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114-487: Causes , or causality , is the relationship between one event and another. It may also refer to: Causes (band) , an indie band based in the Netherlands Causes (company) , an online company See also [ edit ] Cause (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

228-442: A n c e r | s m o k i n g ) {\displaystyle P(cancer|smoking)} , and interventional probabilities , as in P ( c a n c e r | d o ( s m o k i n g ) ) {\displaystyle P(cancer|do(smoking))} . The former reads: "the probability of finding cancer in a person known to smoke, having started, unforced by

342-546: A 4×4 matrix depending on spacetime position . Minkowski space is thus a comparatively simple special case of a Lorentzian manifold . Its metric tensor is in coordinates with the same symmetric matrix at every point of M , and its arguments can, per above, be taken as vectors in spacetime itself. Introducing more terminology (but not more structure), Minkowski space is thus a pseudo-Euclidean space with total dimension n = 4 and signature (1, 3) or (3, 1) . Elements of Minkowski space are called events . Minkowski space

456-506: A progression of events following one after the other as cause and effect. Incompatibilism holds that determinism is incompatible with free will, so if determinism is true, " free will " does not exist. Compatibilism , on the other hand, holds that determinism is compatible with, or even necessary for, free will. Causes may sometimes be distinguished into two types: necessary and sufficient. A third type of causation, which requires neither necessity nor sufficiency, but which contributes to

570-418: A 'substance', as distinct from an action. Since causality is a subtle metaphysical notion, considerable intellectual effort, along with exhibition of evidence, is needed to establish knowledge of it in particular empirical circumstances. According to David Hume , the human mind is unable to perceive causal relations directly. On this ground, the scholar distinguished between the regularity view of causality and

684-511: A causal ordering. The system of equations must have certain properties, most importantly, if some values are chosen arbitrarily, the remaining values will be determined uniquely through a path of serial discovery that is perfectly causal. They postulate the inherent serialization of such a system of equations may correctly capture causation in all empirical fields, including physics and economics. Some theorists have equated causality with manipulability. Under these theories, x causes y only in

798-429: A cause is incorrectly identified. Counterfactual theories define causation in terms of a counterfactual relation, and can often be seen as "floating" their account of causality on top of an account of the logic of counterfactual conditionals . Counterfactual theories reduce facts about causation to facts about what would have been true under counterfactual circumstances. The idea is that causal relations can be framed in

912-441: A cause, and what kind of entity can be an effect?" One viewpoint on this question is that cause and effect are of one and the same kind of entity, causality being an asymmetric relation between them. That is to say, it would make good sense grammatically to say either " A is the cause and B the effect" or " B is the cause and A the effect", though only one of those two can be actually true. In this view, one opinion, proposed as

1026-586: A definite time. Such a process can be regarded as a cause. Causality is not inherently implied in equations of motion , but postulated as an additional constraint that needs to be satisfied (i.e. a cause always precedes its effect). This constraint has mathematical implications such as the Kramers-Kronig relations . Causality is one of the most fundamental and essential notions of physics. Causal efficacy cannot 'propagate' faster than light. Otherwise, reference coordinate systems could be constructed (using

1140-408: A known causal effect or to test a causal model than to generate causal hypotheses. For nonexperimental data, causal direction can often be inferred if information about time is available. This is because (according to many, though not all, theories) causes must precede their effects temporally. This can be determined by statistical time series models, for instance, or with a statistical test based on

1254-403: A mathematical definition of "confounding" and helps researchers identify accessible sets of variables worthy of measurement. While derivations in causal calculus rely on the structure of the causal graph, parts of the causal structure can, under certain assumptions, be learned from statistical data. The basic idea goes back to Sewall Wright 's 1921 work on path analysis . A "recovery" algorithm

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1368-443: A metaphysical account of what it is for there to be a causal relation between some pair of events. If correct, the analysis has the power to explain certain features of causation. Knowing that causation is a matter of counterfactual dependence, we may reflect on the nature of counterfactual dependence to account for the nature of causation. For example, in his paper "Counterfactual Dependence and Time's Arrow," Lewis sought to account for

1482-444: A metaphysical principle in process philosophy , is that every cause and every effect is respectively some process, event, becoming, or happening. An example is 'his tripping over the step was the cause, and his breaking his ankle the effect'. Another view is that causes and effects are 'states of affairs', with the exact natures of those entities being more loosely defined than in process philosophy. Another viewpoint on this question

1596-595: A one-to-one manner with a directional derivative operator on the set of smooth functions. This is promoted to a definition of tangent vectors in manifolds not necessarily being embedded in R . This definition of tangent vectors is not the only possible one, as ordinary n -tuples can be used as well. A tangent vector at a point p may be defined, here specialized to Cartesian coordinates in Lorentz frames, as 4 × 1 column vectors v associated to each Lorentz frame related by Lorentz transformation Λ such that

1710-411: A process and a pseudo-process . As an example, a ball moving through the air (a process) is contrasted with the motion of a shadow (a pseudo-process). The former is causal in nature while the latter is not. Salmon (1984) claims that causal processes can be identified by their ability to transmit an alteration over space and time. An alteration of the ball (a mark by a pen, perhaps) is carried with it as

1824-407: A process can have multiple causes, which are also said to be causal factors for it, and all lie in its past . An effect can in turn be a cause of, or causal factor for, many other effects, which all lie in its future . Some writers have held that causality is metaphysically prior to notions of time and space . Causality is an abstraction that indicates how the world progresses. As such it

1938-418: A real number. One has to be careful in the use of the word cause in physics. Properly speaking, the hypothesized cause and the hypothesized effect are each temporally transient processes. For example, force is a useful concept for the explanation of acceleration, but force is not by itself a cause. More is needed. For example, a temporally transient process might be characterized by a definite change of force at

2052-480: A rotation matrix in four-dimensional space, which rotates the four-vector around a particular axis. x 2 + y 2 + z 2 + ( i c t ) 2 = constant . {\displaystyle x^{2}+y^{2}+z^{2}+(ict)^{2}={\text{constant}}.} Rotations in planes spanned by two space unit vectors appear in coordinate space as well as in physical spacetime as Euclidean rotations and are interpreted in

2166-402: A triangle. Nonetheless, even when interpreted counterfactually, the first statement is true. An early version of Aristotle's "four cause" theory is described as recognizing "essential cause". In this version of the theory, that the closed polygon has three sides is said to be the "essential cause" of its being a triangle. This use of the word 'cause' is of course now far obsolete. Nevertheless, it

2280-536: A wave packet travels at the phase velocity; since phase is not causal, the phase velocity of a wave packet can be faster than light. Causal notions are important in general relativity to the extent that the existence of an arrow of time demands that the universe's semi- Riemannian manifold be orientable, so that "future" and "past" are globally definable quantities. Minkowski space#Causal structure In physics , Minkowski space (or Minkowski spacetime ) ( / m ɪ ŋ ˈ k ɔː f s k i , - ˈ k ɒ f -/ )

2394-404: A window and it breaks. If Alice hadn't thrown the brick, then it still would have broken, suggesting that Alice wasn't a cause; however, intuitively, Alice did cause the window to break. The Halpern-Pearl definitions of causality take account of examples like these. The first and third Halpern-Pearl conditions are easiest to understand: AC1 requires that Alice threw the brick and the window broke in

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2508-511: Is η ( u 1 , u 2 ) = u 1 ⋅ u 2 = c 2 t 1 t 2 − x 1 x 2 − y 1 y 2 − z 1 z 2 . {\displaystyle \eta (u_{1},u_{2})=u_{1}\cdot u_{2}=c^{2}t_{1}t_{2}-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}.} Positivity of scalar product : An important property

2622-472: Is a basic concept; it is more apt to be an explanation of other concepts of progression than something to be explained by other more fundamental concepts. The concept is like those of agency and efficacy . For this reason, a leap of intuition may be needed to grasp it. Accordingly, causality is implicit in the structure of ordinary language, as well as explicit in the language of scientific causal notation . In English studies of Aristotelian philosophy ,

2736-556: Is a concern of the subject known as metaphysics . Kant thought that time and space were notions prior to human understanding of the progress or evolution of the world, and he also recognized the priority of causality. But he did not have the understanding that came with knowledge of Minkowski geometry and the special theory of relativity , that the notion of causality can be used as a prior foundation from which to construct notions of time and space. A general metaphysical question about cause and effect is: "what kind of entity can be

2850-559: Is a coordinate-invariant property of η . The space of bilinear maps forms a vector space which can be identified with M ∗ ⊗ M ∗ {\displaystyle M^{*}\otimes M^{*}} , and η may be equivalently viewed as an element of this space. By making a choice of orthonormal basis { e μ } {\displaystyle \{e_{\mu }\}} , M := ( V , η ) {\displaystyle M:=(V,\eta )} can be identified with

2964-439: Is a process that is varied from occasion to occasion. The occurrence or non-occurrence of subsequent bubonic plague is recorded. To establish causality, the experiment must fulfill certain criteria, only one example of which is mentioned here. For example, instances of the hypothesized cause must be set up to occur at a time when the hypothesized effect is relatively unlikely in the absence of the hypothesized cause; such unlikelihood

3078-462: Is a smoker") probabilistically causes B ("The person has now or will have cancer at some time in the future"), if the information that A occurred increases the likelihood of B s occurrence. Formally, P{ B | A }≥ P{ B } where P{ B | A } is the conditional probability that B will occur given the information that A occurred, and P{ B } is the probability that B will occur having no knowledge whether A did or did not occur. This intuitive condition

3192-1255: Is another consequence of the convexity of either light cone. For two distinct similarly directed time-like vectors u 1 and u 2 this inequality is η ( u 1 , u 2 ) > ‖ u 1 ‖ ‖ u 2 ‖ {\displaystyle \eta (u_{1},u_{2})>\left\|u_{1}\right\|\left\|u_{2}\right\|} or algebraically, c 2 t 1 t 2 − x 1 x 2 − y 1 y 2 − z 1 z 2 > ( c 2 t 1 2 − x 1 2 − y 1 2 − z 1 2 ) ( c 2 t 2 2 − x 2 2 − y 2 2 − z 2 2 ) {\displaystyle c^{2}t_{1}t_{2}-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}>{\sqrt {\left(c^{2}t_{1}^{2}-x_{1}^{2}-y_{1}^{2}-z_{1}^{2}\right)\left(c^{2}t_{2}^{2}-x_{2}^{2}-y_{2}^{2}-z_{2}^{2}\right)}}} From this,

3306-526: Is called the Poincaré group . Minkowski's model follows special relativity, where motion causes time dilation changing the scale applied to the frame in motion and shifts the phase of light. Spacetime is equipped with an indefinite non-degenerate bilinear form , called the Minkowski metric , the Minkowski norm squared or Minkowski inner product depending on the context. The Minkowski inner product

3420-640: Is closely associated with Einstein's theories of special relativity and general relativity and is the most common mathematical structure by which special relativity is formalized. While the individual components in Euclidean space and time might differ due to length contraction and time dilation , in Minkowski spacetime, all frames of reference will agree on the total interval in spacetime between events. Minkowski space differs from four-dimensional Euclidean space insofar as it treats time differently than

3534-457: Is current nowadays, although the older view involving imaginary time has also influenced special relativity. In the English translation of Minkowski's paper, the Minkowski metric, as defined below, is referred to as the line element . The Minkowski inner product below appears unnamed when referring to orthogonality (which he calls normality ) of certain vectors, and the Minkowski norm squared

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3648-402: Is defined so as to yield the spacetime interval between two events when given their coordinate difference vector as an argument. Equipped with this inner product, the mathematical model of spacetime is called Minkowski space. The group of transformations for Minkowski space that preserves the spacetime interval (as opposed to the spatial Euclidean distance) is the Poincaré group (as opposed to

3762-611: Is equivalent to the definition given above under a canonical isomorphism. For some purposes, it is desirable to identify tangent vectors at a point p with displacement vectors at p , which is, of course, admissible by essentially the same canonical identification. The identifications of vectors referred to above in the mathematical setting can correspondingly be found in a more physical and explicitly geometrical setting in Misner, Thorne & Wheeler (1973) . They offer various degrees of sophistication (and rigor) depending on which part of

3876-459: Is more basic than causal interaction. But describing manipulations in non-causal terms has provided a substantial difficulty. The second criticism centers around concerns of anthropocentrism . It seems to many people that causality is some existing relationship in the world that we can harness for our desires. If causality is identified with our manipulation, then this intuition is lost. In this sense, it makes humans overly central to interactions in

3990-409: Is not adequate as a definition for probabilistic causation because of its being too general and thus not meeting our intuitive notion of cause and effect. For example, if A denotes the event "The person is a smoker," B denotes the event "The person now has or will have cancer at some time in the future" and C denotes the event "The person now has or will have emphysema some time in the future," then

4104-453: Is not covered here. For an overview, Minkowski space is a 4 -dimensional real vector space equipped with a non-degenerate, symmetric bilinear form on the tangent space at each point in spacetime, here simply called the Minkowski inner product , with metric signature either (+ − − −) or (− + + +) . The tangent space at each event is a vector space of the same dimension as spacetime, 4 . In practice, one need not be concerned with

4218-451: Is often denoted R or R to emphasize the chosen signature, or just M . It is an example of a pseudo-Riemannian manifold . Then mathematically, the metric is a bilinear form on an abstract four-dimensional real vector space V , that is, η : V × V → R {\displaystyle \eta :V\times V\rightarrow \mathbf {R} } where η has signature (+, -, -, -) , and signature

4332-457: Is possible to have other combinations of vectors. For example, one can easily construct a (non-orthonormal) basis consisting entirely of null vectors, called a null basis . Vector fields are called timelike, spacelike, or null if the associated vectors are timelike, spacelike, or null at each point where the field is defined. Time-like vectors have special importance in the theory of relativity as they correspond to events that are accessible to

4446-426: Is referred to (somewhat cryptically, perhaps this is a translation dependent) as "sum". Minkowski's principal tool is the Minkowski diagram , and he uses it to define concepts and demonstrate properties of Lorentz transformations (e.g., proper time and length contraction ) and to provide geometrical interpretation to the generalization of Newtonian mechanics to relativistic mechanics . For these special topics, see

4560-744: Is specifically characteristic of quantal phenomena that observations defined by incompatible variables always involve important intervention by the experimenter, as described quantitatively by the observer effect . In classical thermodynamics , processes are initiated by interventions called thermodynamic operations . In other branches of science, for example astronomy , the experimenter can often observe with negligible intervention. The theory of "causal calculus" (also known as do-calculus, Judea Pearl 's Causal Calculus, Calculus of Actions) permits one to infer interventional probabilities from conditional probabilities in causal Bayesian networks with unmeasured variables. One very practical result of this theory

4674-482: Is that the scalar product of two similarly directed time-like vectors is always positive. This can be seen from the reversed Cauchy–Schwarz inequality below. It follows that if the scalar product of two vectors is zero, then one of these, at least, must be space-like. The scalar product of two space-like vectors can be positive or negative as can be seen by considering the product of two space-like vectors having orthogonal spatial components and times either of different or

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4788-480: Is the Born coordinates . Another useful set of coordinates is the light-cone coordinates . The Minkowski inner product is not an inner product , since it is not positive-definite , i.e. the quadratic form η ( v , v ) need not be positive for nonzero v . The positive-definite condition has been replaced by the weaker condition of non-degeneracy. The bilinear form is said to be indefinite . The Minkowski metric η

4902-718: Is the characterization of confounding variables , namely, a sufficient set of variables that, if adjusted for, would yield the correct causal effect between variables of interest. It can be shown that a sufficient set for estimating the causal effect of X {\displaystyle X} on Y {\displaystyle Y} is any set of non-descendants of X {\displaystyle X} that d {\displaystyle d} -separate X {\displaystyle X} from Y {\displaystyle Y} after removing all arrows emanating from X {\displaystyle X} . This criterion, called "backdoor", provides

5016-512: Is the main mathematical description of spacetime in the absence of gravitation . It combines inertial space and time manifolds into a four-dimensional model. The model helps show how a spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Mathematician Hermann Minkowski developed it from the work of Hendrik Lorentz , Henri Poincaré , and others said it "was grown on experimental physical grounds". Minkowski space

5130-577: Is the metric tensor of Minkowski space. It is a pseudo-Euclidean metric, or more generally, a constant pseudo-Riemannian metric in Cartesian coordinates. As such, it is a nondegenerate symmetric bilinear form, a type (0, 2) tensor. It accepts two arguments u p , v p , vectors in T p M , p ∈ M , the tangent space at p in M . Due to the above-mentioned canonical identification of T p M with M itself, it accepts arguments u , v with both u and v in M . As

5244-477: Is the more classical one, that a cause and its effect can be of different kinds of entity. For example, in Aristotle's efficient causal explanation, an action can be a cause while an enduring object is its effect. For example, the generative actions of his parents can be regarded as the efficient cause, with Socrates being the effect, Socrates being regarded as an enduring object, in philosophical tradition called

5358-500: Is to be established by empirical evidence. A mere observation of a correlation is not nearly adequate to establish causality. In nearly all cases, establishment of causality relies on repetition of experiments and probabilistic reasoning. Hardly ever is causality established more firmly than as more or less probable. It is most convenient for establishment of causality if the contrasting material states of affairs are precisely matched, except for only one variable factor, perhaps measured by

5472-456: Is velocity, x , y , and z are Cartesian coordinates in 3-dimensional space, c is the constant representing the universal speed limit, and t is time, the four-dimensional vector v = ( ct , x , y , z ) = ( ct , r ) is classified according to the sign of c t − r . A vector is timelike if c t > r , spacelike if c t < r , and null or lightlike if c t = r . This can be expressed in terms of

5586-497: Is within the scope of ordinary language to say that it is essential to a triangle that it has three sides. A full grasp of the concept of conditionals is important to understanding the literature on causality. In everyday language, loose conditional statements are often enough made, and need to be interpreted carefully. Fallacies of questionable cause, also known as causal fallacies, non-causa pro causa (Latin for "non-cause for cause"), or false cause, are informal fallacies where

5700-635: The Galilean group ). In his second relativity paper in 1905, Henri Poincaré showed how, by taking time to be an imaginary fourth spacetime coordinate ict , where c is the speed of light and i is the imaginary unit , Lorentz transformations can be visualized as ordinary rotations of the four-dimensional Euclidean sphere. The four-dimensional spacetime can be visualized as a four-dimensional space, with each point representing an event in spacetime. The Lorentz transformations can then be thought of as rotations in this four-dimensional space, where

5814-463: The Lorentz transform of special relativity ) in which an observer would see an effect precede its cause (i.e. the postulate of causality would be violated). Causal notions appear in the context of the flow of mass-energy. Any actual process has causal efficacy that can propagate no faster than light. In contrast, an abstraction has no causal efficacy. Its mathematical expression does not propagate in

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5928-449: The counterfactual conditional , has a stronger connection with causality, yet even counterfactual statements are not all examples of causality. Consider the following two statements: In the first case, it would be incorrect to say that A's being a triangle caused it to have three sides, since the relationship between triangularity and three-sidedness is that of definition. The property of having three sides actually determines A's state as

6042-428: The dot product in R to R × C . This works in the flat spacetime of special relativity, but not in the curved spacetime of general relativity, see Misner, Thorne & Wheeler (1973 , Box 2.1, Farewell to ict ) (who, by the way use (− + + +) ). MTW also argues that it hides the true indefinite nature of the metric and the true nature of Lorentz boosts, which are not rotations. It also needlessly complicates

6156-596: The light cone of that event. Given a timelike vector v , there is a worldline of constant velocity associated with it, represented by a straight line in a Minkowski diagram. Once a direction of time is chosen, timelike and null vectors can be further decomposed into various classes. For timelike vectors, one has Null vectors fall into three classes: Together with spacelike vectors, there are 6 classes in all. An orthonormal basis for Minkowski space necessarily consists of one timelike and three spacelike unit vectors. If one wishes to work with non-orthonormal bases, it

6270-415: The skeletons (the graphs stripped of arrows) of these three triplets are identical, the directionality of the arrows is partially identifiable. The same distinction applies when X {\displaystyle X} and Z {\displaystyle Z} have common ancestors, except that one must first condition on those ancestors. Algorithms have been developed to systematically determine

6384-512: The (mentioned above) regularity, probabilistic , counterfactual, mechanistic , and manipulationist views. The five approaches can be shown to be reductive, i.e., define causality in terms of relations of other types. According to this reading, they define causality in terms of, respectively, empirical regularities (constant conjunctions of events), changes in conditional probabilities , counterfactual conditions, mechanisms underlying causal relations, and invariance under intervention. Causality has

6498-436: The Euclidean three-space to a quasi-Euclidean four-space that included time, Einstein was already aware that this is not valid, because it excludes the phenomenon of gravitation . He was still far from the study of curvilinear coordinates and Riemannian geometry , and the heavy mathematical apparatus entailed. For further historical information see references Galison (1979) , Corry (1997) and Walter (1999) . Where v

6612-498: The Minkowski inner product are all the same object; it is a bilinear function that accepts two (contravariant) vectors and returns a real number. In coordinates, this is the 4×4 matrix representing the bilinear form. For comparison, in general relativity , a Lorentzian manifold L is likewise equipped with a metric tensor g , which is a nondegenerate symmetric bilinear form on the tangent space T p L at each point p of L . In coordinates, it may be represented by

6726-404: The absence of firefighters. Together these are unnecessary but sufficient to the house's burning down (since many other collections of events certainly could have led to the house burning down, for example shooting the house with a flamethrower in the presence of oxygen and so forth). Within this collection, the short circuit is an insufficient (since the short circuit by itself would not have caused

6840-473: The actual work. AC3 requires that Alice throwing the brick is a minimal cause (cf. blowing a kiss and throwing a brick). Taking the "updated" version of AC2(a), the basic idea is that we have to find a set of variables and settings thereof such that preventing Alice from throwing a brick also stops the window from breaking. One way to do this is to stop Bob from throwing the brick. Finally, for AC2(b), we have to hold things as per AC2(a) and show that Alice throwing

6954-938: The algebraic definition with the reversed Cauchy inequality: ‖ u + w ‖ 2 = ‖ u ‖ 2 + 2 ( u , w ) + ‖ w ‖ 2 ≥ ‖ u ‖ 2 + 2 ‖ u ‖ ‖ w ‖ + ‖ w ‖ 2 = ( ‖ u ‖ + ‖ w ‖ ) 2 . {\displaystyle {\begin{aligned}\left\|u+w\right\|^{2}&=\left\|u\right\|^{2}+2\left(u,w\right)+\left\|w\right\|^{2}\\[5mu]&\geq \left\|u\right\|^{2}+2\left\|u\right\|\left\|w\right\|+\left\|w\right\|^{2}=\left(\left\|u\right\|+\left\|w\right\|\right)^{2}.\end{aligned}}} The result now follows by taking

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7068-417: The antecedent to precede or coincide with the consequent in time, whereas conditional statements do not require this temporal order. Confusion commonly arises since many different statements in English may be presented using "If ..., then ..." form (and, arguably, because this form is far more commonly used to make a statement of causality). The two types of statements are distinct, however. For example, all of

7182-478: The asymmetry of the causal relation is unrelated to the asymmetry of any mode of implication that contraposes. Rather, a causal relation is not a relation between values of variables, but a function of one variable (the cause) on to another (the effect). So, given a system of equations, and a set of variables appearing in these equations, we can introduce an asymmetric relation among individual equations and variables that corresponds perfectly to our commonsense notion of

7296-419: The ball goes through the air. On the other hand, an alteration of the shadow (insofar as it is possible) will not be transmitted by the shadow as it moves along. These theorists claim that the important concept for understanding causality is not causal relationships or causal interactions, but rather identifying causal processes. The former notions can then be defined in terms of causal processes. A subgroup of

7410-434: The brick breaks the window. (The full definition is a little more involved, involving checking all subsets of variables.) Interpreting causation as a deterministic relation means that if A causes B , then A must always be followed by B . In this sense, war does not cause deaths, nor does smoking cause cancer or emphysema . As a result, many turn to a notion of probabilistic causation. Informally, A ("The person

7524-503: The case that one can change x in order to change y . This coincides with commonsense notions of causations, since often we ask causal questions in order to change some feature of the world. For instance, we are interested in knowing the causes of crime so that we might find ways of reducing it. These theories have been criticized on two primary grounds. First, theorists complain that these accounts are circular . Attempting to reduce causal claims to manipulation requires that manipulation

7638-442: The conceptual frame of the scientific method , an investigator sets up several distinct and contrasting temporally transient material processes that have the structure of experiments , and records candidate material responses, normally intending to determine causality in the physical world. For instance, one may want to know whether a high intake of carrots causes humans to develop the bubonic plague . The quantity of carrot intake

7752-696: The counterfactual notion. According to the counterfactual view , X causes Y if and only if, without X, Y would not exist. Hume interpreted the latter as an ontological view, i.e., as a description of the nature of causality but, given the limitations of the human mind, advised using the former (stating, roughly, that X causes Y if and only if the two events are spatiotemporally conjoined, and X precedes Y ) as an epistemic definition of causality. We need an epistemic concept of causality in order to distinguish between causal and noncausal relations. The contemporary philosophical literature on causality can be divided into five big approaches to causality. These include

7866-428: The derivation of a cause-and-effect relationship from observational studies must rest on some qualitative theoretical assumptions, for example, that symptoms do not cause diseases, usually expressed in the form of missing arrows in causal graphs such as Bayesian networks or path diagrams . The theory underlying these derivations relies on the distinction between conditional probabilities , as in P ( c

7980-434: The effect, is called a "contributory cause". J. L. Mackie argues that usual talk of "cause" in fact refers to INUS conditions ( i nsufficient but n on-redundant parts of a condition which is itself u nnecessary but s ufficient for the occurrence of the effect). An example is a short circuit as a cause for a house burning down. Consider the collection of events: the short circuit, the proximity of flammable material, and

8094-455: The experimenter, to do so at an unspecified time in the past", while the latter reads: "the probability of finding cancer in a person forced by the experimenter to smoke at a specified time in the past". The former is a statistical notion that can be estimated by observation with negligible intervention by the experimenter, while the latter is a causal notion which is estimated in an experiment with an important controlled randomized intervention. It

8208-423: The fire) but non-redundant (because the fire would not have happened without it, everything else being equal) part of a condition which is itself unnecessary but sufficient for the occurrence of the effect. So, the short circuit is an INUS condition for the occurrence of the house burning down. Conditional statements are not statements of causality. An important distinction is that statements of causality require

8322-412: The first time in this context. From his reformulation, he concluded that time and space should be treated equally, and so arose his concept of events taking place in a unified four-dimensional spacetime continuum . In a further development in his 1908 "Space and Time" lecture, Minkowski gave an alternative formulation of this idea that used a real time coordinate instead of an imaginary one, representing

8436-509: The following definition of the notion of causal dependence : Causation is then analyzed in terms of counterfactual dependence. That is, C causes E if and only if there exists a sequence of events C, D 1 , D 2 , ... D k , E such that each event in the sequence counterfactually depends on the previous. This chain of causal dependence may be called a mechanism . Note that the analysis does not purport to explain how we make causal judgements or how we reason about causation, but rather to give

8550-417: The following statements are true when interpreting "If ..., then ..." as the material conditional: The first is true since both the antecedent and the consequent are true. The second is true in sentential logic and indeterminate in natural language, regardless of the consequent statement that follows, because the antecedent is false. The ordinary indicative conditional has somewhat more structure than

8664-410: The following three relationships hold: P{ B | A } ≥ P{ B }, P{ C | A } ≥ P{ C } and P{ B | C } ≥ P{ B }. The last relationship states that knowing that the person has emphysema increases the likelihood that he will have cancer. The reason for this is that having the information that the person has emphysema increases the likelihood that the person is a smoker, thus indirectly increasing the likelihood that

8778-494: The form of "Had C not occurred, E would not have occurred." This approach can be traced back to David Hume 's definition of the causal relation as that "where, if the first object had not been, the second never had existed." More full-fledged analysis of causation in terms of counterfactual conditionals only came in the 20th century after development of the possible world semantics for the evaluation of counterfactual conditionals. In his 1973 paper "Causation," David Lewis proposed

8892-404: The former convention include "continuity" from the Euclidean case corresponding to the non-relativistic limit c → ∞ . Arguments for the latter include that minus signs, otherwise ubiquitous in particle physics, go away. Yet other authors, especially of introductory texts, e.g. Kleppner & Kolenkow (1978) , do not choose a signature at all, but instead, opt to coordinatize spacetime such that

9006-410: The four variables ( x , y , z , t ) of space and time in the coordinate form in a four-dimensional real vector space . Points in this space correspond to events in spacetime. In this space, there is a defined light-cone associated with each point, and events not on the light cone are classified by their relation to the apex as spacelike or timelike . It is principally this view of spacetime that

9120-498: The idea of Granger causality , or by direct experimental manipulation. The use of temporal data can permit statistical tests of a pre-existing theory of causal direction. For instance, our degree of confidence in the direction and nature of causality is much greater when supported by cross-correlations , ARIMA models, or cross-spectral analysis using vector time series data than by cross-sectional data . Nobel laureate Herbert A. Simon and philosopher Nicholas Rescher claim that

9234-428: The invariance of the spacetime interval. This structure provides the background setting of all present relativistic theories, barring general relativity for which flat Minkowski spacetime still provides a springboard as curved spacetime is locally Lorentzian. Minkowski, aware of the fundamental restatement of the theory which he had made, said The views of space and time which I wish to lay before you have sprung from

9348-410: The material conditional. For instance, although the first is the closest, neither of the preceding two statements seems true as an ordinary indicative reading. But the sentence: intuitively seems to be true, even though there is no straightforward causal relation in this hypothetical situation between Shakespeare's not writing Macbeth and someone else's actually writing it. Another sort of conditional,

9462-420: The material one chooses to read. The metric signature refers to which sign the Minkowski inner product yields when given space ( spacelike to be specific, defined further down) and time basis vectors ( timelike ) as arguments. Further discussion about this theoretically inconsequential but practically necessary choice for purposes of internal consistency and convenience is deferred to the hide box below. See also

9576-430: The notion of causality is metaphysically prior to the notions of time and space. In practical terms, this is because use of the relation of causality is necessary for the interpretation of empirical experiments. Interpretation of experiments is needed to establish the physical and geometrical notions of time and space. The deterministic world-view holds that the history of the universe can be exhaustively represented as

9690-532: The observer at (0, 0, 0, 0) with a speed less than that of light. Of most interest are time-like vectors that are similarly directed , i.e. all either in the forward or in the backward cones. Such vectors have several properties not shared by space-like vectors. These arise because both forward and backward cones are convex, whereas the space-like region is not convex. The scalar product of two time-like vectors u 1 = ( t 1 , x 1 , y 1 , z 1 ) and u 2 = ( t 2 , x 2 , y 2 , z 2 )

9804-440: The one nearest to the concerns of the present article is the "efficient" one. David Hume , as part of his opposition to rationalism , argued that pure reason alone cannot prove the reality of efficient causality; instead, he appealed to custom and mental habit, observing that all human knowledge derives solely from experience . The topic of causality remains a staple in contemporary philosophy . The nature of cause and effect

9918-426: The ordinary sense of the word, though it may refer to virtual or nominal 'velocities' with magnitudes greater than that of light. For example, wave packets are mathematical objects that have group velocity and phase velocity . The energy of a wave packet travels at the group velocity (under normal circumstances); since energy has causal efficacy, the group velocity cannot be faster than the speed of light. The phase of

10032-442: The ordinary sense. The "rotation" in a plane spanned by a space unit vector and a time unit vector, while formally still a rotation in coordinate space, is a Lorentz boost in physical spacetime with real inertial coordinates. The analogy with Euclidean rotations is only partial since the radius of the sphere is actually imaginary, which turns rotations into rotations in hyperbolic space (see hyperbolic rotation ). This idea, which

10146-544: The page treating sign convention in Relativity. In general, but with several exceptions, mathematicians and general relativists prefer spacelike vectors to yield a positive sign, (− + + +) , while particle physicists tend to prefer timelike vectors to yield a positive sign, (+ − − −) . Authors covering several areas of physics, e.g. Steven Weinberg and Landau and Lifshitz ( (− + + +) and (+ − − −) respectively) stick to one choice regardless of topic. Arguments for

10260-415: The person will have cancer. However, we would not want to conclude that having emphysema causes cancer. Thus, we need additional conditions such as temporal relationship of A to B and a rational explanation as to the mechanism of action. It is hard to quantify this last requirement and thus different authors prefer somewhat different definitions. When experimental interventions are infeasible or illegal,

10374-452: The positive property of the scalar product can be seen. For two similarly directed time-like vectors u and w , the inequality is ‖ u + w ‖ ≥ ‖ u ‖ + ‖ w ‖ , {\displaystyle \left\|u+w\right\|\geq \left\|u\right\|+\left\|w\right\|,} where the equality holds when the vectors are linearly dependent . The proof uses

10488-441: The process theories is the mechanistic view on causality. It states that causal relations supervene on mechanisms. While the notion of mechanism is understood differently, the definition put forward by the group of philosophers referred to as the 'New Mechanists' dominate the literature. For the scientific investigation of efficient causality, the cause and effect are each best conceived of as temporally transient processes. Within

10602-478: The properties of antecedence and contiguity. These are topological, and are ingredients for space-time geometry. As developed by Alfred Robb , these properties allow the derivation of the notions of time and space. Max Jammer writes "the Einstein postulate ... opens the way to a straightforward construction of the causal topology ... of Minkowski space." Causal efficacy propagates no faster than light. Thus,

10716-403: The referenced articles, as the presentation below will be principally confined to the mathematical structure (Minkowski metric and from it derived quantities and the Poincaré group as symmetry group of spacetime) following from the invariance of the spacetime interval on the spacetime manifold as consequences of the postulates of special relativity, not to specific application or derivation of

10830-439: The rotation axis corresponds to the direction of relative motion between the two observers and the rotation angle is related to their relative velocity. To understand this concept, one should consider the coordinates of an event in spacetime represented as a four-vector ( t , x , y , z ) . A Lorentz transformation is represented by a matrix that acts on the four-vector, changing its components. This matrix can be thought of as

10944-726: The same signs. Using the positivity property of time-like vectors, it is easy to verify that a linear sum with positive coefficients of similarly directed time-like vectors is also similarly directed time-like (the sum remains within the light cone because of convexity). The norm of a time-like vector u = ( ct , x , y , z ) is defined as ‖ u ‖ = η ( u , u ) = c 2 t 2 − x 2 − y 2 − z 2 {\displaystyle \left\|u\right\|={\sqrt {\eta (u,u)}}={\sqrt {c^{2}t^{2}-x^{2}-y^{2}-z^{2}}}} The reversed Cauchy inequality

11058-423: The second basis vector identification is referred to as parallel transport . The first identification is the canonical identification of vectors in the tangent space at any point with vectors in the space itself. The appearance of basis vectors in tangent spaces as first-order differential operators is due to this identification. It is motivated by the observation that a geometrical tangent vector can be associated in

11172-467: The sign of η ( v , v ) , also called scalar product , as well, which depends on the signature. The classification of any vector will be the same in all frames of reference that are related by a Lorentz transformation (but not by a general Poincaré transformation because the origin may then be displaced) because of the invariance of the spacetime interval under Lorentz transformation. The set of all null vectors at an event of Minkowski space constitutes

11286-664: The skeleton of the underlying graph and, then, orient all arrows whose directionality is dictated by the conditional independencies observed. Alternative methods of structure learning search through the many possible causal structures among the variables, and remove ones which are strongly incompatible with the observed correlations . In general this leaves a set of possible causal relations, which should then be tested by analyzing time series data or, preferably, designing appropriately controlled experiments . In contrast with Bayesian Networks, path analysis (and its generalization, structural equation modeling ), serve better to estimate

11400-444: The soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. Though Minkowski took an important step for physics, Albert Einstein saw its limitation: At a time when Minkowski was giving the geometrical interpretation of special relativity by extending

11514-753: The space R 1 , 3 := ( R 4 , η μ ν ) {\displaystyle \mathbf {R} ^{1,3}:=(\mathbf {R} ^{4},\eta _{\mu \nu })} . The notation is meant to emphasize the fact that M and R 1 , 3 {\displaystyle \mathbf {R} ^{1,3}} are not just vector spaces but have added structure. η μ ν = diag ( + 1 , − 1 , − 1 , − 1 ) {\displaystyle \eta _{\mu \nu }={\text{diag}}(+1,-1,-1,-1)} . An interesting example of non-inertial coordinates for (part of) Minkowski spacetime

11628-409: The square root on both sides. It is assumed below that spacetime is endowed with a coordinate system corresponding to an inertial frame . This provides an origin , which is necessary for spacetime to be modeled as a vector space. This addition is not required, and more complex treatments analogous to an affine space can remove the extra structure. However, this is not the introductory convention and

11742-623: The tangent spaces defined by e μ | p = ∂ ∂ x μ | p  or  e 0 | p = ( 1 0 0 0 ) , etc . {\displaystyle \left.\mathbf {e} _{\mu }\right|_{p}=\left.{\frac {\partial }{\partial x^{\mu }}}\right|_{p}{\text{ or }}\mathbf {e} _{0}|_{p}=\left({\begin{matrix}1\\0\\0\\0\end{matrix}}\right){\text{, etc}}.} Here, p and q are any two events, and

11856-1690: The tangent spaces. The vector space structure of Minkowski space allows for the canonical identification of vectors in tangent spaces at points (events) with vectors (points, events) in Minkowski space itself. See e.g. Lee (2003 , Proposition 3.8.) or Lee (2012 , Proposition 3.13.) These identifications are routinely done in mathematics. They can be expressed formally in Cartesian coordinates as ( x 0 , x 1 , x 2 , x 3 )   ↔   x 0 e 0 | p + x 1 e 1 | p + x 2 e 2 | p + x 3 e 3 | p ↔   x 0 e 0 | q + x 1 e 1 | q + x 2 e 2 | q + x 3 e 3 | q {\displaystyle {\begin{aligned}\left(x^{0},\,x^{1},\,x^{2},\,x^{3}\right)\ &\leftrightarrow \ \left.x^{0}\mathbf {e} _{0}\right|_{p}+\left.x^{1}\mathbf {e} _{1}\right|_{p}+\left.x^{2}\mathbf {e} _{2}\right|_{p}+\left.x^{3}\mathbf {e} _{3}\right|_{p}\\&\leftrightarrow \ \left.x^{0}\mathbf {e} _{0}\right|_{q}+\left.x^{1}\mathbf {e} _{1}\right|_{q}+\left.x^{2}\mathbf {e} _{2}\right|_{q}+\left.x^{3}\mathbf {e} _{3}\right|_{q}\end{aligned}}} with basis vectors in

11970-459: The three spatial dimensions. In 3-dimensional Euclidean space , the isometry group (maps preserving the regular Euclidean distance ) is the Euclidean group . It is generated by rotations , reflections and translations . When time is appended as a fourth dimension, the further transformations of translations in time and Lorentz boosts are added, and the group of all these transformations

12084-413: The time coordinate (but not time itself!) is imaginary. This removes the need for the explicit introduction of a metric tensor (which may seem like an extra burden in an introductory course), and one needs not be concerned with covariant vectors and contravariant vectors (or raising and lowering indices) to be described below. The inner product is instead affected by a straightforward extension of

12198-428: The time-directedness of counterfactual dependence in terms of the semantics of the counterfactual conditional. If correct, this theory can serve to explain a fundamental part of our experience, which is that we can causally affect the future but not the past. One challenge for the counterfactual account is overdetermination , whereby an effect has multiple causes. For instance, suppose Alice and Bob both throw bricks at

12312-451: The title Causes . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Causes&oldid=980376781 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Causality In general,

12426-418: The use of tools of differential geometry that are otherwise immediately available and useful for geometrical description and calculation – even in the flat spacetime of special relativity, e.g. of the electromagnetic field. Mathematically associated with the bilinear form is a tensor of type (0,2) at each point in spacetime, called the Minkowski metric . The Minkowski metric, the bilinear form, and

12540-671: The vector v in a frame related to some frame by Λ transforms according to v → Λ v . This is the same way in which the coordinates x transform. Explicitly, x ′ μ = Λ μ ν x ν , v ′ μ = Λ μ ν v ν . {\displaystyle {\begin{aligned}x'^{\mu }&={\Lambda ^{\mu }}_{\nu }x^{\nu },\\v'^{\mu }&={\Lambda ^{\mu }}_{\nu }v^{\nu }.\end{aligned}}} This definition

12654-486: The word "cause" is used as a specialized technical term, the translation of Aristotle 's term αἰτία, by which Aristotle meant "explanation" or "answer to a 'why' question". Aristotle categorized the four types of answers as material, formal, efficient, and final "causes". In this case, the "cause" is the explanans for the explanandum , and failure to recognize that different kinds of "cause" are being considered can lead to futile debate. Of Aristotle's four explanatory modes,

12768-455: The world. Some attempts to defend manipulability theories are recent accounts that do not claim to reduce causality to manipulation. These accounts use manipulation as a sign or feature in causation without claiming that manipulation is more fundamental than causation. Some theorists are interested in distinguishing between causal processes and non-causal processes (Russell 1948; Salmon 1984). These theorists often want to distinguish between

12882-712: Was developed by Rebane and Pearl (1987) which rests on Wright's distinction between the three possible types of causal substructures allowed in a directed acyclic graph (DAG): Type 1 and type 2 represent the same statistical dependencies (i.e., X {\displaystyle X} and Z {\displaystyle Z} are independent given Y {\displaystyle Y} ) and are, therefore, indistinguishable within purely cross-sectional data . Type 3, however, can be uniquely identified, since X {\displaystyle X} and Z {\displaystyle Z} are marginally independent and all other pairs are dependent. Thus, while

12996-664: Was mentioned only briefly by Poincaré, was elaborated by Minkowski in a paper in German published in 1908 called "The Fundamental Equations for Electromagnetic Processes in Moving Bodies". He reformulated Maxwell equations as a symmetrical set of equations in the four variables ( x , y , z , ict ) combined with redefined vector variables for electromagnetic quantities, and he was able to show directly and very simply their invariance under Lorentz transformation. He also made other important contributions and used matrix notation for

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