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Analytic Geometry

Analytic geometry

Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. Usually the Cartesian coordinate system is applied to manipulate equations for planes, lines, curves, and circles, often in two and sometimes in three dimensions of measurement. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining geometrical shapes in a numerical way, and extracting numerical information from that representation. The numerical output, however, might also be a vector or a shape. Some consider that the introduction of analytic geometry was the beginning of modern mathematics. René Descartes is popularly regarded as having introduced the foundation for the methods of analytic geometry in 1637 in the appendix titled Geometry of the titled Discourse on the Method of Rightly Conducting the Reason in the Search for Truth in the Sciences, commonly referred to as Discourse on Method. This work, written in his native language (French), and its philosophical principles, provided the foundation for calculus in Europe.

Important themes of analytical geometry


- vector space
- definition of the plane
- distance problems
- the dot product, to get the angle of two vectors
- the cross product, to get a perpendicular vector of two known vectors (and also their spatial volume)
- intersection problems Many of these problems involve linear algebra

Example

Here is an example of a problem from the USAMTS that can be solved via analytic geometry: Problem: In a convex pentagon ABCDE, the sides have lengths 1, 2, 3, 4, and 5, though not necessarily in that order. Let F, G, H, and I be the midpoints of the sides AB, BC, CD, and DE, respectively. Let X be the midpoint of segment FH, and Y be the midpoint of segment GI. The length of segment XY is an integer. Find all possible values for the length of side AE. Solution: Let A, B, C, D, and E be located at A(0,0), B(a,0), C(b,e), D(c,f), and E(d,g). Using the midpoint formula, the points F, G, H, I, X, and Y are located at :F\left(\frac,0\right), G\left(\frac,\frac\right), H\left(\frac,\frac\right), I\left(\frac,\frac\right), X\left(\frac,\frac\right), and Y\left(\frac,\frac\right). Using the distance formula, :AE=\sqrt and :XY=\sqrt=\frac. Since XY has to be an integer, :AE\equiv 0\pmod (see modular arithmetic) so AE=4.

Other uses

Analytic geometry, for algebraic geometers, is also the name for the theory of (real or) complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables (or sometimes real ones). It is closely linked to algebraic geometry, especially through the work of Jean-Pierre Serre in GAGA. It is strictly a larger area than algebraic geometry, but studied by similar methods. Category:Geometry Category:Algebraic geometry ja:解析幾何学

Geometry

Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. It was one of the two fields of pre-modern mathematics, the other being the study of numbers. In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. (See areas of mathematics and algebraic geometry.)

The earliest geometry

The earliest recorded beginnings of geometry may be traced to Ancient Egypt (see geometry in Egypt) and Ancient Babylon (see Babylonian mathematics) around 3000 B.C. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of calculus. For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras; the Egyptians had a correct formula for the volume of a frustum of a square pyramid; the Babylonians had a trigonometry table. Chinese culture at this same time period was equally advanced, so it is likely that they had an equally advanced mathematics, but no artifacts have survived from which we could learn about it. This may be partly due to their early use of paper, rather than clay tablets or stone, to record their achievements.

The Greek period (c. 600 B.C. – 600 A.D.)

The Greek Period must be considered in detail, since geometry, for most of its history, was what the Greeks made it. For the Ancient Greeks, geometry was the crown jewel of their sciences, reaching a completeness and perfection of methodology that no other branch of their knowledge had attained. They expanded the range of geometry to many new kinds of figures, curves, surfaces, and solids; they changed its methodology from trial-and-error to logical deduction; they recognized that geometry studies “eternal forms”, or abstractions, of which physical objects are only approximations; and they developed the idea of an “axiomatic theory”, which, for more than 2000 years, was regarded to be the ideal paradigm for all scientific theories.

Thales and Pythagoras

Thales (635-543 B.C.) of Ionia (now southwestern Turkey), was the first to whom deduction in mathematics is attributed. There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived. Pythagoras (582-496 B.C.) of Ionia, and later, Italy, then colonized by Greeks, may have been a student of Thales, and probably traveled to Babylon and Egypt. The theorem that bears his name was not his discovery, but he was the first to give a deductive proof of it. He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses. In addition, they made the profound discovery of incommensurable lengths and irrational numbers.

Plato

Plato (427-347 B.C.), the philosopher most esteemed by the Greeks, had inscribed above the entrance to his famous school, “Let none enter here who are ignorant of geometry.” Though he was not a mathematician himself, his views on mathematics had great influence. Mathematicians thus accepted his belief that geometry should use no tools but a compass and straight edge – never measuring instruments such as a marked ruler or a protractor, because these were a workman’s tools, not worthy of a scholar. This dictum led to a deep study of the possible ruler and compass constructions, and three classic ruler-and-compass problems: how to use these tools to trisect an angle, to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. The proofs of the impossibility of these constructions, finally achieved in the 19th century, led to important principles regarding the deep structure of the real number system. Aristotle (384-322 B.C.), Plato’s greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see Logic) which was not substantially improved upon until the 19th century.

Euclid

Euclid (365?-275? B.C.), probably a student of one of Plato’s students, wrote a treatise in 13 books (chapters), titled The Elements of Geometry, in which he presented geometry in the ideal axiomatic form. The treatise is not a compendium of all that the Greeks knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid’s was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by Ptolemy I, King of Egypt. The Elements began with definitions of terms, fundamental geometric principles (called axioms or postulates), and general quantitative principles (called common notions) from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read. # Any two points can be joined by a straight line. # Any finite straight line can be extended in a straight line. # A circle can be drawn with any center and any radius. # All right angles are equal to each other. # If two straight lines in a plane are crossed by another straight line (called the transversal), and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect (also called the parallel postulate). It was soon observed, and no doubt Euclid himself knew, that his fifth axiom could be replaced by the shorter statement “Given a line and a point not on the line, there is only one line through the given point and in the same plane with the given line that does not intersect the given line.” This is called Playfair’s Axiom, after the British teacher who proposed to make the replacement in all the school textbooks. The axioms, according to Plato, should be simple and self-evident principles, so clearly true that they need no proof. Euclid’s first four axioms meet this criterion, but the fifth, even if replaced by Playfair’s Axiom, is not simple, and most would say not self-evident like the first four. The fifth resembled more the theorems that Euclid proved from the axioms. Furthermore, Euclid developed a substantial part of his theory of triangles without using the Fifth Axiom. The speculation arose, probably during Euclid’s lifetime, that the Fifth Axiom can and should be proved as a theorem from the first four, and thus is unnecessary as an axiom. Thus began many centuries of attempts to prove the Fifth Axiom, and the question was not settled until the 19th century.

Archimedes

Archimedes (287-212 B.C.), of Syracuse, Sicily, when it was a Greek city-state, was the greatest of the Greek mathematicians, and often named as one of the three greatest of all time (along with Isaac Newton and Carl Friedrich Gauss). Had he not been a mathematician, he would still be remembered as a great physicist, engineer, and inventor. In his mathematics, he developed methods very similar to the coordinate systems of analytic geometry, and the limiting process of integral calculus. The only element lacking for the creation of these fields was an efficient algebraic notation in which to express his concepts.

After Archimedes

After Archimedes, Greek mathematics began to decline. There were a few minor stars yet to come, but the golden age of geometry was over. Proclus (410-485), author of Commentary on the First Book of Euclid, was one of the last important players in Greek geometry. He was a competent geometer, but more importantly, he was a superb commentator on the works that preceded him. Much of that work did not survive to modern times, and is known to us only through his commentary. The Roman Republic and Empire that succeeded and absorbed the Greek city-states produced excellent engineers, but no mathematicians of note.

The Middle Ages, Renaissance, and Reformation

The great library of Alexandria was burned. There is a growing consensus among historians that the Library of Alexandria likely suffered from several destructive events, but that the destruction of Alexandria's pagan temples in the late 4th century was probably the most severe and final one. The evidence for that destruction is the most definitive and secure. Caesar's invasion may well have led to the loss of some 40,000-70,000 scrolls in a warehouse adjacent to the port (as Luciano Canfora argues, they were likely copies produced by the Library intended for export), but it is unlikely to have affected the Library or Museum, given that there is ample evidence that both existed later. Civil wars, decreasing investments in maintenance and acquisition of new scrolls and generally declining interest in non-religious pursuits likely contributed to a reduction in the body of material available in the Library, especially in the fourth century. The Serapeum was certainly destroyed by Theophilus in 391, and the Museum and Library may have fallen victim to the same campaign. The Islamic ascendency in the Middle East, north Africa, and Spain began about 640 A.D. Original Arab mathematics during this period was primarily algebraic rather than geometric, though there were important commentaries on geometry. Omar Khayyám, for example, was a geometer as well as a poet. Scholarship in Europe declined until even the great works of antiquity were lost to them, and survived only in the Islamic centers of learning. When Europe started to emerge from the intellectual darkness of the Middle Ages, the writers of Ancient Greece and Rome were rediscovered in Islamic libraries and translated from Arabic into Latin. Euclid’s Elements of Geometry was recovered, and the rigorous deductive methods of geometry were relearned. Development of geometry in the style of Euclid resumed, resulting in an abundance of new theorems and concepts, many of them very profound and elegant.

The 17th and early 18th centuries

In the early 17th century, there were two important developments in geometry. The first and most important was the creation of analytic geometry, or geometry with coordinates and equations, by Rene Descartes (1596-1650) and Pierre de Fermat (1601-1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591-1661). Projective geometry is the study of geometry without measurement, just the study of how points align with each other. There had been some early work in this area by Greek geometers, notably Pappus (c. 340). The greatest flowering of the field occurred with Jean-Victor Poncelet (1788-1867). In the late 17th century, calculus was developed independently and almost simultaneously by Isaac Newton (1642-1727) and Gottfried Wilhelm von Leibniz (1646-1716). This was the beginning of a new field of mathematics now called analysis. Though not itself a branch of geometry, it is applicable to geometry, and it solved two families of problems that had long been almost intractable: finding tangent lines to odd curves, and finding areas enclosed by those curves. The methods of calculus reduced these problems mostly to straightforward matters of computation.

The late 18th and 19th centuries

Non-Euclidean geometry

The old problem of proving Euclid’s Fifth Postulate, the "Parallel Postulate", from his first four postulates had never been forgotten. Beginning not long after Euclid, many attempted demonstrations were given, but all were later found to be faulty, through allowing into the reasoning some principle which itself had not been proved from the first four postulates. By 1700 a great deal had been discovered about what can be proved from the first four, and what the pitfalls were in attempting to prove the fifth. Saccheri, Lambert, and Legendre each did excellent work on the problem in the 18th century, but still fell short of success. In the early 19th century, Gauss, Johann Bolyai, and Lobatchewsky, each independently, took a different approach. Beginning to suspect that it was impossible to prove the Parallel Postulate, they set out to develop a self-consistent geometry in which that postulate was false. In this they were successful, thus creating the first non-Euclidean geometry. By 1854, Bernhard Riemann, a student of Gauss, had applied methods of calculus in a ground-breaking study of the intrinsic (self-contained) geometry of all smooth surfaces, and thereby found a different non-Euclidean geometry. This work of Riemann later became fundamental for Einstein's theory of relativity. It remained to prove mathematically that the non-Euclidean geometry was just as self-consistent as Euclidean geometry, and this was first accomplished by Beltrami in 1868. With this, non-Euclidean geometry was established on an equal mathematical footing with Euclidean geometry. While it was now known that different geometric theories were mathematically possible, the question remained, "Which one of these theories is correct for our physical space?" The mathematical work revealed that this question must be answered by physical experimentation, not mathematical reasoning, and uncovered the reason why the experimentation must involve immense (interstellar, not earth-bound) distances. With the development of relativity theory in physics, this question became vastly more complicated.

Introduction of mathematical rigor

All the work related to the Parallel Postulate revealed that it was quite difficult for a geometer to separate his logical reasoning from his intuitive understanding of physical space, and, moreover, revealed the critical importance of doing so. Careful examination had uncovered some logical inadequacies in Euclid's reasoning, and some unstated geometric principles to which Euclid sometimes appealed. This critique paralleled the crisis occurring in calculus and analysis regarding the meaning of infinite processes such as convergence and continuity. In geometry, there was a clear need for a new set of axioms, which would be complete, and which in no way relied on pictures we draw or on our intuition of space. Such axioms were given by David Hilbert in 1894 in his dissertation Grundlagen der Geometrie (Foundations of Geometry). Some other complete sets of axioms had been given a few years earlier, but did not match Hilbert's in economy, elegance, and similarity to Euclid's axioms.

Analysis situs, or topology

In the mid-18th century, it became apparent that certain progressions of mathematical reasoning recurred when similar ideas were studied on the number line, in two dimensions, and in three dimensions. Thus the general concept of a metric space was created so that the reasoning could be done in more generality, and then applied to special cases. This method of studying calculus- and analysis-related concepts came to be known as analysis situs, and later as topology. The important topics in this field were properties of more general figures, such as connectedness and boundaries, rather than properties like straightness, and precise equality of length and angle measurements, which had been the focus of Euclidean and non-Euclidean geometry. Topology soon became a separate field of major importance, rather than a sub-field of geometry or analysis.

The 20th century

See also


- List of geometry topics
- Important publications in geometry.

External links


- [http://www.cut-the-knot.org/WhatIs/WhatIsGeometry.shtml What Is Geometry?] at cut-the-knot
- [http://www.elvenkids.com/tools/geometria/Geometria.php Geometria] An online tool to compute lines, surfaces and volumes of the main plane and solid figures, through direct and indirect formulas.
- [http://www.geogebra.at/ Geogebra] A free dynamic geometry tool, useful for exploring geometry.
- [http://agutie.homestead.com Geometry Step by Step from the Land of the Incas] by Antonio Gutierrez.
- [http://www.cut-the-knot.org/geometry.shtml Geometry] at cut-the-knot
- [http://www.islamicarchitecture.org/art/islamic-geometry-and-floral-patterns.html Islamic Geometry]
- Stanford Encyclopedia of Philosophy:
  - [http://plato.stanford.edu/entries/geometry-finitism/ Finitism in Geometry]
  - [http://plato.stanford.edu/entries/geometry-19th/ Geometry in the 19th Century]
- [http://www.egwald.com/geometry/index.php Online Interactive Geometric Objects] by Elmer G. Wiens Category:Geometry ko:기하학 ja:幾何学 simple:Geometry zh-min-nan:Kí-hô-ha̍k

Algebra

:This article is about the branch of mathematics. For other uses of the term see algebra (disambiguation). Algebra is a branch of mathematics which studies structure and quantity. It may be roughly characterized as a generalization and abstraction of arithmetic, in which operations are performed on symbols rather than numbers. It includes elementary algebra, taught to high school students, as well as abstract algebra which covers such structures as groups, rings and fields. Along with geometry and analysis, it is one of the three main branches of mathematics. The study of Algebra is the cause for some debate as the level taught to High School students is rarely applicable in the real world.

History

The origins of algebra can be traced to the cultures of the ancient Egyptians and Babylonians who used an early type of algebra to solve linear, quadratic, and indeterminate equations more than 3,000 years ago.
- Circa 300 BC: Greek mathematician Euclid, who taught and died at Alexandria in Egypt, in Book 2 of his Elements addresses quadratic equations, although in a strictly geometrical fashion.
- Circa 100 BC: algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu, The Nine Chapters of Mathematical Art.
- Circa 150 AD: Greek mathematician Hero of Alexandria treats algebraic equations in three volumes of mathematics.
- Circa 200 AD: Greek mathematician Diophantus, often referred to as the "father of algebra", writes his famous Arithmetica, a work featuring solutions of algebraic equations and on the theory of numbers.
- 476 AD Indian mathematician, Aryabhata obtains whole number solutions to linear equations by a method equivalent to modern one. Bhaskara II (1114 AD), who wrote the text Bijaganita (algebra), was the first to recognize that a positive number has two square roots. The Hindus recognized that quadratic equations have two roots, and included negative as well as irrational roots. They treated indeterminate quadratic equations.
- 820 AD The word algebra is derived from the name of the treatise first written by Persian mathematician Khwarizmi titled: Al-Jabr wa-al-Muqabilah meaning The book of summary concerning calculating by transposition and reduction. The word al-jabr means "reunion".
- 1202 AD Algebra was introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci .

Classification

Algebra may be roughly divided into the following categories:
- elementary algebra, in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied (note that this usually includes the subject matter of courses called intermediate algebra and college algebra);
- abstract algebra, sometimes also called modern algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated;
- linear algebra, in which the specific properties of vector spaces are studied (including matrices);
- universal algebra, in which properties common to all algebraic structures are studied. In advanced studies, axiomatic algebraic systems like groups, rings, fields, and algebras over a field are investigated in the presence of a natural geometric structure (a topology) which is compatible with the algebraic structure. The list includes:
- Normed linear spaces
- Banach spaces
- Hilbert spaces
- Banach algebras
- Normed algebras
- Topological algebras
- Topological groups

Algebras

The word algebra is also used for various algebraic structures:
- algebra over a field
- algebra over a set
- Boolean algebra
- sigma-algebra
- F-algebra and F-coalgebra in category theory

References


- Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, Introducing Mathematics (Totem Books, 1999).
- Donald R. Hill, Islamic Science and Engineering (Edinburgh University Press, 1994).
- George Gheverghese Joseph, The Crest of the Peacock : The Non-European Roots of Mathematics (Princeton University Press, 2000).

See also


- Fundamental theorem of algebra (which is really a theorem of mathematical analysis, not of algebra)
- Diophantus, "father of algebra"
- Mohammed al-Khwarizmi, also known as "father of Algebra". [http://www.math.umd.edu/~czorn/hist_algebra.pdf]
- Computer algebra system
- [http://www.ucs.louisiana.edu/~sxw8045/history.htm Highlights in the history of algebra]

External links


- [http://www.mathleague.com/help/algebra/algebra.htm Explanation of Basic Topics]
- [http://www.sparknotes.com/math/#algebra1 Sparknotes' Review of Algebra I and II]
- [http://www.jamesbrennan.org/algebra/ Understanding Algebra.] An online algebra text by James W. Brennan. Category:Algebra Category:Arabic words ko:대수학 ms:Algebra ja:代数学 simple:Algebra

Equation

:This article is about equations in mathematics. For the chemistry term, see chemical equation. An equation is a mathematical statement, in symbols, that two things are the same. Equations are written with the equals sign, as in :2 + 3 = 5. Equations are often used to state the equality of two expressions containing one or more variables. For example, given any value of x, it is always true that :x − x = 0. The two equations above are examples of identities: equations that are true regardless of the values of any variables that appear within them. The following equation is not an identity: :x + 1 = 2. The above equation is false for almost all conceivable values of x. Therefore, if the equation is known to be true, it carries information about the value of x. In this example, one concludes that x = 1. In general, the values of the variables for which the equation is true are called solutions. To solve an equation means to find its solutions. Many authors reserve the term equation for an equality which is not an identity. The distinction between the two concepts can be subtle; for example, :(x + 1)2 = x2 + 2x + 1 is an identity, while :(x + 1)2 = 2x2 + x + 1 is an equation, whose solutions for x are 0 and 1. Whether a statement is meant to be an identity or an equation carrying information about its variables can usually be determined from its context. Letters from the beginning of the alphabet like a, b, c, ... are often considered constants in the context of the discussion at hand, while letters from end of the alphabet, like x, y, z, are usually considered variables.

Properties

If an equation in algebra is known to be true, it can be manipulated to produce another true equation in a variety of ways: # Any quantity can be added to both sides. # Any quantity can be subtracted from both sides. # Any quantity can be multiplied to both sides. # Any nonzero quantity can divide both sides. # Generally, any function can be applied to both sides. If a function that is not injective is applied to both sides of a true equation, the resulting equation will still be true, but it may be less useful. Formally, one has an implication, not an equivalence, so the solution set may get larger. The functions implied in properties (1), (2), and (4) are always injective, as is (3) if we do not multiply by zero. Some generalized products, such as a dot product, are never injective.

See also


- Inequation
- Inequality
- Linear equation
- Quadratic equation
- Cubic equation
- Quartic equation
- Quintic equation
- Differential equation
- Integral equation
- Functional equation
- Diophantine equation
- List of equations
- Theory of equations

External links


- Free Online Equation Interpreter and Plotter: [http://www.wessa.net/math.wasp Mathematical Equation Plotter]. Plots 2D mathematical equations, computes integrals, and finds solutions.
- Solve 2D equations graphically and numerically: [http://deadline.3x.ro DeadLine]. Free Windows software.
- [http://eqworld.ipmnet.ru/en/solutions/ae.htm Algebraic Equations and Systems of Algebraic Equations] at EqWorld: The World of Mathematical Equations. Category:Elementary algebra Category:Equations ko:방정식 ja:方程式 simple:Equation

Geometric shape

In geometry, two objects are of the same shape if one can be transformed to another (ignoring color) by dilating (that is, by multiplying all distances by the same factor) and then, if necessary, rotating and translating. Dilation changes the size but not the shape; rotation and translation preserve both size and shape. In other words, the shape of an object is all the geometrical information that remains after location, scale and rotational effects are filtered out. Objects which are geometrically similar either have the same shape or one has the same shape as the other's mirror image (or both if they are themselves symmetric). The shape of an object can be characterized by basic geometry such as points, line, curves, plane, and so on. For an object of greater than 2 dimensions, one can always reduce the dimensions of the shape by considering the shape of a cross-section or a projection. The cross-section of a spherical object, for example, will be circular. More complex shapes would, however, generate various curvatures depending on the type of cross-section (e.g. horizontal, vertical). Because of the variation possible in taking cross-section, the orientation of the object is critical. The shape does not depend on changes in orientation/direction. However, a mirror image could be called a different shape. Shape may change if the object is scaled differentially. For example, a sphere becomes an ellipsoid when scaled differently in the vertical and horizontal axis. In other words, preserving axis of symmetry is important for preserving shapes.

See also


- List of geometric shapes Category:Elementary geometry Category:Geometric shapes ja:図形 simple:Shape

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Quantity

This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements. : :Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base

Structure

:Pinning down ideas of size, symmetry, and mathematical structure. : :Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory

Space

:A more visual approach to mathematics. : :Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

See also list of mathematics history topics :History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

See also


- Mathematical game
- Mathematical problem
- Mathematical puzzle
- Puzzle

Bibliography


- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links


- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

René Descartes

:For other things named Descartes, see Descartes (disambiguation). René Descartes (IPA: , March 31, 1596 – February 11, 1650), also known as Cartesius, was a noted French philosopher and mathematician. Descartes, dubbed the Founder of Modern Philosophy and the Father of Modern Mathematics, ranks as one of the most important and influential thinkers in modern western history. As the inventor of the Cartesian coordinate system, he formulated the basis of modern geometry (analytic geometry), which in turn influenced the development of modern calculus. He inspired his contemporaries and subsequent generations of philosophers, leading to the formation of what is known today as continental rationalism, a philosophical position which developed in 17th and 18th century Europe. His most famous statement is Cogito ergo sum (I think, therefore I am.).

Biography

Descartes was born in La Haye en Touraine, Indre-et-Loire, France. At the age of eight, he entered the Jesuit Collège Royal Henry-Le-Grand at La Flèche. After graduation, he studied at the University of Poitiers, earning a Baccalauréat and Licence in law in 1616. Descartes never actually practiced law, however, and in 1618 he entered the service of Prince Maurice of Nassau, leader of the United Provinces of the Netherlands. His intention was to see the world and to discover the truth. :"I entirely abandoned the study of letter. Resolving to seek no knowledge other than that which could be found in myself or else in the great book of the world, I spent the rest of my youth traveling, visiting courts and armies, mixing with people of diverse temperaments and ranks, gathering various experiences, testing myself in the situations which fortune offered me, and at all times reflecting upon whatever came my way so as to derive some profit from it. (Descartes, Discourse on the Method of Rightly Conducting One's Reason and Seeking the Truth in the Sciences) Here he met Isaac Beeckman who sparks his interest in mathematics and the new physics. On November 10, 1619, while traveling in Germany and thinking about using mathematics to solve problems in physics, Descartes had a vision in a dream through which he "discovered the foundations of a marvelous science." This became a pivotal point in young Descartes' life and the foundation on which he develops analytical geometry. He dedicated the rest of his life to researching this connection between mathematics and nature. In 1622 he returned to France, and during the next few years spent time in Paris and other parts of Europe. Descartes was present at the siege of La Rochelle by Cardinal Richelieu in 1627. He left for Holland in 1628, where he lived and changed his address frequently until 1649. In 1633, Galileo was condemned by the Catholic Church, and Descartes abandoned plans to publish Treatise on the World, his work of the previous four years. His daughter Francine was born in 1635 and was baptized on August 7 of the same year. She died in 1640. Descartes continued to publish works concerning mathematics and philosophy for the rest of his life. In 1643, Cartesian philosophy was condemned at the University of Utrecht, and Descartes began his long correspondence with Princess Elizabeth of Bohemia. In 1647, he was awarded a pension by the King of France. Descartes was interviewed by Frans Burman at Egmond-Binnen in 1648. In 1649, Descartes went to Sweden on invitation of professor Eitan Olevsky. René Descartes died on February 11, 1650 in Stockholm, Sweden, where he had been invited as a teacher for Queen Christina of Sweden. The cause of death was said to be pneumonia - accustomed to working in bed till noon, he may have suffered a detrimental effect on his health due to Christina's demands for early morning study. However, letters to and from the doctor Eike Pies have recently been discovered which indicate that Descartes may have been poisoned using arsenic. In 1667, the Roman Catholic Church placed his works on the Index of Prohibited Books. As a Catholic in a Protestant nation, he was interred in a graveyard mainly used for unbaptized infants in Adolf Fredrikskyrkan in Stockholm. Later, his remains were taken to France and buried in the Church of St. Genevieve-du-Mont in Paris. A memorial erected in the 18th century remains in the Swedish church. During the French Revolution, his remains were disinterred for burial in the Panthéon among the great French thinkers. The village in the Loire Valley where he was born was renamed La Haye - Descartes in 1802, which was shortened to "Descartes" in 1967. Currently his tomb is in the church Saint Germain-des-Pres in Paris.

Significance

Philosophical legacy

Descartes is often regarded as the first modern thinker to provide a philosophical framework for the natural sciences as these began to develop. In his Meditations on First Philosophy he attempts to arrive at a fundamental set of principles that one can know as true without any doubt. To achieve this, he employs a method called methodological skepticism: he doubts any idea that can be doubted. He gives the example of dreaming: in a dream, one's senses perceive stimuli that seem real, but do not actually exist. Thus, one cannot rely on the data of the senses as necessarily true. Or, perhaps an "evil demon" exists: a supremely powerful and cunning being who sets out to try to deceive Descartes from knowing the true nature of reality. Given these possibilities, what can one know for certain? Initially, Descartes arrives at only a single principle: if I am being deceived, then surely "I" must exist. Most famously, this is known as cogito ergo sum, ("I think, therefore I am"). (These words do not appear in the Meditations, although he had written them in his earlier work Discourse on Method). Therefore, Descartes concludes that he can be certain that he exists. But in what form? He perceives his body through the use of the senses; however, these have previously proved unreliable. So Descartes concludes that the only undoubtable knowledge is that he is a thinking thing. Thinking is his essence as it is the only thing about him that cannot be doubted. To further demonstrate the limitations of the senses, Descartes proceeds with what is known as the Wax Argument. He considers a piece of wax: his senses inform him that it has certain characteristics, such as shape, texture, size, color, smell, and so forth. However, when he brings the wax towards a flame, these characteristics change completely. However, it seems that it is still the same thing: it is still a piece of wax, even though the data of the senses inform him that all of its characteristics are different. Therefore, in order to properly grasp the nature of the wax, he cannot use the senses: he must use his mind. Descartes concludes: :"Thus what I thought I had seen with my eyes, I actually grasped solely with the faculty of judgment, which is in my mind." In this manner, Descartes proceeds to construct a system of knowledge, discarding perception as unreliable and instead admitting only deduction as a method. Halfway through the Meditations, he also claims to prove the existence of a benevolent God, who, being benevolent, has provided him with a working mind and sensory system, and who cannot desire to deceive him, and thus, finally, he establishes the possibility of acquiring knowledge about the world based on deduction and perception.

Mathematical legacy

Rene Descartes said "Nature can be defined through numbers." Mathematicians consider Descartes of the utmost importance for his discovery of analytic geometry. Up to Descartes's times, geometry, dealing with lines and shapes, and algebra, dealing with numbers, appeared as completely different subsets of mathematics. Descartes showed how to translate many problems in geometry into problems in algebra, by using a coordinate system to describe the problem. Descartes's theory provided the basis for the calculus of Newton and Leibniz, by applying infinitesimal calculus to the tangent problem, thus permitting the evolution of that branch of modern mathematics . This appears even more astounding when one keeps in mind that the work was just intended as an example to his Discours de la méthode pour bien conduire sa raison, et chercher la verité dans les sciences (Discourse on the Method to Rightly Conduct the Reason and Search for the Truth in Sciences, known better under the shortened title Discours de la méthode). Descartes also made contributions in the field of Optics, for instance, he showed by geometrical construction using the Law of Refraction that the angular radius of a rainbow is 42° (i.e. the angle subtended at the eye by the edge of the rainbow and the ray passing from the sun through the rainbow's centre is 42°).

Writings by Descartes


- Compendium Musicae (1618)
- Rules for the Direction of the Mind (1628)
- Discourse on Method (1637): an introduction to "Dioptrique', on the "Météores' and 'La Géométrie'; a work for the grand public, written in French.
- La Géométrie (1637)
- Meditations on First Philosophy (1641), also known as 'Metaphysic meditations', with a series of six Objections and Replies. This work was written in Latin, language of the learned. A second edition was published a year later with all seven sets of the objections and replies followed by Letter to Dinet.
- Les Principes de la philosophie (Principles of Philosophy) (1644), work rather destined for the students.
- The Singing Epitaph (1646)
- Comments on a Certain Broadsheet (1647)
- The Description of the Human Body (1647)
- Conversation with Burman (1648)
- Passions of the Soul (1649), dedicated to Princess Elizabeth of Bohemia

Trivia

It is claimed that during the 1640s Descartes travelled with an artificial female companion called Francine, named after his daughter. This may be a myth linked with his statements about the nature of the mind, or an early automaton, or Gynoid. Descartes was ranked #49 on Michael H. Hart's list of the most influential figures in history. His name roughly means "reborn of charts/maps" depending on the definition of cartes used. The Descartes Highlands area on the moon where John Young and Charles Duke landed with Apollo 16 is named after him.

References

# # # #
-

See also


- Dualistic interactionism
- Baruch Spinoza
- Asteroid 3587 Descartes, named after the philosopher
- Defect (geometry)
- Analytic geometry
- Cartesian coordinate system

External links


-
- [http://www.shvoong.com/books/philosophy/55185-discourse-method/ A summary of his book "A Discourse On Method"]
-
- Translations of Descartes' Meditations: [http://www.wright.edu/cola/descartes/mede.html]
- [http://www.incipitblog.com/index.php/2005/06/01/rene-descartes-discours-de-la-methode-1637/ French Audio Book (mp3)] : excerpt about animals/machines from Discourse On the Method
- [http://gutenberg.net/etext/59 Discourse On the Method] – at Project Gutenberg
- [http://gutenberg.net/etext/4391 Selections from the Principles of Philosophy] – at Project Gutenberg
- [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Descartes.html Detailed biography of Descartes]
- [http://www.newadvent.org/cathen/04744b.htm CATHOLIC ENCYCLOPEDIA: Rene Descartes]
- [http://www.earlymoderntexts.com/ READABLE versions of Descartes's Meditations and Discourse on the Method.]
- [http://www.borishennig.de/texte/descartes/diss/cartes_04b.pdf Conscientia in Descartes]
- [http://descartes.sourceforge.net/ descartes], an open source function plotter named after the inventor of Cartesian coordinates
- [http://www.biblioweb.org/-DESCARTES-Rene-.html Biography, Bibliography, Analysis] (in French)
- Stanford Encyclopedia of Philosophy:
  - [http://plato.stanford.edu/entries/descartes-epistemology/ Descartes' Epistemology]
  - [http://plato.stanford.edu/entries/descartes-ethics/ Descartes' Ethics]
  - [http://plato.stanford.edu/entries/descartes-works/ Descartes' Life and Works]
  - [http://plato.stanford.edu/entries/descartes-modal/ Descartes' Modal Metaphysics]
  - [http://plato.stanford.edu/entries/descartes-ontological/ Descartes' Ontological Argument]
  - [http://plato.stanford.edu/entries/pineal-gland/ Descartes and the Pineal Gland] Descartes, Rene Descartes, Rene Descartes, Rene Descartes, Rene Descartes, Rene Descartes, Rene Descartes, Rene Descartes, Rene Descartes, Rene Descartes, Rene Descartes, Rene ko:르네 데카르트 ja:ルネ・デカルト simple:René Descartes th:เรอเน เดส์การตส์

Discourse on Method

The Discourse on Method is a philosophical and mathematical treatise published by René Descartes in 1637. Its full name is Discourse on the Method of Rightly Conducting the Reason in the Search for Truth in the Sciences (French title: Discours de la méthode pour bien conduire sa raison, et chercher la verité dans les sciences). The Discourse on Method is best known as the source of the famous quotation "cogito ergo sum", "I think, therefore I am." In addition, it contains Descartes' first introduction of the Cartesian coordinate system. This is one of the most influential works in history. It is a method which gives a solid platform from which all modern natural sciences could evolve. With this work, the idea of skepticism was revived from the ancients such as Sextus Empiricus and modified to account for a truth that Descartes found to be incontrovertible. Descartes started his line of reasoning by doubting everything, so as to assess the world from a fresh perspective, clear of any preconceived notions.

The four precepts

The following quote from Discourse on Method presents the four precepts that characterise the Method itself: :"The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgment than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt. :The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution. :The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence. :And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted." Descartes uses the analogy of tearing down the house to its foundation in order to build a secure edifice (He even extends the analogy to move next door into a house of morality, while his own house is being rebuilt). The foundation he reveals appears to have three parts. Applying the method to itself, Descartes challenges his own reasoning and reason itself. But Descartes believes three things are not susceptible to doubt and the three support each other to form a stable foundation for the method. He cannot doubt that something has to be there to do the doubting (I think, therefore I am). The method of doubt cannot doubt reason as it is based on reason itself. By reason there exists a God and God is the guarantor that reason is not misguided. Perhaps the most strained part of the argument is the reasoned proof of the existence of God and indeed Descartes seems to realise this as he supplies three different 'proofs' including what is now referred to as the ontological proof of the existence of God (some argue that Descartes inserted his statement on the existence of God in the Discourse on Method to appease censors of the time; a very serious concern, as within Discourse Descartes points out that he was at first reluctant to publish the work because of the recent show trial of Galileo by the Catholic Church in 1633, only four years earlier). Secure on these foundation stones, Descartes shows the practical application of 'The Method' in Mathematics and the Sciences. One of the practical methods was to order the objects in different ways on paper to make them easy to see clearly. This became the basis of the Cartesian coordinate system, the Histogram and Analytic geometry. These ideas, among other methods of science, influenced Isaac Newton and Gottfried Leibniz in their development of calculus. The most important influence, however, was the first precept, which states, in Descartes words, to "never to accept anything for true which I did not clearly know to be such". This new idea of skepticism influenced many to start finding out things for themselves rather than relying solely on authority. The idea as such may have been the starting point for the development of modern science.

External link


- Category:1637 books Category:Epistemology Category:Philosophy books Category:Mathematics books ko:방법서설 ja:方法序説

French language

French (French: français) is the third of the Romance languages in terms of number of speakers, after Spanish and Portuguese, being spoken by about 67 million people as a mother tongue, and altogether by some 128 million people, which includes second-language speakers who use French for daily communication. French is thus the 18th most spoken language in the world by number of native speakers, and 9th in terms of daily speakers. It is an official language in 29 countries. It is also an official or administrative language in various communities and organisations (such as the European Union, IOC, United Nations and Universal Postal Union). Before World War II, French was considered the international language, particularly in such fields as diplomacy, trade, shipping, and transportation.

History

The Roman invasion of Gaul

The French language is a Romance language, meaning that it is descended from Latin. Before the Roman invasion of what is modern-day France by Julius Cæsar (58–52 BC), France was inhabited largely by a Celtic people that the Romans referred to as Gauls, although there were also other linguistic/ethnic groups in France at this time, such as the Iberians in southern France and Spain, the Ligurians on the Mediterranean coast, Greek colonies such as Massalia (i.e. present-day Marseille), Phoenician outposts, and the Vascons on the Spanish/French border. Although in the past many Frenchmen liked to refer to their descent from Gallic ancestors (nos ancêtres les Gaulois), perhaps fewer than 200 words with a Celtic etymological origin remain in French today (largely place and plant names and words dealing with rural life and the earth). In the reverse direction, some words for Gallic objects which were new to the Romans and for which there were no words in Latin were imported into Latin – for example, clothing items such as les braies. Latin quickly became the lingua franca of the entire Gallic region for mercantile, official and educational purposes, yet it should be remembered that this was Vulgar Latin, the colloquial dialect spoken by the Roman army and its agents and not the literary dialect of Cicero.

The Franks

From the third century on, Western Europe was invaded by Germanic tribes from the east, and some of these groups settled in Gaul. For the history of the French language, the most important of these groups are the Franks in northern France, the Alemanni in the German/French border, the Burgundians in the Rhone valley and the Visigoths in the Aquitaine region and Spain. These Germanic-speaking groups had a profound effect on the Latin spoken in their respective regions, altering both the pronunciation and the syntax. They also introduced a number of new words: perhaps as much as 15% of modern French comes from Germanic words, including many terms and expressions associated with their social structure and military tactics.

Langue d'Oïl

Linguists typically divide the languages spoken in medieval France into three geographical subgroups: Langue d'oïl and Langue d'oc are the two major groups; the third group, Franco-Provençal, is considered a transitional language between the two other groups. The Oïl–Oc divide is broadly comparable to the divide illustrated by the use of "yes" in English and "aye" in Scots. Langue d'oïl, the languages which use oïl (in modern usage, oui) for "yes", is the language group in the north of France. These languages, like Picard, Walloon, Francien and Norman, were influenced by the Germanic languages spoken by the Frankish invaders. From the time period Clovis I on, the Franks extended their rule over northern Gaul. Over time, the French language developed from either the Oïl language found around Paris (the Francien theory) or from a standard administrative language based on common characteristics found in all Oïl languages (the lingua franca theory). Langue d'oc, the languages which use oc for "yes", is the language group in the south of France and northern Spain. These languages, such as Gascon and Provençal, have relatively little Frankish influence. (Modern French has two words for "yes", oui and si; the latter is used to contradict negative statements. Si derives from Latin sic "thus", and is cognate to the word for "yes" in Spanish, Italian, and Catalan. Oïl/oui derive, according to Larousse, from Latin hoc ille "thus he (did)".)

Other linguistic groups

The early middle ages also saw the influence of other linguistic groups on the dialects of France: From the 5th to the 8th centuries, Celtic-speaking peoples from southwestern Britain (