P(B)} For example,
6560-407: 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
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6720-446: Is formally undefined by this expression. In this case A {\displaystyle A} and B {\displaystyle B} are independent, since P ( A ∩ B ) = P ( A ) P ( B ) = 0. {\displaystyle P(A\cap B)=P(A)P(B)=0.} However, it is possible to define a conditional probability for some zero-probability events, for example by using
6880-578: Is given by P (not A ) = 1 − P ( A ) . As an example, the chance of not rolling a six on a six-sided die is 1 – (chance of rolling a six) = 1 − 1 / 6 = 5 / 6 . For a more comprehensive treatment, see Complementary event . If two events A and B occur on a single performance of an experiment, this is called the intersection or joint probability of A and B , denoted as P ( A ∩ B ) . {\displaystyle P(A\cap B).} If two events, A and B are independent then
7040-493: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as
7200-530: Is in France's Auvergne region , by the Massif Central . He lost his mother, Antoinette Begon, at the age of three. His father, Étienne Pascal , also an amateur mathematician, was a local judge and member of the " Noblesse de Robe ". Pascal had two sisters, the younger Jacqueline and the elder Gilberte . In 1631, five years after the death of his wife, Étienne Pascal moved with his children to Paris. The newly arrived family soon hired Louise Delfault,
7360-479: Is known still today as Pascal's theorem . It states that if a hexagon is inscribed in a circle (or conic) then the three intersection points of opposite sides lie on a line (called the Pascal line). Pascal's work was so precocious that René Descartes was convinced that Pascal's father had written it. When assured by Mersenne that it was, indeed, the product of the son and not the father, Descartes dismissed it with
7520-586: 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
7680-404: 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,
7840-1192: 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
8000-547: 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
8160-433: 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
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#17327727333848320-542: Is referred to as theoretical probability (in contrast to empirical probability , dealing with probabilities in the context of real experiments). For example, tossing a coin twice will yield "head-head", "head-tail", "tail-head", and "tail-tail" outcomes. The probability of getting an outcome of "head-head" is 1 out of 4 outcomes, or, in numerical terms, 1/4, 0.25 or 25%. However, when it comes to practical application, there are two major competing categories of probability interpretations, whose adherents hold different views about
8480-420: Is simply the ratio of the probabilities of the two events. When arbitrarily many events A {\displaystyle A} are of interest, not just two, the rule can be rephrased as posterior is proportional to prior times likelihood , P ( A | B ) ∝ P ( A ) P ( B | A ) {\displaystyle P(A|B)\propto P(A)P(B|A)} where
8640-567: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after
8800-639: Is the probability of some event A , given the occurrence of some other event B . Conditional probability is written P ( A ∣ B ) {\displaystyle P(A\mid B)} , and is read "the probability of A , given B ". It is defined by P ( A ∣ B ) = P ( A ∩ B ) P ( B ) {\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}\,} If P ( B ) = 0 {\displaystyle P(B)=0} then P ( A ∣ B ) {\displaystyle P(A\mid B)}
8960-418: 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,
9120-405: 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
9280-399: Is there—purity of language, nobility of thought, solidity in reasoning, finesse in raillery, and throughout an agrément not to be found anywhere else." Pascal is arguably best known as a philosopher, considered by some the second greatest French mind behind René Descartes . He was a dualist following Descartes. However, he is also remembered for his opposition to both the rationalism of
9440-508: 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
9600-442: Is used to design games of chance so that casinos can make a guaranteed profit, yet provide payouts to players that are frequent enough to encourage continued play. Another significant application of probability theory in everyday life is reliability . Many consumer products, such as automobiles and consumer electronics, use reliability theory in product design to reduce the probability of failure. Failure probability may influence
9760-586: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It
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#17327727333849920-549: The Copenhagen interpretation , it deals with probabilities of observing, the outcome being explained by a wave function collapse when an observation is made. However, the loss of determinism for the sake of instrumentalism did not meet with universal approval. Albert Einstein famously remarked in a letter to Max Born : "I am convinced that God does not play dice". Like Einstein, Erwin Schrödinger , who discovered
10080-768: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during
10240-507: The Kolmogorov formulation and the Cox formulation. In Kolmogorov's formulation (see also probability space ), sets are interpreted as events and probability as a measure on a class of sets. In Cox's theorem , probability is taken as a primitive (i.e., not further analyzed), and the emphasis is on constructing a consistent assignment of probability values to propositions. In both cases,
10400-511: The Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity
10560-536: 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
10720-446: 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
10880-680: The laws of probability are the same, except for technical details. There are other methods for quantifying uncertainty, such as the Dempster–Shafer theory or possibility theory , but those are essentially different and not compatible with the usually-understood laws of probability. Probability theory is applied in everyday life in risk assessment and modeling . The insurance industry and markets use actuarial science to determine pricing and make trading decisions. Governments apply probabilistic methods in environmental regulation , entitlement analysis, and financial regulation . An example of
11040-426: The odds of event A 1 {\displaystyle A_{1}} to event A 2 , {\displaystyle A_{2},} before (prior to) and after (posterior to) conditioning on another event B . {\displaystyle B.} The odds on A 1 {\displaystyle A_{1}} to event A 2 {\displaystyle A_{2}}
11200-431: The probable error of a single observation, is well known. In the nineteenth century, authors on the general theory included Laplace , Sylvestre Lacroix (1816), Littrow (1833), Adolphe Quetelet (1853), Richard Dedekind (1860), Helmert (1872), Hermann Laurent (1873), Liagre, Didion and Karl Pearson . Augustus De Morgan and George Boole improved the exposition of the theory. In 1906, Andrey Markov introduced
11360-402: The theory of probability is a representation of its concepts in formal terms – that is, in terms that can be considered separately from their meaning. These formal terms are manipulated by the rules of mathematics and logic, and any results are interpreted or translated back into the problem domain. There have been at least two successful attempts to formalize probability, namely
11520-400: 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
11680-405: 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
11840-532: 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
12000-584: 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 ,
12160-671: The arithmetical triangle, but is now called Pascal's triangle . The triangle can also be represented: He defined the numbers in the triangle by recursion : Call the number in the ( m + 1)th row and ( n + 1)th column t mn . Then t mn = t m –1, n + t m , n –1 , for m = 0, 1, 2, ... and n = 0, 1, 2, ... The boundary conditions are t m ,−1 = 0, t −1, n = 0 for m = 1, 2, 3, ... and n = 1, 2, 3, ... The generator t 00 = 1. Pascal concluded with
12320-399: The ball, variations in hand speed during the turning, and so forth. A probabilistic description can thus be more useful than Newtonian mechanics for analyzing the pattern of outcomes of repeated rolls of a roulette wheel. Physicists face the same situation in the kinetic theory of gases , where the system, while deterministic in principle , is so complex (with the number of molecules typically
12480-583: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During
12640-468: The book be shredded and burnt in 1660. In 1661, in the midst of the formulary controversy , the Jansenist school at Port-Royal was condemned and closed down; those involved with the school had to sign a 1656 papal bull condemning the teachings of Jansen as heretical. The final letter from Pascal, in 1657, had defied Alexander VII himself. Even Pope Alexander, while publicly opposing them, nonetheless
12800-428: The case of a roulette wheel, if the force of the hand and the period of that force are known, the number on which the ball will stop would be a certainty (though as a practical matter, this would likely be true only of a roulette wheel that had not been exactly levelled – as Thomas A. Bass' Newtonian Casino revealed). This also assumes knowledge of inertia and friction of the wheel, weight, smoothness, and roundness of
12960-1116: The chance of rolling a 1 or 2 on a six-sided die is P ( 1 or 2 ) = P ( 1 ) + P ( 2 ) = 1 6 + 1 6 = 1 3 . {\displaystyle P(1{\mbox{ or }}2)=P(1)+P(2)={\tfrac {1}{6}}+{\tfrac {1}{6}}={\tfrac {1}{3}}.} If the events are not (necessarily) mutually exclusive then P ( A or B ) = P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A and B ) . {\displaystyle P\left(A{\hbox{ or }}B\right)=P(A\cup B)=P\left(A\right)+P\left(B\right)-P\left(A{\mbox{ and }}B\right).} Rewritten, P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) {\displaystyle P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)} For example, when drawing
13120-443: The circumstances." However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence. The sixteenth-century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes). Aside from
13280-500: The concept of a measure. The opposite or complement of an event A is the event [not A ] (that is, the event of A not occurring), often denoted as A ′ , A c {\displaystyle A',A^{c}} , A ¯ , A ∁ , ¬ A {\displaystyle {\overline {A}},A^{\complement },\neg A} , or ∼ A {\displaystyle {\sim }A} ; its probability
13440-511: 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,
13600-480: The course of future events." Pascal, in the Pensées , used a probabilistic argument, Pascal's wager , to justify belief in God and a virtuous life. However, Pascal and Fermat, though doing important early work in probability theory, did not develop the field very far. Christiaan Huygens , learning of the subject from the correspondence of Pascal and Fermat, wrote the first book on the subject. Later figures who continued
13760-423: The course of the following year. Pascal fell away from this initial religious engagement and experienced a few years of what some biographers have called his "worldly period" (1648–54). His father died in 1651 and left his inheritance to Pascal and his sister Jacqueline, for whom Pascal acted as conservator. Jacqueline announced that she would soon become a postulant in the Jansenist convent of Port-Royal . Pascal
13920-399: 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
14080-552: The curve equals 1. He gave two proofs, the second being essentially the same as John Herschel 's (1850). Gauss gave the first proof that seems to have been known in Europe (the third after Adrain's) in 1809. Further proofs were given by Laplace (1810, 1812), Gauss (1823), James Ivory (1825, 1826), Hagen (1837), Friedrich Bessel (1838), W.F. Donkin (1844, 1856), and Morgan Crofton (1870). Other contributors were Ellis (1844), De Morgan (1864), Glaisher (1872), and Giovanni Schiaparelli (1875). Peters 's (1856) formula for r ,
14240-442: The cycloid. His toothache disappeared, and he took this as a heavenly sign to proceed with his research. Eight days later he had completed his essay and, to publicize the results, proposed a contest. Pascal proposed three questions relating to the center of gravity , area and volume of the cycloid, with the winner or winners to receive prizes of 20 and 40 Spanish doubloons . Pascal, Gilles de Roberval and Pierre de Carcavi were
14400-569: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely
14560-556: 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
14720-415: The development of the theory include Abraham de Moivre and Pierre-Simon Laplace . The work done by Fermat and Pascal into the calculus of probabilities laid important groundwork for Leibniz 's formulation of the calculus . Pascal's Traité du triangle arithmétique , written in 1654 but published posthumously in 1665, described a convenient tabular presentation for binomial coefficients which he called
14880-460: The early development of the very concept of mathematical probability. The theory of errors may be traced back to Roger Cotes 's Opera Miscellanea (posthumous, 1722), but a memoir prepared by Thomas Simpson in 1755 (printed 1756) first applied the theory to the discussion of errors of observation. The reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable, and that certain assignable limits define
15040-478: The effect of such groupthink on pricing, on policy, and on peace and conflict. In addition to financial assessment, probability can be used to analyze trends in biology (e.g., disease spread) as well as ecology (e.g., biological Punnett squares ). As with finance, risk assessment can be used as a statistical tool to calculate the likelihood of undesirable events occurring, and can assist with implementing protocols to avoid encountering such circumstances. Probability
15200-511: The elementary work by Cardano, the doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal (1654). Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject. Jakob Bernoulli 's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre 's Doctrine of Chances (1718) treated the subject as a branch of mathematics. See Ian Hacking 's The Emergence of Probability and James Franklin's The Science of Conjecture for histories of
15360-435: The end of October in 1651, a truce had been reached between brother and sister. In return for a healthy annual stipend, Jacqueline signed over her part of the inheritance to her brother. Gilberte had already been given her inheritance in the form of a dowry. In early January, Jacqueline left for Port-Royal. On that day, according to Gilberte concerning her brother, "He retired very sadly to his rooms without seeing Jacqueline, who
15520-551: The events {1,6}, {3}, and {2,4}), the probability that at least one of the events will occur is given by the sum of the probabilities of all the individual events. The probability of an event A is written as P ( A ) {\displaystyle P(A)} , p ( A ) {\displaystyle p(A)} , or Pr ( A ) {\displaystyle {\text{Pr}}(A)} . This mathematical definition of probability can extend to infinite sample spaces, and even uncountable sample spaces, using
15680-428: 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
15840-426: The expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems . When dealing with random experiments – i.e., experiments that are random and well-defined – in a purely theoretical setting (like tossing a coin), probabilities can be numerically described by the number of desired outcomes, divided by the total number of all outcomes. This
16000-409: The experiment five times with care...each at different points on the summit...found the same height of quicksilver...in each case... Pascal replicated the experiment in Paris by carrying a barometer up to the top of the bell tower at the church of Saint-Jacques-de-la-Boucherie , a height of about 50 metres. The mercury dropped two lines. He found with both experiments that an ascent of 7 fathoms lowers
16160-409: The fields of hydrodynamics and hydrostatics centered on the principles of hydraulic fluids . His inventions include the hydraulic press (using hydraulic pressure to multiply force) and the syringe . He proved that hydrostatic pressure depends not on the weight of the fluid but on the elevation difference. He demonstrated this principle by attaching a thin tube to a barrel full of water and filling
16320-478: The first published proof. Pascal contributed to several fields in physics, most notably the fields of fluid mechanics and pressure. In honour of his scientific contributions, the name Pascal has been given to the SI unit of pressure and Pascal's law (an important principle of hydrostatics). He introduced a primitive form of roulette and the roulette wheel in his search for a perpetual motion machine. His work in
16480-567: 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",
16640-411: The fundamental nature of probability: The word probability derives from the Latin probabilitas , which can also mean " probity ", a measure of the authority of a witness in a legal case in Europe, and often correlated with the witness's nobility . In a sense, this differs much from the modern meaning of probability , which in contrast is a measure of the weight of empirical evidence , and
16800-411: The impetus for the mathematical study of probability, fundamental issues are still obscured by superstitions. According to Richard Jeffrey , "Before the middle of the seventeenth century, the term 'probable' (Latin probabilis ) meant approvable , and was applied in that sense, univocally, to opinion and to action. A probable action or opinion was one such as sensible people would undertake or hold, in
16960-491: 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
17120-400: 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
17280-688: The joint probability is P ( A and B ) = P ( A ∩ B ) = P ( A ) P ( B ) . {\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)=P(A)P(B).} For example, if two coins are flipped, then the chance of both being heads is 1 2 × 1 2 = 1 4 . {\displaystyle {\tfrac {1}{2}}\times {\tfrac {1}{2}}={\tfrac {1}{4}}.} If either event A or event B can occur but never both simultaneously, then they are called mutually exclusive events. If two events are mutually exclusive , then
17440-465: The judges, and neither of the two submissions (by John Wallis and Antoine de Lalouvère ) were judged to be adequate. While the contest was ongoing, Christopher Wren sent Pascal a proposal for a proof of the rectification of the cycloid; Roberval claimed promptly that he had known of the proof for years. Wallis published Wren's proof (crediting Wren) in Wallis's Tractus Duo , giving Wren priority for
17600-538: The language and understood by everyone because they naturally designate their referent. The second type would be characteristic of the philosophy of essentialism . Pascal claimed that only definitions of the first type were important to science and mathematics, arguing that those fields should adopt the philosophy of formalism as formulated by Descartes. In De l'Art de persuader ("On the Art of Persuasion"), Pascal looked deeper into geometry's axiomatic method , specifically
17760-400: The law of facility of error, ϕ ( x ) = c e − h 2 x 2 {\displaystyle \phi (x)=ce^{-h^{2}x^{2}}} where h {\displaystyle h} is a constant depending on precision of observation, and c {\displaystyle c} is a scale factor ensuring that the area under
17920-413: The likes of Descartes and simultaneous opposition to the main countervailing epistemology, empiricism , preferring fideism . In terms of God, Descartes and Pascal disagreed. Pascal wrote that "I cannot forgive Descartes. In all his philosophy he would have been quite willing to dispense with God, but he couldn't avoid letting him put the world in motion; afterwards he didn't need God anymore". He opposed
18080-409: 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
18240-409: The mercury by half a line. Note: Pascal used pouce and ligne for "inch" and "line", and toise for "fathom". In a reply to Étienne Noël , who believed in the plenum, Pascal wrote, echoing contemporary notions of science and falsifiability : "In order to show that a hypothesis is evident, it does not suffice that all the phenomena follow from it; instead, if it leads to something contrary to
18400-400: 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
18560-409: 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 a foundation for all mathematics). Mathematics involves
18720-482: The next decade, and he refers to some 50 machines that were built to his design. He built 20 finished machines over the following 10 years. In 1654, prompted by his friend the Chevalier de Méré , Pascal corresponded with Pierre de Fermat on the subject of gambling problems, and from that collaboration was born the mathematical theory of probability . The specific problem was that of two players who want to finish
18880-435: The notion of Markov chains , which played an important role in stochastic processes theory and its applications. The modern theory of probability based on measure theory was developed by Andrey Kolmogorov in 1931. On the geometric side, contributors to The Educational Times included Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin . See integral geometry for more information. Like other theories ,
19040-536: 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
19200-518: The order of magnitude of the Avogadro constant 6.02 × 10 ) that only a statistical description of its properties is feasible. Probability theory is required to describe quantum phenomena. A revolutionary discovery of early 20th century physics was the random character of all physical processes that occur at sub-atomic scales and are governed by the laws of quantum mechanics . The objective wave function evolves deterministically but, according to
19360-598: The original as the Discourse on the Machine , a fideistic probabilistic argument for why one should believe in God. In that year, he also wrote an important treatise on the arithmetical triangle. Between 1658 and 1659, he wrote on the cycloid and its use in calculating the volume of solids. Following several years of illness, Pascal died in Paris at the age of 39. Pascal was born in Clermont-Ferrand , which
19520-521: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000 BC , when
19680-2599: The possibilities included in the "3 that are both" are included in each of the "13 hearts" and the "12 face cards", but should only be counted once. This can be expanded further for multiple not (necessarily) mutually exclusive events. For three events, this proceeds as follows: P ( A ∪ B ∪ C ) = P ( ( A ∪ B ) ∪ C ) = P ( A ∪ B ) + P ( C ) − P ( ( A ∪ B ) ∩ C ) = P ( A ) + P ( B ) − P ( A ∩ B ) + P ( C ) − P ( ( A ∩ C ) ∪ ( B ∩ C ) ) = P ( A ) + P ( B ) + P ( C ) − P ( A ∩ B ) − ( P ( A ∩ C ) + P ( B ∩ C ) − P ( ( A ∩ C ) ∩ ( B ∩ C ) ) ) P ( A ∪ B ∪ C ) = P ( A ) + P ( B ) + P ( C ) − P ( A ∩ B ) − P ( A ∩ C ) − P ( B ∩ C ) + P ( A ∩ B ∩ C ) {\displaystyle {\begin{aligned}P\left(A\cup B\cup C\right)=&P\left(\left(A\cup B\right)\cup C\right)\\=&P\left(A\cup B\right)+P\left(C\right)-P\left(\left(A\cup B\right)\cap C\right)\\=&P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)+P\left(C\right)-P\left(\left(A\cap C\right)\cup \left(B\cap C\right)\right)\\=&P\left(A\right)+P\left(B\right)+P\left(C\right)-P\left(A\cap B\right)-\left(P\left(A\cap C\right)+P\left(B\cap C\right)-P\left(\left(A\cap C\right)\cap \left(B\cap C\right)\right)\right)\\P\left(A\cup B\cup C\right)=&P\left(A\right)+P\left(B\right)+P\left(C\right)-P\left(A\cap B\right)-P\left(A\cap C\right)-P\left(B\cap C\right)+P\left(A\cap B\cap C\right)\end{aligned}}} It can be seen, then, that this pattern can be repeated for any number of events. Conditional probability
19840-494: The principle of the maximum product of the probabilities of a system of concurrent errors. Adrien-Marie Legendre (1805) developed the method of least squares , and introduced it in his Nouvelles méthodes pour la détermination des orbites des comètes ( New Methods for Determining the Orbits of Comets ). In ignorance of Legendre's contribution, an Irish-American writer, Robert Adrain , editor of "The Analyst" (1808), first deduced
20000-399: The probability of both occurring is denoted as P ( A ∩ B ) {\displaystyle P(A\cap B)} and P ( A and B ) = P ( A ∩ B ) = 0 {\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)=0} If two events are mutually exclusive , then the probability of either occurring
20160-552: The probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These concepts have been given an axiomatic mathematical formalization in probability theory , which is used widely in areas of study such as statistics , mathematics , science , finance , gambling , artificial intelligence , machine learning , computer science , game theory , and philosophy to, for example, draw inferences about
20320-412: The procedure used in geometry was as perfect as possible, with certain principles assumed and other propositions developed from them. Nevertheless, there was no way to know the assumed principles to be true. Pascal also used De l'Esprit géométrique to develop a theory of definition . He distinguished between definitions which are conventional labels defined by the writer and definitions which are within
20480-658: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been
20640-452: The proof, In the same treatise, Pascal gave an explicit statement of the principle of mathematical induction . In 1654, he proved Pascal's identity relating the sums of the p -th powers of the first n positive integers for p = 0, 1, 2, ..., k . That same year, Pascal had a religious experience, and mostly gave up work in mathematics. In 1658, Pascal, while suffering from a toothache, began considering several problems concerning
20800-646: The proportionality symbol means that the left hand side is proportional to (i.e., equals a constant times) the right hand side as A {\displaystyle A} varies, for fixed or given B {\displaystyle B} (Lee, 2012; Bertsch McGrayne, 2012). In this form it goes back to Laplace (1774) and to Cournot (1843); see Fienberg (2005). In a deterministic universe, based on Newtonian concepts, there would be no probability if all conditions were known ( Laplace's demon ) (but there are situations in which sensitivity to initial conditions exceeds our ability to measure them, i.e. know them). In
20960-419: The range of all errors. Simpson also discusses continuous errors and describes a probability curve. The first two laws of error that were proposed both originated with Pierre-Simon Laplace . The first law was published in 1774, and stated that the frequency of an error could be expressed as an exponential function of the numerical magnitude of the error – disregarding sign. The second law of error
21120-464: The rationalism of people like Descartes as applied to the existence of a God, preferring faith as "reason can decide nothing here". For Pascal the nature of God was such that such proofs cannot reveal God. Humans "are in darkness and estranged from God" because "he has hidden Himself from their knowledge". He cared above all about the philosophy of religion. Pascalian theology has grown out of his perspective that humans are, according to Wood, "born into
21280-486: The results that actually occur fall in a given event, the event is said to have occurred. A probability is a way of assigning every event a value between zero and one, with the requirement that the event made up of all possible results (in our example, the event {1,2,3,4,5,6}) is assigned a value of one. To qualify as a probability, the assignment of values must satisfy the requirement that for any collection of mutually exclusive events (events with no common results, such as
21440-425: The sample space is formed by considering all different collections of possible results. For example, rolling a die can produce six possible results. One collection of possible results gives an odd number on the die. Thus, the subset {1,3,5} is an element of the power set of the sample space of dice rolls. These collections are called "events". In this case, {1,3,5} is the event that the die falls on some odd number. If
21600-657: 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
21760-568: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and
21920-672: 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,
22080-487: 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
22240-476: The tube and what filled the space above the mercury in the tube. At the time, most scientists including Descartes believed in a plenum, i. e. some invisible matter filled all of space, rather than a vacuum (" Nature abhors a vacuum )." This was based on the Aristotelian notion that everything in motion was a substance, moved by another substance. Furthermore, light passed through the glass tube, suggesting
22400-415: The tube with water up to the level of the third floor of a building. This caused the barrel to leak, in what became known as Pascal's barrel experiment. By 1647, Pascal had learned of Evangelista Torricelli 's experimentation with barometers . Having replicated an experiment that involved placing a tube filled with mercury upside down in a bowl of mercury, Pascal questioned what force kept some mercury in
22560-485: The tubes to the vessel and marked the height of the quicksilver and...asked Father Chastin, one of the Minim Brothers...to watch if any changes should occur through the day...Taking the other tube and a portion of the quick silver...I walked to the top of Puy-de-Dôme, about 500 fathoms higher than the monastery, where upon experiment...found that the quicksilver reached a height of only 23" and 2 lines...I repeated
22720-508: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in
22880-496: The use of probability theory in equity trading is the effect of the perceived probability of any widespread Middle East conflict on oil prices, which have ripple effects in the economy as a whole. An assessment by a commodity trader that a war is more likely can send that commodity's prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are neither assessed independently nor necessarily rationally. The theory of behavioral finance emerged to describe
23040-470: The wave function, believed quantum mechanics is a statistical approximation of an underlying deterministic reality . In some modern interpretations of the statistical mechanics of measurement, quantum decoherence is invoked to account for the appearance of subjectively probabilistic experimental outcomes. Mathematics Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for
23200-452: Was also a pioneer in the natural and applied sciences. Pascal wrote in defense of the scientific method and produced several controversial results. He made important contributions to the study of fluids , and clarified the concepts of pressure and vacuum by generalising the work of Evangelista Torricelli . Following Torricelli and Galileo Galilei , in 1647 he rebutted the likes of Aristotle and Descartes who insisted that nature abhors
23360-591: Was at this point immediately after his conversion when he began writing his first major literary work on religion, the Provincial Letters . In literature, Pascal is regarded as one of the most important authors of the French Classical Period and is read today as one of the greatest masters of French prose. His use of satire and wit influenced later polemicists . Beginning in 1656–57, Pascal published his memorable attack on casuistry ,
23520-461: Was back in good graces with the Cardinal and in 1639 had been appointed the king's commissioner of taxes in the city of Rouen —a city whose tax records, thanks to uprisings, were in utter chaos. In 1642, in an effort to ease his father's endless, exhausting calculations, and recalculations, of taxes owed and paid (into which work the young Pascal had been recruited), Pascal, not yet 19, constructed
23680-579: Was chancy last Saturday...[but] around five o'clock that morning...the Puy-de-Dôme was visible...so I decided to give it a try. Several important people of the city of Clermont had asked me to let them know when I would make the ascent...I was delighted to have them with me in this great work... ...at eight o'clock we met in the gardens of the Minim Fathers, which has the lowest elevation in town....First I poured 16 pounds of quicksilver ...into
23840-497: Was deeply affected and very sad, not because of her choice, but because of his chronic poor health; he needed her just as she had needed him. Suddenly there was war in the Pascal household. Blaise pleaded with Jacqueline not to leave, but she was adamant. He commanded her to stay, but that didn't work, either. At the heart of this was...Blaise's fear of abandonment...if Jacqueline entered Port-Royal, she would have to leave her inheritance behind...[but] nothing would change her mind. By
24000-420: Was eventually forced to flee Paris because of his opposition to the fiscal policies of Richelieu, leaving his three children in the care of his neighbour Madame Sainctot, a great beauty with an infamous past who kept one of the most glittering and intellectual salons in all France. It was only when Jacqueline performed well in a children's play with Richelieu in attendance that Étienne was pardoned. In time, Étienne
24160-462: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until
24320-414: 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"
24480-476: Was invested in a government bond which provided, if not a lavish, then certainly a comfortable income which allowed the Pascal family to move to, and enjoy, Paris, but in 1638 Cardinal Richelieu , desperate for money to carry on the Thirty Years' War , defaulted on the government's bonds. Suddenly Étienne Pascal's worth had dropped from nearly 66,000 livres to less than 7,300. Like so many others, Étienne
24640-437: 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
24800-430: Was persuaded by Pascal's arguments. Aside from their religious influence, the Provincial Letters were popular as a literary work. Pascal's use of humor, mockery, and vicious satire in his arguments made the letters ripe for public consumption, and influenced the prose of later French writers like Voltaire and Jean-Jacques Rousseau . It is in the Provincial Letters that Pascal made his oft-quoted apology for writing
24960-467: Was proposed in 1778 by Laplace, and stated that the frequency of the error is an exponential function of the square of the error. The second law of error is called the normal distribution or the Gauss law. "It is difficult historically to attribute that law to Gauss, who in spite of his well-known precocity had probably not made this discovery before he was two years old." Daniel Bernoulli (1778) introduced
25120-571: 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
25280-475: Was thus given a copy of Euclid's Elements . Particularly of interest to Pascal was a work of Desargues on conic sections . Following Desargues' thinking, the 16-year-old Pascal produced, as a means of proof, a short treatise on what was called the Mystic Hexagram , Essai pour les coniques ( Essay on Conics ) and sent it — his first serious work of mathematics — to Père Mersenne in Paris; it
25440-432: Was unpublished until over a century after his death. Here, Pascal looked into the issue of discovering truths, arguing that the ideal of such a method would be to found all propositions on already established truths. At the same time, however, he claimed this was impossible because such established truths would require other truths to back them up—first principles, therefore, cannot be reached. Based on this, Pascal argued that
25600-408: Was waiting in the little parlor..." In early June 1653, after what must have seemed like endless badgering from Jacqueline, Pascal formally signed over the whole of his sister's inheritance to Port-Royal, which, to him, "had begun to smell like a cult." With two-thirds of his father's estate now gone, the 29-year-old Pascal was now consigned to genteel poverty. For a while, Pascal pursued the life of
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