# Jason Rosenhouse and “Mathematical Proof”

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.

A common rhetorical ploy is to overstate an opponent’s position so much that it becomes untenable and even ridiculous. Jason Rosenhouse deploys this tactic repeatedly throughout his book. Design theorists, for instance, argue that there’s good evidence to think that the bacterial flagellum is designed, and they see mathematics as relevant to making such an evidential case. Yet with reference to the flagellum, Rosenhouse writes, “Anti-evolutionists make bold, sweeping claims that some complex system [here, the flagellum] could not have arisen through evolution. They tell the world they have conclusive mathematical proof of this.” (p. 152) I am among those who have made a mathematical argument for the design of the flagellum. And so, Rosenhouse levels that charge specifically against me: “Dembski claims his methods allow him to prove mathematically that evolution has been refuted …” (p. 136)

Rosenhouse, as a mathematician, must at some level realize that he’s prevaricating. It’s one thing to use mathematics in an argument. It’s quite another to say that one is offering a mathematical proof. The latter is much, much stronger than the former, and Rosenhouse knows the difference. I’ve never said that I’m offering a mathematical proof that systems like the flagellum are designed. Mathematical proofs leave no room for fallibility or error. Intelligent design arguments use mathematics, but like all empirical arguments they fall short of the deductive certainty of mathematical proof. I can prove mathematically that 6 is a composite number by pointing to 2 and 3 as factors. I can prove mathematically that 7 is a prime number by running through all the numbers greater than 1 and less than 7, showing that none of them divide it. But no mathematical proof that the flagellum is designed exists, and no design theorist that I know has ever suggested otherwise.

## Rosenhouse’s Agenda

So, how did Rosenhouse arrive at the conclusion that I’m offering a mathematical proof of the flagellum’s design? I suspect the problem is Rosenhouse’s agenda, which is to discredit my work on intelligent design irrespective of its merit. Rosenhouse has no incentive to read my work carefully or to portray it accurately. For instance, he seizes on a probabilistic argument that I make for the flagellum’s design in my 2002 book No Free Lunch, characterizing it as a mathematical proof, and a failed one at that. But he has no possible justification for calling what I do there a mathematical proof. Note how I wrap up that argument — the very language used is as far from a mathematical proof as one can find (and I’ve proved my share of mathematical theorems, so I know):

Although it may seem as though I have cooked these numbers, in fact I have tried to be conservative with all my estimates. To be sure, there is plenty of biological work here to be done. The big challenge is to firm up these numbers and make sure they do not cheat in anybody’s favor. Getting solid, well-confirmed estimates for perturbation tolerance and perturbation identity factors [used to estimate probabilities gauging evolvability] will require careful scientific investigation. Such estimates, however, are not intractable. Perturbation tolerance factors can be assessed empirically by random substitution experiments where one, two, or a few substitutions are made.

No Free Lunch, pp. 301–302

Obviously, I’ve used mathematics here to make an argument. But equally obviously, I’m not claiming to have provided a mathematical proof. In the section where this quote appears, I’m laying out various mathematical and probabilistic techniques that can be used to make an evidential case for the flagellum’s design. It’s not a mathematical proof but an evidential argument, and not even a full-fledged evidential argument so much as a template for such an argument. In other words, I’m laying out what such an argument would look like if one filled in the biological and probabilistic details.

## All or Nothing

As such, the argument falls short of deductive certainty. Mathematical proof is all or nothing. Evidential support comes in degrees. The point of evidential arguments is to increase the degree of support for a claim, in this case for the claim that the flagellum is intelligently designed. A dispassionate reader would regard my conclusion here as measured and modest. Rosenhouse’s refutation, by contrast, is to set up a strawman, so overstating the argument that it can’t have any merit.

The reference to perturbation tolerance and perturbation identity factors here refers to the types of neighborhoods that are relevant to evolutionary pathways. Such neighborhoods and pathways were the subject of the two previous posts in this review series. These perturbation factors are probabilistic tools for investigating the evolvability of systems like the flagellum. They presuppose some technical sophistication, but their point is to try honestly to come to terms with the probabilities that are actually involved with real biological systems.

At this point, Rosenhouse might feign shock, suggesting that I give the impression of presenting a bulletproof argument for the design of the flagellum, but that I’m now backpedaling, only to admit that the probabilistic evidence for the design of the flagellum is tentative. But here’s what’s actually happening. Mike Behe, in defining irreducible complexity, has identified a class of biological systems (those that are irreducibly complex) that resist Darwinian explanations and that implicate design. At the same time, there’s also this method for inferring design developed by Dembski. What happens if that method is applied to irreducibly complex systems? Can it infer design for such systems? That’s the question I’m trying to answer, and specifically for the flagellum.

## Begging the Question?

Since the design inference, as a method, infers design by identifying what’s called specified complexity (more on this is coming up), Rosenhouse claims that my argument begs the question. Thus, I’m supposed to be presupposing that irreducible complexity makes it impossible for a system to evolve by Darwinian means. And from there I’m supposed to conclude that it must be highly improbable that it could evolve by Darwinian means (if it’s impossible, then it’s improbable). But that’s not what I’m doing. Instead, I’m using irreducible complexity as a signpost of where to look for biological improbability. Specifically, I’m using particular features of an irreducibly complex system like the bacterial flagellum to estimate probabilities related to its evolvability. I conclude, in the case of the flagellum, that those probabilities seem low and warrant a design inference.

Now I might be wrong (that’s why I say the numbers need to be firmed up and we need to make sure no one is cheating). To this day, I’m not totally happy with the actual numbers in the probability calculation for the bacterial flagellum as presented in my book No Free Lunch. But that’s no reason for Rosenhouse and his fellow Darwinists to celebrate. The fact is that they have no probability estimates at all for the evolution of these systems. Worse yet, because they are so convinced that these systems evolved by Darwinian means, they know in advance, simply from their armchairs, that the probabilities must be high. The point of that section in No Free Lunch was less to do a definitive calculation for the flagellum as to lay out the techniques for calculating probabilities in such cases (such as the perturbation probabilities).

In his book, Rosenhouse claims that I have “only once tried to apply [my] method to an actual biological system” (p. 137), that being to the flagellum in No Free Lunch. And, obviously, he thinks I failed in that regard. But as it is, I have applied the method elsewhere, and with more convincing numbers. See, for instance, my analysis of Doug Axe’s investigation into the evolvability of enzyme folds in my 2008 book The Design of Life (co-authored with Jonathan Wells; see chapter seven). My design inferential method yields much firmer conclusions there than for the flagellum for two reasons: (1) the numbers come from the biology as calculated by biologists (in this case, the biologist is Axe), and (2) the systems in question (small enzymatic proteins with 150 or so amino acids) are much easier to analyze than big molecular machines like the flagellum, which have tens of thousands of protein subunits.

## Hiding Behind Complexities

Darwinists have always hidden behind the complexities of biological systems. Instead of coming to terms with the complexities, they turn the tables and say: “Prove us wrong and show that these systems didn’t evolve by Darwinian means.” As always, they assume no burden of proof. Given the slipperiness of the Darwinian mechanism, in which all interesting evolution happens by co-option and coevolution, where structures and functions must both change in concert and crucial evolutionary intermediates never quite get explicitly identified, Darwinists have essentially insulated their theory from challenge. So the trick for design theorists looking to apply the design inferential method to actual biological systems is to find a Goldilocks zone in which a system is complex enough to yield design if the probabilities can be calculated and yet simple enough for the probabilities actually to be calculated. Doug Axe’s work is, in my view, the best in this respect. We’ll return to it since Axe also comes in for criticism from Rosenhouse.

Next, “Jason Rosenhouse and Specified Complexity.”

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