Mathematics is the study of numbers, quantity, space, pattern, structure, and change. Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. 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, and practical applications for what began as pure mathematics are often discovered.
Georg Ferdinand Ludwig Philipp Cantor (December 3, 1845, St. Petersburg, Russia – January 6, 1918, Halle, Germany) was a German mathematician who is best known as the creator of set theory. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities." He defined the cardinal and ordinal numbers, and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.
Cantor's work encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré, and later from Hermann Weyl and L.E.J. Brouwer. Ludwig Wittgenstein raised philosophical objections. Nowadays, the vast majority of mathematicians who are neither constructivists nor finitists accept Cantor's work on transfinite sets and arithmetic, recognizing it as a major paradigm shift.
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Conway's Game of Life is a cellular automaton devised by the British mathematician John Horton Conway in 1970. It is an example of a zero-player game, meaning that its evolution is completely determined by its initial state, requiring no further input as the game progresses. After an initial pattern of filled-in squares ("live cells") is set up in a two-dimensional grid, the fate of each cell (including empty, or "dead", ones) is determined at each step of the game by considering its interaction with its eight nearest neighbors (the cells that are horizontally, vertically, or diagonally adjacent to it) according to the following rules: (1) any live cell with fewer than two live neighbors dies, as if caused by under-population; (2) any live cell with two or three live neighbors lives on to the next generation; (3) any live cell with more than three live neighbors dies, as if by overcrowding; (4) any dead cell with exactly three live neighbors becomes a live cell, as if by reproduction. By repeatedly applying these simple rules, extremely complex patterns can emerge. In this animation, a breeder (in this instance called a puffer train, colored red in the final frame of the animation) leaves guns (green) in its wake, which in turn "fire out" gliders (blue). Many more complex patterns are possible. Conway developed his rules as a simplified model of a hypothetical machine that could build copies of itself, a more complicated version of which was discovered by John von Neumann in the 1940s. Variations on the Game of Life use different rules for cell birth and death, use more than two states (resulting in evolving multicolored patterns), or are played on a different type of grid (e.g., a hexagonal grid or a three-dimensional one). After making its first public appearance in the October 1970 issue of Scientific American, the Game of Life popularized a whole new field of mathematical research called cellular automata, which has been applied to problems in cryptography and error-correction coding, and has even been suggested as the basis for new discrete models of the universe.
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