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Lettris is a curious tetris-clone game where all the bricks have the same square shape but different content. Each square carries a letter. To make squares disappear and save space for other squares you have to assemble English words (left, right, up, down) from the falling squares.
Boggle gives you 3 minutes to find as many words (3 letters or more) as you can in a grid of 16 letters. You can also try the grid of 16 letters. Letters must be adjacent and longer words score better. See if you can get into the grid Hall of Fame !
Change the target language to find translations.
Tips: browse the semantic fields (see From ideas to words) in two languages to learn more.
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Causality is the relationship between causes and effects. It is considered to be fundamental to all natural science, especially physics. Causality is also a topic studied from the perspectives of philosophy and statistics.
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Humans have a practical interest in their surroundings, and tend to be resistant to the idea that things "just happen." If one or more sheep die, humans will attempt to discover why. Learning what has killed the sheep is an important step in protecting the flock. The question can be phrased as, "What caused the sheep to die?" The answer may be "The wolves broke their necks," or "Eating too much clover caused them to bloat." These explanations assume the presence of an agent of some kind. In cases where an obvious cause is not discovered, humans may attribute the events to miracles or to evil supernatural agencies, or create an ungrounded theory to present something as a causative agent. There is a learned preference for some alternative to saying that something occurred without there being a reason for it. Any event that stands as 'uncaused' may motivate us to understand the salient events in the environment of that event.
When trying to answer questions such as, "Why is the water boiling?" it is tempting to search for a single responsible figure: "Mother lit the gas burner under the tea kettle." However, closer examination shows that the lighting of a flame is not the only feature that must be accounted for. Closer study shows that the ambient air pressure is a controlling factor, the Earth's gravity holding the water in the pot is a controlling factor, the temperature to which the water has been heated is a controlling factor, etc.
Given a situation in which water is already sitting in a tea kettle on top of a gas burner, then someone lighting the gas under the kettle may be perceived as the "cause" of the water's boiling. But in the same situation the water would also boil if the air pressure were sufficiently reduced. In more complicated situations, more factors that influence outcomes may be involved. Underlying the expectations that most people hold in regard to the interactions of these several factors are everyday experiences in which someone succeeds in producing a desired result. "If I turn this key, the motor will start." This may be a true statement, but underlying it is a very complicated set of conditions that must all be in place. So the idea of "cause" tends to focus on foreground events and leave out necessary factors that reside in the background.
Real world factors can intervene to challenge simplistic ideas of causation. Water in a pressure cooker will not boil at 100 degrees Celsius. The key is turned but the car motor will not start. So at this level of sophistication, it is not difficult to gain acceptance of a more refined idea of causation. And, in practice, few individuals expect to get a desired outcome without having fulfilled the conditions necessary to earn it. People may wish for a million dollars, but they do not wish for hot toast to emerge from a cold and empty toaster.
The most abstract form of the argument from experience is: (1) Do something and get a certain result. (2) Do not do it and do not get the result. Hence, a cause is both sufficient (1) and necessary (2) in order that the event happen.
Another lesson consistent with everyday experience is that these sequences do not work in reverse. For instance, if a cup of boiling water is allowed to sit at room temperature then it will cool to room temperature, but if a cup of room temperature water is allowed to sit at room temperature it will not warm to the point that it boils. So there is a 'sequentiality' in events familiar to humans from their everyday experience. Whenever this 'sequentiality' is detected or believed to be present, the temporally prior change is said to be a cause, and the temporally posterior change is said to be an effect.
One crack in this belief system has been produced by radioactivity. An atom of some radioactive substance such as radium will eventually decay, and in the process it will emit energy. But there is no known triggering event that could serve as the cause of this decay event. In a large collection of radium atoms the rate of decay can be accurately predicted, but the identity of the decayed atoms cannot be determined beforehand. Their decay is random and was considered to be uncaused in an older framework of quantum theory. Under the new formulation, spontaneous emission for example, is initiated by vacuum fluctuations.
Another crack in this belief system has been produced by quantum mechanical events such that the same sequence of causal events (or causal factors) regularly produces different effects (i.e., results), but the results may repeat themselves in some random (unknowable) sequence. Furthermore, the percentages of results of each kind can be calculated and they are highly predictable.
Results of this kind are seen in the macro world of human beings only in the case of crooked roulette wheels or other such crooked gambling devices since legitimate roulette wheels should have an equal probability of stopping at each position.
In physics it is useful to interpret certain terms of a physical theory as causes and other terms as effects. Thus, in classical (Newtonian) mechanics a cause may be represented by a force acting on a body, and an effect by the acceleration which follows as quantitatively explained by Newton's second law. For different physical theories the notions of cause and effect may be different. For instance, in Aristotelian physics the effect is not said to be acceleration but to be velocity (one must push a cart twice as hard in order to have its velocity doubled). In the general theory of relativity, too, acceleration is not an effect (since it is not a generally relativistic vector); the general relativistic effects comparable to those of Newtonian mechanics are the deviations from geodesic motion in curved spacetime. Also, the meaning of "uncaused motion" is dependent on the theory being employed: for Aristotle it is (absolute) rest, for Newton it is inertial motion (constant velocity with respect to an inertial frame of reference), in the general theory of relativity it is geodesic motion (to be compared with frictionless motion on the surface of a sphere at constant tangential velocity along a great circle). So what constitutes a "cause" and what constitutes an "effect" depends on the total system of explanation in which the putative causal sequence is embedded.
A formulation of physical laws in terms of cause and effect is useful for the purposes of explanation and prediction. For instance, in Newtonian mechanics an observed acceleration can be explained by reference to an applied force. So Newton's second law can be used to predict the force necessary to realize a desired acceleration.
In classical physics a cause should always precede its effect. In relativity theory this requirement is strengthened[why?] so as to limit causes to the back (past) light cone of the event to be explained (the "effect"); nor can an event be a cause of any event outside the former event's front (future) light cone. These restrictions are consistent with the grounded belief (or assumption) that causal influences cannot travel faster than the speed of light and/or backwards in time.
Another requirement, at least valid at the level of human experience, is that cause and effect be mediated across space and time (requirement of contiguity). This requirement has been very influential in the past, in the first place as a result of direct observation of causal processes (like pushing a cart), in the second place as a problematic aspect of Newton's theory of gravitation (attraction of the earth by the sun by means of action at a distance) replacing mechanistic proposals like Descartes' vortex theory; in the third place as an incentive to develop dynamic field theories (e.g., Maxwell's electrodynamics and Einstein's general theory of relativity) restoring contiguity in the transmission of influences in a more successful way than did Descartes' theory.
The empiricists' aversion to metaphysical explanations (like Descartes' vortex theory) lends heavy influence against the idea of the importance of causality. Causality has accordingly sometimes been downplayed (e.g., Newton's "Hypotheses non fingo"). According to Ernst Mach the notion of force in Newton's second law was pleonastic, tautological and superfluous. Indeed it is possible to consider the Newtonian equations of motion of the gravitational interaction between the sun (s) and a planet (p),
as two coupled equations describing the positions and of planet and sun, without interpreting the right hand sides of these equations as forces; the equations just describe a process of interaction, without any necessity to interpret the sun as the cause of the motion of the planet (or vice versa), and allow one to predict the states of the system s+p at later (as well as earlier) times.
The ordinary situations in which humans singled out some factors in a physical interaction as being prior and therefore supplying the "because" of the interaction were often ones in which humans decided to bring about some state of affairs and directed their energies to producing that state of affairs—a process that took time to establish and left a new state of affairs that persisted beyond the time of activity of the actor. It would be difficult and pointless, however, to explain the motions of binary stars with respect to each other in that way.
The possibility of such a time-independent view is at the basis of the deductive-nomological (D-N) view of scientific explanation, considering an event to be explained if it can be subsumed under a scientific law. In the D-N view, a physical state is considered to be explained if, applying the (deterministic) law, it can be derived from given initial conditions. (Such initial conditions could include the momenta and distance from each other of binary stars at any given moment.) Such 'explanation by determinism' is sometimes referred to as causal determinism. A disadvantage of the D-N view is that causality and determinism are more or less identified. Thus, in classical physics, it was assumed that all events are caused by earlier ones according to the known laws of nature, culminating in Pierre-Simon Laplace's claim that if the current state of the world were known with precision, it could be computed for any time in the future or the past (see Laplace's demon). However, this is usually referred to as Laplace determinism (rather than `Laplace causality') because it hinges on determinism in mathematical models as dealt with in the mathematical Cauchy problem. Confusion of causality and determinism is particularly acute in quantum mechanics, this theory being acausal (in consequence of its inability to provide descriptions of the causes of all actually observed effects) but deterministic in the mathematical sense.
In modern physics, the notion of causality had to be clarified. The insights of the theory of special relativity confirmed the assumption of causality, but they made the meaning of the word "simultaneous" observer-dependent. Consequently, the relativistic principle of causality says that the cause must precede its effect according to all inertial observers. This is equivalent to the statement that the cause and its effect are separated by a timelike interval, and the effect belongs to the future of its cause. If a timelike interval separates the two events, this means that a signal could be sent between them at less than the speed of light. On the other hand, if signals could move faster than the speed of light, this would violate causality because it would allow a signal to be sent across spacelike intervals, which means that at least to some inertial observers the signal would travel backward in time. For this reason, special relativity does not allow communication faster than the speed of light.
In the theory of general relativity, the concept of causality is generalized in the most straightforward way: the effect must belong to the future light cone of its cause, even if the spacetime is curved. New subtleties must be taken into account when we investigate causality in quantum mechanics and relativistic quantum field theory in particular. In quantum field theory, causality is closely related to the principle of locality. However, the principle of locality is disputed: whether it strictly holds depends on the interpretation of quantum mechanics chosen, especially for experiments involving quantum entanglement that satisfy Bell's Theorem.
Despite these subtleties, causality remains an important and valid concept in physical theories. For example, the notion that events can be ordered into causes and effects is necessary to prevent (or at least outline) causality paradoxes such as the grandfather paradox, which asks what happens if a time-traveler kills his own grandfather before he ever meets the time-traveler's grandmother. See also Chronology protection conjecture.
"Small variations of the initial condition of a nonlinear dynamical system may produce large variations in the long term behavior of the system."
This opens up the opportunity to understand a distributed causality.
A related way to interpret the Butterfly effect is to see it as highlighting the difference between the application of the notion of causality in physics and a more general use of causality as represented by Mackie's INUS conditions. In physics, in general, only those conditions are (explicitly) taken into account, that are both necessary and sufficient. For instance, when a massive sphere is caused to roll down a slope starting from a point of unstable equilibrium, then its velocity is assumed to be caused by the force of gravity accelerating it; the small push that was needed to set it into motion is not explicitly dealt with as a cause. In order to be a physical cause there must be a certain proportionality with the ensuing effect. A distinction is drawn between triggering and causation of the ball's motion. By the same token the butterfly can be seen as triggering a tornado, its cause being assumed to be seated in the atmospherical energies already present beforehand, rather than in the movements of a butterfly.
Causal dynamical triangulation (abbreviated as "CDT") invented by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz, and popularized by Fotini Markopoulou and Lee Smolin, is an approach to quantum gravity that like loop quantum gravity is background independent. This means that it does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves. The Loops '05 conference, hosted by many loop quantum gravity theorists, included several presentations which discussed CDT in great depth, and revealed it to be a pivotal insight for theorists. It has sparked considerable interest as it appears to have a good semi-classical description. At large scales, it re-creates the familiar 4-dimensional spacetime, but it shows spacetime to be 2-d near the Planck scale, and reveals a fractal structure on slices of constant time. Using a structure called a simplex, it divides spacetime into tiny triangular sections. A simplex is the generalized form of a triangle, in various dimensions. A 3-simplex is usually called a tetrahedron, and the 4-simplex, which is the basic building block in this theory, is also known as the pentatope, or pentachoron. Each simplex is geometrically flat, but simplices can be "glued" together in a variety of ways to create curved spacetimes. Where previous attempts at triangulation of quantum spaces have produced jumbled universes with far too many dimensions, or minimal universes with too few, CDT avoids this problem by allowing only those configurations where cause precedes any event. In other words, the timelines of all joined edges of simplices must agree.
Thus, maybe, causality lies in the foundation of the spacetime geometry.