Mathematics is the study of all self-consistent structures using numbers, symbols, operations, and logical reasoning.
The field is sometimes divided into pure and applied mathematics. Almost all fields of science require at least some familiarity with mathematics, but this is especially true of theoretical sciences like physics, and of applied fields like engineering and infotech.
Applied Mathematics - Text by M. Alan Kazlev Use of mathematical techniques in an attempt to resolve scientific and engineering problems. Contrast with Pure Mathematics.
Attractor - Text by M. Alan Kazlev An archetype or state that is characterization of the long-term behaviour of a dissipative dynamic system. Over long periods of time, the state space of some dynamical systems will contract toward this region. The Archailects are said to be dynamic systems that characterize particular attractors. Likewise, certain biological forms, certain memes, certain technological solutions, and so on, are known to have emerged independently on completely different planets or among different alien races.
Bit - Text by M. Alan Kazlev The smallest unit of information a computer or virtual processing system can hold. The value of a bit is 1 or 0.
Bode's Law - Text by M. Alan Kazlev (also known as the Titus-Bode Law) is an interesting mathematical coincidence, and not a physical law. However, the general form of geometric spacing is valid in Sol-type stable stellar systems, hence the numerical series that matches planetary distances from Sol and many other stars. Because of resonance, planets form with periods that are rational multiples of each other. As some resonances are stable and others are not, modelling requires complex calculations. The Titus-Bode series predicts the positions of all the planets in the Sol System except Neptune (plus it predicts a planet where the asteroid belt is). In this instance, start with 0.4 (A.U.), then form the series by adding 0, 0.3, 0.6, 1.2, 2.4, etc. (doubling each time after the first). The results it predicts are: Mercury: 0.4 A.U., Venus 0.7 A.U., Earth 1.0 A.U., Mars 1.6 A.U., Asteroid Belt 2.8 A.U., Jupiter 5.2 A.U., Saturn 10.0 A.U., Uranus 19.6 A.U. [named for Johann Titus and the German astronomer Elert Bode, who did their work in the late 1700's.]
Cellular Automaton - Text by M. Alan Kazlev A vir dynamic system that is composed of an array of cells. All interactions are local, with the next state of a cell being a function of the current state of itself and its neighbors. Cellular Automa are a non-sentient common environment for alife experiments. Today some alife species (most notably the AKD) exist in cellular automata. Cellular Automata are also important in the creation of virchworlds.
Chaos (physics) - Text by M. Alan Kazlev Irregular motion of a dynamic system that is impossible to predict in the long term. The subject of chaos theory and a defining quality of most complex (whether living or non-living) systems.
Chaos Theory - Text by M. Alan Kazlev The branch of mathematics, simulation, and applied cosmology that deals with chaos.
Complex Number - Text by M. Alan Kazlev A number that has a real and an imaginary (e.g. square root of -1) component and is characterized as a point on a plane (instead of the real number line). Complex numbers are important in many forms of computation, simulation, and higher mathematics.
Computation - Text by M. Alan Kazlev Basically, what a computer does; which is mapping one set of numbers to another. The actual process of computing can be defined in terms of a very small number of very simple operations, such as addition, multiplication, recursion, and so on. Computing devices can also make statements about other computing devices.
Computer Science - Text by Adapted by M. Alan Kazlev from the original write-up by Robert J. Hall The theoretical and practical application of computational devices, including hardware, firmware and software architecture, programming skills, simulation techniques, customized algorithms and dedicated aioids, virtualics, networking, parallel programming, intelligent subroutines, information storage, comparative machine-level low level, and intermediate level operating systems, and subturing artificial intelligences.
Diverge, Divergence - Text by M. Alan Kazlev Algorithms that continue forever, iterative systems, Mandelbrot-like fractals, virchworlds or alife environments that reaching a state such that all future states increase (sometimes exponentially) in size.
Edge of Chaos - Text by M. Alan Kazlev The (computational/mathematical) region between static patterns of order and a chaotic regime. Many natural systems - including biological, alife, ai evolutionary, social, organizational, and so on - tend towards a dynamic behavior in this region.
Formal System - Text by M. Alan Kazlev A mathematical formalism in which statements can be constructed and manipulated with logical rules. Some formal systems are built around a few basic axioms and can be expanded with theorems that can be deduced through proofs.
Fractal - Text by M. Alan Kazlev An object with a fractal dimension. Fractals are self-similar and recursive; they may be deterministic or stochastic (random). They are important in creating rl-like virch-universes and simulations with only a relatively limited degree of processing. Many phenomena in nature have a fractal form - e.g. clouds, geographical features (coastlines, mountains, etc), snowflakes, plants, metabolic rhythms (e.g. heartbeat), economic cycles, and so on. Well-known fractals include the Cantor Set, Julia Set, and Mandelbrot Set.
Fractal Dimension - Text by M. Alan Kazlev A fractional or non-integer dimension. A fractal may be more than a line (1 dimension) but less than a plane (2 dimensions), or alternatively more than a plane but less than a sphere (3 dimensions). Hence fractal dimensions are defined in terms of decimal or fractional numbers. There are a number of ways of computing a fractal dimension, including some unusual but popular algorithms employed by transingularitan intelligences.
Game Theory - Text by Anders Sandberg A mathematical theory that deals with strategies for maximizing gains and minimizing losses within certain prescribed constraints, incredibly sophisticated variants are widely used by AIs in the solution of various decision-making problems, as those of military strategy, business policy, economic theories, and sociology. Many military, ethical and economical applications.
Gaussian - Text by M. Alan Kazlev Bell-shaped curve, corresponding to the normal (Gaussian) distribution in probability.
Group (mathematics) - Text by M. Alan Kazlev A mathematical object, consisting of a set of elements and a multiplication rule such that the product of two elements is also an element of the group, there exists an element called the identity that when multiplied with another element just produces that element, and that for each element there is an inverse element so that the product of them is the identity.
Gödel's Incompleteness Theorem - Text by M. Alan Kazlev from Anders Andberg'sTranshumanist Terminology_ Proves that any proposed axiom set for arithmetic is either consistent (no contradictions can be derived) or complete (it will say yes or no to every arithmetic proposition). In other words, any system or axiom set strong enough to include arithmetic which is complete will be inconsistent (it will say yes and no to at least one question). The theorem is named after Kurt Gödel, Czech mathematician, Atomic Age Old Earth.
Hyperbola (mathematics) - Text by M. Alan Kazlev Conic section (the intersection of a cone with a plane) that has two mirror-image branches. Hyperbolas have an eccentricity greater than 1.
Julia Set - Text by M. Alan Kazlev Complex fractal derived from iterating a set of non-diverging values. There is an infinite number of possible Julia sets, each of which corresponds to a particular complex number that appears as a constant in the iterative procedure. A number of hyperturing clades and aioids find experimenting with Julia Sets aesthetically pleasing. All Julia sets are related to, and included in, the Mandelbrot set.
Local Minimum / Local Maximum - Text by M. Alan Kazlev In mathematical modelling, the bottom of a valley or the top of a peak; such that all nearby points are either higher (for a minimum) or lower (for a maximum). The particular valley or peak may not necessarily be the lowest or highest location in the space, see global minimum / maximum.
Metacomplexity - Text by M. Alan Kazlev Complexity in which even the relations between the various complexities that make up the whole is complex.
Metrology - Text by M. Alan Kazlev, from the original by Robert J. Hall The science of precise measurement, dealing with a variety of physical properties of materials and structures and both simple and complex systems, on both the nano (nanometrology) and macro scale, and the various associated tools and techniques for this. An important element in engineering design and manufacturing process.
Parabola - Text by M. Alan Kazlev; amended by Stephen Inniss A conic section, a curve that is a set of points (P) such that the distance from a line (the directrix) to P is equal to the distance from P to focus F. A parabolic mirror will concentrate incoming light at a single point, or send out light from a source at that point in a collimated beam. In celestial mechanics parabolas have an eccentricity of 1, and an object in a parabolic orbit will swing past and change course but will not return since it is moving at escape velocity.
Pythagoras - Text by M. Alan Kazlev Presocratic philosopher, Old Earth, circa 2550/2540 to 2470 BT (580/570-500 b.c.e.). Founder of a major school of religious philosophy that emphasized the mystical interconnections in numbers, nature, and the human soul, on the basis of geometric ratios, musical chords, etc. He considered the natural and the ethical world to be inseparable. Pythagoras had a great influence on later thinkers, including Plato and Kepler. His vision of correspondences in the natural and spiritual world, albeit greatly modified, are still influential in parts of the Sophic League today.
Recursion - Text by M. Alan Kazlev from KurzweilAI The process of defining or expressing a function or procedure in terms of itself. Typically, each iteration of a recursive-solution procedure produces a simpler (or possibly smaller) version of the problem than the previous iteration. This process continues until a subproblem whose answer is already known (or that can be readily computed without recursion) is obtained. A surprisingly large number of symbolic and numerical problems lend themselves to recursive formulations. Recursion is typically used by game-playing programs, fractal aioids, and many AIs.
Saddle - Text by M. Alan Kazlev A type of surface that is neither a peak nor a valley but still has a zero gradient. Saddles are used in modeling a wide range of topologies - from the curvature of space-time to certain membranotoposophies. The Keterist Institute of Transapient Saddle Studies on the "B" Ring Band of the Pidelo Megastructure, Pidelo [Keter dominion], is an artistic SI:1 community dedicated to various interpretations of Saddle Simulations, especially in the area of Applied Mathematical Art.
Scalar - Text by M. Alan Kazlev based on original by Gary William Flake A single number, as opposed to a multidimensional vector or matrix.
Self-Organized Criticality - Text by M. Alan Kazlev base on original Gary William Flake A mathematical theory (information age and later) that describes how a self-organized system may arise. Although transapient powers and minds long since developed more sophisticated patterns of understanding the emergence of complexity, subminds and superbright sapients find simulations using these protocols a useful way to map and understand dynamic behavior that is neither stable nor unstable but at a region near a phase transition.
Set - Text by M. Alan Kazlev A collection of things, usually numbers. Sets may be finite or infinite in size.
Singularity - Text by M. Alan Kazlev  A point where the known laws of mathematics or physics no longer apply  A state impossible to predict or comprehend by those that have not attained it.  A toposophic grade of creative problem-solving, incomprehensible to those that have not attained that state. See also, S (Singularity Level).
Statistics - Text by M. Alan Kazlev Field of mathematics dealing with evaluating sampled data to find mathematical or other patterns. Includes probability theory, the application of statistical methods for performing studies of complex systems, and sampling techniques for measuring specific information. Has many applications including white noise aesthetics, random walk simms, economics, materials studies, medicine, psychology, sociology, market research, anakalyptics, and cliology.
Strange Attractor - Text by Gary William Flake An attractor of a dynamical system that is usually fractal in dimension and is indicative of chaos.
Structic - Text by Anders Sandberg Generalization of mathematics to encompass indeterminate structures.
Trapdoor Function - Text by Anders Sandberg in his Transhuman Terminology A function that is easily computable, but whose inverse is very hard to compute unless an extra bit of information is provided. The term is used in cryptography.
Vector  - Text by M. Alan Kazlev A one-dimensional array of numbers that can be used to represent a point in a multidimensional space. Commonly in 3-space a vector is viewed as a number (a magnitude) plus a direction (compare with scalar). A vector can be represented by an arrow whose length represents the magnitude and the direction represents the direction. For example, velocity is a vector; velocity tells you how fast something is traveling, and its direction.