CHANCE

CHANCE , in the most general sense of the word, is the negation of necessity and the opposite of determinism. The word "chance," derived from the Latin cadere ("to fall"), has a wide spectrum of meanings encompassing randomness, probability, coincidence, contingence, fluke, accident, incident, fortuity, serendipity, hazard, risk, opportunity, luck, fortune, and fate. Many words related to chance, such as coincidence, contingence, or the German Zufall, indicate a binary structure, the coming together of two causally independent series of events. Something happens, or a certain situation or person is encountered by chance. (The word "incident" derives from Latin incidere, "to befall, to fall out.")

The awareness of chance is an integral part of worldviews, both indeterministic and deterministic. Chance may be regarded positively as "an essential aspect of any real process" (Bohm, p. 141); negatively as the lack of causality or knowledge of such; and neutrally as the law of probability.

To some, chance denotes human freedom, but to others, fate. Chance can be haphazard; it can be fortunate or unfortunate. It is a highly equivocal, bifacial term, in that one meaning can easily turn into its opposite. This ambivalence may be traced back to the essential unpredictability and unknowability of any happening. The insurance business, for instance, rests on its customers' belief in chance (in the sense of unpredictability), but itself uses the theory of chance—that is, probability—to calculate its risks and price its policies (see Knight).

Although in the early twenty-first century the theory of probability predicts the course of class events to a great extent, the ultimate unknowability and uncertainty of individual events can never vanish from the realm of human experience (see Von Mises). This persistent presence of chance elements can be argued from the contingent nature of one's existence or from free will. Again, the uncertainty and indeterminateness of reality can be the source of inspiration for art or enterprise. The spirit of gambling, for instance, deliberately creates uncertain situations for the enjoyment of the risks themselves (see Rothbard, p. 500).

Chance, Greek views

Dante noted that Democritus "ascribes the world to chance" (Inferno 4.136; cf. Cioffari, chap. 1). Aristotle also observed that for Democritus the cosmos was ordered by chance (automaton ), that is, out of itself (auto ) without any reason or purpose (maton ). (For the etymology of automaton, see Physics 197b.) In opposition to this view of chance as a spontaneous event, or as "a cause that is inscrutable to human intelligence, as being a divine thing and full of mystery" (Physics 196b), Aristotle considered chance (tuchē and automaton ) as an accidental cause of the "efficient order" and what happens "by accident" (kata sumbebēkos ). Chance is indeterminate, changeful, and unstable. It is whatever comes about, neither always nor usually, but rarely (Metaphysics 1026b–1027a, 1065a; Physics 196b–198a).

Aristotle moreover distinguished two types of chance events, tuchē and automaton. Illustrating this distinction, Alexander of Aphrodisias, a third-century commentator, gave the example of a lost horse recovered by chance by his former owner. For the owner, the event is fortunate (tuchē ), but for the horse it is simply fortuitous (automaton ). Automaton has a broader range of meanings than tuchē, as it is applicable both to the natural and human worlds, whereas tuchē applies only to the latter (see Kuki, pp. 63–67).

Regarding luck and chance, Aristotle observed: "We speak of 'good luck' when luck brings us something good, and 'bad luck' in the opposite event, or, in serious cases, of 'good fortune' [eutuchia ] or 'misfortune' [dustuchia ]" (Physics 197a). Good fortune or chance for Aristotle comprised such qualities as noble birth, good children, wealth, political power, friends, and beauty (cf. Rhetoric 1389a). Fortune or chance is the cause of these external goods (Politics 1322b). The ethical virtues of justice, courage, temperance, and wisdom, however, lie outside the realm of chance, that is, within human control (cf. Politics 1323a).

The Greek word for chance, tuchē, contains the long history of poets' and writers' reflections on the subjects of luck, fate, the vicissitudes of life, and the gods' share in such human events. For Pindar, Soteira Tyche (Fortune the Savior) is "heaven-sent good fortune," the "kindly power who may crown the efforts of man" (Greene, pp. 72–73). Plato talked about a theia tuchē, a divine chance (Timaeus 25e), who comes to save human beings from their folly. Many Greeks worshipped Agathe Tyche, the goddess of good fortune (Timaeus 26e; Greene, p. 299). Aristotle admitted that chance has a religio-ethical significance in that fortune and happiness (eudaimonia ) are often synonymous, that "happiness is a divine gift" (Nicomachean Ethics 1099b; Greene, p. 325), and that "the lucky seem to succeed owing to God" (Ethica Eudemia 1248b; Cioffari, p. 27).

But Goddess Tyche is not always benevolent or dependable. Archilochus (c. 700–650 bce) is said to have introduced the idea of tuchē into the discourse, along with the already familiar Homeric notion of moira (fate), to account for what controlled human destiny. According to Orphic doctrines, fate was the law that controlled the conditions of human beings' birth, death, and reincarnation, but by the fifth and fourth centuries bce, goddess Tyche became increasingly important. An anonymous poet wrote: "Fortune [Tyche], beginning and end of human beings. Thou sittest in the seats of wisdom, and grantest honor to human deeds … thou most excellent of gods" (Loeb ed., Lyra Graeca, vol. 3, p. 477). In Greek tragedies, the role of tuchē was considerable. Euripides's Ion exclaims: "O Tyche, thou who hast brought change to myriads of human beings, causing them now to suffer misfortune, and now to fare well, by what a narrow margin have I escaped slaying my mother!" (Euripides, Ion 1512–1515). Tyche, as the goddess of chance, was associated with Lachesis, one of the Moirai (Fates) and the "dispenser of human lots" (Hesiod), and took on a fickle, unpredictable character.

Chance, the Roman view

The cult of the native Italian goddess Fortuna was revived when she was identified with Tyche. Pliny the Elder noted:

Everywhere in the whole world, at every hour by everyone's voices Fortuna alone is invoked and named, alone accused, alone impeached, alone pondered, alone applauded, alone rebuked and visited with reproaches; deemed volatile and indeed by most people blind as well, wayward, inconstant, uncertain, fickle in her favors and favoring the unworthy.… We are so much at the mercy of chance that Chance herself takes the place of god. (Natural History 2.22)

The belief in Fortuna persisted well into Renaissance Europe; she was often depicted with wings, bearing a rudder and wheel, symbolizing swiftly changing fortune.

Chance in Christianity and rationalist philosophy

Christian views on chance vary somewhat. Whereas Augustine denied any possibility of chance or fortune in view of all-controlling providence (City of God 5.1), Aquinas admitted chance (contingens ) within the providential scheme. Things "happen necessarily or contingently according to God's will" (Summa theologiae 1.19.8).

Spinoza spoke of chance "with reference to a deficiency in our knowledge [of the cause]" (Ethics 1.33.1); likewise, Laplace took it as the expression of "our ignorance as to the causes of phenomena." Hume declared that "there is no such thing as chance," but "our ignorance of the real cause of any event begets this sort of belief or opinion" (Concerning Human Understanding 6). Chance thus understood has merely a subjective reality. Leibniz, on the other hand, considered the world as "the whole assemblage of contingent things" that has its necessary and eternal substance (i.e., God) for its existence ("Essays on the Justice of God and the Freedom of Man in the Origin of Evil," 1.7); also he distinguished two kinds of truths and held that "truths of fact are contingent," while "truths of reasoning are necessary" (Monadology 33).

Chance as serendipity

In opposition to the mechanical necessitarianists of the late nineteenth century, C. S. Peirce developed a philosophical position that he called "tychism." It preserves the necessary presence of chance (Gr., tuchē), "a spontaneity which is to some degree regular," in the evolutionary process of the world, and this accounts for the individual specification (1923, pp. 200–201). Max Born, from the standpoint of quantum mechanics, likewise took chance to be mixed with "certain regularities," and nature to be "ruled by laws of cause and laws of chance." Distinguishing causality from determinism, Born incorporated chance into the consideration of causality, and thereby gave quantum mechanics indeterministic foundations (cf. Heisenberg's "principle of indeterminacy" or Niels Bohr's "principle of complementarity"). This indeterministic position was rejected by Einstein, who was convinced that God was not a "dice-playing God" (Born, pp. 3, 109, 122–123). The Nobel laureate biologist Jacques Monod declared that "chance alone is at the source of every innovation, of all creation in the biosphere" (p. 112). Objectors to this view hold that Monod's equation of "chance and man's freedom to choose his own ethical value" is erroneous (see MacKay, p. 31), or that "physico-chemical determinism" is not synonymous with the "absence of choice and freedom" (Schoffeniels, p. xix).

Not only the old question of divine providence, human freedom and chance, but the question of scientific discoveries and their philosophical implications occupy the contemporary mind. The current trend is in agreement with the worldview that is fast moving towards indeterminism, and chance, understood as serendipity, is considered instrumental in biological and other scientific discoveries and breakthroughs (see works by J. H. Austin and A. Kantorovich, for instance).

Radical contingency: A Buddhist view

The Buddhist doctrine of dependent co-origination (pratītya-samutpāda ) may be interpreted as a theory of radical contingency. It holds that there are "no accidental occurrences" and that everything in the world is produced "causally conditioned." Buddhists deny any theory of creation by a transcendental agent or anything such as fate. Moreover, things, causally produced in this fashion, have no "self-nature" (svabhāva ). This view diametrically opposed the determinism of the Indian materialists, the Ājīvikas, as well as the syncretic view of the theory of inner and outer causation held by Jains (see Kalupahana).

From a certain perspective, this Buddhist doctrine appears to be a deterministic view in that it asserts that everything is subject to the law of causation. But from a reverse perspective, the convergence of causal factors is thoroughly indeterminate; it rests on a radical contingence of various factors, both of the spatio-temporal and psycho-mental nature. Innumerable conditioning elements come together in the arising of a single event at each moment.

Jung's view on synchronicity

C. G. Jung coined the term "synchronicity" to designate the phenomenon of the coincidence of events and subjective psychic states. It "takes the coincidence of events in space and time as meaning something more than mere chance, namely, a peculiar interdependence of objective events among themselves as well as with the subjective (psychic) states of the observer or observers" (1967, p. xxiv). Jung was inclined to value the "practical result of chance" more highly than the "theoretical considerations of cause and effect" (p. xxiii), and hence, "we must admit that there is something to be said for the immense importance of chance" (ibid., p. xxiv).

Divination

Belief in fortune opens the way for divination. Throughout the history of humankind, recourse to divination has been practiced in times of trouble or uncertainty. Divination was originally a means to obtain answers to questions that are insoluble by rational reasoning. A story is recorded in Plutarch of the successor to the throne of the Thessalian kingdom being chosen by casting lots at Delphi. In Shang China, divination originated in a human attempt to fathom the mind of the deity; during the Zhou period the art of divination was given philosophical foundation (see The Book of Changes or Yijing ). In Japan well into the thirteenth century, shrine virgins known as saigū, who served at the most auspicious shrine of all, the Ise Shrine, were chosen from among eligible princesses by divination.

It appears that only later did divination come to be interpreted as dealing with chance or randomness. It is noteworthy in this connection that Apollo, the Greek god of knowledge, despised the uncertainty of the lot and handed over the divination dealing with the chances of the dice to Hermes, who thus became the gambler's god.

Belief in chance has a double role to play in the practice of divination—in the method (as the principle of randomness) and in the interpretation (as the principle of coincidence). A deterministic worldview that negates chance can nevertheless employ divination. For example, an African system of divination, Ifa, is based on the assumption that individuals basically cannot change their own destiny, but just as they can spoil it to a degree, so can the practice of Ifa improve it. Even Stoics, who were thoroughgoing determinists, eagerly sought knowledge of the future that fell outside the prediction of scientists, physicians, and other experts. The harmony between the human soul and the divine soul provided them with the basis for divination as a means of communication with God in order that human beings "might know the divine will in advance and obey it" (William A. Falconer, introduction to Cicero's De divinatione, Loeb ed., 1923, p. 216).

Like the widely practiced throwing of pebbles or stones for divinatory purposes, the method of the Chinese yijing divination consists in casting yarrow stalks (or coins) to yield randomly determined odd or even numbers. The philosophers of the later Song period maintained that this randomness was essential, for "some truths could only be sought by means of the random cast of the stalks and the evolution of the all-informing hexagram; this was achieved by means that were anything but systematic or responsive to reason" (Loewe, in Loewe and Blacker, p. 52).

Be it bibliomancy, a random opening of books such as the Bible, the Qurʾān, or Vergil's Aeneid; rhapsodomancy, which consists in writing out passages from books on separate slips and drawing one of them at random; or kledonomancy, appealing to a chance word overheard—all rest on randomness as the vehicle. (Incidentally, the Latin word for fate, fatum, comes from for, "to speak," "to say." Fatum is "what is said.") As chance is unknowable in essence, so does randomness, a form of chance, appear as an appropriate means to grasp the unknown. The mathematical doctrine of chance can be applied to calculating the outcome of random throwing of dice, for instance, but it does not replace the purpose of divination, which is to provide an answer to a question brought to it.

A skilled interpretation of such signs as those mentioned above is of central importance for divination and may be said to rely on the principle of coincidence or correspondence, according to which signs are somehow related to the human situation under consideration. It is assumed not only that there is a certain correspondence between the method of divination and the meaning obtained through it but that there is a correspondence between human affairs and the larger cosmic movement (as in, for example, The Book of Changes ) or the divine will. "The casting of lots is familiar in the Old and New Testaments as a method of ascertaining divine will" (Halliday, p. 206; cf. Jos. 6:14, Jon. 1:17, Acts 1:26, Prv. 16:33), and a divinatory message was regarded as sacred and mysterious (Prv. 16:10).

Miracle

An extremely rare or unusual occurrence may be considered a miracle. Aquinas summarized the traditional Christian understanding of miracle as: "When anything is done outside the order of created nature by a power unknown to us, it is called a miracle as regards ourselves" (Summa theologiae 1.110.4.2). He argued that just as ignorance of the cause is the source of amazement, so also when the cause is completely hidden, as God is, a thing is wondrous in an unqualified way, and this is a miracle—"what is of itself filled with admirable wonder" (Summa contra gentiles 3.101; cf. Augustine, City of God 21.8). For Hume, who denied chance, a miracle is "a violation of the laws of nature" supported by human testimony and sustained by belief (Concerning Human Understanding 10). Over against the Humean interpretation, Peirce found Butler's position that "the order of nature is a law to the doctrine of miracles" to affirm miracles and to be "in consonance with the higher teachings of modern science" ("Hume on Miracles," Collected Papers 6.546–547). Contemporary theists argue that a "dynamically stable world," which embraces chance, affords the possibility of miracles.

Chance and the unknown

Chance events, beyond human ratiocination and calculations, disclose the radical uncertainty present at the heart of reality. The interpretation of chance depends on whether one's worldview is religious or nonreligious. The fundamental unknowability of events—their mystery—can inspire awe. The religious mind has perceived in chance something sacred or a manifestation of the divine will. Some have placed chance within the governance of divine providence. Others reject it in deference to the same divine providence, arguing that what happens has already been determined by the transcendent scheme. Hence a seemingly chance occurrence, either fortunate or unfortunate, takes on the meaning of fate. In contrast, chance seen as pointing out the utter indeterminateness of things would signify the presence of free will. From a strictly fatalistic point of view, of course, there is no room for chance, for everything is already predetermined prior to the occurrence of events, and everything is already fated. Chance and fate—these initially contradictory notions are but two counter-interpretations of the experience of unexpected coincidence or happenings that seem arbitrary but nevertheless have a decisive impact on one's life and in some cases totally change it.

See Also

Divination; Fate; Gambling; Miracles; Pratītya-samutpāda.

Bibliography

Comprehensive works on chance in English are few. In other languages, one may profitably consult Kuki Shūzō's Gūzensei no mondai [The problem of contingency] (1935; Tokyo, 1976), translated as Le problème de la contingence (Tokyo, 1966), and Wilhelm Windelband's Die Lehren vom Zufall (Berlin, 1870).

On the economic theory of risk, probability, and uncertainty, see Frank H. Knight's Risk, Uncertainty and Profit (New York, 1921). On the distinction between "class probability" and "actual singular events," see Ludwig Von Mises's Human Action, 3d rev. ed. (Chicago, 1963), and M. N. Rothbard's Man, Economy, and State, 2 vols. (Princeton, 1962).

For a popular, readable introduction to the laws of chance and probability, see Darrell Huff's How to Take a Chance (New York, 1959), and Deborah J. Bennett, Randomness (Cambridge, Massachusetts & London, 1998). For a philosophical treatment of this subject, see D. H. Mellor's The Matter of Chance (Cambridge, 1971).

On the ancient Greek view of chance and fate, see William C. Greene's Moira: Fate, Good and Evil in Greek Thought (Cambridge, Mass., 1944). On Aristotle and the Scholastics, see Vincenzo Cioffari's Fortune and Fate: From Democritus to St. Thomas Aquinas (New York, 1935). On Leibniz's view of contingency, see Theodicy (London, 1951); also "Monadology" in Leroy E. Loemker, trans. & ed., Philosophical Papers and Letters (Chicago, 1956). For C. S. Peirce's philosophy of chance, see his Chance, Love and Logic (1923; New York, 1949).

On the Buddhist view of chance and causation, see David J. Kalupahana's Causality: The Central Philosophy of Buddhism (Honolulu, 1975); G. C. Pande, "Causality in Buddhist Philosophy," Eliot Deutsche & Ron Bontekoe, ed., A Companion to World Philosophies (Oxford, 1997), pp. 370–380.

For a contemporary view of chance from a scientific perspective, see Max Born's Natural Philosophy of Cause and Chance (Oxford, 1951) and David Bohm's Causality and Chance in Modern Physics (Princeton, 1957). Jacques Monod's position is stated in his Chance and Necessity (New York, 1971), and Ernest Schoffeniels's critique is in his Anti-Chance (Oxford, 1976). On the role of serendipity in scientific discoveries, see James H. Austin, Chase, Chance, and Creativity: The Lucky Art of Novelty (New York, 1978), and Aharon Kantorovich, Scientific Discovery: Logic and Tinkering (Albany, N.Y., 1993).

For a theistic position on chance, see Donald M. MacKay's Science, Chance, and Providence (Oxford, 1978), and William G. Pollard's Chance and Providence (New York, 1958).

On divination, see Greek Divination by W. R. Halliday (1913; Chicago, 1967); Oracles and Divination, edited by Michael Loewe and Carmen Blacker (New York, 1981), contains a wide range of material from many cultures.

On the idea of synchronicity, see C. G. Jung's foreword to The I Ching [Yijing ], or Book of Changes, 3d ed., translated by Cary F. Baynes (Princeton, 1967), and Jung's essays "Synchronicity: An Acausal Connecting Principle" and "On Synchronicity," in The Structure and Dynamics of the Psyche, 2d ed. (Princeton, 1969), vol. 8 of The Collected Works of C. G. Jung.

On miracles, see Antony Flew's "Miracles," in The Encyclopedia of Philosophy, edited by Paul Edwards (New York, 1967), vol. 5, and C. S. Peirce's "Hume on Miracles," in the Collected Papers of Charles Sanders Peirce, edited by Charles Hartshorne and Paul Weiss (Cambridge, Mass., 1960), vol. 6. Richard Swinburne's The Concept of Miracle (London, 1970) deals with the problem from the standpoint of philosophy of religion.

Michiko Yusa (1987 and 2005)

CHANCE

Much is asked of the concept of chance. It has been thought to play various roles, some in tension, or even incompatible, with others. Chance has been characterized negatively as the absence of causation; yet also positively—the ancient Greek "tyche" reifies it—as a cause of events not governed by laws of nature, or as a feature of laws of nature. Chance events have been understood epistemically as those whose causes are unknown; yet also objectively as a distinct ontological kind, sometimes called "pure" chance events. Chance gives rise to individual unpredictability and disorder; yet it yields collective predictability and order: stable long-run statistics and, in the limit, aggregate behavior susceptible to precise mathematical theorems. Some authors believe that to posit chances is to abjure explanation; yet others think that chances are themselves explanatory. During the Enlightenment, talk of chance was regarded as unscientific, unphilosophical, the stuff of superstition or ignorance; yet at the beginning of the twenty-first century it is often taken to be a fundamental notion of our most successful scientific theory, quantum mechanics, and a central concept of contemporary metaphysics.

Chance has both negative and positive associations in daily life. The old word in English for it, "hazard," which derives from French and originally from Arabic, still has unwelcome connotations of risk; "chance" evokes uncertainty, uncontrollability, and chaos. Yet chance is also allied with luck, fortune, freedom from constraint, and diversity. And it apparently has various practical uses and benefits. It forms the basis of randomized trials in statistics, and of mixed strategies in decision theory and game theory; it is appealed to in order to resolve problems of fair division and other ethical stalemates; and it is even thought to underpin biological and cultural adaptation. Throughout history, "chance" devices have been a source of entertainment, as well as of scorn.

A Brief History of Theories of Chance

The study of gambling games motivated the first serious mathematical study of chance by Blaise Pascal and Pierre de Fermat in the mid-seventeenth century, culminating in the Port Royal Logic. But inchoate ideas about chance date back to antiquity. Epicurus, and later Lucretius, believed that atoms occasionally underwent uncaused, indeterministic swerves—an early doctrine of pure chance. Aristotle, by contrast, believed that all events are necessary and regarded what we call coincidences (as in "We met at the market place by chance") as the intersections of independent deterministic causal chains—a view later shared by Thomas Aquinas, Antoine Augustin Cournot, and John Stuart Mill. Augustine believed that God's will controls everything, and thus that nothing happens by chance. In the middle ages, Averroes had a notion of "equipotency" that arguably resonated with Gottfried Wilhelm Leibniz's and later Pierre Simon de Laplace's ideas about "equipossibility," which undergirded their classical interpretation of probability: The probability of an event is the ratio of the number of equipossible cases in which it occurs to the total number of such cases. Girolamo Cardano, Galileo, Fermat, and Pascal also anticipated this interpretation.

Throughout the development of probability theory during the seventeenth through nineteenth centuries by authors such as Christian Huygens, Jakob Bernoulli, Thomas Bayes, Pierre Simon de Laplace, the Marquis de Condorcet, Abraham de Moivre, and John Venn, the fortunes of chance were at best mixed. De Moivre called chance "a mere word." David Hume captured the attitude of his time when he wrote, "'Tis commonly allowed by philosophers that what the vulgar call chance is nothing but a secret and conceal'd cause" (Hume 1975, p. 130). The triumphs of Newtonian mechanics engendered great confidence in determinism, personified by Laplace's image of an intelligent being (the so-called "Laplacean demon") for whom "nothing would be uncertain and the future, as the past, would be present to its eyes" (Laplace 1951, p. 4). Eliminativism about chance in nature had, moreover, good theological credentials: God's omniscience apparently made the world safe for determinism. But even the atheist Bertrand Russell insisted that a chance event is merely one whose cause is unknown. F. H. Bradley found the very notion of chance unintelligible.

Nonetheless, other intellectual developments set the stage for a revival of chance. With the burgeoning of social statistics in the nineteenth century came a realization that various social phenomena—births, deaths, crime rates, etc.—while unpredictable on an individual basis, conformed to large-scale statistical regularities. A somewhat analogous pattern of collective order from individual chaos appeared in statistical mechanics. The social sciences and then the physical sciences thus admitted statistical laws into their conceptual repertoire. This culminated in the early twentieth century with the advent of quantum mechanics, which appeared to show that chance was irreducible and ineradicable. Andrey Kolmogorov's axiomatization of probability came soon after Werner Heisenberg and Erwin Schrödinger brought quantum mechanics to its apogee.

Meanwhile, chance was also making a comeback in philosophy. Charles Sanders Peirce defended pure chance on the basis of empirical evidence. William James saw the postulation of chance as a way to resolve the apparent conflict between determinism and free will. To be sure, philosophers such as John Stuart Mill, Moritz Schlick, and C. D. Broad thought that capricious chance could provide no ground for genuine freedom. Nevertheless, chance had regained its respectability. In the 1950s Hans Reichenbach's work on probabilistic causation placed chance in the limelight in the philosophy of science.

The Mathematics of Chance

The mathematics of chance, unlike its philosophy, is relatively uncontroversial. That mathematics is widely taken to be probability theory. In Kolmogorov's theory (1933/1950), events are assigned numerical values between 0 and 1 inclusive:

P (X ) ≥ 0

P (Ω) = 1

(Here Ω is the universal set of all possible outcomes.) The probability of one of two mutually exclusive events occurring is the sum of their probabilities:

P (X ∪ Y ) = P (X ) + P (Y ) if X ∩ Y = ∅

(This law has an infinite generalization.) And the conditional probability of A given B is as follows:

P (A |B ) = P (A ∩ B )/P (B ) for P (B ) > 0

While Kolmogorov's theory remains the orthodoxy, some philosophers (e.g., James Fetzer, Paul Humphreys, Karl Popper) question its appropriateness for chance.

Chance in Science

Probability was introduced into physics in the late nineteenth century, when James Clerk Maxwell and Ludwig Boltzmann grounded thermodynamics in statistical mechanics. The status of this probability was an important interpretive issue, but it was not universally regarded as objective chance. Statistical mechanics was based on Newtonian particle mechanics, which was apparently deterministic. There are profound and ongoing controversies over the existence and nature of chance in both statistical mechanics and quantum mechanics.

In nonrelativistic quantum mechanics, according to the canonical Copenhagen interpretation, there are two rules for the evolution of a physical system:

• Schrödinger's equation prescribes a deterministic evolution for the state of the system. Typically, the state is a superposition (combined state) of the various definite-property states that the system might possess (e.g., definite position, definite momentum, etc.). While the system is in a superposition, it has no single value for such quantities.
• The collapse postulate is where chance enters quantum mechanics. Upon measurement of such a superposition, the state instantaneously collapses to one of the quantity's eigenstates (definite-property states). Which one is a matter of chance, the probability for each being derivable by Born's rule.

Albert Einstein considered this intrusion of chance into microphysics an unacceptable violation of causality and hoped for an underlying deterministic theory, with hidden variables, that explains the apparently chancy behavior of quantum systems. In 1935, Einstein, Boris Podolsky, and Nathan Rosen (EPR) insisted that there must be such an underlying theory, arguing that the quantum-mechanical description of a certain two-particle system is incomplete. Neils Bohr and Werner Heisenberg effectively criticized the EPR argument, and since an experimental test of an EPR pair of particles appeared to be physically unrealizable, most physicists quickly forgot the debate.

In 1952 David Bohm proposed a variant of the EPR setup using two coupled particles with correlated spins. Bohm's variant was both immune to the criticisms of Bohr and Heisenberg and physically realizable. In 1965 John Bell proved a now-legendary theorem stating that no local hidden-variable theory, of the type desired by Einstein, could replicate the statistical predictions of quantum mechanics for the correlated spins. Contrary to what the EPR paper had assumed, an underlying hidden-variable theory that assigned definite local values of spin to individual particles was incompatible with the predictions of quantum mechanics. Physicists then realized that a decisive experimental test was possible, and numerous experiments were performed in the 1970s, culminating in Alain Aspect's 1982 experiments, widely regarded as decisive. Nature sided with Bohr and Heisenberg, not Einstein.

Ironically, however, this confirmation of the predictions of quantum mechanics did not definitively show that God plays dice, to use Einstein's memorable phrase. In 1952 Bohm also formulated a hidden-variable variant of quantum mechanics that ascribes definite positions to all particles at all times, reproduces all the experimental predictions of standard quantum mechanics, and is perfectly deterministic. This is consistent with Bell's theorem. No local hidden-variable theory can match the predictions of quantum mechanics for coupled particles, but Bohm's version of quantum mechanics is nonlocal : A particle in one place may be affected, instantaneously, by distant events. Einstein would have approved of Bohm's theory for its deterministic microphysics and disapproved of it for violating the even more cherished precept of no nonlocal interactions.

There are other versions of quantum mechanics besides Bohm's that reject chancy collapses. It is thus unclear whether the success of quantum-mechanical theories implies a fundamental indeterminism in nature, and whether future experiments can resolve the issue.

Evolutionary biology is another area of science in which the existence and role of chance has been sharply debated. Evolutionary fitness is held by some philosophers and biologists to be fundamentally chancy, while others disagree.

Philosophical Accounts of Chance

Now, at the beginning of the twenty-first century, "chance" is typically taken to be synonymous with "objective probability," as distinguished from epistemic or subjective probability. Frequentists, originating with Venn, identify chance with relative frequency. For example, the chance that a particular coin lands heads is the frequency of tosses on which it so lands, divided by the total number of tosses. If we restrict ourselves to actual outcomes, then such frequencies will presumably be finite. A concern is that the outcomes may ill-reflect the true chances; a fair coin may land heads nine times out of ten. At the extreme, the problem of the single case, various events are unrepeatable, yet arguably have nontrivial chances (e.g., the outcome of the next presidential election). In such cases, mismatch between chance and relative frequency is guaranteed. Sometimes we might include in the reference class for a given event various other events. For example, regarding your chance of getting cancer, the class might include various other people like you. But there may be competing classes that yield different relative frequencies. You may belong both to the class of smokers and the class of those with no family history of cancer. What, then, is the real chance? This is the problem of the reference class.

Some frequentists follow Richard von Mises in requiring the sequences of trials that ground chances to be infinite, and thus presumably hypothetical. Then the chance of an outcome type is identified with its limiting relative frequency. (Further randomness constraints might also be imposed on the sequences.) Counterintuitively, such "chances" are then sensitive to the ordering of the trials (a sequence with infinitely many heads and tails can be rearranged to give whatever limiting relative frequency we like). Moreover, the appeal to hypothetical trials, let alone infinitely many of them, may betray the empiricist and scientific scruples that made frequentism initially appear attractive, for such "chances" are not constrained by anything in our experience.

Historically associated with Peirce and Popper, propensity accounts of chance postulate primitive dispositions, or tendencies, possessed by various physical systems. Propensity theories fall into two broad categories. According to single-case propensity theories, propensities measure the tendencies of a system to produce given outcomes; according to long-run propensity theories, propensities are tendencies to produce long-run outcome frequencies over repeated trials. The former have been advocated by the later Popper, David Miller, and James Fetzer; the latter by the early Popper, Paul Humphreys, and Donald Gillies.

Adopting a long-run view answers a need for testability of propensity attributions, one arguably found wanting for single-case propensity attributions. A long-run attribution may be held falsified if the long-run statistics diverge too much from those expected. However, defining propensities in terms of long-run relative frequencies may render single-case chance attributions problematic. This poses a dilemma for the long-run propensity theorist. If propensities are linked too closely to long-run frequencies, the view risks collapsing into a variant of frequentism. But if the view is cast so as to make single-case chance attributions possible, it risks collapsing into a variant of the single-case propensity view.

Long-run propensity theories may be motivated by the worry that in a single case there can be factors present that are not part of the description of the chance setup but that affect the chances of various outcomes. If the long-run propensity theorist responds by, in effect, falling back on long-run frequentism, the single-case propensity theorist goes the other way, embracing all causally or physically relevant details as part of the chance setup, determining the single-case chance (though we cannot measure it) for any given trial. The chance of each outcome is determined by everything that might influence the evolution of the setup. Propensity theories of this type respect some of our physical and causal intuitions, but pay a price epistemically. Since each single-case setup is presumably unique, we cannot use frequencies to estimate the chances or to falsify hypotheses about them.

A final problem, specifically for conditional propensities, is Humphreys' paradox. If Pr(A |B ) is a propensity, it seems to have a built-in causal direction, from B to A ; the "inverse" conditional probability Pr(B |A ) can often be calculated, but it appears to get the causal direction wrong. Various authors argue that inverse probabilities cannot be considered propensities, earlier events not having propensities to arise from later events. Thus, not all conditional probabilities may be interpretable as propensities.

While frequentist and propensity theories have dominated philosophical accounts of chance, a recent recurring proposal is that "chance" be viewed as a theoretical term similar to others in the sciences, such as "mass" or "fitness." In this post-positivist era, philosophers mostly agree that such terms cannot be reduced to non-theoretical terms. Instead, we may view theoretical terms as implicitly defined by their roles in scientific and philosophical theories. This approach avoids many of the difficulties discussed above, but it may not satisfy philosophers who find something troubling about the very notion of chance (see below). It also renounces giving a philosophical account of chance with normative status—claiming, for example, that theorists should admit objective chances into quantum mechanics but not into economics.

Pioneering work by David Lewis on the connections between chance and credence (subjective probability) has inspired Humean best-system theories. They share these tenets:

• Chances are defined so that their distinctive connection with credences is rendered transparent (see "Chance and Credence" below).
• Chances supervene on (are determined by) the entire history of actual events, and not on anything modal that does not itself supervene on the actual.
• Chances are determined by the laws of nature: the regularities of a best system (theory) that optimizes the balance of simplicity, strength (covering as many phenomena as possible), and fit (how typical actual events are, given the chances posited by the system).

Humean best-system accounts aim to be as acceptable to empiricists as finite frequentism, while avoiding the defects of that account.

Chance and Credence

Perhaps the most crucial demand we make of chances is that they guide our bets, expectations, and predictions—that they be guides to life in the face of uncertainty. This role is captured by some chance-credence principle or other, the most common coinage recently being Lewis's Principal Principle (Lewis 1986, p. 83–132):

(PP) Cr (A |ch (A ) = x & E ) = x

Here Cr is one's credence function, A is a proposition, ch (A ) is the chance of A (presumably time-indexed), and E is further evidence that one may have. For (PP) to be applicable, E cannot be relevant to whether A is true or false, other than by bearing on the chance of A. (PP) codifies something crucial about chance. A touchstone for any theory of chance is that it should underwrite (PP). There is considerable controversy over which theory (if any) can meet this challenge.

Chance and Determinism

Determinism is the thesis that any complete past or present state of the world, conjoined with the laws of nature, entails all future events. In a deterministic world, some insist, chance has no work left to do, the entire future being already determined by past events. Philosophers are divided over whether determinism rules out (nontrivial) chances. Since the definition of determinism says nothing about chance, more is needed to argue that determinism rules out chances.

D. H. Mellor, Popper, and others who view propensities as fundamental physical loci of indeterminism, see an immediate inference from determinism to the nonexistence of chances. Frequentists such as Venn and Reichenbach see no such inference: intermediate frequencies can exist in both deterministic and indeterministic worlds. The Humean best-system approach leaves open whether a deterministic system of laws can include chance laws (although Lewis rejects this possibility). And on the implicit-definition approach, intermediate chances and determinism coexist just in case our fundamental physical theories are deterministic but some scientific theory postulates objective probabilities. Statistical mechanics uses chances, but its underpinnings are deterministic, and typical uses of chance in biology and the social sciences involve no presumption for or against determinism, as Isaac Levi (1990) and others have argued.

Nor does indeterminism guarantee the existence of chances. Fundamental physical laws may fail to entail a unique future without being probabilistic. However, if these laws are probabilistic, as some interpretations of quantum mechanics contend, then chances are apparently guaranteed on any but a skeptical/subjectivist view.

Subjectivism, Skepticism about Chance, and Exchangeability

Chance is meant to play a certain theoretical role. It is a further matter what, if anything, actually plays this role. According to Bruno de Finetti, nothing does. "Probability does not exist," he said (1990, p. x), meaning that chance does not exist and that all probability is subjective. Skepticism about chance is easily assimilated to skepticism about kindred modal notions—possibility, counterfactuals, causation, laws of nature—that seem not to be straightforwardly reducible to nonmodal notions, in particular, notions congenial to an empiricist. And skepticism specifically about chance can be based on further arguments, for one can be skeptical not just about its modality, but also about its putative degrees. Subjectivists have also argued that chance is redundant, its alleged role being completely discharged by credences. Richard Jeffrey, Bas van Fraassen, Brian Skyrms, and others have developed subjectivist positions in the spirit of de Finetti.

Moreover, the mathematics of chance (unlike the other modal notions) permits a particular eliminativist gloss. A sequence of trials is said to be exchangeable with respect to a probability function if the probabilities of trial outcomes are invariant under finite permutations of trials; probabilities may be sensitive to the numbers of outcomes of each kind, but not to their ordering. De Finetti (1990) showed that when this condition is met, there is a unique representation of the probability distribution over the trials as an expectation of simpler probability distributions according to which the trials are independent and identically distributed. For example, if your credences over the results of repeated coin tossing are exchangeable, then it is as if you treat the trials as tosses of a coin of unknown bias, with credences over the possible biases. Subjectivists have argued that this delivers some of the supposed benefits of chance, without any questionable metaphysics.

Conclusion

Many of the perplexities about chance—its controversial metaphysics, its seeming resistance to reduction, its epistemological recalcitrance, etc.—are familiar from other modal notions. But chance has been handled in mathematics and philosophy with more precision than those other notions. In the process, still further perplexities have been born. For the foreseeable future, at least in the writings of philosophers and philosophically minded scientists, chance is probably here to stay.

See also Probability and Chance.

Bibliography

a brief history of theories of chance

David, F. N. Games, Gods, and Gambling: A History of Probability and Statistical Ideas. Mineola, NY: Dover, 1962.

Franklin, James. The Science of Conjecture: Evidence and Probability before Pascal. Baltimore, MD: Johns Hopkins University Press, 2001.

Hacking, Ian. The Emergence of Probability. Cambridge, U.K.: Cambridge University Press, 1975.

Hacking, Ian. The Taming of Chance. Cambridge, U.K.: Cambridge University Press, 1990.

Hume, David. A Treatise of Human Nature, edited by L. A. Selby-Bigge, 2nd ed. Oxford: Clarendon Press, 1975.

Laplace, Pierre Simon. A Philosophical Essay on Probabilities. English edition 1951 (originally published 1814). New York: Dover Publications Inc.

the mathematics of chance

Billingsley, Patrick. Probability and Measure. 3rd ed. New York: John Wiley and Sons, 1995.

Kolmogorov, Andrei N. Grundbegriffe der Wahrscheinlichkeitrechnung. Berlin: J. Springer, 1933. Translated as Foundations of Probability. New York: Chelsea, 1950.

Skyrms, Brian. Choice and Chance: An Introduction to Inductive Logic. 4th ed. Belmont, CA: Wadsworth, 1999.

chance in science

Albert, David. Time and Chance. Cambridge, MA: Harvard University Press, 2000.

Bell, John S. Speakable and Unspeakable in Quantum Mechanics. Cambridge, U.K.: Cambridge University Press, 1987.

Einstein, Albert, Boris Podolsky, and Nathan Rosen. "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" Physical Review 47 (1935): 777–780.

Sklar, Lawrence. Physics and Chance. Cambridge, U.K.: Cambridge University Press, 1993.

Sober, Elliott. Philosophy of Biology. 2nd ed. Boulder, CO: Westview Press, 2000.

philosophical accounts of chance

Fetzer, James H. Scientific Knowledge: Causation, Explanation, and Corroboration. Dordrecht, Netherlands: D. Reidel, 1981.

Fine, Terrence. Theories of Probability. New York: Academic Press, 1973.

Gillies, Donald. "Varieties of Propensity." British Journal for the Philosophy of Science 51 (2000): 807–835

Loewer, Barry. "David Lewis's Humean Theory of Objective Chance." Philosophy of Science 71 (2004): 1115–1125.

Mellor, D. H. The Matter of Chance. Cambridge, U.K.: Cambridge University Press, 1971.

Miller, David. Critical Rationalism: A Restatement and Defence. La Salle, IL: Open Court Press, 1994.

Sober, Elliott. "Evolutionary Theory and the Reality of Macro Probabilities." In Probability in Science, edited by Ellery Eells and James H. Fetzer. La Salle, IL: Open Court, 2005.

chance and credence

Lewis, David, "A Subjectivist's Guide to Objective Chance." In Studies in Inductive Logic and Probability. Vol II., University of California Press, 263-293, 1980; reprinted with postscripts in Philosophical Papers, Vol. II. Oxford: Oxford University Press, 83-132, 1986. Lewis, David. "Humean Supervenience Debugged." Mind 103 (1994): 473–490.

Popper, Karl. "The Propensity Interpretation of Probability." British Journal of the Philosophy of Science 10 (1959): 25-42.

Vranas, Peter. "Who's Afraid of Undermining?" Erkenntnis 57 (2002): 151–174.

chance and determinism

Levi, Isaac. "Chance." Philosophical Topics 18 (2) (1990): 117–148.

Loewer, Barry. "Determinism and Chance." Studies in History and Philosophy of Modern Physics 32 (2001): 609–620.

Strevens, Michael. Bigger than Chaos: Understanding Complexity Through Probability. Cambridge, MA: Harvard University Press, 2003.

subjectivism, skepticism about chance, and exchangeability

de Finetti, Bruno. Theory of Probability, Vol. 1. Chichester: Wiley Classics Library, John Wiley & Sons, 1990

Skyrms, Brian. "Bayesian Projectibility." In Grue: The New Riddle of Induction, edited by Douglas Stalker, 241–262. Chicago: Open Court, 1994.

web resources

"Chance." The Chance Project, Mathematics Department, Dartmouth College. Available from http://www.dartmouth.edu/~chance/.

Hájek, Alan. "Probability, Interpretations of." In The Stanford Encyclopedia of Philosophy, summer 2003 ed., edited by Edward N. Zalta. Available from http://plato.stanford.edu/archives/sum2003/entries/probability-interpret/.

Alan Hájek (2005)

Carl Hoefer (2005)

CHANCE

The term chance (Lat. casus ) is used in a variety of ways. In some contexts it is considered as that which is entirely without cause; this was the view of democritus and lucretius. Other writers count chance as a cause, but differ as to the kind of causality it exercises. Thus some modern scientists, such as Max Born, maintain that chance is the cause of all things; A. einstein, on the other hand, protested against this thesis by saying that God does not play dice. Others call chance a cause, but insist that it is indeterminate, either because it is the result of a basic indeterminism in nature or because the human intellect cannot encompass the various lines of causality that exist. What these various notions have in common can be clarified by a proper definition of chance, and this is the burden of the present article.

Aristotle's Analysis. aristotle attempted such a clarification in bk. 2 of the Physics (195b 30–198a 13), where he made use of several distinctions in his search for a definition of chance. Of things that come to be, some come to be always in the same way, whereas others do not. Of the latter, some come to be often, whereas others come to be seldom. Chance is found among those things that happen seldom; however, since not everything that happens seldom is by chance, other divisions are necessary to manifest the definition. A further division considers events that happen for a purpose and those that do not. Of the former, some are the result of an intention— whether this be the intention of an intelligent agent or simply what is intended by nature—whereas others are not.

Apart from these distinctions, Aristotle also proposes a division based on causes, since most thinkers agree that chance is in some way a cause. Thus he holds that just as beings are either per se or per accidens, so also are causes. For example, assuming that a white, musical builder constructs a house, the builder is the per se cause of the house, whereas white and musical are its per accidens causes. Among per accidens causes, some are such by reason of something accidentally associated with the cause, as in the example mentioned, and others are such by reason of something accidentally associated with the effect—for example, an argument that might arise over the house already built. The difference is shown in the accompanying diagram. Chance itself is a kind of per accidens cause that results from something accidentally associated with an effect, as the builder just chances to be the cause of the argument over the house. (Notice that in this case one per se cause is also a per accidens cause; in the case of a per accidens cause that is such by reason of something accidentally associated with a per se cause, the latter cause is itself composite, namely, the white builder.)

Utilizing these divisions, Aristotle defines chance as a per accidens cause in things that are for an end and that happen seldom. As something happening seldom, the effect in chance is something neither intended nor expected by the agent. Aristotle's example is a man who collects money by going to market for some purpose other than collecting money. If such a man always or usually collected money by going to market, this event would not be by chance.

A further clarification of the notion of chance is achieved by Aristotle's contrasting the chance with the vain. An action is vain when that which was intended does not happen. Aristotle shows that actions can be (1) vain and chance, (2) vain and not chance, (3) chance and not vain, and (4) neither vain nor chance. Suppose that Socrates goes to market to buy cabbage. It might happen that the store is out of cabbage but that Socrates does meet his friend who owed him a debt: vain and chance. Again, he might neither get the cabbage nor meet his friend: vain and not chance. Yet again, he might get the cabbage and meet his friend: chance and not vain. Finally, he might get the cabbage and not meet his friend: neither vain nor chance.

The failure to distinguish between the chance and the vain has led some to hold that chance happens only when the intended end is not achieved. However, as has been seen, there can be chance whether the intended end is achieved or not. What is necessary is that some end be intended. If an agent who acts by intelligence and will attains the unintended end, this is usually called fortune. Among Aristotelians, the term chance is reserved for agents who act by nature.

Causal Intersections. From this definition of chance, it is possible to explain the various positions held concerning it. In the first place, philosophers who hold that all things happen of necessity deny that chance exists. Even among philosophers who admit the existence of chance, there are those who hold that chance causes nothing since it is a per accidens cause. It is certainly true that there is an accidental unity in whatever results from chance. It is also true that two or more per se causes will be found to have been acting in the production of such an event. St. thomas aquinas thus says that "a cause which hinders the action of a cause so ordered to its effect as to produce it in the majority of cases, clashes sometimes with this cause by accident: and the clashing of these two causes, inasmuch as it is accidental, has no cause. Consequently what results from this clashing of causes is not to be reduced to a further pre-existing cause, from which it follows of necessity" (Summa theologiae 1a, 115.6). The last statement, that the per accidens intersection of two lines of causality is not to be reduced to a further preexisting cause, must be understood of a cause preexisting in nature. Aquinas notes in another place: "Let us suppose that a man is prompted to dig a grave by the influence of a celestial body, working through his emotions, as was said. Now the grave and the location of the treasure are united only accidentally, for they have no intrinsic relation to each other. Thus, the power of the celestial body cannot directly give an inclination to this entire result, namely, that this man should dig this grave and that it should be done at the place where the treasure is. But an agent working by intellect can be the cause of an inclination to this entire result, for it is proper for an intelligent being to order many things toward one" (C. gent. 3.92). Aquinas further observes that man's intellect can cause an event that in nature would be by chance. He continues, "Fortuitous events of this kind, when referred to their divine cause, lose their fortuitous aspect; but when referred to a celestial cause, they do not" (ibid. ). Thus chance remains even when the combined effect might be caused by the ordering of a higher cause. The reason is that nature, in this case the celestial body as a natural cause, produces effects that are per se one. It cannot have, as a proper effect, something that is only accidentally one. Of such it can be only the per accidens cause. This also shows that chance is more than mere ignorance of the concatenation of causes and that chance results from the inability of the lower cause to control causal intersections.

Accidental Causality. The notion that chance is the cause of all things results from a different kind of confusion over the per accidens. In the Metaphysics (1013b 34–1014a 20) Aristotle again discusses the causes and their division into per se and per accidens. St. Thomas's commentary on this point is illuminating (In 5 meta. 3.789). He states that the per se cause can become a per accidens cause by reason of something happening to the effect in one of three ways. (1) It may come about in such a way that what is added to the per se effect has a necessary order to it, as happens when the primary effect removes an obstacle to the secondary effect. This may happen when a contrary is removed, as when food is spoiled by removing it from a refrigerator, not because heat itself spoils the food, but because the refrigerator's cold opposed the growth of bacteria that is a cause of the food's spoiling. There can also be a necessary connection of effects when there is no contrariety, as when an arch falls because a pillar is removed. When the secondary effect follows the primary in this way, the per accidens cause is not called chance, since such added effects follow always or often. (2) Again, the secondary effect can follow the primary effect, not as something necessary or often, but as happening seldom, as the argument over the house or the finding of a treasure by one digging a grave. The per accidens cause of such a secondary effect is called chance or fortune. (3) Finally, the connection between two events may be only in the mind, as one might imagine that his opening a door was the cause of an earthquake, because a tremor occurred just as he was opening the door.

Chance and Luck. Thus not every intersection of lines of causality is to be attributed to chance. If a person decides to cross a muddy street, he should not attribute the soiling of his shoes to chance merely because he did not intend this effect. Such would be chance only if it happened seldom to one who crossed a muddy street. In spite of this, many use the term chance in such indiscriminate fashion. They speak of taking a chance on the horses or of luck in a dice game. Chance in a strict sense is not found in such actions. Suppose, for example, a person bets on a horse and loses. This is not chance but vain. Similarly, if he bets on a horse and wins, to call this chance is to overlook the fact that the winning was what was intended, whereas chance is something that is not intended but is accidentally associated with a primary effect. There is justification for the use of the term chance in such instances, however, because the mind, seeing the general rule, counts what departs from this only slightly as something that has already happened. For example, a person calls the lost wager bad luck because he has carefully considered the factors and come to the firm belief that the possibility of this horse's losing the race is so small that it can be ignored. In other words, he considers the connection of primary and secondary effects to be that of (1) above. The winning is attributed to chance in a similar way. The person bets on the horse, keenly aware that he seldom wins; considering this, he in effect forgets or ignores the fact that he actually intends to win. When he does win, it is something that happens seldom and is, in a way, unintended.

Randomness and Probability. Chance is used improperly in another way when applied to randomness or probability. For example, it might be said that an even distribution of sand and cement comes about by chance since it is the result of a random mixing. Again, the killing of a bird by one or two of the many shot pellets fired is said to be accounted for by the laws of chance. This overlooks the fact that the end was intended and, more important in this example, is something probable, whereas chance is what happens seldom. Yet nature is also said to use chance in this way to accomplish her ends. In her production of great numbers of seeds and of many individuals of each species, she intends the preservation of such species. In the circumstances, this seems to be the most economical means of achieving her ends.

That such a use of the term chance is that of Democritus, of Lucretius, and of many modern scientists seems further evidenced by the latters' reference to the laws of chance as laws of probability. Even the term law, when used here, indicates a regularity that is foreign to the proper definition of chance. On the other hand, Einstein's maintaining that God does not play dice is well founded. If God is throwing dice to achieve His effects, He does not do so as a casual player awaiting a fortunate turn of a seven or eleven. Rather, He is more like the scientist investigating probabilities, who throws the dice countless times with the firm assurance that these numbers will occur with a definite frequency.

This last consideration seems to be the basis of the denial of chance by such thinkers as B. spinoza, and G.W. leibniz. They hold that chance results only from the fact that man's intellect cannot encompass the causes at work in any event. Thus, for a greater intellect, chance would not exist. However, although it is true that for a greater intellect there are fewer effects owed to chance and that for the divine intellect nothing is by chance, chance is nonetheless a reality. In effect, these last thinkers are denying indeterminism in nature. Such a solution ignores the fact that something ordained with certainty by a higher cause can still be contingent when considered in its relation to lower causes.

See Also: fate and fatalism; contingency; necessity.

Bibliography: h. j. freeman, The Problem of Chance (Doctoral diss. unpub. River Forest, Ill. 1963). m. born, Natural Philosophy of Cause and Chance (Oxford 1949). c. de koninck, "Abstraction from Matter, III," Laval Théologique et Philosophique 16 (1960) 169–188. a. aliotta, Enciclopedia filosofica (Venice-Rome 1957) 1:921–927. r. eisler, Wörterbuch der philosophischen Begriffe (Berlin 1027–30) 3:667–670. m. j. adler, ed., The Great Ideas: A Syntopicon of Great Books of the Western World (Chicago 1952) 1:179–192.

[r. a. kocourek]

Chance

In both science and religion there is a lively debate about the role of chance in the universe. In science, this debate usual takes the form of a discussion deciding between determinism (all events follow of necessity from prior initial conditions) and physical indeterminism (some events, at least, are not so determined). In religion, the dispute is between those who accept total predestination (the view that God unilaterally ordains everything that happens) and theological indeterminism (God leaves some things to chance or to determination by finite agents). Most religious views deny any role for pure chance, but many allow some role for chance even in a providentially-governed universe. Debate is often clouded by a failure to define what "chance" is.

Different senses of chance

In its most radical sense, chance is the occurrence of an event without any cause or reason. Thus the universe may be said to come into existence for no reason and without any antecedent cause—by chance. In this sense, absolutely anything might happen at any time, and there is no point in seeking reasons for what happens. If everything happened by chance, in this sense, science would be impossible.

Another, more common, sense of chance is involved in gambling or lotteries. When a gambler throws a die, the side that lands uppermost is a matter of chance. It is not that there are no causes for the position of the die, but that the causes are far too difficult, complex, or tedious to be detected. The roll of the die could be determined in every particular by applying the laws of mechanics, but it would still be considered a matter of chance because the system is set up so that no human can predict the outcome. In this case, chance primarily refers to unpredictability; whether something is chance or not depends on the knowledge available to the observer.

Another sense is that in which something happens "by chance" because it is not intended by any agent. A person may meet a long-lost friend by chance if neither the person nor the friend nor God had intended the meeting to happen, or tried to bring it about. Genetic mutations are said to be random, to occur by chance, in this sense. They have causes, but they are not intended to happen as they do.

This sense can be extended to events that are not parts of any directional process or propensity. Thus, many geneticists would say that genetic mutations do not tend in any particular direction (they do not, for example, always occur so as to maximize the chances of survival for some organism). This view is contentious, for some argue that there are propensities in organic mutation; the process does tend to realize consciousness eventually, and this tendency is inbuilt in the system from the beginning. If this were true, particular mutations could happen by chance (they would not each be determined to increase the chances of consciousness coming into being), but the process as a whole (the whole set of mutations in their environmental context) might have a propensity to terminate in consciousness.

This introduces yet another sense of chance, for which particular events have a specific probability of occurring, but are not sufficiently determined. An event is sufficiently determined when, given its initial conditions and the laws of nature, it could not happen in any other way. An event is not sufficiently determined when, from the very same initial conditions and laws, there are a number of possible effects that could result. In other words, the same cause in the same situation can have different effects. Some physicists have denied this possibility, but the Copenhagen Interpretation of quantum mechanics asserts precisely that particular subatomic events have a highly specific probability of occurring in a specific way, but they may not do so. When large numbers of quantum events occur, however, this probability will turn into a predictable certainty—thus the equations of quantum mechanics are deterministic, though they refer to events that are to some extent indeterminate. Such processes are called "stochastic"; there is a high probability that specific types of events will occur, but particular events may be unpredictable and not sufficiently determined.

Implications for freedom

There are thus two main components of the idea of chance—lack of predictability and lack of sufficient causality. For some philosophers, human freedom requires chance, since humans could not be held responsible for their actions if they were sufficiently caused (if they were determined by some cause, whether natural or divine) to act as they do. According to this view, chance is a necessary condition of responsible freedom. A free act is distinguished from a purely arbitrary (non-caused) act by being intentional, initiated for a purpose.

A believer in God may say that the creation of the universe is the primary instance of a free act. Creation is not caused by any prior initial state or by some general laws, but it is brought about for a reason. God has some value or values in mind, and realizes them by creating the universe. A free act is thus a form of causality for the sake of realizing some envisaged value. This causality distinguishes such an act sharply from pure chance, even though the act may appear unpredictable and undetermined from the point of view of physical laws and prior physical or mental states.

Some theologians have proposed that quantum mechanics shows the fundamental laws of the universe to be stochastic, or statistical, rather than deterministic. This, they claim, would permit both human free acts to occur, and would also allow God to act freely within the statistical probabilities of the physical system without "breaking" any laws of nature. For others, it is much too restrictive to confine God's free actions to scrabbling around in the sub-atomic basement. In any case, quantum indeterminacies cancel out because of the large numbers of probabilistic events involved in supra-atomic events, which means that the overall statistical distribution is virtually uncertain.

The existence of dynamic systems far from equilibrium allows quantum fluctuations to be amplified to produce macrocosmic effects. Thus in the right circumstances (in the brain, for example) quantum indeterminacies could produce huge observable indeterminacies in nature. Or it could be held that, quantum considerations apart, laws of nature are in themselves probabilistic, operating on an "other things being equal" basis, and they do not exclude free, or teleological causality, at all.

Religious views cannot easily live with any supposition of pure chance, in the radical sense. Most classical theistic views are deterministic (all is determined by God), seeing freedom as compatible with determinism. But in the twentieth century there has been an increase in the number of people holding nondeterministic views, for which chance (as probabilistic indeterminism) allows free creative activity both of creatures and of God, and a mutual responsiveness of creaturely and divine acts that may be held to be close to a biblical perspective.

See also Complexity; Contingency; Convergence

Bibliography

bartholomew, david j. god of chance. london: s.c.m. press, 1990.

murphy, nancey. "divine action in the natural order." in chaos and complexity: scientific perspectives on divine action, eds. robert john russell, nancey murphy, and arthur peacocke. vatican city state: vatican observatory, 1995.

plantinga, alvin. god, freedom, and evil. london: allen, unwin, 1974.

polkinghorne, john. "chaos theory and divine action." in religion and science, eds. mark richardson and wesley wildman. new york: routledge, 1996.

russell, robert j., murphy, nancey; and peacocke, arthur, eds. perspectives on divine action, notre dame, ind.: university of notre dame press, 1997.

saunders, nicholas t. "does god cheat at dice? divine action and quantum possibilities." zygon: journal of religion and science 35, no. 3 (2000): 517–544.

swinburne, richard. responsibility and atonement. new york: oxford university press, 1989.

ward, keith. god, chance, and necessity. oxford: oneworld, 1996.

ward, keith. god, faith, and the new millenium. oxford: oneworld, 1998.

keith ward

chance / chans/ • n. 1. a possibility of something happening: there is little chance of his finding a job. ∎  (chances) the probability of something happening: he played down his chances of becoming chairman. ∎  [in sing.] an opportunity to do or achieve something: I gave her a chance to answer. ∎  a ticket in a raffle or lottery.2. the occurrence and development of events in the absence of any obvious design: he met his brother by chance. ∎  the unplanned and unpredictable course of events regarded as a power: chance was offering me success.• adj. fortuitous; accidental: a chance meeting.• v. 1. [intr.] do something by accident or without design: if they chanced to meet. ∎  (chance upon/on) find or see by accident: he chanced upon an interesting advertisement.2. [tr.] inf. do (something) despite its being dangerous or of uncertain outcome: she chanced another look.PHRASES: by any chance possibly (used in tentative inquiries or suggestions): were you looking for me by any chance?no chance inf. there is no possibility of that: I asked if we could leave early and she said, “No chance.”on the (off) chance just in case: Joan phoned at noon on the off chance that he’d be home.stand a chance have a prospect of success or survival: his rivals don't stand a chance.take a chance (or chances) behave in a way that leaves one vulnerable to danger or failure. ∎  (take a chance on) put one's trust in (something or someone) knowing that it may not be safe or certain.take one's chances do something risky with the hope of success.