What Is Chance?

The term chance in its scientific sense is tied to the notion of probability.⁹ Probability theory is now widely used in physical, biological, and social sciences. While there is almost no disagreement about the formulation of probability theory, its interpretation has been challenging since its birth in the seventeenth century.¹⁰ There are three main approaches to interpreting probabilities: the subjective interpretation, the frequency interpretation, and the propensity interpretation. According to the subjective interpretation, the term probability refers to a person’s degree of belief—or credence—quantified as a real number between 0 and 1. The degree of belief is determined by both the world and the epistemic position of the agent in the world.

According to the frequency interpretation, associated with Richard von Mises ([1928] 1957) and Hans Reichenbach (1949), probability is a theoretical term that its referent is the stable frequency in the sequence of outcomes in the long run, where stable frequency means the frequency that as the number of observations or experiments increases, fluctuations in the value of that frequency become smaller and smaller. According to the propensity interpretation, associated mostly with Karl R. Popper (1959), probability is a theoretical term that its referent is a tendency in systems of a special kind, i.e., systems capable of generating repeatable outcomes, to yield a stable frequency in the long run.¹¹ The frequency and propensity interpretations both belong to the category of ontological interpretations, while the subjective interpretation is restricted to the epistemological ones.

By chance, we mean ontological probabilities, or, more precisely, “whatever objective property in the world the formal concept [of probability] picks out” (Suárez 2022, 645). In this sense, chance, if it exists, is determined entirely by the world and independent of the epistemic positions of agents.¹² The existence of chance in the world means the existence of some kind of indeterminacy, arbitrariness, or degrees of freedom inherent in objects or systems.¹³

Evolutionary theory, like many scientific theories, employs probabilities to describe and explain the biological world. The probabilities used in evolutionary theory can also be considered objective.¹⁴ For example, consider genetic mutations. Genetic mutations occur by chance, meaning that, roughly speaking, they do not occur in response to environmental threats or opportunities; “there is no physical mechanism (either inside organisms or outside of them) that detects which mutations would be beneficial and causes those mutations to occur” (Sober 2011, 192).¹⁵ More precisely, there is no (statistical) correlation between the usefulness of a possible mutation and the probability of its occurrence.¹⁶ Here, probabilities are interpreted ontologically as a property of the biological systems.

It is noteworthy while natural selection is driven by environmental factors, it still involves chance. However, this chance is different from the chance in genetic mutations. Natural selection operates over small random variations that can be transmitted to the next generation. Those heritable variations are in the fitness of an organism or a trait, consisting of two components, viability and fertility, both are probabilistic concepts. Roughly speaking, viability is the probability of surviving to reproductive age, and fertility is the expected, i.e., the average, number of offspring, which in turn includes probabilities of having exactly i (i = 1, 2, 3 . . .) offspring.¹⁷ If these probabilities are considered objective, they represent the chance involved in the process of natural selection.

Blind Chance and the Argument against Compatibilism

We mentioned the scientific meaning of chance in the previous subsection. However, the term chance originally belongs to the natural language and is widely used in everyday contexts in addition to the scientific ones.¹⁸ Sometimes an everyday meaning of chance finds its way into scientific texts.

Blind chance is a familiar phrase that can be found everywhere, for example, as the title of a movie directed by Krzysztof Kieślowski or in a letter from Charles Darwin (1958, 162) to J. D. Hooker: “My theology is a simple muddle; I cannot look at the universe as the result of blind chance, yet I can see no evidence of beneficent design, or indeed of design of any kind, in the details.” Or, similarly, in Darwin’s (1958, 92) autobiography, where he mentions “the extreme difficulty or rather impossibility of conceiving this immense and wonderful universe including man with his capacity of looking far backwards and far into futurity, as the result of blind chance or necessity.” Blind chance is a key notion to the extent that some consider it the basis of the evolutionary process. Jacques Monod (1974, 112–13) puts it in these words:

[C]hance alone is at the source of every innovation, of all creation in the biosphere. Pure chance, absolutely free but blind, at the very root of the stupendous edifice of evolution: this central concept of modern biology is no longer one among other possible or even conceivable hypotheses. It is today the sole conceivable hypothesis, the only one that squares with observed and tested fact.

Different everyday meanings of chance can somehow be combined with its scientific meaning if this combination does not lead to a bare inconsistency. Blind chance does not seem incompatible with the scientific meaning of chance as ontological probability. So, chance can be considered blind, or at least we assume so for the sake of the argument, because many have done so, especially among incompatibilists (to whom compatibilists should reply).

Incompatibilists believe that evolutionary theory and interventionist theism are incompatible. They can defend their position in several ways (see the introduction of this article). One possible way is to argue that chance processes cannot be guided by God because chance processes, like genetic mutations, are blind and aimless. Such an argument is as follows:

• (Premise 1) Every event in the history of the biosphere is a chance event.¹⁹

• (Premise 2) Every chance event occurs blindly and aimlessly.

• (Conclusion 1) Every event in the history of the biosphere occurs blindly and aimlessly.

• (Premise 3) If every event in the history of the biosphere occurs blindly and aimlessly, then the set of events that make up the history of the biosphere is blind and aimless.

• (Conclusion 2) Therefore, the set of events that make up the history of the biosphere is blind and aimless.

Conclusion 2 can be interpreted as follows: the history of the biosphere is not guided.

The Bartholomew-Bradley Model

Bartholomew (1984, 2008) argues that the existence of blind chance is compatible with the world being guided by God. He illustrates this point through instances where orders are built on chance arrangements. The attractive feature of these orders is that they are very difficult to achieve by design and fairly easy to extract from chaos. Additionally, he presents instances where chance occurrences stem from order. He concludes that chance and order are interconnected in their nature.

To do this, he employs the notion of “level” and states that, while there is blind chance at the level of individuals, statistical orders emerge at the aggregate level in accordance with mathematical rules, particularly the central limit theorem. This theorem states that, under some conditions, the average of many independent samples of a random variable, regardless of the original distribution of the individual samples, will itself be a random variable with a distribution that converges to a Gaussian (also known as normal) distribution as the number of samples increases (Figure 1). The central limit theorem guarantees that as the size of the population increases, statistical patterns emerge.

Figure 1

Whatever the initial probability distribution is, as the number of samples increases, the mean probability distribution tends to the Gaussian distribution (redrawn from Manfred Borovcnik and Ramesh Kapadia (2011, 30), with changes).

Bartholomew intends to demonstrate that God has used both order and chance in creating and governing the world. He claims that divine sovereignty varies at different levels: while at the level of individuals, God uses chance; at the aggregate level, He uses order. According to Bartholomew, if God wants to produce the existing orders at the aggregate level, the best way is to use chance at the level of individuals.

Bartholomew argues that if there is such a simple and beautiful way of producing order, then God will use it and not need to bother himself with the details of individuals. It may be objected that it is not difficult at all for God, as the omniscient and omnipotent, to deal with details, and He therefore need not choose the easier way. Bartholomew (2008, 128) replies that:

[t]he profound theological question is not so much whether God could handle the enormous complexity of [those] scenarios . . . but whether it is a God-like enough thing for him to be doing.

Whether we agree with Bartholomew or not does not make a difference in our discussion concerning whether blind chance can be combined with being guided. Bartholomew has tried to show that the existence of chance in the world does not contradict the world being guided by God and that chance should be considered within the divine providence, not outside of it.

Bradley (2012)²¹ takes a relatively similar approach. However, he points out that the notion of “level” is too simplistic; the real world does not consist of two levels, on one of which God acts through chance and on the other through order. Bradley’s objection seems nothing more than that “level” is a vague notion. A vague notion, roughly speaking, is a notion that “its applicability is tolerant with respect to very small changes” (Priest 2000, 72); if a vague notion is applicable once, it is still applicable despite a small change in the situation. Most of our everyday notions are vague. For instance, “being a child” is a vague notion; if Sarah is considered a child, she can be considered a child just a second later. However, being vague does not require being unreal or inapplicable. That the notion of “level” is vague does not mean that it is unreal or inapplicable, just as the notion of “being a child” being vague does not mean it is not a real or applicable. There are clear-cut cases of the level at which chance appears and the level at which orders appear. This would be enough to justify the application of this notion and the belief in its reality. So, the vagueness of the notion of “level” does not cause serious problems with Bartholomew’s approach.

Bartholomew’s and Bradley’s position can thus be summarized as follows: God regulates and supervises chance events through statistical orders. This position implies a model that includes both chance and order. In this model, the world consists of levels, some dominated by chance and others by order, and the existence of one requires the existence of the other. We call this model the Bartholomew-Bradley model.

The Bartholomew-Bradley model demonstrates the compatibility of blind chance with statistical orders. However, it is not clear whether being regulated and supervised through statistical orders is the proper articulation of the world being guided by God. In other words, it is debatable whether “being guided” can be reduced to being regulated and supervised through statistical orders. The notion of “divine guidance” seems to imply more than simply being controlled by statistical orders. It suggests leading towards an intentional direction or goal as well as intervention and (fine-) tuning of some features of the universe, like its initial conditions or fundamental constants.

Theists and compatibilists²² usually consider the concepts of divine sovereignty and guidance to be more than regulation and supervision through statistical orders. According to them, God should guide the universe towards a specific goal, for example, to the point where a rational being who can know and worship Him comes into existence. They consider the notion of “being guided” as having an intended direction or goal, which is more than following statistical regulations. To clarify this, consider the Gaussian distribution. The Gaussian pattern simply expresses that, in large enough populations, a certain percentage will exhibit a specific feature. This pattern does not imply any direction or purpose in the development of that feature. The Gaussian distribution is indifferent to any specific direction, making it neutral with respect to any particular goal.

Furthermore, for theists who believe God guides the world, this often means that God actively intervenes in the world’s evolution, that is, it requires special divine action. Also, those theists who accept fine-tuning of some features of the universe, like its initial conditions or fundamental constants, consider it a part of divine guidance.²³ However, intervention and (fine-) tuning are absent from the Bartholomew-Bradly model. Moreover, incompatibilists acknowledge the existence of statistical orders but find them insufficient to prove the biosphere is guided.

We do not intend to fully analyze the meaning of “being guided” since, for our purposes, it is enough to offer a partial articulation: leading towards an intentional direction or goal, intervention, and (fine-) tuning. Consequently, it shows that the Bartholomew-Bradley model is inadequate for demonstrating the compatibility of blind chance with being guided.

The van Inwagen Model

Van Inwagen (2003) defends compatibilism, arguing that the existence of chance in the world is compatible with the idea that the biosphere is guided and designed by God. While he has not explicitly provided a model of chance events that collectively form a guided whole, we believe such a model can be extracted from his work, especially when he discusses the “area-measuring device.”

There is a marvelous device for calculating the areas surrounded by irregular closed curves. It is an electronic realization of what is sometimes called the dartboard technique. To simplify somewhat: you draw the curve on a screen; then the device selects points on the screen at random, and looks at each point to see whether it falls inside or outside the curve; as the number of points chosen increases, the ratio of the chosen points that fall inside the curve to the total number of points chosen tends to the ratio of the area enclosed by the curve to the area of the screen. For a large class of curves, including all that you could draw by hand, and probably all that would be of practical interest to scientists or engineers, the convergence of ratios is quite rapid. Because of this, such devices are useful and have been built. (van Inwagen 2003, 353)

Van Inwagen points out that this device is designed based on a random process that results in selecting random points, and yet it is designed to serve a specific purpose: to measure the area of irregular shapes.

Now the properties of each point that is chosen—its coordinates—are products of chance . . . But the whole assemblage of points chosen in the course of solving a given area problem has an important property that is not due to chance: its capacity to represent the area of a curve that had been drawn before any of the points were chosen. Indeed, since the device was built by purposive beings, there can be no objection to saying that the whole assemblage of points has the purpose of representing the area of that curve—despite the fact that the coordinates of each individual point have no purpose whatsoever. (van Inwagen 2003, 353)

He goes on to state that the coordinates of each point can be considered derivatively purposive.

It is also true that the fact that each point has coordinates that are due to chance is not due to chance and has a purpose: its purpose is the elimination of bias, to insure that the probability of a given point’s falling inside the curve depends on the proportion of the screen enclosed by the curve and on nothing else. (van Inwagen 2003, 353–54)

Let us clarify his point by giving very similar example: consider the method of approximating the value of π using a random process. Consider a circle inscribed within a square. The ratio of the circle’s area to the square’s area is proportional to π (Figure 2).

Figure 2

Approximating π with random numbers. The ratio of the number of points falling within the circle to the number of those falling within the square is approximately $\frac{\pi r^2}{{(2r)}^2}\hspace{0.17em}=\hspace{0.17em}\frac{\pi }{4}$ , where r is the radius of the circle.

Now, consider two processes of generating random numbers, one for the x-component and the other for the y-component, constrained within the square. These processes result in selecting random points in the square. The ratio of the number of points falling within the circle to the number of those falling within the square approximates the value of π. While the coordinate of each point in this scenario is due to chance, as van Inwagen points out, the entire scenario has a purpose—to approximate the value of π—and is designed to achieve this goal. One can justifiably attribute this model to van Inwagen. Therefore, we call it the van Inwagen model.

While one might think the van Inwagen model demonstrates the compatibility of chance and being guided, it does not. This model incorporates an additional element: a geometric component that acts as a constraint on the selection of points. It enjoys an additional technique beyond merely collecting chance events. In this model, we do not simply collect chance events and accumulate them. Instead, we first embed them within a geometric space and then collect them. Therefore, the van Inwagen model demonstrates that being guided is not compatible with mere chance but with geometrically constrained chance. This suggests that chance and being guided can be compatible if some form of constraint is present.

One might object that evolutionary theory is free of such constraints, rendering models with extra constraints inappropriate within its framework. Van Inwagen (2003, 354) himself seems informed of this objection:

How might an advocate of the thesis that Darwinism is incompatible with design respond to these points? One way might be to argue that the features of the biosphere are in a very important respect unlike the features of an assemblage of points produced by our area-measuring device. Each time we draw a curve on the screen of the area-measurer and turn the thing on, it is for all practical purposes determined, foreordained, that the assemblage of points it produces will have the property of representing the area enclosed by the curve. But, it might be argued, the properties of the biosphere are not like that.

He replies:

But the reasoning does show that if someone wants to construct an argument for the conclusion that Darwinism is in any sense incompatible with the thesis that some features of the biosphere are not products of chance, he will have to employ some premise in addition to “Darwinism implies that all events of evolutionary significance are due to chance.” (van Inwagen 2003, 354)

Indeed, van Inwagen believes that this claim that “the evolutionary theory is free of such constraints” goes beyond what evolutionary theory actually says and that the onus is on the opponent to demonstrate otherwise. He believes that the burden of proof lies with those who claim that evolutionary theory explicitly excludes such geometric constraints. Whether we agree with him or not, his model does not demonstrate that mere chance is compatible with being guided.

The Polkinghorne Model

Although Polkinghorne has not addressed the problem of the compatibility of chance and being guided, one can extract a compatibilist model from his works. Polkinghorne (2000b) intends to provide a metaphysics for divine action based on:²⁴

• rejecting reductionism

• applying top-down causation

• the existence of ontological gaps

• the leading role of active information for physical systems

• embracing holism

• adopting dual-aspect monism

• objective becoming.

According to the Polkinghorne model, the world consists of parts knowing the interactions between them is not enough to know the world. If we examine the properties of, say, a glass of water by considering them the result of collisions between the water molecules themselves and the water molecules and those of the glass, i.e., as the bottom-up causal relations between its parts, then we enjoy a reductionist point of view. However, to understand the world, the glass of water should be considered as a whole. Reductionism fails.²⁵ If so, the space opens for other kinds of causality, including top-down causation.

Top-down causation occurs when higher levels of a hierarchical structure exert causal influence over lower levels. This can be done, roughly speaking, by setting a context within which the dynamics of the lower levels have been constrained.²⁶ An illustration of top-down causation can be seen in the three-dimensional folding of an RNA or protein molecule. The overall limitations set by the network of interactions among its constituent parts lead the molecule to adopt a particular shape (See Auletta et al. 2008). So, many believe that top-down causation enables genuine emergence, with higher levels having genuine causal powers, and that this should be considered as the basis for true complexity, life, and human mind and agency.²⁷

However, Polkinghorne (2000b, 151) points out that applying top-down causality when there is no guarantee that explanations based solely on bottom-up causality are inadequate does not truly provide an explanation because the appropriate application of top-down causality requires the use of something more “non-local,” “something much more open and dynamic than simply the generation of long-range order or the propagation of boundary effects . . . [because] they are often fully explicable in terms of a bottom-up approach, generating long-range correlations between localized constituents.”²⁸ He points out that there must be room for top-down causality to operate, i.e., a kind of indeterminacy, arbitrariness, or inherent degrees of freedom, which he calls ontological gaps.²⁹

The existence of ontological gaps is a sign of a different kind of causal influence, not through energy exchange but via active information (more on this soon) transfer that can affect the whole system’s behavior. So, the mechanism of this different kind of causation, namely, top-down causation, is not local at all. Rather, its effect shows itself only in a holistic picture of the world, that is, examining the world as an integrated whole (see Polkinghorne 2000b, 2009). Polkinghorne considers the transfer of active information a sign of the existence of a complementarity of the mental and material aspects of the same reality. According to his dual-aspect monism, reality has both a material and a mental aspect, just as we humans have. And, the divine agency in the world is through top-down causation, as per the human agency in our bodies (see Polkinghorne 1996, 2000a, 2000b, 2006, 2009).

Polkinghorne takes the similarity of the relationship between the mind and the body and the relationship between God and the world very seriously and thinks it clarifies why God is not just an invisible cause among and equal to other causes. Other causes are effective through the exchange of energy, but divine action is effective through the transmission of active information (see Polkinghorne 1996).

Chance is the paradigmatic instance of Polkinghorne’s ontological gaps because, as defined in the second section, chance is a kind of indeterminacy, arbitrariness, or some degrees of freedom inherent in systems. Also, since ontological gaps are direct consequences of top-down causation, the existence of chance is the outcome that directly stems from top-down causation. As mentioned earlier, Polkinghorne allows the application of top-down causality when there is a guarantee that explanations based solely on bottom-up causality are inadequate. Now, it turns out the existence of chance in the world provides such a guarantee. It also indicates some kind of openness in the future. Polkinghorne believes that a world that includes top-down causality is the world of objective becoming. In such a world, the future is open and cannot be determined by the past (see Polkinghorne 1996, 2000b).

According to Polkinghorne’s model, chance exists in the world. At the same time, divine agency is also involved and guides the world, including the biosphere and the process of evolution. The connecting link between chance and being guided is top-down causality: the mechanism of divine agency in the world is top-down causality, and the appearance of chance is the direct consequence of top-down causation. So, his model employs a strong meaning of “being guided” that includes operation via top-down causation, intervention, and even (fine-) tuning. This presents a significant advantage over the Bartholomew-Bradley model because anyone contemplating the concept of “being guided” may anticipate its realization through a causal intervention. Also, the Polkinghorne model has an advantage over the van Inwagen model in that it does not include any constraint on chance events—at least not within any specific region or area.

Although the Polkinghorne model seeks to integrate chance and guidance, its reliance on concepts such as active information, dual-aspect monism, and objective becoming presents a distinct set of problems that must be addressed.³⁰ He points out that while using the notion of active information is promising and aligned with our everyday experience of human agency, active information is a conjectural and heuristic notion. No one knows what exactly active information is. In contrast to the passive information familiar in physics and computer sciences, active information is not a clarified notion.³¹ Polkinghorne borrows this notion from the theory of Bohmian mechanics, in which active information affects the behavior of quantum entities in a way that it is as if that entity is somehow aware of its surroundings.³² However, this notion is left unexplained even in Bohmian mechanics. Applying active information might be a good conjecture but nothing more.

The same can be said about dual-aspect monism. What exactly is meant by the mental aspect of the world? Polkinghorne appeals to an analogy with Bohr’s complementarity principle, which proposes that quantum entities like electrons have several inconsistent though complementary properties, like being both a particle and a wave at the same time.³³ As uncertainties of quantum systems are hints for a complementarity of two incompatible aspects of those systems, the uncertainties in the world are similarly hints for the existence of a complementarity of two aspects of the world: mental and material. However, this analogy, though craftily designed, is too weak to be the basis of a solid argument. In fact, in answering the aforementioned question, Polkinghorne has nothing to do except resort to active information. And, as said, active information remains a poorly defined notion. So, dual-aspect monism is as clarified a notion as active information; if we do not know exactly what active information is, then we will not know what dual-aspect monism means.³⁴

A more serious objection to Polkinghorne’s approach, we believe, is due to the assumption that top-down causation is the mechanism of divine agency. There are fully naturalistic interpretations of top-down causation. For example, Sara Imari Walker (2014) explains the nature and origin of life using the concept of top-down causality in a completely naturalistic sense.³⁵ She believes that information processing distinguishes between non-living and living beings. The biosphere has a hierarchical structure in which the highest amount of information processing occurs at its upper levels. This information flows to lower levels; thereby, higher levels causally affect lower levels. This proposal aims to explain the emergence of life in a way wherein it is not necessary to consider it as the result of the action of an intelligent agent. It is the nature itself, not a supernatural entity, that is located at the top when applying top-down casaulity.³⁶

What is the best interpretation of top-down causation, naturalistic or supernaturalistic? The answer is controversial and goes beyond the scope of this article. What is important here is that Polkinghorne’s model presupposes a rejection of naturalism. This is because one can accept holism, the existence of top-down causality, ontological gaps and chance, active information, and objective becoming without necessarily concluding that top-down causation is a mechanism of divine action. So, his model successfully demonstrates the compatibility of chance and being guided only if the naturalistic interpretation of top-down causation is false. If naturalism accurately explains the world, then Polkinghorne’s model becomes less compelling. The degree to which naturalism provides a satisfactory explanation of the universe, therefore, directly impacts the persuasiveness of Polkinghorne’s model as a potential underlying metaphysics of nature.