I am reviewing Jason Rosenhouse’s new book, The Failures of Mathematical Anti-Evolutionism (Cambridge University Press), serially. For the full series so far, go here.

Jason Rosenhouse weighs the mathematical merits of intelligent design as a sophist intent on destroying it, not as a serious thinker intent on gaining real insight. Nor does he show any inclination to understand evolution as it really is. It therefore helps to lay out his evolutionary presuppositions and the lengths to which he will go to defend them. That has now been accomplished. With the stage thus set, let’s next turn to the mathematical details of Rosenhouse’s critique.

In the interest of helping his case, Rosenhouse appoints himself as math cop, stipulating the rules by which design proponents may use mathematics against Darwinism and for intelligent design. But there’s a problem: What’s good for the goose isn’t good for the gander. Rosenhouse insists that intelligent design proponents obey his rules, but happily flouts them himself. The device he uses to play math cop is to separate mathematical usage into two tracks: track 1 and track 2. Track 1 refers to math used intuitively, with few if any details or formalism. Track 2 refers to math used with full rigor, filling in all the details and explicitly identifying the underlying formalism. According to Rosenhouse, math that’s taken seriously needs to operate on track 2.

An Artificial Distinction

Rosenhouse’s distinction is artificial because most mathematics happens somewhere between tracks 1 and 2, not totally informal but not obsessively rigorous. The fact is that mathematicians, especially when working in their areas of specialization, can assume a lot of background knowledge in common with their fellow mathematicians. I remember an algebraic topology course I once took and thinking “Where are the proofs?” — the justification of theorems in the course seemed so visual, so evocative, so abbreviated. But this level of rigor (or lack thereof) seemed not to hamper the intellectual vitality of the course. Thus, mathematicians can seem to be merely on track 1 when in fact they are tacitly filling in the details needed to satisfy track 2. 

No matter, to play math cop effectively, Rosenhouse needs a sharp distinction between track 1 and track 2. Specifically, to debunk intelligent design’s use of mathematics, he levels the following accusation: You say you’re on track 2, but really you’re on track 1, and so you haven’t made your case or established anything. Over and over again he makes this accusation against my work as well as that of my colleagues in the ID community. He makes the accusation even when I have operated on track 2 by his standards. And he makes the accusation at other times even when I’ve supplied enough details so that track 2 can be readily achieved. 

Considering Probability

As it is, Rosenhouse doesn’t meet his own exacting standards. With probability, for instance, he insists that track 2 requires fully specifying the underlying probability space as well as the probability distribution over it, and also any pertinent geometry of the space. Frankly, that can be overkill when the spaces and probabilities are given empirically, or when all the interesting probabilistic action is in some corner of the space not requiring full details for the entire space. What’s more, estimates of probabilities are often easy and suffice to make an argument even when exact probabilities may be difficult to calculate. It may, for instance, be enough to see that a probability is less than 1 in 10^100, and thus suitably “small,” without doing any further work to show that it really is the much smaller 1 in 10^243. 

Even so, having set the standard, Rosenhouse should meet it. But he doesn’t. For instance, when he describes a standard statistical mechanical set up of gas molecules in a box, he remarks: “We are far more likely than not to find the molecules evenly distributed.” (p. 234) I would agree, but what exactly is the probability space and probability distribution here? And what level of probability does he mean by “far more likely.” Rosenhouse doesn’t say. His entire treatment of the topic here, even in context, resides on track 1 rather than on track 2 (assuming we’re forced to play his game of choosing tracks). Rosenhouse might reply that he intended the argument to reside on track 1. But given the weight he puts on statistical mechanics in refuting appeals to the Second Law of Thermodynamics, one could argue that he had no business confining himself to track 1. Note that I don’t fault Rosenhouse for the substance of what he’s saying here. I fault him for the double standard.

Or consider his claim that evolution faces no obstacle from the sparsity or improbability of viable biological systems so long as there are both gradualistic pathways that connect these systems and local areas around these systems (neighborhoods) that can readily be explored by evolution to find new steps along the pathways. Rosenhouses illustrates this claim with a two-dimensional diagram showing dots with circular neighborhoods around them, where overlapping neighborhoods suggest an evolutionary path. (p. 128) He even identifies one of the dots as “origin of life” and captions the diagram with “searching protein space.” 

The point of Rosenhouse’s “searching protein space” example is that new proteins can evolve by Darwinian means irrespective of how improbable proteins may be when considered in isolation; instead, the important thing for Darwinian processes to evolve new proteins is that proteins be connected by gradual evolutionary paths. Accordingly, what’s needed is for protein space to contain highly interconnected gradualistic evolutionary pathways (they can be but the merest tendrils) that from the vantage of the large-scale structure of the protein space may be highly improbable. Darwinian processes can then still traverse such pathways.

Sparsity or Improbability

I’m largely in agreement with the mathematical point Rosenhouse is making in this example. Even so, it does seem that the sparsity or improbability of proteins with respect to the large-scale structure of the protein space may contribute to a lack of interconnectivity among proteins, making it difficult for evolutionary pathways to access far-flung proteins. Moreover, in Rosenhouse’s model, it is necessary to get on an evolutionary path in the first place. So brute improbability may become a challenge for getting the evolutionary process started. It’s therefore ironic that he starts his model from the origin of life, for which no widely accepted naturalistic theory exists and for which the absence of causal details is even worse than for Darwinism.

Whatever the merits of Rosenhouse’s argument in proposing this model, it is nonetheless the case that if forced to choose between his two tracks, we’d need to assign what he’s doing here to track 1 rather than track 2. And unlike the statistical mechanics example, this example is central to Rosenhouse’s defense of Darwinism and to his attack on “mathematical anti-evolutionism.” So there’s really no excuse for him to develop this model without the full specificity of track 2.

Next, “Rosenhouse and Discrete Hypercube Evolution.”

Editor’s note: This review is cross-posted with permission of the author from BillDembski.com.