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Explanatory Indispensability Argument
The explanatory indispensability argument is an argument in the philosophy of mathematics for the existence of mathematical objects. It claims that rationally we should believe in mathematical objects such as numbers because they are indispensable to scientific explanations of empirical phenomena. An altered form of the Quine–Putnam indispensability argument, it differs from that argument in its increased focus on specific explanations instead of whole theories and in its shift towards inference to the best explanation as a justification for belief in mathematical objects rather than confirmational holism. Specific explanations proposed as examples of mathematical explanations in science include why periodical cicadas have prime-numbered life cycles, why bee honeycomb has a hexagonal structure, and the solution to the Seven Bridges of Königsberg problem. Objections to the argument include the idea that mathematics is only used as a representational device, even when it featur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophy Of Mathematics
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathematical objects are purely abstract entities or are in some way concrete, and in what the relationship such objects have with physical reality consists. Major themes that are dealt with in philosophy of mathematics include: *''Reality'': The question is whether mathematics is a pure product of human mind or whether it has some reality by itself. *''Logic and rigor'' *''Relationship with physical reality'' *''Relationship with science'' *''Relationship with applications'' *''Mathematical truth'' *''Nature as human activity'' (science, the arts, art, game, or all together) Major themes Reality Logic and rigor Mathematical reasoning requires Mathematical rigor, rigor. This means that the definitions must be absolutely unambiguous and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up quark, up and down quark, down quarks. Electrons are extremely lightweight particles that orbit the positively charged atomic nucleus, nucleus of atoms. Their negative charge is balanced by the positive charge of protons in the nucleus, giving atoms their overall electric charge#Charge neutrality, neutral charge. Ordinary matter is composed of atoms, each consisting of a positively charged nucleus surrounded by a number of orbiting electrons equal to the number of protons. The configuration and energy levels of these orbiting electrons determine the chemical properties of an atom. Electrons are bound to the nucleus to different degrees. The outermost or valence electron, valence electrons are the least tightly bound and are responsible for th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antipodes
In geography, the antipode () of any spot on Earth is the point on Earth's surface diametrically opposite to it. A pair of points ''antipodal'' () to each other are situated such that a straight line connecting the two would pass through Earth's center. Antipodal points are as far away from each other as possible. The North and South Poles are antipodes of each other. In the Northern Hemisphere, "the Antipodes" may refer to Australia and New Zealand, and Antipodeans to their inhabitants. Geographically, the antipodes of the British Isles are in the Pacific Ocean, south of New Zealand. This gave rise to the name of the Antipodes Islands of New Zealand, which are close to the antipode of London. The antipodes of Australia are in the North Atlantic Ocean, while parts of Spain, Portugal, France and Morocco are antipodal to New Zealand. Approximately 15% of land territory is antipodal to other land, representing approximately 4.4% of Earth's surface. Another source estimates th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a Set (mathematics), set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', then this graph is directed, because owing mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonhard Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and Mathematical notation, notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Kingdom of Prussia, Prussia. Euler is credited for popularizing the Greek letter \pi (lowercase Pi (letter), pi) to denote Pi, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Königsberg
Königsberg (; ; ; ; ; ; , ) is the historic Germany, German and Prussian name of the city now called Kaliningrad, Russia. The city was founded in 1255 on the site of the small Old Prussians, Old Prussian settlement ''Twangste'' by the Teutonic Knights during the Northern Crusades, Baltic Crusades. It was named in honour of King Ottokar II of Bohemia, who led a campaign against the pagan Old Prussians, a Baltic tribe. A Baltic Sea, Baltic port city, it successively became the capital of the State of the Teutonic Order, the Duchy of Prussia and the provinces of East Prussia and Province of Prussia, Prussia. Königsberg remained the coronation city of the Prussian monarchy from 1701 onwards, though the capital was Berlin. From the thirteenth to the twentieth centuries on, the inhabitants spoke predominantly German language, German, although the city also had a profound influence upon the Lithuanian and Polish cultures. It was a publishing center of Lutheranism, Lutheran literatu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Darwin
Charles Robert Darwin ( ; 12 February 1809 – 19 April 1882) was an English Natural history#Before 1900, naturalist, geologist, and biologist, widely known for his contributions to evolutionary biology. His proposition that all species of life have descended from a Common descent, common ancestor is now generally accepted and considered a fundamental scientific concept. In a joint presentation with Alfred Russel Wallace, he introduced his scientific theory that this Phylogenetics, branching pattern of evolution resulted from a process he called natural selection, in which the struggle for existence has a similar effect to the artificial selection involved in selective breeding.. Darwin has been described as one of the most influential figures in human history and was honoured by Burials and memorials in Westminster Abbey, burial in Westminster Abbey. Darwin's early interest in nature led him to neglect his medical education at the University of Edinburgh Medical Schoo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Honeycomb Conjecture
The honeycomb theorem, formerly the honeycomb conjecture, states that a regular hexagonal grid or honeycomb has the least total perimeter of any subdivision of the plane into regions of equal area. The conjecture was proven in 1999 by mathematician Thomas C. Hales. Theorem Let \Gamma be any system of smooth curves in \mathbb^2, subdividing the plane into regions (connected components of the complement of \Gamma) all of which are bounded and have unit area. Then, averaged over large disks in the plane, the average length of \Gamma per unit area is at least as large as for the hexagon tiling. The theorem applies even if the complement of \Gamma has additional components that are unbounded or whose area is not one; allowing these additional components cannot shorten \Gamma. Formally, let B(0,r) denote the disk of radius r centered at the origin, let L_r denote the total length of \Gamma\cap B(0,r), and let A_r denote the total area of B(0,r) covered by bounded unit-area components. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Cicada Number Line
Periodicity or periodic may refer to: Mathematics * Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups * Periodic function, a function whose output contains values that repeat periodically * Periodic mapping Physical sciences * Periodic table of chemical elements * Periodic trends, relative characteristics of chemical elements observed * Redshift periodicity, astronomical term for redshift quantization Other uses * Fokker periodicity blocks, which mathematically relate musical intervals * Periodic acid, a compound of iodine * Principle of periodicity, a concept in generally accepted accounting principles * Quasiperiodicity, property of a system that displays irregular periodicity See also * Aperiodic (other) * Cycle (other) * Frequency (other) * Period (other) * Periodical * Seasonality In time series data, seasonality refers to the trends that occur at specif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
