(Cross Posted at PowerUp.)
[Okay folks, this is the neat thing I mentioned from a while back. The blogger with whom I am collaborating on it is Blake Stacey. The basic idea is to provide a series of posts debunking specific misuses of science among cranks, in a way that is analogous to the Hoofnagle's Denialist Deck of Cards. For our game, we go with a Clue Format, and we call this series Cranks Cluedo. The Dossier has three parameters: "Who?", describing the perpetrating crank; "What Fallacy?" describing the rhetorical device or logical fallacy involved; and finally, "What Field?", describing the field of science the perpetrating crank abuses. Blake will be doing quantum woo for his first post in the series, I focus on creationists in abuses of probability and related concepts.]
DOSSIER
Who?: Creationists.
What Fallacy?: The Argument from Augmented Ignorance.
What Field?: Probability and Optimization Theory.
Ignorance, though much maligned, is a natural aspect of the human condition. We are all ignorant of more subjects than we can ever hope to be knowledgeable in. As the saying goes, “ask me how to fix my car, and I'm completely in the dark.” But it is equally true that ignorance is by no means a reason to declare something a priori impossible. For instance, if my car mechanic informs me that my car is not starting because of a loose spark plug, I cannot dismiss this simply because I do not understand how such a small part can disrupt the functioning of such a complex device. But this utterly fallacious argument, or at least something similar to it, underlies the claims of many varieties of creationist when they assert that evolution involves prohibitively improbable events.
Creationists use a special case of the argument from ignorance. Here, I informally denote it as the argument from augmented ignorance. Whereas the garden variety appeal to ignorance exploits genuine (which is not to say necessarily excusable) ignorance in a given instance, appeals to augmented ignorance use a myriad of clever rhetorical ploys to amplify ignorance beyond proportion. When creationists appeal to the “low probability” of evolutionary outcomes, they typically use unnecessarily turgid mathematical jargon and notation to gloss over important details of evolution. Nowhere is this better illustrated than in the classical creationist appeal to probability, conveniently provided by the TalkOrigins Index to Creationist Claims:
The proteins necessary for life are very complex. The odds of even one simple protein molecule forming by chance are 1 in 10113, and thousands of different proteins are needed to form life.
Setting aside the usual conflation of evolution and abiogenesis, even this concise assertion manages to replicate every fallacy in the creationist probability playbook. It is nothing but a regurgitation of numbers without any consideration of whether the mathematical model that contextualizes them provides accurate predictions in the physical world. Unfortunately, it doesn't. The two basic false assumptions that render it moot are as follows:
The Assumption of Uniformly Distributed Outcomes: Or less formally, assuming that all outcomes in the sample space of a random experiment are equally likely. The process of assembling proteins above is assumed to be a completely unguided and effectively random process, thus ignoring its underlying physics and chemistry. An analogous but more basic scenario in high school level chemistry would be to “randomly” pop balloons of hydrogen and oxygen and calculate the odds of them “randomly” assembling the same molecule (H2O). Such an assumption is patently absurd, as the probability density is clearly concentrated in a specific outcome.
The Assumption of Strictly Sequential, Unweighted Trials: The assumption that the process of creating the proteins is a one-dimensional sequence of unweighted trials. In a realistic biological scenario, concurrency would have to play a significant role, as would contingency and building on past successes.
Expanding on the invocation of these fallacies in more refined instances of the argument from augmented ignorance helps to illuminate its problematic nature. In one form or another, almost all creationist probability arguments replicate them.
Uniformity is a Force That Gives Us Meaning
Probability theory uses methods analogous to measuring the dispersal and concentration of physical objects. This is enabled by the important concept of a “random variable”, which is a function X that maps each outcome in a random experiment S to the real numbers. Two broad classes of random variables are discrete random variables, where X is conceived of as a stepwise function, and continuous random variables, where X is conceived of as a continuous function able to assume any real numbered value. In the continuous case (chosen here because of its intuitive value) a uniform probability distribution in an arbitrary interval [a, b] assumes a value 1/x for all a \<= x \<= b (and a value of 0 for any argument outside the interval). A more substantial statement to this effect would be that a probability distribution function of 1/b-a represents a uniform distribution if it yields a derivative of unity across the entire interval. The latter bit is known as the probability density function. Intuitively, it should be obvious that a continuous function assuming a constant value of 1 represents no mass concentrations of probability in favor of any outcome, as illustrated by the second graph below.
(Images Courtesy of Wikimedia Commons.)
Uniformity is a useful assumption in many scenarios encountered in probability theory, but as a concept it often fails to accurately map the real world. Usually it is tantamount to assuming pure randomness, while it should be obvious that many measurable processes can be constrained to a large degree. One can always describe an arbitrary non-uniform distribution function, and several recur often enough to be given names of their own (e.g., binomial distributions, Poisson distributions, etc.). In an example considered above, the outcomes of the experiment wherein one pops balloons of hydrogen and oxygen are certainly not uniformly distributed. One could in the abstract conceive of individual atoms as discrete objects whose combinatorial outcomes are uniformly distributed across all possible combinations of n atoms. This would be absurd, for obvious reasons. A mathematical model which has no predictive value in its specialized scenario fails, and ignoring known principles of physics and chemistry is guaranteed to engender such a failure.
However, whereas such a thing would be meaningless to scientists interested in accurately describing reality, the crank often uses it as a tactic in declaring that the scenarios postulated by mainstream science are impossible. This is usually a perfect replication of the argument from augmented ignorance, because it uses unfounded assumptions to amplify doubts and give a semi-convincing aura of legitimacy to their own notions. Consider this bit of pablum from one William Dembski:
Most searches that come up in scientific investigation occur over spaces that are far too large to be searched exhaustively. Take the search for a very modest protein, one that is, say, 100 amino acids in length (most proteins are at least 250 to 300 amino acids in length). The space of all possible protein sequences that are 100 amino acids in length has size 20100, or approximately 1.27×10130. Exhaustively searching a space this size to find a target this small is utterly beyond not only present computational capacities but also the computational capacities of the universe as we know it. Seth Lloyd (2002), for instance, has argued that 10120 is the maximal number of bit operations that the known, observable universe could have performed throughout its entire multi-billion year history.
Dembski implicitly invokes uniform distribution in the above paragraph, but if you follow the link you will see that he explicitly invokes the concept for much of the paper, in addition to rambling on about searches and optimization problems (which we'll get to soon). But even setting aside the assumptions of computational searches, one doesn't have to assume that the space of proteins 100 amino acids in length is a uniformly distributed space of combinations. Biochemistry, like all chemistry, is not random but constrained by the physics underlying its operation. In addition, there is no reason to assume an exhaustive traversal of such a space is necessary at all. In evolution, what you want is a functional protein, which hardly requires an exhaustive evaluation of all possible forms that said protein can take.
The latter flaw alludes to our discussion of flaws in the equivalence drawn by Dembski between real world evolutionary biology and the various algorithms inspired by that process in solving optimization problems. His reasoning is fundamentally flawed on several levels.
Searching For an Accurate Analogy
Since the 2002 publication of No Free Lunch, Dembski has been harping on a particular invocation of optimization theoretic concepts. His claim is essentially that the results known as the No Free Lunch (NFL) theorems, originally derived by David Wolpert and William Macready, show that evolutionary processes do not perform better than randomness. The actual result of the NFL theorems is that given the task of optimizing (i.e., locating minima or maxima in) an arbitrary function f(x), the space of potential domains where any particular technique outperforms another is less than or equal to the potential domains where it does not. Thus, averaged over all possible domains, no optimization technique will ever be guaranteed to perform better than a random walk.
While the technical statement above may seem haughty, it can actually be understood as relatively mundane statement of mathematical reality. As Olle Haggastrom has pointed out in his own review of Dembski, an intuitive way of understanding the NFL theorems is through the following process: shuffle a deck of cards and spread them face down on a table. The NFL theorems are essentially another way of saying that, when completely ignorant of the structure or layout of the cards on the table, no sequential, deterministic process is guaranteed to find the ace of spades in fewer steps than randomly checking the cards manually.
Even assuming that evolution can be reasonably modeled as a search as considered by the NFL theorems, the analogy ignores one of the principal reasons that, despite the NFL results being well established, research in optimization theory is still incredibly lucrative. In most practical optimization problems, there is some degree of known structure than can be exploited to produce a useful optimization method. In the case of evolution, almost all scenarios involve existing templates and highly constrained criteria for selection (at least, more highly than randomness). For instance, in an environment where food is scarce, organisms that rely on the ubiquity of energy sources are almost automatically winnowed out. The environment is simply unfavorable to populations containing such organisms, weighting particular outcomes in favor of smaller, more resilient organisms.
It is also worth noting that the analogy between evolutionary algorithms and real world evolution is in a sense itself faulty. While evolutionary algorithms are usually specialized for optimization scenarios where a global optimum must be approximated, the results of real world evolution are far more analogous to what would be expected through traditional local search and hill-climbing methods. Biological and ecological scenarios only require a functional solutions, which in optimization theoretic terms can at best be described locally optimal. There is no necessity for a global optimum in biological evolution. While this doesn't in itself negate the applicability of NFL (this will become obvious in the next section), it does show that the two scenarios are not always as conceptually contiguous as it would seem.
However, the first flaw is far more instructive when stated in a more mathematical fashion. When uprooted it becomes obvious just how absurd Dembski's claims regarding NFL and real world evolution are.
Uniformity Revisited: Pure Noise
Olle Haggastrom provides us with this keen insight in a later reply to Dembski (with the latter being joined at the time by Robert Marks): the NFL is only applicable when the “Pure Noise” condition is met. “Pure Noise” stipulates that for each function f: V --\> S, each element v of V is chosen randomly according to uniform distribution on V, and each f(v) is chosen randomly according to uniform distribution on S. In search specific lingo, this is another way of saying that all permutations of an evaluation function are equally likely. The core assumption in both statements is that of pure randomness, under which it would be correct to say that no algorithm outperforms another.
The retort to Dembski is thus obvious, and one that often has been made to less sophisticated creationists: evolution is not randomness, and thus NFL is not applicable (at least no directly). But just to take the argument further, when taking Dembski's logic to its proper conclusion we would have to toss away more than simply evolution. The “Pure Noise” model is applicable to all algorithms designed to operate over search spaces, and evolution isn't the only natural process to inspire an algorithm of this sort. Two more that come to mind are simulated annealing, which simulates a thermodynamic system (more specifically, cooling and crystallization) and ACO, which simulates the behavior of ant-colonies.
In the most general form, simulated annealing used an initially high parameter T which gradually decreases upon successive evaluations of better solutions to an optimization problem. T here is usually used to represent temperature, and the goal of simulated annealing is to reach an abstract analogue of the “ground state” in a thermodynamic system (i.e., the global optimum). Its counterpart in local search problems is often called “rapid quenching”, an analogy to the fact that rapid cooling of crystals usually preserves irregularities in the structure (i.e., you only get a local optimum). Ant-Colony Optimization (ACO) uses simulated populations of ants which leave simulated pheromone trails to narrow down the space of feasible solutions (analogous to how real ant-colonies narrow down supply routes). If “Pure Noise” were satisfied in either case, you lose all structure. Temperature never decreases and pheromone trails never accumulate. Everything would “melt” in your simulated annealing algorithm, and your abstract ants would all lose their way and die in the absence of mutual support from the nest. Or, if we're considering all algorithms, it would seem that the universe has no structure whatsoever.
Behold, the power of unfounded assumptions.
|