Misconceptions about Randomness
Note: Originally posted April 7, 2010.
Christians often equate randomness with an atheistic worldview, but randomness is an essential feature of many God-ordained biological processes, from the union of egg and sperm during reproduction to the generation of antibodies by the immune system. In fact, based on its prevalence in the natural world, one might conclude that randomness is one of God’s favorite mechanisms for creating life!
Here I want to clarify a few misconceptions about randomness before moving on, in future posts, to describing other biological processes that make use of it.
Randomness is like “God of the Gaps”. With time, advancements in science will allow us to make accurate predictions in previously “random” systems.
Isaac Newton’s famous three laws of motion, described in his 1687 classic Principia Mathematica, have empowered physics students for centuries. Using these and Newton’s universal law of gravitation, you can predict the trajectory of everything from pool balls to planets. By the early nineteenth century, the idea of a “clockwork universe” was firmly established, and scientists believed that with time, science would be unlimited in its predictive power.
Although it could be true that we live in a “clockwork universe,”1 two developments in the twentieth century shattered our hopes of having a fully predictable universe. The first was quantum mechanics, which describes how things work at an extremely small scale. One of the major discoveries in quantum mechanics was Werner Heisenberg’s “uncertainty principle,” which holds that the more certain one is about the location of a particle, the less certain in principle one can be about its momentum, and vice versa. At the quantum level, then, our predictive powers are ultimately limited.
Another discovery that destroyed all hope for a fully predictable universe was chaos. Mathematically chaotic systems are those which are extremely sensitive to changes in their initial conditions. Even fully deterministic systems can exhibit chaotic behavior and act in unpredictable ways. Consider a famous function, the logistic map2:
I know equations make people nervous, but stay with me! This one does some fascinating things. Here’s how it works: start with some initial value for x at time t = 0, and plug that in for x(t). Let’s start with = 0.2. R is just some constant value; let it be 2. Now we use the function to calculate the outcome in the next time step, t = 1:
We can use this value as input for the next step, and repeat this process over and over to find the output at each time point.
What happens? The answer is plotted in the figure below on the left. If we follow along the x-axis, which represents time, we see that the value of x goes toward 0.5 and stays there forever.
What if we start with the same , but increase R to 3.1? Following the same process as before, we get a very different result! The middle graph shows that the outcome oscillates between two values over time.
If you make R = 4, the function does something very strange. In the right-most plot, the function still fluctuates up and down, but it begins to look irregular. And if we change the initial condition, , just slightly, from .20 (blue solid line) to 0.2000000001 (red dotted line), we see they are virtually the same until somewhere around t = 14. After that point, they exhibit completely different behavior.
Several observations can be made here. First, the same equation can produce three different classes of behavior, simply by changing R and . These classes are called fixed-point (left), periodic (middle), and chaotic (right). Below the values of R that lead to chaos, the system is not sensitive to the initial value of . Over time, the system will either become a flat line or oscillate.
When R is greater than approximately 3.569946, however, the system becomes chaotic, and the outcome is extremely sensitive to changes in the initial value of . What this means is you would have to know the value of to infinite precision to predict its long term behavior. Since this is impossible in any kind of real-world application, the detailed behavior of a chaotic system is impossible to predict.
If this is true even for a simple, completely deterministic equation, how much more difficult is it to predict the behavior of a more complicated chaotic system, like a hurricane! Even the poor weathermen here in San Diego get it wrong sometimes, and the weather here doesn’t change very much.
So, between the uncertainty principle in quantum mechanics, and the sensitivity of chaotic systems, we now know that we are fundamentally limited in our predictive power––not just temporarily. Whether the systems we study are truly indeterministic is another (interesting) question, which of course has implications for divine action.
Randomness means anything can happen, and all possibilities are equally likely.
People often think randomness means the outcome is completely open-ended, but you can’t roll a 7 on a 6-sided dice, nor draw a red marble from a bag of blue ones. Even random processes function according to rules. (The logistic map in the last section is another good example.)
Sometimes, the word random is used to mean unbiased. If you want to know who will win a political election, you make sure to poll a random sample of people, not just those hanging around a Tea Party rally. But the word random doesn’t have to mean that all possibilities are equally likely. When maternal and paternal chromosomes get together during conception, they exchange long sequences of DNA in a process called recombination. We now know that recombination happens more often in some places of the genome than others, but the specific sites where it will occur in a given embryo are impossible to predict. So recombination is random in the sense that it is unpredictable, but not in the sense that all outcomes are equally likely.
Randomness always leads to disorder.
On the contrary, randomness often leads to exquisitely ordered and complex outcomes. In my next post, we'll watch a simulation of viral self-assembly from individual proteins bouncing around in a jar. You could repeat the simulation a thousand times and always get the same result, even though the particular assembly pathway would look different each time. That is, if the starting materials are present and the conditions (temperature, pH, etc) are right, you will always get a beautiful, highly symmetric virus. Random motion is the mechanism used to search “solution space” for a favorable outcome.
Fractals provide another great example of patterns emerging from randomness. Fractals are chaotic patterns with the same basic property: no matter how much you “zoom in,” the overall structure is maintained. Clouds, trees, crystals, and snow flakes are naturally-occurring fractals.
You can construct a fractal like the Sierpinski triangle shown at left by rolling a die and following simple rules.3 If 100 people in a room independently rolled a die 100 times and followed the rules, they would all have different sequences of rolls, but all would end up with the same pattern!
Thus, for many random processes, the fine details may be unpredictable along the way, but the macro-level outcome is foreseeable.
“Randomness,” when taken to mean unconquerable unpredictability, is inherent in many processes created by God, from hurricanes to viral assembly to genetic recombination to antibody production. Randomness means that the details of the future are unpredictable, and will stay that way regardless of scientific progress. That said, randomness is constrained by rules and often leads to complex patterns and macro-level order. More misconceptions about randomness no doubt lurk in all our minds, leading to suspicion when we hear phrases like, “evolution is random.” But hopefully, this post can help to clarify some of the confusion.
1. Philosophers of physics still debate whether there is some underlying deterministic structure to the universe, or whether events at the quantum level are indeterministic. See http://plato.stanford.edu/entries/determinism-causal/#QuaMec. In either case, we are fundamentally limited in our ability to make predictions about the outcomes of quantum events.
2. The logistic map is one of the best-studied equations in dynamical systems theory. The particular values used in the figure were taken from Melanie Mitchell’s excellent book, Complexity: A Guided Tour, and were created using MATLAB.
3. Thanks to Isaac Yonemoto for pointing this out.