1.
Filosofie
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Philosophy is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language. The term was coined by Pythagoras. Philosophical methods include questioning, critical discussion, rational argument and systematic presentation, classic philosophical questions include, Is it possible to know anything and to prove it. However, philosophers might also pose more practical and concrete questions such as, is it better to be just or unjust. Historically, philosophy encompassed any body of knowledge, from the time of Ancient Greek philosopher Aristotle to the 19th century, natural philosophy encompassed astronomy, medicine and physics. For example, Newtons 1687 Mathematical Principles of Natural Philosophy later became classified as a book of physics, in the 19th century, the growth of modern research universities led academic philosophy and other disciplines to professionalize and specialize. In the modern era, some investigations that were part of philosophy became separate academic disciplines, including psychology, sociology. Other investigations closely related to art, science, politics, or other pursuits remained part of philosophy, for example, is beauty objective or subjective. Are there many scientific methods or just one, is political utopia a hopeful dream or hopeless fantasy. Major sub-fields of academic philosophy include metaphysics, epistemology, ethics, aesthetics, political philosophy, logic, philosophy of science, since the 20th century, professional philosophers contribute to society primarily as professors, researchers and writers. Traditionally, the term referred to any body of knowledge. In this sense, philosophy is related to religion, mathematics, natural science, education. This division is not obsolete but has changed, Natural philosophy has split into the various natural sciences, especially astronomy, physics, chemistry, biology and cosmology. Moral philosophy has birthed the social sciences, but still includes value theory, metaphysical philosophy has birthed formal sciences such as logic, mathematics and philosophy of science, but still includes epistemology, cosmology and others. Many philosophical debates that began in ancient times are still debated today, colin McGinn and others claim that no philosophical progress has occurred during that interval. Chalmers and others, by contrast, see progress in philosophy similar to that in science, in one general sense, philosophy is associated with wisdom, intellectual culture and a search for knowledge. In that sense, all cultures and literate societies ask philosophical questions such as how are we to live, a broad and impartial conception of philosophy then, finds a reasoned inquiry into such matters as reality, morality and life in all world civilizations. Socrates was an influential philosopher, who insisted that he possessed no wisdom but was a pursuer of wisdom
2.
Natuurwetenschap
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Natural science is a branch of science concerned with the description, prediction, and understanding of natural phenomena, based on observational and empirical evidence. Mechanisms such as review and repeatability of findings are used to try to ensure the validity of scientific advances. Natural science can be divided into two branches, life science and physical science. Physical science is subdivided into branches, including physics, space science, chemistry and these branches of natural science may be further divided into more specialized branches. Modern natural science succeeded more classical approaches to natural philosophy, usually traced to ancient Greece, galileo, Descartes, Francis Bacon, and Newton debated the benefits of using approaches which were more mathematical and more experimental in a methodical way. Still, philosophical perspectives, conjectures, and presuppositions, often overlooked, systematic data collection, including discovery science, succeeded natural history, which emerged in the 16th century by describing and classifying plants, animals, minerals, and so on. Today, natural history suggests observational descriptions aimed at popular audiences, philosophers of science have suggested a number of criteria, including Karl Poppers controversial falsifiability criterion, to help them differentiate scientific endeavors from non-scientific ones. Validity, accuracy, and quality control, such as peer review and this field encompasses a set of disciplines that examines phenomena related to living organisms. The scale of study can range from sub-component biophysics up to complex ecologies, biology is concerned with the characteristics, classification and behaviors of organisms, as well as how species were formed and their interactions with each other and the environment. The biological fields of botany, zoology, and medicine date back to periods of civilization. However, it was not until the 19th century that became a unified science. Once scientists discovered commonalities between all living things, it was decided they were best studied as a whole, modern biology is divided into subdisciplines by the type of organism and by the scale being studied. Molecular biology is the study of the chemistry of life, while cellular biology is the examination of the cell. At a higher level, anatomy and physiology looks at the internal structures, constituting the scientific study of matter at the atomic and molecular scale, chemistry deals primarily with collections of atoms, such as gases, molecules, crystals, and metals. The composition, statistical properties, transformations and reactions of these materials are studied, chemistry also involves understanding the properties and interactions of individual atoms and molecules for use in larger-scale applications. Most chemical processes can be studied directly in a laboratory, using a series of techniques for manipulating materials, chemistry is often called the central science because of its role in connecting the other natural sciences. Early experiments in chemistry had their roots in the system of Alchemy, the science of chemistry began to develop with the work of Robert Boyle, the discoverer of gas, and Antoine Lavoisier, who developed the theory of the Conservation of mass. The success of science led to a complementary chemical industry that now plays a significant role in the world economy
3.
Heelal
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The Universe is all of time and space and its contents. It includes planets, moons, minor planets, stars, galaxies, the contents of intergalactic space, the size of the entire Universe is unknown. The earliest scientific models of the Universe were developed by ancient Greek and Indian philosophers and were geocentric, over the centuries, more precise astronomical observations led Nicolaus Copernicus to develop the heliocentric model with the Sun at the center of the Solar System. In developing the law of gravitation, Sir Isaac Newton built upon Copernicuss work as well as observations by Tycho Brahe. Further observational improvements led to the realization that our Solar System is located in the Milky Way galaxy and it is assumed that galaxies are distributed uniformly and the same in all directions, meaning that the Universe has neither an edge nor a center. Discoveries in the early 20th century have suggested that the Universe had a beginning, the majority of mass in the Universe appears to exist in an unknown form called dark matter. The Big Bang theory is the prevailing cosmological description of the development of the Universe, under this theory, space and time emerged together 13. 799±0.021 billion years ago with a fixed amount of energy and matter that has become less dense as the Universe has expanded. After the initial expansion, the Universe cooled, allowing the first subatomic particles to form, giant clouds later merged through gravity to form galaxies, stars, and everything else seen today. Some physicists have suggested various multiverse hypotheses, in which the Universe might be one among many universes that likewise exist, the Universe can be defined as everything that exists, everything that has existed, and everything that will exist. According to our current understanding, the Universe consists of spacetime, forms of energy, the Universe encompasses all of life, all of history, and some philosophers and scientists suggest that it even encompasses ideas such as mathematics and logic. The word universe derives from the Old French word univers, which in turn derives from the Latin word universum, the Latin word was used by Cicero and later Latin authors in many of the same senses as the modern English word is used. Another synonym was ὁ κόσμος ho kósmos, synonyms are also found in Latin authors and survive in modern languages, e. g. the German words Das All, Weltall, and Natur for Universe. The same synonyms are found in English, such as everything, the cosmos, the world, the prevailing model for the evolution of the Universe is the Big Bang theory. The Big Bang model states that the earliest state of the Universe was extremely hot and dense, the model is based on general relativity and on simplifying assumptions such as homogeneity and isotropy of space. The Big Bang model accounts for such as the correlation of distance and redshift of galaxies, the ratio of the number of hydrogen to helium atoms. The initial hot, dense state is called the Planck epoch, after the Planck epoch and inflation came the quark, hadron, and lepton epochs. Together, these epochs encompassed less than 10 seconds of time following the Big Bang, the observed abundance of the elements can be explained by combining the overall expansion of space with nuclear and atomic physics. As the Universe expands, the density of electromagnetic radiation decreases more quickly than does that of matter because the energy of a photon decreases with its wavelength
4.
Wiskunde
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
5.
Natuurkunde
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Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
6.
Symbool
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A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different concepts and experiences, all communication is achieved through the use of symbols. Symbols take the form of words, sounds, gestures, ideas or visual images and are used to other ideas. For example, a red octagon may be a symbol for STOP, on a map, a blue line might represent a river. Alphabetic letters may be symbols for sounds, personal names are symbols representing individuals. A red rose may symbolize love and compassion, the variable x, in a mathematical equation, may symbolize the position of a particle in space. In cartography, a collection of symbols forms a legend for a map The word derives from the Greek symbolon meaning token or watchword. It is an amalgam of syn- together + bole a throwing, a casting, the sense evolution in Greek is from throwing things together to contrasting to comparing to token used in comparisons to determine if something is genuine. The meaning something which stands for something else was first recorded in 1590, later, expanding on what he means by this definition Campbell says, a symbol, like everything else, shows a double aspect. We must distinguish, therefore between the sense and the meaning of the symbol. The term meaning can only to the first two but these, today, are in the charge of science – which is the province as we have said, not of symbols. The ineffable, the unknowable, can be only sensed. Heinrich Zimmer gives an overview of the nature, and perennial relevance. Concepts and words are symbols, just as visions, rituals, through all of these a transcendent reality is mirrored. They are so many metaphors reflecting and implying something which, though thus variously expressed, is ineffable, though thus rendered multiform, Symbols hold the mind to truth but are not themselves the truth, hence it is delusory to borrow them. Each civilisation, every age, must bring forth its own, in the book Signs and Symbols, it is stated that A symbol. Is a visual image or sign representing an idea -- a deeper indicator of a universal truth, Symbols are a means of complex communication that often can have multiple levels of meaning. This separates symbols from signs, as signs have only one meaning, human cultures use symbols to express specific ideologies and social structures and to represent aspects of their specific culture
7.
Kardinaalgetal
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In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a set is a natural number, the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite sets, cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, in the case of finite sets, this agrees with the intuitive notion of size. In the case of sets, the behavior is more complex. It is also possible for a subset of an infinite set to have the same cardinality as the original set. There is a sequence of cardinal numbers,0,1,2,3, …, n, …, ℵ0, ℵ1, ℵ2, …, ℵ α, …. This sequence starts with the natural numbers including zero, which are followed by the aleph numbers, the aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number, If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs. Cardinality is studied for its own sake as part of set theory and it is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra, and mathematical analysis. In category theory, the numbers form a skeleton of the category of sets. The notion of cardinality, as now understood, was formulated by Georg Cantor, cardinality can be used to compare an aspect of finite sets, e. g. the sets and are not equal, but have the same cardinality, namely three. Cantor applied his concept of bijection to infinite sets, e. g. the set of natural numbers N =, thus, all sets having a bijection with N he called denumerable sets and they all have the same cardinal number. This cardinal number is called ℵ0, aleph-null and he called the cardinal numbers of these infinite sets transfinite cardinal numbers. Cantor proved that any unbounded subset of N has the same cardinality as N and he later proved that the set of all real algebraic numbers is also denumerable. His proof used an argument with nested intervals, but in an 1891 paper he proved the result using his ingenious. The new cardinal number of the set of numbers is called the cardinality of the continuum. His continuum hypothesis is the proposition that c is the same as ℵ1 and this hypothesis has been found to be independent of the standard axioms of mathematical set theory, it can neither be proved nor disproved from the standard assumptions
8.
Geheel getal
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
9.
Verzameling (wiskunde)
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In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2,4, and 6 are distinct objects when considered separately, Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a part of mathematics. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, the German word Menge, rendered as set in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a collection of distinct objects. The objects that make up a set can be anything, numbers, people, letters of the alphabet, other sets, Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. Cantors definition turned out to be inadequate, instead, the notion of a set is taken as a notion in axiomatic set theory. There are two ways of describing, or specifying the members of, a set, one way is by intensional definition, using a rule or semantic description, A is the set whose members are the first four positive integers. B is the set of colors of the French flag, the second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets, one often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D, there are two important points to note about sets. First, in a definition, a set member can be listed two or more times, for example. However, per extensionality, two definitions of sets which differ only in one of the definitions lists set members multiple times, define, in fact. Hence, the set is identical to the set. The second important point is that the order in which the elements of a set are listed is irrelevant and we can illustrate these two important points with an example, = =. For sets with many elements, the enumeration of members can be abbreviated, for instance, the set of the first thousand positive integers may be specified extensionally as, where the ellipsis indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members, thus the set of positive even numbers can be written as
10.
Deelverzameling
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In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained inside B, that is, all elements of A are also elements of B. The relationship of one set being a subset of another is called inclusion or sometimes containment, the subset relation defines a partial order on sets. The algebra of subsets forms a Boolean algebra in which the relation is called inclusion. For any set S, the inclusion relation ⊆ is an order on the set P of all subsets of S defined by A ≤ B ⟺ A ⊆ B. We may also partially order P by reverse set inclusion by defining A ≤ B ⟺ B ⊆ A, when quantified, A ⊆ B is represented as, ∀x. So for example, for authors, it is true of every set A that A ⊂ A. Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively and this usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may or may not equal y, but if x < y, then x definitely does not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if A ⊆ B, then A may or may not equal B, the set A = is a proper subset of B =, thus both expressions A ⊆ B and A ⊊ B are true. The set D = is a subset of E =, thus D ⊆ E is true, any set is a subset of itself, but not a proper subset. The empty set, denoted by ∅, is also a subset of any given set X and it is also always a proper subset of any set except itself. These are two examples in both the subset and the whole set are infinite, and the subset has the same cardinality as the whole. The set of numbers is a proper subset of the set of real numbers. In this example, both sets are infinite but the set has a larger cardinality than the former set. Another example in an Euler diagram, Inclusion is the partial order in the sense that every partially ordered set is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identified with the set of all ordinals less than or equal to n, then a ≤ b if and only if ⊆. For the power set P of a set S, the partial order is the Cartesian product of k = |S| copies of the partial order on for which 0 <1. This can be illustrated by enumerating S = and associating with each subset T ⊆ S the k-tuple from k of which the ith coordinate is 1 if and only if si is a member of T