Attractor

From Wikipedia, the free encyclopedia

Visual representation of a strange attractor

In the mathematical field of dynamical systems, an attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed. 

In finite-dimensional systems, the evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space. In physical systems, the n dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in economic systems, they may be separate variables such as the inflation rate and the unemployment rate. 

If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented geometrically in two or three dimensions, (as for example in the three-dimensional case depicted to the right). An attractor can be a point, a finite set of points, a curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor (see strange attractor below). If the variable is a scalar, the attractor is a subset of the real number line. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory. 

A trajectory of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor, forward in time. The trajectory may be periodic or chaotic. If a set of points is periodic or chaotic, but the flow in the neighborhood is away from the set, the set is not an attractor, but instead is called a repeller (or repellor).

Motivation of attractors

A dynamical system is generally described by one or more differential or difference equations. The equations of a given dynamical system specify its behavior over any given short period of time. To determine the system's behavior for a longer period, it is often necessary to integrate the equations, either through analytical means or through iteration, often with the aid of computers. 

Dynamical systems in the physical world tend to arise from dissipative systems: if it were not for some driving force, the motion would cease. (Dissipation may come from internal friction, thermodynamic losses, or loss of material, among many causes.) The dissipation and the driving force tend to balance, killing off initial transients and settle the system into its typical behavior. The subset of the phase space of the dynamical system corresponding to the typical behavior is the attractor, also known as the attracting section or attractee. 

Invariant sets and limit sets are similar to the attractor concept. An invariant set is a set that evolves to itself under the dynamics. Attractors may contain invariant sets. A limit set is a set of points such that there exists some initial state that ends up arbitrarily close to the limit set (i.e. to each point of the set) as time goes to infinity. Attractors are limit sets, but not all limit sets are attractors: It is possible to have some points of a system converge to a limit set, but different points when perturbed slightly off the limit set may get knocked off and never return to the vicinity of the limit set.

For example, the damped pendulum has two invariant points: the point x₀ of minimum height and the point x₁ of maximum height. The point x₀ is also a limit set, as trajectories converge to it; the point x₁ is not a limit set. Because of the dissipation due to air resistance, the point x₀ is also an attractor. If there was no dissipation, x₀ would not be an attractor. Aristotle believed that objects moved only as long as they were pushed, which is an early formulation of a dissipative attractor. 

Exponential divergence of trajectories complicates detailed predictions, but the world is knowable due to the existence of robust attractors.

Mathematical definition

Let t represent time and let f(t, •) be a function which specifies the dynamics of the system. That is, if a is a point in an n-dimensional phase space, representing the initial state of the system, then f(0, a) = a and, for a positive value of t, f(t, a) is the result of the evolution of this state after t units of time. For example, if the system describes the evolution of a free particle in one dimension then the phase space is the plane R² with coordinates (x,v), where x is the position of the particle, v is its velocity, a = (x,v), and the evolution is given by 

Attracting period-3 cycle and its immediate basin of attraction for a certain parametrization of f(z) = z² + c. The three darkest points are the points of the 3-cycle, which lead to each other in sequence, and iteration from any point in the basin of attraction leads to (usually asymptotic) convergence to this sequence of three points.

${\displaystyle f(t,(x,v))=(x+tv,v).\ }$

An attractor is a subset A of the phase space characterized by the following three conditions:

• A is forward invariant under f: if a is an element of A then so is f(t,a), for all t > 0.
• There exists a neighborhood of A, called the basin of attraction for A and denoted B(A), which consists of all points b that "enter A in the limit t → ∞". More formally, B(A) is the set of all points b in the phase space with the following property:

For any open neighborhood N of A, there is a positive constant T such that f(t,b) ∈ N for all real t > T.

• There is no proper (non-empty) subset of A having the first two properties.

Since the basin of attraction contains an open set containing A, every point that is sufficiently close to A is attracted to A. The definition of an attractor uses a metric on the phase space, but the resulting notion usually depends only on the topology of the phase space. In the case of Rⁿ, the Euclidean norm is typically used.

Many other definitions of attractor occur in the literature. For example, some authors require that an attractor have positive measure (preventing a point from being an attractor), others relax the requirement that B(A) be a neighborhood. 

Types of attractors

Attractors are portions or subsets of the phase space of a dynamical system. Until the 1960s, attractors were thought of as being simple geometric subsets of the phase space, like points, lines, surfaces, and simple regions of three-dimensional space. More complex attractors that cannot be categorized as simple geometric subsets, such as topologically wild sets, were known of at the time but were thought to be fragile anomalies. Stephen Smale was able to show that his horseshoe map was robust and that its attractor had the structure of a Cantor set.

Two simple attractors are a fixed point and the limit cycle. Attractors can take on many other geometric shapes (phase space subsets). But when these sets (or the motions within them) cannot be easily described as simple combinations (e.g. intersection and union) of fundamental geometric objects (e.g. lines, surfaces, spheres, toroids, manifolds), then the attractor is called a strange attractor.

Fixed point

Weakly attracting fixed point for a complex number evolving according to a complex quadratic polynomial. The phase space is the horizontal complex plane; the vertical axis measures the frequency with which points in the complex plane are visited. The point in the complex plane directly below the peak frequency is the fixed point attractor.

 

A fixed point of a function or transformation is a point that is mapped to itself by the function or transformation. If we regard the evolution of a dynamical system as a series of transformations, then there may or may not be a point which remains fixed under each transformation. The final state that a dynamical system evolves towards corresponds to an attracting fixed point of the evolution function for that system, such as the center bottom position of a damped pendulum, the level and flat water line of sloshing water in a glass, or the bottom center of a bowl contain a rolling marble. But the fixed point(s) of a dynamic system is not necessarily an attractor of the system. For example, if the bowl containing a rolling marble was inverted and the marble was balanced on top of the bowl, the center bottom (now top) of the bowl is a fixed state, but not an attractor. This is equivalent to the difference between stable and unstable equilibria. In the case of a marble on top of an inverted bowl (a hill), that point at the top of the bowl (hill) is a fixed point (equilibrium), but not an attractor (stable equilibrium). 

In addition, physical dynamic systems with at least one fixed point invariably have multiple fixed points and attractors due to the reality of dynamics in the physical world, including the nonlinear dynamics of stiction, friction, surface roughness, deformation (both elastic and plasticity), and even quantum mechanics. In the case of a marble on top of an inverted bowl, even if the bowl seems perfectly hemispherical, and the marble's spherical shape, are both much more complex surfaces when examined under a microscope, and their shapes change or deform during contact. Any physical surface can be seen to have a rough terrain of multiple peaks, valleys, saddle points, ridges, ravines, and plains. There are many points in this surface terrain (and the dynamic system of a similarly rough marble rolling around on this microscopic terrain) that are considered stationary or fixed points, some of which are categorized as attractors.

Finite number of points

In a discrete-time system, an attractor can take the form of a finite number of points that are visited in sequence. Each of these points is called a periodic point. This is illustrated by the logistic map, which depending on its specific parameter value can have an attractor consisting of 2ⁿ points, 3×2ⁿ points, etc., for any value of n.

Limit cycle

A limit cycle is a periodic orbit of a continuous dynamical system that is isolated. Examples include the swings of a pendulum clock, and the heartbeat while resting. (The limit cycle of an ideal pendulum is not an example of a limit cycle attractor because its orbits are not isolated: in the phase space of the ideal pendulum, near any point of a periodic orbit there is another point that belongs to a different periodic orbit, so the former orbit is not attracting). 

Van der Pol phase portrait: an attracting limit cycle

Limit torus

There may be more than one frequency in the periodic trajectory of the system through the state of a limit cycle. For example, in physics, one frequency may dictate the rate at which a planet orbits a star while a second frequency describes the oscillations in the distance between the two bodies. If two of these frequencies form an irrational fraction (i.e. they are incommensurate), the trajectory is no longer closed, and the limit cycle becomes a limit torus. This kind of attractor is called an N_t-torus if there are N_t incommensurate frequencies. For example, here is a 2-torus: 

A time series corresponding to this attractor is a quasiperiodic series: A discretely sampled sum of N_t periodic functions (not necessarily sine waves) with incommensurate frequencies. Such a time series does not have a strict periodicity, but its power spectrum still consists only of sharp lines.

Strange attractor

A plot of Lorenz's strange attractor for values ρ = 28, σ = 10, β = 8/3

 

An attractor is called strange if it has a fractal structure. This is often the case when the dynamics on it are chaotic, but strange nonchaotic attractors also exist. If a strange attractor is chaotic, exhibiting sensitive dependence on initial conditions, then any two arbitrarily close alternative initial points on the attractor, after any of various numbers of iterations, will lead to points that are arbitrarily far apart (subject to the confines of the attractor), and after any of various other numbers of iterations will lead to points that are arbitrarily close together. Thus a dynamic system with a chaotic attractor is locally unstable yet globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor.

The term strange attractor was coined by David Ruelle and Floris Takens to describe the attractor resulting from a series of bifurcations of a system describing fluid flow. Strange attractors are often differentiable in a few directions, but some are like a Cantor dust, and therefore not differentiable. Strange attractors may also be found in the presence of noise, where they may be shown to support invariant random probability measures of Sinai–Ruelle–Bowen type.

Examples of strange attractors include the double-scroll attractor, Hénon attractor, Rössler attractor, Tamari attractor, and the Lorenz attractor.

Effect of parameters on the attractor

Bifurcation diagram of the logistic map. The attractor for any value of the parameter r is shown on the vertical line at that r.

 

A particular functional form of a dynamic equation can have various types of attractor depending on the particular parameter values used in the function. An example is the well-studied logistic map, ${\displaystyle x_{t+1}=rx_{t}(1-x_{t}),}$ whose basins of attraction for various values of the parameter r are shown in the diagram. At some values of the parameter the attractor is a single point, at others it is two points that are visited in turn, at others it is 2ⁿ points or k × 2ⁿ points that are visited in turn, for any value of n depending on the value of the parameter r, and at other values of r an infinitude of points are visited.

Basins of attraction

An attractor's basin of attraction is the region of the phase space, over which iterations are defined, such that any point (any initial condition) in that region will eventually be iterated into the attractor. For a stable linear system, every point in the phase space is in the basin of attraction. However, in nonlinear systems, some points may map directly or asymptotically to infinity, while other points may lie in a different basin of attraction and map asymptotically into a different attractor; other initial conditions may be in or map directly into a non-attracting point or cycle.

Linear equation or system

A single-variable (univariate) linear difference equation of the homogeneous form ${\displaystyle x_{t}=ax_{t-1}}$ diverges to infinity if |a| > 1 from all initial points except 0; there is no attractor and therefore no basin of attraction. But if |a| < 1 all points on the number line map asymptotically (or directly in the case of 0) to 0; 0 is the attractor, and the entire number line is the basin of attraction. 

Likewise, a linear matrix difference equation in a dynamic vector X, of the homogeneous form ${\displaystyle X_{t}=AX_{t-1}}$ in terms of square matrix A will have all elements of the dynamic vector diverge to infinity if the largest eigenvalue of A is greater than 1 in absolute value; there is no attractor and no basin of attraction. But if the largest eigenvalue is less than 1 in magnitude, all initial vectors will asymptotically converge to the zero vector, which is the attractor; the entire n-dimensional space of potential initial vectors is the basin of attraction.

Similar features apply to linear differential equations. The scalar equation ${\displaystyle dx/dt=ax}$ causes all initial values of x except zero to diverge to infinity if a > 0 but to converge to an attractor at the value 0 if a < 0, making the entire number line the basin of attraction for 0. And the matrix system ${\displaystyle dX/dt=AX}$ gives divergence from all initial points except the vector of zeroes if any eigenvalue of the matrix A is positive; but if all the eigenvalues are negative the vector of zeroes is an attractor whose basin of attraction is the entire phase space.

Nonlinear equation or system

Equations or systems that are nonlinear can give rise to a richer variety of behavior than can linear systems. One example is Newton's method of iterating to a root of a nonlinear expression. If the expression has more than one real root, some starting points for the iterative algorithm will lead to one of the roots asymptotically, and other starting points will lead to another. The basins of attraction for the expression's roots are generally not simple—it is not simply that the points nearest one root all map there, giving a basin of attraction consisting of nearby points. The basins of attraction can be infinite in number and arbitrarily small. For example, for the function ${\displaystyle f(x)=x^{3}-2x^{2}-11x+12}$, the following initial conditions are in successive basins of attraction: 

Basins of attraction in the complex plane for using Newton's method to solve x⁵ − 1 = 0. Points in like-colored regions map to the same root; darker means more iterations are needed to converge.

2.35287527 converges to 4;

2.35284172 converges to −3;

2.35283735 converges to 4;

2.352836327 converges to −3;

2.352836323 converges to 1.

Newton's method can also be applied to complex functions to find their roots. Each root has a basin of attraction in the complex plane; these basins can be mapped as in the image shown. As can be seen, the combined basin of attraction for a particular root can have many disconnected regions. For many complex functions, the boundaries of the basins of attraction are fractals.

Partial differential equations

Parabolic partial differential equations may have finite-dimensional attractors. The diffusive part of the equation damps higher frequencies and in some cases leads to a global attractor. The Ginzburg–Landau, the Kuramoto–Sivashinsky, and the two-dimensional, forced Navier–Stokes equations are all known to have global attractors of finite dimension. 

For the three-dimensional, incompressible Navier–Stokes equation with periodic boundary conditions, if it has a global attractor, then this attractor will be of finite dimensions.

Numerical localization (visualization) of attractors: self-excited and hidden attractors

Chaotic hidden attractor (green domain) in Chua's system. Trajectories with initial data in a neighborhood of two saddle points (blue) tend (red arrow) to infinity or tend (black arrow) to stable zero equilibrium point (orange).

 

From a computational point of view, attractors can be naturally regarded as self-excited attractors or hidden attractors. Self-excited attractors can be localized numerically by standard computational procedures, in which after a transient sequence, a trajectory starting from a point on an unstable manifold in a small neighborhood of an unstable equilibrium reaches an attractor, such as the classical attractors in the Van der Pol, Belousov–Zhabotinsky, Lorenz, and many other dynamical systems. In contrast, the basin of attraction of a hidden attractor does not contain neighborhoods of equilibria, so the hidden attractor cannot be localized by standard computational procedures.

Argument from nonbelief

From Wikipedia, the free encyclopedia

An argument from nonbelief is a philosophical argument that asserts an inconsistency between the existence of God and a world in which people fail to recognize him. It is similar to the classic argument from evil in affirming an inconsistency between the world that exists and the world that would exist if God had certain desires combined with the power to see them through.

 

There are two key varieties of the argument. The argument from reasonable nonbelief (or the argument from divine hiddenness) was first elaborated in J. L. Schellenberg's 1993 book Divine Hiddenness and Human Reason. This argument says that if God existed (and was perfectly good and loving) every reasonable person would have been brought to believe in God; however, there are reasonable nonbelievers; therefore, this God does not exist.

Theodore Drange subsequently developed the argument from nonbelief, based on the mere existence of nonbelief in God. Drange considers the distinction between reasonable (by which Schellenberg means inculpable) and unreasonable (culpable) nonbelief to be irrelevant and confusing. Nevertheless, the overwhelming majority of academic discussion is concerned with Schellenberg's formulation.

Historical references to the problem of divine hiddenness

The theme of divine hiddenness, silence or darkness has a long history in Judeo-Christian theology. The roots of the Judeo-Christian description of God as hidden are in the Bible, for example in the Psalms, "My God, my God, why have you forsaken me?....I cry by day, but you do not answer....", and in Isaiah: "Truly you are a God who hides himself, O God of Israel, the Savior."

One of the first philosophers to write on the theme of divine hiddenness was Anselm of Canterbury, who in his Proslogion links it to an existential or spiritual concern:

I have never seen thee, O Lord my God; I do not know thy form. What, O most high Lord, shall this man do, an exile far from thee? What shall thy servant do, anxious in his love of thee, and cast out afar from thy face? He pants to see thee, and thy face is too far from him. He longs to come to thee, and thy dwelling place is inaccessible. He is eager to find thee, and knows not thy place. He desires to seek thee, and does not know thy face. Lord, thou art my God, and thou art my Lord, yet never have I seen thee. It is thou that hast made me, and hast made me anew, and hast bestowed upon me all the blessings I enjoy; and not yet do I know thee. Finally, I was created to see thee and not yet have I done that for which I was made.

Daniel Howard-Snyder and Paul Moser, in the introduction to a volume of papers on the idea of divine hiddenness as evidence against theism, cite Nietzsche's question as anticipating this contemporary theme: "a god who is all-knowing and all-powerful and who does not even make sure his creatures understand his intentions — could that be a god of goodness?"

Schellenberg's hiddenness argument

Discussion of Schellenberg's argument has made explicit a non-theological use of the term 'hiddenness', which is now commonly used simply as a way of talking about the subjective condition of nonbelief in God. In his first presentation of the argument Schellenberg emphasized inculpable or reasonable nonbelief, but he has since shifted to speaking more specifically about nonresistant nonbelief. The first presentation is often given by commentators as follows, based on Schellenberg's own summing up:

1. If there is a God, he is perfectly loving.
2. If a perfectly loving God exists, reasonable nonbelief does not occur.
3. Reasonable nonbelief occurs.
4. No perfectly loving God exists (from 2 and 3).
5. Hence, there is no God (from 1 and 4).

Schellenberg has stated that this formulation is misleading, when taken on its own, because it does not make explicit the reason why a perfectly loving God would want to prevent nonbelief. His deepest claim, he says, is "about the connection between love and openness to relationship -- a personal and positively meaningful and explicit sort of relationship of the sort that logically presupposes each party's belief in the other's existence." A later presentation of the argument by Schellenberg, which aims at accessibility for students, includes this element:

1. If no perfectly loving God exists, then God does not exist.
2. If a perfectly loving God exists, then there is a God who is always open to personal relationship with each human person.
3. If there is a God who is always open to personal relationship with each human person, then no human person is ever non-resistantly unaware that God exists.
4. If a perfectly loving God exists, then no human person is ever non-resistantly unaware that God exists (from 2 and 3).
5. Some human persons are non-resistantly unaware that God exists.
6. No perfectly loving God exists (from 4 and 5).
7. God does not exist (from 1 and 6).

In an article revisiting the argument ten years after it was originally proposed, Schellenberg observes that criticism has mainly centered around the idea that God would prevent inculpable nonbelief. He asserts that there are relatively few criticisms questioning the existence of inculpable nonbelief, and almost no theistic philosopher objects to the idea that God is perfectly loving.

God is perfectly loving

Schellenberg says he has not seen any serious objections to this premise by theistic philosophers, but there certainly are other conceptions of God. Daniel Howard-Snyder writes about the possibility of believing in an unsurpassably great personal god that is nevertheless dispassionate towards its creatures. Drawing on the Stoic concept of Eudaimonia, he says one can think of a god more akin to a wise sage than the loving parent that Schellenberg envisions.

Theodore Drange, in his attempt to improve the argument (see below), states that there are many theists who do not view God as perfectly loving, and "some Christians think of him as an angry deity bent on punishing people for their sins." Drange concludes that the argument should be put forward only in relation to theists who already accept the first premise and believe in a god who is perfectly loving.

Most theists, in fact, do admit that love is a central concept in almost all of the world's religions. God is often directly associated with love, especially with agape. Theologians such as N.T. Wright suggest that our experience of love is itself a proof of God's existence. However, there are a few others (e.g. Brian Davies in the Thomist tradition) who suggest that the modern interpretation of what it means to say God loves human beings is incorrect, and so that God is able to be loving in a sense while actually willing disbelief.

Nonresistant nonbelief, lack of evidence, and sin

When asked what he would say when facing God on judgment day, Bertrand Russell famously replied that he would say "Not enough evidence, God! Not enough evidence!" Some nonbelievers may have hidden from themselves what seems to them to be possible evidence of the divine, but the view of the hiddenness argument is that others have tried hard to believe in God. Schellenberg addresses this difference with his distinction between culpable and inculpable nonbelief, with the latter defined as "non-belief that exists through no fault of the non-believer."

Historically, the Calvinist tradition has placed the blame on nonbelievers. Calvin's religious epistemology is based on the sensus divinitatis (Sense of Divinity), the view that the presence of God is universally perceived by all humans. Paul Helm explains, "Calvin’s use of the term 'sense' signals that the knowledge of God is a common human endowment; mankind is created not only as capable of knowing God, but as actually knowing him." According to this tradition, there is no inculpable or nonresistant nonbelief. Jonathan Edwards, the 18th century American theologian, claimed that while every human being has been granted the capacity to know God, successful use of these capacities requires an attitude of "true benevolence", a willingness to be open to the truth about God. Thus, the failure of non-believers to see "divine things" is in his view due to "a dreadful stupidity of mind, occasioning a sottish insensibility of their truth and importance."

Demographics of theism and the problem of natural nonbelief

In modern times, there are fewer proponents of these views. One reason is that, as Stephen Maitzen argues, anthropology has long established that while religious belief in general is essentially universal, belief in what Calvin would recognize as God is very unevenly distributed among cultures (consider for example God in Buddhism, Jain cosmology, or non-theistic animism). If God exists, then why, Maitzen asks, does the prevalence of belief in God vary so dramatically with cultural and national boundaries? Jason Marsh has extended this kind of demographic challenge by focusing on human evolution and cognitive science of religion. Why is theistic belief apparently non-existent among early humans but common at later times, at least in some regions? According to Marsh, the hiddenness problem is harder to answer once we appreciate that much nonbelief is 'natural', owing to the kinds of minds people naturally possess and to their place in evolutionary and cultural history.

Another reason why many philosophers no longer attribute nonbelief to human sinfulness has to do with respect. In fact, modern critics, such as Howard-Snyder, who praised Schellenberg's book for being "religiously sensitive," are similarly sensitive towards the nonbeliever. Howard-Snyder wrote:

Even though some nonbelievers lack true benevolence, the empirical evidence strongly suggests that others possess it since they really do earnestly seek the truth about God, love the Good, assess evidence judiciously, and, if anything, display a prejudice for God, not against Him.

Would a perfectly loving God prevent nonresistant nonbelief?

The most serious criticisms of the hiddenness argument have been leveled against the idea that a perfectly loving God would prevent nonresistant nonbelief. Schellenberg argues in two steps, by first claiming that a loving God would enable humans to partake in a relationship with it, and then, assuming that belief in that god is a necessary condition for such relationships to occur, inferring that a loving God would not permit nonbelief. He states:

There is, first of all, the claim that if there is a personal God who is perfectly loving, creatures capable of explicit and positively meaningful relationship with God, who have not freely shut themselves off from God, are always in a position to participate in such relationship – able to do so just by trying to.

He justifies this claim by arguing that a conception of divine love can best be formed by extrapolating the best aspects of love in human relations, and draws an analogy with perfect parental love:

The perfectly loving parent, for example, from the time the child can first respond to her at all until death separates them, will, insofar as she can help it, see to it that nothing she does ever puts relationship with herself out of reach for her child.

But, says Schellenberg, belief in God's existence is necessary for engaging in such a meaningful relationship with God. He therefore concludes that if there is a perfectly loving God, such creatures will always believe in it. He further argues that since belief is involuntary, these creatures should always have evidence "causally sufficient" for such belief:

The presence of God will be for them like a light that – however much the degree of its brightness may fluctuate – remains on unless they close their eyes.

Objections and counterarguments

Skeptical theism

Skeptical theism is the view that we should remain skeptical of claims that our perceptions about God's purposes can reasonably be considered good evidence of what they are . The central thesis of skeptical theism is that it would not be surprising for an infinitely intelligent and knowledgeable being's reasons for permitting a perception of evil or alleged hiddenness to be beyond human comprehension. That is, what is perceived as hiddeness may be necessary for a greater good or to prevent equal or even greater evils. 

Schellenberg has responded to skeptical theism (i.e. noseeum/unknown-purpose defense). First, Schellenberg says that he has given known reasons to think that a perfectly loving being would always be open to a personal relationship; therefore, God would not sacrifice some time in the relationship for the sake of unknown greater goods, and if the greatest good for finite creatures is to be in a relationship with God, then God would not sacrifice that for the sake of unknown greater goods. Finally, Schellenberg's position is that all known and unknown goods are ultimately in God; hence, God can bring about unknown greater goods without hiddenness.

Noseeum defense

The philosophers Michael Bergmann and Michael Rea described the philosopher William Rowe's justification for the second premise of the argument from evil, which is equally applicable to a perception of hiddenness:

Some evidential arguments ... rely on a “noseeum” inference of the following sort: NI: If, after thinking hard, we can’t think of any God-justifying reason for permitting some horrific evil then it is likely that there is no such reason. (The reason NI is called a ‘noseeum’ inference is that it says, more or less, that because we don’t see ‘um, they probably ain’t there.)

Various analogies are offered to show that the noseeum inference is logically unsound. For example, a novice chess player's inability to discern a chess master's choice of moves cannot be used to infer that there is no good reason for the move. The skeptical theist and noseum defense place the burden of proof on the atheist to prove that their intuitions about God are trustworthy.

Unreasonable demands on God

This argument is sometimes seen as demanding God to prove his existence, for example by performing miracles. Critics have argued that even in Schellenberg's more refined version, the nonbeliever is imposing her own epistemological expectations on the will of God. A detailed discussion of these kinds of demands, and their moral and spiritual implications, is provided by Paul Moser, who says that such demands amount to cognitive idolatry. He defines idolatry as "our not letting the true God be Lord in our lives" and instead committing to something other than God by pursuing a quest for self-realization in our own terms. If this is idolatry in our actions, then idolatry in our knowing, he says, is as follows:

Cognitive idolatry relies on a standard for knowledge that excludes the primacy of the morally self-transforming knowledge of God central to knowing God as Lord. It rests on an epistemological standard, whether empiricist, rationalist, or some hybrid, that does not let God be Lord. Such idolatry aims to protect one's lifestyle from serious challenge by the God who calls, convicts, and reconciles. It disallows knowledge of God as personal subject and Lord to whom we are morally and cognitively responsible. It allows at most for knowledge of God as an undemanding object of human knowledge.

Schellenberg considers this criticism irrelevant to the argument, which in his opinion, does not impose any demands for demonstrations of God's power, but rather looks for evidence that "need only be such as will be causally sufficient for belief in the absence of resistance... This result might be effected through the much more spiritually appropriate means of religious experience, interpreted in the sensitive manner of a Pascal or a Kierkegaard." Schellenberg then expresses a certain frustration that theistic writers who otherwise extol the value of religious experiences deny non-theists the right to do so.

Soul-making theodicy

John Hick used the term "soul-making" in his theodicy Evil and the God of Love to describe the kind of spiritual development that he believes justifies the existence of evil. This defense is employed by Michael Murray, who explains how, in his view, divine hiddenness is essential to soul-making. It may seem that it is not hard to imagine a world where God is known and yet believers act freely with ample opportunities for spiritual development. But Murray gives a deep and careful analysis of the argument, concluding that if God's existence were revealed in such a way as to remove reasonable non-belief, then "any desire that we might have to believe or act in ways contrary to that which has been revealed would be overwhelmed." 

Critics note here that, for example, in Christianity (and even more in Judaism, where God is represented as talking to Job and explaining why he is just), God is already believed to have exposed himself very distinctly: for example to the Apostles who saw his resurrection. One theistic explanation of this might be that God knows some people would not believe anyway but if God knows this before creating, there is a problem about God's liability for what is created. More fundamentally in relation to Murray's argument, there is the problem for orthodox believers of explaining the existence of Satan, a fallen angel who is obviously aware of God and yet, according to theistic scriptures, freely chose to rebel against God.

Unknown purpose defense

Alvin Plantinga writes that the statement "We can see no good reason for God to do X" only implies "There is no good reason for God to do X" on the assumption that "If there were a good reason for God to do X, we would be able to see it," which he suggests is absurd. This point might be applied to versions of the argument from nonbelief that suggest without support that there is no good reason for God to permit nonbelief.

There really are no atheists defense

This is the argument that all true atheists are at heart lying so that they may live in a way that is contrary to God's commands (as seen in particular interpretations of Romans 1:18-25). Critics note that there are atheists who are not lying and are not using their atheism as an escape to sin. Proponents note, however, that they could just as easily still be lying, perhaps not to others anymore but themselves (i.e. loving the wrong woman argument). Some have claimed this argument, however, fails to account for Stephen Maitzen's point on the demographics of theism. If all atheists are liars, why are people in some societies so much more likely to lie than in others? Finally, some have also claimed this argument fails to account for Jason Marsh's point on natural nonbelief in early humans. Since there was quite plausibly such a thing as natural nonbelief in early humans, then it does not make much sense to say that said nonbelief is self-deceptive. That is because natural nonbelief entails nonresistant nonbelief.

Drange's argument from nonbelief

Theodore Drange proposed a version of the nonbelief argument in 1996. He considers the distinction between culpable and inculpable nonbelief to be unhelpful in the argument, arguing instead that the mere existence of nonbelief is evidence against the existence of God. A semi-formal presentation of the argument is as follows:

1. If God exists, God:
   1. wants all humans to believe God exists before they die;
   2. can bring about a situation in which all humans believe God exists before they die;
   3. does not want anything that would conflict with and be at least as important as its desire for all humans to believe God exists before they die; and
   4. always acts in accordance with what it most wants.
2. If God exists, all humans would believe so before they die (from 1).
3. But not all humans believe God exists before they die.
4. Therefore, God does not exist (from 2 and 3).

Drange's argument is directed primarily to Christians, and the philosopher Laura Garcia has replied from that perspective. She says that Drange's argument hinges on the idea that belief in God's existence is, according to Christians, necessary for salvation. According to Garcia this idea is mistaken: "many Christians deny this claim and the Catholic Church explicitly rejects it." But as Garcia notes, Drange has answered that for many Christians—in particular, evangelical Christians—his point should remain convincing, and that there are in any case other good things that belief in God can bring for humans, which a good God would desire, such as peace of mind and a sense of meaning in life.

Teleology

From Wikipedia, the free encyclopedia

Plato and Aristotle, depicted here in The School of Athens, both developed philosophical arguments addressing the universe's apparent order (logos)

 

Teleology or finality is a reason or explanation for something in function of its end, purpose, or goal. It is derived from two Greek words: telos (end, goal, purpose) and logos (reason, explanation). A urpose that is imposed by a human use, such as that of a fork, is called extrinsic. Natural teleology, common in classical philosophy but controversial today, contends that natural entities also have intrinsic purposes, irrespective of human use or opinion. For instance, Aristotle claimed that an acorn's intrinsic telos is to become a fully grown oak tree.

Though ancient atomists rejected the notion of natural teleology, teleological accounts of non-personal or non-human nature were explored and often endorsed in ancient and medieval philosophies, but fell into disfavor during the modern era (1600–1900). In the late 18th century, Immanuel Kant used the concept of telos as a regulative principle in his Critique of Judgment. Teleology was also fundamental to the philosophy of G. W. F. Hegel. 

Contemporary philosophers and scientists are still discussing whether teleological axioms are useful or accurate in proposing modern philosophies and scientific theories. Example of reintroducing of teleology in modern language is notion of attractor. For another instance in 2012, Thomas Nagel, who is not a biologist, proposed a non-Darwinian account of evolution that incorporates impersonal and natural teleological laws to explain the existence of life, consciousness, rationality, and objective value. Regardless, the accuracy can also be considered independently from the usefulness: it is a common experience in pedagogy that a minimum of apparent teleology can be useful in thinking about and explaining Darwinian evolution even if there is no true teleology driving evolution. Thus it is easier to say that evolution "gave" wolves sharp canine teeth because those teeth "serve the purpose of" predation regardless of whether there is an underlying nonteleologic reality in which evolution is not an actor with intentions. In other words, because human cognition and learning often rely on the narrative structure of stories (with actors, goals, and proximal rather than distal causation), some minimal level of teleology might be recognized as useful or at least tolerable for practical purposes even by people who reject its cosmologic accuracy.

Etymology

The word teleology builds on the Greek τέλος, telos (root: τελε-, "end, purpose") and -λογία, logia, "speak of, study of, a branch of learning". The German philosopher Christian von Wolff coined the term (in the Latin form "teleologia") in 1728 in his work Philosophia rationalis, sive logica.

Historical overview

In western philosophy, the term and concept of teleology originated in the writings of Plato and Aristotle. Aristotle's Four Causes give special place to each thing's telos or "final cause." In this, he followed Plato in seeing purpose in both human and sub-human nature.

Platonic

In the Phaedo, Plato through Socrates argues that true explanations for any given physical phenomenon must be teleological. He bemoans those who fail to distinguish between a thing's necessary and sufficient causes, which he identifies respectively as material and final causes (Phaedo 98–99):

Imagine not being able to distinguish the real cause, from that without which the cause would not be able to act, as a cause. It is what the majority appear to do, like people groping in the dark; they call it a cause, thus giving it a name that does not belong to it. That is why one man surrounds the earth with a vortex to make the heavens keep it in place, another makes the air support it like a wide lid. As for their capacity of being in the best place they could be at this very time, this they do not look for, nor do they believe it to have any divine force, but they believe that they will some time discover a stronger and more immortal Atlas to hold everything together more, and they do not believe that the truly good and 'binding' binds and holds them together.

— Plato, Phaedo 99

Plato here argues that while the materials that compose a body are necessary conditions for its moving or acting in a certain way, they nevertheless cannot be the sufficient condition for its moving or acting as it does. For example, (given in Phaedo 98), if Socrates is sitting in an Athenian prison, the elasticity of his tendons is what allows him to be sitting, and so a physical description of his tendons can be listed as necessary conditions or auxiliary causes of his act of sitting (Phaedo 99b; Timaeus 46c9–d4, 69e6). However, these are only necessary conditions of Socrates' sitting. To give a physical description of Socrates' body is to say that Socrates is sitting, but it does not give us any idea why it came to be that he was sitting in the first place. To say why he was sitting and not not sitting, we have to explain what it is about his sitting that is good, for all things brought about (i.e., all products of actions) are brought about because the actor saw some good in them. Thus, to give an explanation of something is to determine what about it is good. Its goodness is its actual cause—its purpose, telos or "reason for which" (Timaeus 27d8–29a).

Aristotelian

Aristotle argued that Democritus was wrong to attempt to reduce all things to mere necessity, because doing so neglects the aim, order, and "final cause", which brings about these necessary conditions:

Democritus, however, neglecting the final cause, reduces to necessity all the operations of nature. Now, they are necessary, it is true, but yet they are for a final cause and for the sake of what is best in each case. Thus nothing prevents the teeth from being formed and being shed in this way; but it is not on account of these causes but on account of the end....

— Aristotle, Generation of Animals V.8, 789a8–b15

In the Physics Aristotle rejected Plato's assumption that the universe was created by an intelligent designer using eternal forms as his model. For Aristotle, natural ends are produced by "natures" (principles of change internal to living things), and natures, Aristotle argued, do not deliberate:

It is absurd to suppose that ends are not present [in nature] because we do not see an agent deliberating.

— Aristotle, Physics 2.8, 199b27-9;

These Platonic and Aristotelian arguments ran counter to those presented earlier by Democritus and later by Lucretius, both of whom were supporters of what is now often called accidentalism:

Nothing in the body is made in order that we may use it. What happens to exist is the cause of its use.

— Lucretius, De rerum natura (On the Nature of Things), IV, 833; cf. 822–56.

Disfavor

Since the Novum Organum of Francis Bacon, teleological explanations in physical science tend to be deliberately avoided in favor of focus on material and efficient explanations. Final and formal causation came to be viewed as false or too subjective. Nonetheless, some disciplines, in particular within evolutionary biology, continue to use language that appears teleological when they describe natural tendencies towards certain end conditions; although some argue that these arguments ought to be, and practicably can be, rephrased in non-teleological forms, others hold that teleological language cannot always be easily expunged from descriptions in the life sciences, at least within the bounds of practical pedagogy.

Economics

A teleology of human aims played a crucial role in the work of Ludwig von Mises especially in the development of his science of praxeology. More specifically he believed that human action, i.e. purposeful behavior, is teleological based on the presupposition that an individual's action is governed or caused by the existence of their chosen ends. Or in other words an individual selects what they believe to be the most appropriate means to achieve a sought after goal or end. Mises however also stressed that teleology with respect to human action was by no means independent of causality as he states "no action can be devised and ventured upon without definite ideas about the relation of cause and effect, teleology presupposes causality"

Modern and postmodern philosophy

Historically, teleology may be identified with the philosophical tradition of Aristotelianism. The rationale of teleology was explored by Immanuel Kant in his Critique of Judgement and, again, made central to speculative philosophy by Hegel and in the various neo-Hegelian schools – proposing a history of our species some consider to be at variance with Darwin, as well as with the dialectical materialism of Karl Marx and Friedrich Engels, and with what is now called analytic philosophy – the point of departure is not so much formal logic and scientific fact but 'identity'. (In Hegel's terminology: 'objective spirit'.) 

Individual human consciousness, in the process of reaching for autonomy and freedom, has no choice but to deal with an obvious reality: the collective identities (such as the multiplicity of world views, ethnic, cultural and national identities) that divide the human race and set (and always have set) different groups in violent conflict with each other. Hegel conceived of the 'totality' of mutually antagonistic world-views and life-forms in history as being 'goal-driven', that is, oriented towards an end-point in history. The 'objective contradiction' of 'subject' and 'object' would eventually 'sublate' into a form of life that leaves violent conflict behind. This goal-oriented, 'teleological' notion of the 'historical process as a whole' is present in a variety of 20th-century authors, although its prominence declined drastically after the Second World War. 

In contrast, teleological based "grand narratives" are eschewed by the postmodern attitude and teleology may be viewed as reductive, exclusionary and harmful to those whose stories are diminished or overlooked.

Against this postmodern position, Alasdair MacIntyre has argued that a narrative understanding of oneself, of one's capacity as an independent reasoner, one's dependence on others and on the social practices and traditions in which one participates, all tend towards an ultimate good of liberation. Social practices may themselves be understood as teleologically oriented to internal goods, for example practices of philosophical and scientific inquiry are teleologically ordered to the elaboration of a true understanding of their objects. MacIntyre's book After Virtue famously dismissed the naturalistic teleology of Aristotle's 'metaphysical biology', but he has cautiously moved from that book's account of a sociological teleology toward an exploration of what remains valid in a more traditional teleological naturalism.

Ethics

Teleology informs the study of ethics.

Business ethics

Business people commonly think in terms of purposeful action as in, for example, management by objectives. Teleological analysis of business ethics leads to consideration of the full range of stakeholders in any business decision, including the management, the staff, the customers, the shareholders, the country, humanity and the environment.

Medical ethics

Teleology provides a moral basis for the professional ethics of medicine, as physicians are generally concerned with outcomes and must therefore know the telos of a given treatment paradigm.

Consequentialism

The broad spectrum of consequentialist ethics, of which utilitarianism is a well-known example, focuses on the end result or consequences, with such principles as utilitarian philosopher John Stuart Mill's "the greatest good for the greatest number", or the Principle of Utility. Hence, this principle is teleological, but in a broader sense than is elsewhere understood in philosophy. In the classical notion, teleology is grounded in the inherent natures of things themselves, whereas in consequentialism, teleology is imposed on nature from outside by the human will. Consequentialist theories justify inherently what most people would call evil acts by their desirable outcomes, if the good of the outcome outweighs the bad of the act. So, for example, a consequentialist theory would say it was acceptable to kill one person in order to save two or more other people. These theories may be summarized by the maxim "the ends can justify the means." 

Consequentialism stands in contrast to the more classical notions of deontological ethics, such as Immanuel Kant's Categorical Imperative, and Aristotle's virtue ethics (although formulations of virtue ethics are also often consequentialist in derivation). In deontological ethics, the goodness or badness of individual acts is primary and a desirable larger goal is insufficient to justify bad acts committed on the way to that goal, even if the bad acts are relatively minor and the goal is major (like telling a small lie to prevent a war and save millions of lives). In requiring all constituent acts to be good, deontological ethics is much more rigid than consequentialism, which varies by circumstances.

Practical ethics are usually a mix of the two. For example, Mill also relies on deontic maxims to guide practical behavior, but they must be justifiable by the principle of utility.

Science

In modern science, explanations that rely on teleology are often, but not always, avoided, either because they are unnecessary or because whether they are true or false is thought to be beyond the ability of human perception and understanding to judge. But using teleology as an explanatory style, in particular within evolutionary biology, is still controversial.

Biology

Apparent teleology is a recurring issue in evolutionary biology, much to the consternation of some writers.

Statements implying that nature has goals, for example where a species is said to do something "in order to" achieve survival, appear teleological, and therefore invalid. Usually, it is possible to rewrite such sentences to avoid the apparent teleology. Some biology courses have incorporated exercises requiring students to rephrase such sentences so that they do not read teleologically. Nevertheless, biologists still frequently write in a way which can be read as implying teleology even if that is not the intention. These issues have recently been discussed by John Reiss. He argues that evolutionary biology can be purged of such teleology by rejecting the analogy of natural selection as a watchmaker; other arguments against this analogy have also been promoted by writers such as Richard Dawkins.

Some authors, like James Lennox, have argued that Darwin was a teleologist, while others like Michael Ghiselin described this claim as a myth promoted by misinterpretations of his discussions and emphasized the distinction between using teleological metaphors and being teleological.

Biologist philosopher Francisco Ayala has argued that all statements about processes can be trivially translated into teleological statements, and vice versa, but that teleological statements are more explanatory and cannot be disposed of. Karen Neander has argued that the modern concept of biological 'function' is dependent upon selection. So, for example, it is not possible to say that anything that simply winks into existence without going through a process of selection has functions. We decide whether an appendage has a function by analysing the process of selection that led to it. Therefore, any talk of functions must be posterior to natural selection and function cannot be defined in the manner advocated by Reiss and Dawkins. Ernst Mayr states that "adaptedness... is an a posteriori result rather than an a priori goal-seeking." Various commentators view the teleological phrases used in modern evolutionary biology as a type of shorthand. For example, S. H. P. Madrell writes that "the proper but cumbersome way of describing change by evolutionary adaptation [may be] substituted by shorter overtly teleological statements" for the sake of saving space, but that this "should not be taken to imply that evolution proceeds by anything other than from mutations arising by chance, with those that impart an advantage being retained by natural selection." J. B. S. Haldane said, "Teleology is like a mistress to a biologist: he cannot live without her but he's unwilling to be seen with her in public."

Selected-effects accounts, like the one Neander suggests, face objections due to their reliance on etiological accounts, which some fields lack the resources to accommodate. Many such sciences, which study the same traits and behaviors regarded by evolutionary biology, still correctly attribute teleological functions without appeal to selection history. Gualtiero Piccinini and Corey J. Maley are a proponent of one such account which focuses instead on goal-contribution. With the objective goals of organisms being survival and inclusive fitness, Piccinini and Maley define teleological functions to be “a stable contribution by a trait (or component, activity, property) of organisms belonging to a biological population to an objective goal of those organisms.”

Cybernetics

Julian Bigelow, Arturo Rosenblueth, and Norbert Wiener have conceived of feedback mechanisms as lending a teleology to machinery. Wiener, a mathematician, coined the term 'cybernetics' to denote the study of "teleological mechanisms." Cybernetics is the study of the communication and control of regulatory feedback both in living beings and machines, and in combinations of the two. In the cybernetic classification presented in "Behavior, Purpose and Teleology", teleology is feedback controlled purpose.

The classification system underlying cybernetics was criticized by Frank Honywill George, who cited the need for an external observability to the purposeful behavior in order to establish and validate the goal-seeking behavior. In this view, the purpose of observing and observed systems is respectively distinguished by the system's subjective autonomy and objective control.

Collective animal behavior

From Wikipedia, the free encyclopedia

Starling flock at sunset in Denmark

 

Collective animal behavior is a form of social behavior involving the coordinated behavior of large groups of similar animals as well as emergent properties of these groups. This can include the costs and benefits of group membership, the transfer of information across the group, the group decision-making process, and group locomotion and synchronization. Studying the principles of collective animal behavior has relevance to human engineering problems through the philosophy of biomimetics. For instance, determining the rules by which an individual animal navigates relative to its neighbors in a group can lead to advances in the deployment and control of groups of swimming or flying micro-robots such as UAVs (Unmanned Aerial Vehicles).

Examples

Examples of collective animal behavior include:

• Flocking birds
• Herding ungulates
• Shoaling and schooling fish
• Schooling Antarctic krill
• Pods of dolphins
• Marching locusts
• Nest building ants

Proposed functions

Many functions of animal aggregations have been proposed. These proposed functions may be grouped into the four following categories: social and genetic, anti-predator, enhanced foraging, and increased locomotion efficiency.

Social interaction

Support for the social and genetic function of aggregations, especially those formed by fish, can be seen in several aspects of their behavior. For instance, experiments have shown that individual fish removed from a school will have a higher respiratory rate than those found in the school. This effect has been partly attributed to stress, although hydrodynamic factors were considered more important in this particular study. The calming effect of being with conspecifics may thus provide a social motivation for remaining in an aggregation. Herring, for instance, will become very agitated if they are isolated from conspecifics. Fish schools have also been proposed to serve a reproductive function since they provide increased access to potential mates.

Protection from predators

School of goldband fusiliers

 

Several anti-predator functions of animal aggregations have been proposed. One potential method by which fish schools or bird flocks may thwart predators is the ‘predator confusion effect’ proposed and demonstrated by Milinski and Heller (1978). This theory is based on the idea that it becomes difficult for predators to pick out individual prey from groups because the many moving targets create a sensory overload of the predator's visual channel. Milinski and Heller's findings have been corroborated both in experiment and computer simulations.

A second potential anti-predator effect of animal aggregations is the "many eyes" hypothesis. This theory states that as the size of the group increases, the task of scanning the environment for predators can be spread out over many individuals. Not only does this mass collaboration presumably provide a higher level of vigilance, it could also allow more time for individual feeding.

A third hypothesis for an anti-predatory effect of animal aggregation is the "encounter dilution" effect. Hamilton, for instance, proposed that the aggregation of animals was due to a "selfish" avoidance of a predator and was thus a form of cover-seeking. Another formulation of the theory was given by Turner and Pitcher and was viewed as a combination of detection and attack probabilities. In the detection component of the theory, it was suggested that potential prey might benefit by living together since a predator is less likely to chance upon a single group than a scattered distribution. In the attack component, it was thought that an attacking predator is less likely to eat a particular animal when a greater number of individuals are present. In sum, an individual has an advantage if it is in the larger of two groups, assuming that the probability of detection and attack does not increase disproportionately with the size of the group.

Enhanced foraging

A third proposed benefit of animal groups is that of enhanced foraging. This ability was demonstrated by Pitcher and others in their study of foraging behavior in shoaling cyprinids. In this study, the time it took for groups of minnows and goldfish to find a patch of food was quantified. The number of fishes in the groups was varied, and a statistically significant decrease in the amount of time necessary for larger groups to find food was established. Further support for an enhanced foraging capability of schools is seen in the structure of schools of predatory fish. Partridge and others analyzed the school structure of Atlantic bluefin tuna from aerial photographs and found that the school assumed a parabolic shape, a fact that was suggestive of cooperative hunting in this species (Partridge et al., 1983).

Increased locomotion efficiency

This theory states that groups of animals moving in a fluid environment may save energy when swimming or flying together, much in the way that bicyclists may draft one another in a peloton. Geese flying in a Vee formation are also thought to save energy by flying in the updraft of the wingtip vortex generated by the previous animal in the formation. Ducklings have also been shown to save energy by swimming in a line. Increased efficiencies in swimming in groups have also been proposed for schools of fish and Antarctic krill.

Group structure

The structure of large animal groups has been difficult to study because of the large number of animals involved. The experimental approach is therefore often complemented by mathematical modeling of animal aggregations.

Experimental approach

Experiments investigating the structure of animal aggregations seek to determine the 3D position of each animal within a volume at each point in time. It is important to know the internal structure of the group because that structure can be related to the proposed motivations for animal grouping. This capability requires the use of multiple cameras trained on the same volume in space, a technique known as stereophotogrammetry. When hundreds or thousands of animals occupy the study volume, it becomes difficult to identify each one. In addition, animals may block one another in the camera views, a problem known as occlusion. Once the location of each animal at each point in time is known, various parameters describing the animal group can be extracted. 

These parameters include:

Density: The density of an animal aggregation is the number of animals divided by the volume (or area) occupied by the aggregation. Density may not be a constant throughout the group. For instance, starling flocks have been shown to maintain higher densities on the edges than in the middle of the flock, a feature that is presumably related to defense from predators.

Polarity: The group polarity describes if the group animals are all pointing in the same direction or not. In order to determine this parameter, the average orientation of all animals in the group is determined. For each animal, the angular difference between its orientation and the group orientation is then found. The group polarity is then the average of these differences (Viscido 2004).

Nearest Neighbor Distance: The nearest neighbor distance (NND) describes the distance between the centroid of one animal (the focal animal) and the centroid of the animal nearest to the focal animal. This parameter can be found for each animal in an aggregation and then averaged. Care must be taken to account for the animals located at the edge of an animal aggregation. These animals have no neighbor in one direction. 

Nearest Neighbor Position: In a polar coordinate system, the nearest neighbor position describes the angle and distance of the nearest neighbor to a focal animal. 

Packing Fraction: Packing fraction is a parameter borrowed from physics to define the organization (or state i.e. solid, liquid, or gas) of 3D animal groups. It is an alternative measure to density. In this parameter, the aggregation is idealized as an ensemble of solid spheres, with each animal at the center of a sphere. The packing fraction is defined as the ratio of the total volume occupied by all individual spheres divided by the global volume of the aggregation (Cavagna 2008). Values range from zero to one, where a small packing fraction represents a dilute system like a gas. Cavagna found that the packing fraction for groups of starlings was 0.012.

Integrated Conditional Density: This parameter measures the density at various length scales and therefore describes the homogeneity of density throughout an animal group.

Pair Distribution Function: This parameter is usually used in physics to characterize the degree of spatial order in a system of particles. It also describes the density, but this measures describes the density at a distance away from a given point. Cavagna et al. found that flocks of starlings exhibited more structure than a gas but less than a liquid.

Modeling approach

The simplest mathematical models of animal aggregations generally instruct the individual animals to follow three rules:

1. Move in the same direction as your neighbor
2. Remain close to your neighbors
3. Avoid collisions with your neighbors

A diagram illustrating the difference between 'metric distance' and 'topological distance' in reference to fish schools

 

An example of such a simulation is the Boids program created by Craig Reynolds in 1986. Another is the Self Propelled Particle model. Many current models use variations on these rules. For instance, many models implement these three rules through layered zones around each animal. In the zone of repulsion very close to the animal, the focal animal will seek to distance itself from its neighbors in order to avoid a collision. In the slightly further away zone of alignment, a focal animal will seek to align its direction of motion with its neighbors. In the outmost zone of attraction, which extends as far away from the focal animal as it is able to sense, the focal animal will seeks to move towards a neighbor. The shape of these zones will necessarily be affected by the sensory capabilities of the animal. For example, the visual field of a bird does not extend behind its body. Fish, on the other hand, rely on both vision and on hydrodynamic signals relayed through its lateral line. Antarctic krill rely on vision and on hydrodynamic signals relayed through its antennae. 

Recent studies of starling flocks have shown, however, that each bird modifies its position relative to the six or seven animals directly surrounding it, no matter how close or how far away those animals are. Interactions between flocking starlings are thus based on a topological rule rather than a metric rule. It remains to be seen whether the same rule can be applied to other animals. Another recent study, based on an analysis of high speed camera footage of flocks above Rome and assuming minimal behavioural rules, has convincingly simulated a number of aspects of flock behaviour.

Collective decision making

Aggregations of animals are faced with decisions which they must make if they are to remain together. For a school of fish, an example of a typical decision might be which direction to swim when confronted by a predator. Social insects such as ants and bees must collectively decide where to build a new nest. A herd of elephants must decide when and where to migrate. How are these decisions made? Do stronger or more experienced 'leaders' exert more influence than other group members, or does the group make a decision by consensus? The answer probably depends on the species. While the role of a leading matriarch in an elephant herd is well known, studies have shown that some animal species use a consensus approach in their collective decision-making process. 

A recent investigation showed that small groups of fish used consensus decision-making when deciding which fish model to follow. The fish did this by a simple quorum rule such that individuals watched the decisions of others before making their own decisions. This technique generally resulted in the 'correct' decision but occasionally cascaded into the 'incorrect' decision. In addition, as the group size increased, the fish made more accurate decisions in following the more attractive fish model. Consensus decision-making, a form of collective intelligence, thus effectively uses information from multiple sources to generally reach the correct conclusion. 

Some simulations of collective decision-making use the Condorcet method to model the way groups of animals come to consensus.