Undeniable packs a powerful punch, but doesn't land a knockout

Undeniable is a beautifully written book, which makes a strong case for Intelligent Design that laypeople will readily grasp. Author Douglas Axe scores some telling points against Darwinism. Unfortunately, the gaps in his argument prevent him from landing a knockout blow.

When I first read Undeniable, I was greatly impressed by its limpid prose, the clarity of its exposition, and the passion with which the author makes his case. Seldom have I seen such an elegantly written book, which people from all walks of life can appreciate. I have no doubt that it will sell well for many years to come, and I have to say that it makes the best case for Intelligent Design at the popular level of any book I've ever seen. The author deliberately keeps his book as non-technical as possible: his stated aim is to free ordinary people from being tied to scientists' apron strings, by giving them the conceptual tools to understand why a naturalistic account of origins makes no sense. Scientific credentials are not required to arrive at such an understanding: all that is needed is common science - the kind of science we all do, as everyday observers of Nature. Undeniable is avowedly anti-elitist: its goal is to give science back to the people.

In his book, molecular biologist Douglas Axe seeks to validate what he calls the Universal Design Intuition: tasks that we would need knowledge to accomplish can be accomplished only by someone who has that knowledge. It was a similar intuition that attracted me to Intelligent Design, nine years ago, when I realized that the complexity of living things was light years ahead of what even our top scientists could create. It seemed obvious to me that Nature could not bring about such feats, and that only a super-Mind was up to the job. But how can you validate an intuition like that, if you're a non-scientist? And how are you supposed to respond when evolutionists cite Orgel's Second Rule: "Evolution is smarter than you are"?

Amazingly, Dr. Axe manages to distil the case for Intelligent Design into a single sentence: functional coherence makes accidental invention fantastically improbable and hence physically impossible. Let me unpack that a little. By "functional coherence," Dr. Axe means: a hierarchical arrangement of parts that are needed to produce a high-level function - each part contributing in a coordinated way to the whole (p. 144). Any functional whole is also a busy whole: it accomplishes a big result, but only by bringing its components together in just the right way (p. 69). By "invention," Axe simply means innovation (p. 222) - in particular, innovation in which the components of the functional whole need to be re-worked, in order to create a new function (p. 193). The term "fantastically improbable" has a very specific meaning in Axe's book: basically, it refers to any probability that falls below 1 in 10116, which Dr. Axe calculates to be the maximal number of atomic-scale physical events that could have occurred during the 14-billion-year history of the universe (p. 282). Any event with a probability below this threshold of 1 in 10116 is one whose realization can only be expected to occur in a universe which is bigger (or older) than our own, and for that reason, it is said to be physically impossible. While its occurrence in our cosmos cannot be ruled out, it would be a fantastic fluke, and thus tantamount to magic.

Readers with a philosophical background will note the parallels between Axe's definition of functional coherence and William Paley's definition of a contrivance as a system possessing the following three features: "relation to an end, relation of parts to one another, and to a common purpose." (Natural Theology. 12th edition. J. Faulder: London, 1809, Chapter XXIII, p. 413.)

The reason why Dr. Axe believes that the accidental invention of a functionally coherent whole is fantastically improbable is that it requires a great many specific actions to be taken, each with a large number of ways of going wrong, which means that "fully appropriate courses of action are utterly lost within the staggeringly large space of raw possibilities" (p. 159). In a nutshell: when it comes to making a functional whole, so many things can go wrong, and in so many ways, that the chances of hitting upon success via a blind process are astronomically low.

This is especially obvious if we look at biological systems. Even the simplest independently living organisms (bacteria) are full of systems that exhibit functional coherence all the way down to the level of their molecular constituents (p. 191). In chapter 10 of his book, Axe discusses photosystem I, one of the two systems used by plants, algae and cyanobacteria when they photosynthesize. In cyanobacteria, this system contains "twelve protein parts and six smaller parts called cofactors, one of which (chlorophyll a) is used 288 times to build the whole photosystem" (p. 169). The cofactors have to be held in their precise positions by the protein framework. Altogether, photosystem I contains a staggering 417 pieces, each of which has to be in just the right place, in order to carry out the function of gathering the sun's photons and converting their energy into chemical energy (p. 169). And photosystem I is just one of many components that make up the whole photosynthetic system of a cyanobacterium. Here's what it looks like (image courtesy of BioWiki & Protein Data Bank):

Now, at last, we can appreciate the full force of Dr. Axe's remark: "Serious invention requires not just a smidgen of functional coherence but extensive amounts arranged over a hierarchy of levels, and this simply can't happen by accident - for any kind of invention" (p. 202).

Could natural selection account for such innovations? Absolutely not: "Natural selection happens only after cells are arranged in ways that work to keep the organism alive, so selection can hardly be the cause of these remarkable arrangements" (p. 192). Natural selection presupposes the existence of living things, which need to execute a host of functions in order to stay alive and reproduce. But surely, once life has been established, natural selection can take over and produce new biological functions and new species of organisms? Not at all, contends Dr. Axe: "Because each new form of life amounts to a new high-level invention, the origin of the thousandth new life form is no more explicable in Darwinian terms than the origin of the first" (p. 194). The massive re-engineering required to produce a new function is physically impossible, and hence beyond the reach of chance.

Astute readers may object that stepping stones from one complex system to the next would enable unguided natural processes to generate new functionally coherent systems without requiring any astronomically improbable events to occur, but Dr. Axe has a ready answer: "Accidental stepping stones leading to these fantastically improbable destinations would themselves be fantastically improbable" (p. 200).

While living things share the property of functional coherence with man-made inventions such as smartphones and robots, they are distinguished by an additional feature which places them in a category of their own: living things are all-or-nothing wholes: "Every cell in the body both sustains the body and is sustained by the body... The body is alive and thriving when all its parts are working, or it is dead and decaying when they are not" (p. 176). The cells in the body are a bit like the Three Musketeers: all for one, and one for all. Human beings are incapable of creating such all-or-nothing wholes: the parts of a smartphone, for instance, are manufactured separately, and when one part fails, the other parts are unaffected by that failure (p. 178).

Who, then, is the Designer of living things? The science underlying Intelligent Design does not yield an answer to that question, but Dr. Axe, to his credit, does not hide his conviction that the Designer is God, and he makes a strong case for God, on philosophical grounds. Attributing the design of life to aliens is theoretically possible, but it only pushes back the question: where did the aliens come from? What's more, persons, with their capabilities for having concepts, for reasoning, for making free moral choices, and for feeling emotions, are categorically irreducible to mere arrangements of matter, for no arrangement of matter can be equated with any of our abstract concepts: for example, the concept of two, or triangle, or true, or time (p. 240). Finally, we can only make sense of the outside world by seeing it as an expression of God's creative thought (p. 242). Persons are not just as fundamental as matter; they are more fundamental, maintains Axe. And only the Creator of the cosmos, a supremely personal being, could create immaterial persons. Thus reason takes us to God.

I have to say that I like Dr. Axe's God: He is a Being with not only a mind but also a heart - someone who invested in the cosmos "not just intellectually but also emotionally, just as we invest in our creations" (p. 250). God has a funny sense of humor, too: "How on earth do we find ourselves on a planet," asks Dr. Axe, "where the great emotive categories of film, story and stage are so beautifully represented by fish, of all things?" (p. 249). The book has an entire set of color plates showing how fish of various species cover the genres of fantasy, drama, romance, comedy, horror and tragedy. God, avers Dr. Axe, meant us to be moved in this way: He designed these creatures to strike a deep emotional chord within us. God is not just a great Inventor, but a great Creator, and as we are the only creatures on this planet capable of knowing their Creator, it is reasonable to conclude that God wants to be our friend. Persons can only have come from a personal God.

In the final chapter of his book, Dr. Axe sketches his vision for the future of biology: instead of asking how life arose, as scientists of the old school of biology did, the new school seeks to capture and formulate the ideas behind life: "those why questions that have no place on the old road become the intellectual core of the new road." Biology's best days still await it: they will happen when the basic constituents of life "come together under a set of organizing principles by which they suddenly make sense" (p. 273).

Up to this point, I have sung the praises of Douglas Axe's book, Undeniable, because it is a genuinely beautiful work of literature: its rhetoric soars, and its message is inspiring and uplifting, as well as being easy to absorb. Make no mistake: this book will change lives.

Nevertheless - and I have to say this - the book contains numerous mathematical, scientific and philosophical blunders, which a sharp-eyed critic could easily spot. I didn't notice them right away: like many avid readers, I skimmed the book for its most interesting bits, when I first received it. But when I perused the book at leisure, making notes of each chapter's key points, the fallacies in the author's arguments leapt out at me. What bothered me most was that the book misrepresented Darwinism, and exaggerated the strength of the case against it, by appealing to a series of poor analogies. In the end, these errors provoked me to such a degree that they made me want to take up the cudgels on behalf of Darwin - something I have not done for a long, long time.

Let me say at the outset that I do not hold Dr. Axe responsible for most of the errors in his book. (The arguments relating to the alleged impossibility of abiogenesis are another matter, and I will discuss those below.) We all have our blind spots - especially when it comes to arguments whose conclusions we passionately embrace. Dr. Axe is a passionate believer in design. To his credit, however, Dr. Axe sent his manuscript out (in whole or part) to no less than fifteen people (mathematicians, scientists, philosophers and writers), soliciting their comments on his book. I won't name them here, as I have no wish to publicly embarrass them, but they are listed in the author's Acknowledgments (pp. 275-276). It is these people whom I hold responsible for the errors in Dr. Axe's book.

(NOTE: In the interests of journalistic accuracy, I feel obliged to state that: (a) at least one reviewer has since contacted me to let me know that some of the changes he proposed were rejected by the author; and (b) not all of the reviewers were eminent people. I should note that this reviewer mentioned in point (a) was overall very positive about Axe's book and said, "overall I strongly agree with Doug's thesis, and I am proud to be listed as a reviewer who would endorse the book.")

Despite its numerous errors, I do not believe Dr. Axe's book is fatally flawed; however, I do think it needs a drastic rewrite. I look forward to seeing the revised second edition of the book.

I should also like to state that I have been in dialogue with Dr. Axe and other people since writing this review, and that I am happy to consider any comments or criticisms they wish to make.

If Dr. Axe ever decides to revise his book, I sincerely hope he changes the title. Attaching the title Undeniable to a book arguing for Intelligent Design is like walking up to the world's heavyweight boxing champion and saying, "Hit me." It's asking for trouble. To undermine the book's thesis, all a Darwinist needs to do is show that Axe's argument in support of the Universal Design Intuition is not an ironclad one. And with a book that's nearly 300 pages long, it's hardly surprising that there are gaps in the argumentation.

Now, if Dr. Axe had titled his book, The Case for Design, or Is Life an Invention?, he could have made a much stronger case. After all, there's no need to disprove a naturalistic account of origins; all Axe really needs to show, in order to render belief in Intelligent Design reasonable, is that the case for a Creator is stronger than the case for naturalism. One thing Axe does very well in his book is to demonstrate that there are no good naturalistic explanations for the origin of systems with a high degree of functional coherence, such as are found in all living organisms. Another thing he does very well is to describe the mind-boggling complexity found in living things, in language accessible to laypeople. He doesn't really need to argue the case for design when he does this: it makes itself.

If he had stuck to his strengths, Dr. Axe could have written a book that evolutionists would have found very hard to attack. Unfortunately, he aimed too high. Reading and reflecting on Axe's book has convinced me that while the case for Intelligent Design is a very powerful one, it is far from being undeniable.

From this point on, I will be playing devil's advocate in the role of a Darwin defender, even though I am deeply skeptical of the possibility of abiogenesis and very much in sympathy with the thinking behind Axe's Universal Design Intuition.

The justification for this intuition is another matter, however. Axe's argument that functional coherence makes accidental invention fantastically improbable and hence physically impossible, doesn't hold water, for several reasons.

Problems with the Universal Design Intuition

The first thing that needs to be said is that Dr. Axe's initial formulation of the Universal Design Intuition is very sloppy. In chapter 2, it reads as follows: "Tasks that we would need knowledge to accomplish can be accomplished only by someone who has that knowledge" - or in plain English, "whenever we think we would be unable to achieve a particular useful result without first learning how, we judge that result to be unattainable by accident" (p. 20).

The fundamental problem with Axe's original formulation of the Universal Design Intuition is that it posits a false dichotomy of design versus accident, ignoring events that happen by natural necessity. But let us be generous, and assume that by "accident," Axe simply means: any blind processes without a long-term goal, regardless of whether they are accidental or driven by natural necessity.

Making an omelet, wrapping a present, and putting sheets on a bed are everyday examples of the sorts of tasks that Axe has in mind (p. 21). We have to be taught how to do these things, and they never happen by accident. Beds don't make themselves; someone has to make them. Likewise, "neither bricks nor shoes get made unless someone makes them" (p. 22).

All right, but what about the task of making rain? Human beings didn't learn how do accomplish this task until the 20th century, yet Nature does it all the time, without using any knowledge. And what about diamonds? The first reproducible success in manufacturing a synthetic diamond didn't occur until 1953, yet diamonds have been forming naturally in the Earth's upper mantle for at least 3.3 billion years. Clearly, Axe has cast his net too wide: using his criteria, we would have to say that rain and diamonds are products of design, since humans are incapable of making them without first learning how.

Perhaps it might be suggested that the Universal Design Intuition applies only to highly complex arrangements of matter, rather than low-complexity structures such as raindrops and diamonds (which have a fairly regular, face-centered cubic crystal structure). Very well, then: but what about individual snowflakes? While making snow isn't too difficult, there's no doubt that a lot of knowledge would be required if I wanted to make a perfect replica of a particular snowflake. Should I then conclude that each snowflake was designed?

So much for the initial formulation of the intuition. On pages 160, 162 and 189 of his book, Axe propounds his core argument in support of the Universal Design Intuition: "Functional coherence makes accidental invention fantastically improbable and therefore physically impossible." Inventions, we are told, "exhibit an organized, functional coherence that can only come from deliberate, intelligent action" (p. 160). Functional coherence is the reason why "inventions can't happen by accident" (p. 160). This formulation of the Universal Design Intuition is a lot sharper than the initial one, because it shifts the focus to a well-defined property (functional coherence), which means that we need not worry about snowflakes, diamonds or rain, as none of them exhibit this singular trait. In Dembskian terminology, Axe has made a clever move here: he has quietly dispensed with the problematic notion of specified complexity (which could apply even to arrangements of pebbles on Chesil beach), and identified functional specified complex information (FCSI) as the hallmark of intelligent design - although curiously, the word "information" is scarcely ever used in Axe's book (he prefers the term "knowledge"). However, Axe's new formulation of the Universal Design Intuition faces a major problem.

Readers will recall that Axe has defined functional coherence as a hierarchical arrangement of parts contributing in a coordinated way to the production of a high-level function (p. 144). If functional coherence is the trait which warrants a design inference, then that would seem to imply that any degree of functional coherence can only be the product of intelligent design. In certain passages in his book, Axe conveys precisely this impression: for instance, on page 162, summing up his case in Figure 9.5, he writes that "chapter 9 develops the idea of functional coherence fully, connecting it to invention and showing insight to be its only possible cause by showing accidental causes to be fantastically improbable" (italics mine). However, on the left-hand side of the figure, he sets out his argument more cautiously, when explaining why origami cranes don't fold by accident: since it is highly unlikely for accidental causes to do the right thing (i.e. fold the paper the right way) at any given step, it is therefore fantastically unlikely for accidental causes to do the right thing at all steps. That would seem to imply that a certain critical level of depth is required in a functional hierarchy, before we can legitimately infer design. Axe himself admits as much in his book: on page 151, he writes that "blind searches can find simple targets ... where very little functional coherence is needed," while on the following page, he insists that "the amount (italics mine) of functional coherence routinely produced by human insight can't be produced by accident." Which invites the obvious question: how much functional coherence warrants a design inference?

The final formulation of Axe's Universal Design Intuition, on page 254, is the most precise one: "No high-level function is ever accomplished without someone thinking up a special arrangement of things and circumstances for that very purpose and then putting those thoughts into action. The hallmark of all these special arrangements is high-level functional coherence, which we now know comes only by insight - never by coincidence" (italics mine). What Axe is saying, then, is that when a functional hierarchy contains a sufficiently large number of levels, laypeople can instantly infer that it couldn't have originated by chance, and their inference is a mathematically justifiable one.

Fine, but what is this critical number? We are never told. Axe might respond that the number is context-sensitive, but that doesn't help us much. How are we supposed to tell? Another response Axe could make is that the critical number of hierarchical levels is simply whatever number yields a likelihood in 1 in 10116. But how are laypeople supposed to figure that out?

To his credit, Axe makes a noble attempt in his book to justify the inference we all share, that certain objects could only have originated through intelligent design, and he states his case with flair and panache, using illustrations that will appeal to readers. In the end, though, his case for the Universal Design Intuition remains unsatisfyingly vague. I don't blame Axe for this: he is, after all, a scientist, not a philosopher. But there are a couple of trained philosophers among the fifteen people who reviewed his book, and I have to say I am very surprised that they didn't notice these gaps in Axe's argumentation and point them out to him.

I'd like to make a modest suggestion. In my humble opinion, Dr. Axe's Universal Design Intuition would be much more persuasive if it were formulated as follows: "If we find a level of functional coherence in living organisms which surpasses anything which our top scientists can create, then we should conclude that the systems displaying this level of functional coherence were designed, and that the accidental invention of these systems is fantastically improbable and therefore physically impossible." This formulation of Axe's Universal Design Intuition borrows from the STOMPS principle which I blogged about, a few years ago:

So far I have explained why some people are open to Intelligent Design, but what makes them believe it? I think the answer consists in what I will call the S.T.O.M.P.S. principle (short for "Smarter Than Our Most Promising Scientists"). Sometimes when people see the solution to a problem, they remark, "I would never have thought of that in a million years." (Often they're right, too.) Most people's reaction, when they watch a science video showing the elegant design of the inner workings of the cell, is along these lines: "Look at that! Not even our best scientists could design a cell as elegantly and efficiently as that. This must be the work of a super-human intelligence." What they recognize at once is that building a cell is very, very hard - and that no human being could come close to designing one as well as the cells we find in Nature. Next, they reason that if even human intelligence cannot come up with a design as elegant as that, non-foresighted processes (commonly known as random mutation plus natural selection) are even less likely to do so. Hand-waving appeals to "deep time" and "billions of years" leave them cold - and rightly so. What these people correctly intuit is that elegance of design isn't the kind of thing that time alone can generate, even when assisted by variation and periodic culling. Could natural selection generate a workable design? Perhaps, in the right kind of environment. But an elegant design? Definitely not.

Dr. Axe does a brilliant job, in chapter 10 of his book, of describing the mind-boggling complexity of multi-level systems such as the visual system. Human inventions don't even come close to the level of skill it embodies. Design is the obvious and sensible inference to make, barring an empirical demonstration that Nature's powers of inventiveness far surpass our own. So far, I am not aware of any.

Axe's arguments relating to fantastically large numbers: why they fail

I now pass to Axe's discussion of fantastically large numbers. Since he is a scientist who expresses himself fluently in the language of mathematics (as anyone who has read his scientific papers will be aware), I would expect him to get this part of his argument right. Unfortunately, there are some gaping holes in his argument.

What Dr. Axe ignores, in his argument for the validity of the Universal Design Intuition, is that there is an important difference between an event which is extremely improbable (but still physically possible) and one which is fantastically improbable (and therefore physically impossible). Axe fails to demonstrate that the origin of new life-forms from a hypothetical Very Simple Replicator (the first life-form on Earth) would have been fantastically improbable, because he overlooks the fact that making something useful by chance is a lot easier than creating something meaningful by chance - say, generating a patent for a new invention by boiling up some alphabet soup (to borrow an example of Axe's). Contrary to what Axe asserts, life is not at all like alphabet soup.

Why life is nothing like alphabet soup

Alphabet blocks illustrate this point perfectly. Let's suppose you had a large box of alphabet blocks - say, about 1,000 of them. It would be pretty easy to make something useful with them, if you wanted to: a large square made of blocks could serve as a playpen for a baby, while a stack of blocks could support a tray or a pot plant. But only a very, very tiny fraction of all the possible arrangements of blocks in a stack (or a square) would spell out a meaningful message. Most of the possible arrangements of alphabetic blocks in a stack don't spell out anything at all. That tells us something: the number of ways in which parts can be arranged to perform a useful function is much, much larger than the number of ways in which letters can be arranged in order to convey a meaning. In other words, the emergence of a system of parts that can perform a function is a much more likely event than the emergence of a sequence that can convey a message. The same point applies to the Chinese characters which Dr. Axe discusses on pages 210-214 of his book: they have a meaning, which is intelligible to anyone who can read Chinese, but they don't actually do anything, so they cannot be said to perform a function. The concepts of meaning and function are quite different, for reasons I shall now explain.

In order for an accidentally generated string of letters to convey a meaningful message, it needs to satisfy three very stringent conditions, each more difficult than the last: first, the letters need to be arranged into meaningful words; second, the sequence of words has to conform to the rules of syntax; and finally, the sequence of words has to make sense at the semantic level: in other words, it needs to express a meaningful proposition. For a string of letters generated at random to meet all of these conditions would indeed be fantastically improbable. But here's the thing: living things don't need to satisfy any of these conditions. Yes, it is true that all living things possess a genetic code. But it is quite impossible for this code to generate anything like nonsense words like "sdfuiop", and additionally, there is nothing in the genome which is remotely comparable to the rules of syntax, let alone the semantics of a meaningful proposition. The sequence of amino acids in a protein needs to do just one thing: it needs to fold up into a shape that can perform a biologically useful task. And that's it. Generating something useful by chance - especially something with enough useful functions to be called alive - is a pretty tall order, but because living things lack the extra dimensions of richness found in messages that carry a semantic meaning, they're going to be a lot easier to generate by chance than (say) instruction manuals or cook books. Hence it may turn out that creating life by chance is extremely improbable, but not fantastically improbable. In practical terms, that means that given enough time, life just might arise.

Functionality: how much warrants a design inference?

Another point that needs to be emphasized here is that functionality is not an all-or-nothing affair: there are many shades of gray. Some protein molecules are only weakly functional, and Axe describes such cases in chapter 7 of his book. Additionally, functionality may be either focused and highly specific, or it may be broad and general: some enzymes, for instance, have a much broader specificity than others. Finally, functionality comes in degrees: some functional hierarchies have more levels than others. Axe scores a very telling point when he describes how even the simplest life-forms contain systems exhibiting functional coherence, all the way down to the level of their molecular constituents. But not all biological systems are equally complex. For instance, the visual system illustrated by Axe on page 177 of his book has no less than nine levels of organization underlying the top level, whereas the photosynthetic system of a cyanobacterium (illustrated on page 173) has "only" three. How many levels of organization were required in the first living organism on planet Earth? If nine were required, then I'd be very much inclined to say that Axe's Universal Design Intuition is correct, and that life must have been designed; but if (say) only two levels were required, then I wouldn't be so sure. A hierarchy with two levels of organization might just lie within the reach of chance - especially if its biological functions were very broad and non-specific (as some scientists believe the first proteins were), or if it were only weakly functional in the beginning, and Darwinian evolution tuned it up later on.

I'd like to make a testable prediction here. If Darwinism is correct, then I would expect that if scientists examined bacteria and drew up hierarchical organization diagrams for all of the functionally coherent systems that they identified in bacteria, they would find that these hierarchies had a minimum depth of say, three levels (like the one for the photosynthetic system in cyanobacteria). Beyond that point, however, I would expect the number of systems at higher levels to taper off dramatically: thus there should be exponentially fewer four-level hierarchies than three-level hierarchies, and likewise, far fewer five-level hierarchies than four-level hierarchies, and so on. For eukaryotes and in particular, for multicellular organisms and especially, plants and animals, I'd expect the minimum threshold for hierarchies to be higher than for bacteria (maybe five or six levels), but once again, I'd expect the number to decline exponentially for higher levels. But if life were designed, there would be no reason to expect that. Would someone like to start counting?

The search for a target: what Axe gets wrong

Axe's argument also misconstrues the nature of the evolutionary search, by focusing on the likelihood of this or that accidental invention. However, evolution isn't a hunt for a single target; it's a search with a fantastically large number of targets: the set of all possible life-forms - by which I mean: all possible configurations of matter in our cosmos that would qualify as being "alive." [By "search," Axe means any process that could potentially find something, regardless of whether there was a goal (p. 114). In an "egg-hunt" search, there are designated special objects that a search might find: these are called "targets." In this sense, we can speak of the set of all possible life-forms as evolution's targets, even if evolution has no built-in goals.] The point I want to make here is that even if we exclude possible life-forms with no plausible evolutionary predecessors (i.e. targets which will never be reached, no matter how much time is available) as well as life-forms whose emergence, on the most optimistic scenario, would require a fantastically long time (a "gazillion" years, as Axe puts it), we may still be left with a very large number of target life-forms which might be reached, over an interval of four billion years. Obviously, these reachable targets would comprise only a minuscule proportion (at best) of the set of all possible life-forms, but even a very small proportion of a fantastically large number of targets may turn out to be quite a large number of targets. All evolution has to do is hit some of these targets.

Finally, even if Axe's argument purporting to show that accidental inventions are fantastically improbable were valid, it would still only apply to accidental inventions in general. A much stronger argument is needed to show that each and every accidental invention is fantastically improbable. By definition, the inventions generated by a blind evolutionary process will tend to be the ones whose emergence is most likely: the creme de la creme, which make up only a tiny proportion of all evolutionary targets. For these targets, the likelihood of success may be very low, but not fantastically improbable.

Stepping stones: not as improbable as you might think

The same goes for Axe's argument that an accidental sequence of stepping-stones leading from one innovation to the next would itself be fantastically improbable. For a typical innovation in the space of all possible life-forms in our cosmos, this may well be the case. But the life-forms that exist on Earth are not typical: they are (supposedly) the ones that a blind evolutionary process was capable of generating, over the span of a few billion years. The vast majority of possible life-forms are incapable of being generated in this fashion. But if a very small proportion - perhaps a fantastically small proportion - of this fantastically large set of possible life-forms lies within the reach of evolution, then by definition, the biochemical systems contained within these life-forms are capable of being generated by a blind evolutionary process, too. A Darwinist would therefore predict that the functional innovations found in terrestrial life-forms would all have a series of predecessors leading up to them, without any fantastically improbable gaps separating these predecessors - otherwise they wouldn't be here.

I might add that Axe dramatically overstates his case when arguing against the possibility of an accidental series of stepping stones leading to a biological target. On pages 97 to 98, he invites his readers to consider the case of a functioning system X, which performs its function by coordinating the functions of its components P, Q and R. Thus a working X requires a working P, Q and R, and similarly, a working P requires a working H, I, J and K, and so on, all the way down to the bottom level. Knowing that a certain species of organism benefits from having a working X gives us no reason to believe that the components needed to build X (e.g. H, I, J and K) would have been beneficial earlier on, argues Axe. Additionally, each of these parts would have had to have been modified in just the right way from their original versions, and at just the right time and place, so that they could support a functioning X. Only in the movies does that sort of thing happen.

Axe is guilty of three oversights here. First, any modifications which are made to a functional hierarchy don't have to benefit each and every component; all they have to do is not harm them (or more precisely, harm them only slightly, at worst). Many of the steps involved in the formation of, say, a nine-level functional hierarchy would be expected to have a neutral (or nearly neutral) impact on most of its components. Neutral modifications, by themselves, cannot build functional hierarchies; but they can allow hierarchies to hold together while they are being built.

Second, when Axe rhetorically asks what blind processes could have coordinated the appearance of each of the necessary components at just the right times and places, he seems to be envisaging the formation of a functional hierarchy as if each step was followed almost immediately by the next one, and as if all the steps leading up to the formation of the higher-level components occurred in the same place. This was very likely not the case. Perhaps Axe is thinking of something like a Lego-style assembly process, where step 188 is followed by step 189. That really would be fantastically improbable. But functional systems don't have to be built in just one way, and they don't have to be built in a day, either. For any functional hierarchy found in living organisms, there are likely to be a multitude (perhaps a fantastically large multitude) of possible ways of building that hierarchy, and the construction would have likely been a very gradual process, which may or may not have occurred at a single location.

Third, Axe is perfectly correct in saying that for any given functional hierarchy that we can imagine, most of its components would have been of no benefit earlier on, before the hierarchy had been put together in its present form. But all that proves is that the vast majority of the fantastically large set of possible functional hierarchies never get built in the first place: they are beyond the reach of evolution. If a functional hierarchy was built by evolution, in a series of steps, then by definition, its components must have performed some biologically useful function when the hierarchy had fewer levels than it does now. The functional hierarchies built by evolution are atypical. But that doesn't make them impossible.

By now, readers will have detected a recurring theme in the foregoing criticisms. The real problem with Axe's argument, on an epistemic level, is that human beings are not good at reasoning about fantastically large numbers - especially when these numbers are unknown, and when they are required to divide one fantastically large number into another fantastically large number (e.g. when calculating the very small proportion of organisms - out of a very large number of possible life-forms - that are reachable by a blind evolutionary search over a four-billion-year period). Additionally, humans are not very good at distinguishing between extremely large numbers and fantastically large numbers. Axe's "layperson's case" for Intelligent Design assumes that most human beings are capable of reasoning soundly about large numbers - and as we've seen, they're not: the holes in Axe's own arguments prove that. For this reason, a skeptic might argue that the widely shared human intuition that living things such as whales, tigers and butterflies are products of design should be taken with a very large grain of salt, and that science should steer well clear of making inferences to the existence of a Designer of life.

Professors A, B and C explain why Axe's mathematical arguments against evolution fall short of being proofs

At this point, Axe might respond that I have been putting forward purely hypothetical scenarios in defense of evolution, and that I have been lazily assuming the possibility of abiogenesis. Both criticisms are valid. Axe could further point out that at various points in his book, he provides substantive, mathematical arguments designed to show that the origin of living things by blind processes is fantastically improbable and hence physically impossible. Fine: so let's have a look at those arguments.

In brief, the three major arguments put forward by Axe are: the astronomically low odds (as calculated by Dr. Eugene Koonin) of a coupled replication-translation system emerging by chance on the primordial Earth (pp. 227-231); the vanishingly small proportion of long amino acid chains (which are essential to life on Earth) that are functional (p. 57); and the enormous difficulty of converting one protein into another, similar-looking protein which folds up just like the first one (pp. 81-86). The first two arguments relate to the physical impossibility of abiogenesis, while the third is supposed to demonstrate the inability of natural selection to produce new forms (after all, if it can't even build a new protein from a very similar one, how much less is it capable of building a new organism from a genetically similar one?) My opinion of these arguments has dramatically changed during the past three weeks, and I have come to conclude that all of these arguments are mathematically dubious, for reasons that I shall now explain. Their conclusion may well turn out to be right, but at the present time, we have no way of knowing that.

At this point, I'd like to introduce three professors, whom I emailed regarding Axe's mathematical arguments. Two of them are biologists and one is an organic chemist, all of them are well-published professors, and all of them are Christians. I promised the first professor I would not reveal his name, and the second professor would prefer to remain anonymous, so I'll simply refer to them as Professors A, B and C. (I'll say more about Professor C anon.) Professor A could fairly be described as a Darwinist, while Professor B has strong doubts regarding the possibility of abiogenesis, but adheres to the neutral theory of evolution. Professor C is sympathetic to the Intelligent Design movement and has vocally criticized theories of abiogenesis. In short: if these three professors doubt the cogency of Axe's mathematical arguments, then it's fair to conclude that the arguments don't work. As it turns out, none of the professors was willing to endorse any of these arguments: Professors A and B disagreed with key premises, while Professor C (whose specialty is synthetic organic chemistry) felt he was unqualified to say whether the arguments do or don't work.

Abiogenesis: so improbable that it requires a multiverse to make it happen?

Let's look at the first argument, put forward by evolutionary biologist Eugene Koonin, relating to the astronomically low odds of life emerging by chance on the primordial Earth. Specifically, Dr. Koonin argues that the odds of life (defined as a coupled translation-replication system) originating by stochastic processes in our observable universe (or O-region, he calls it) can be roughly estimated at 1 in 101,018, using a toy model. He makes this argument in the Appendix to his peer-reviewed article, "The Cosmological Model of Eternal Inflation and the Transition from Chance to Biological Evolution in the History of Life" (Biology Direct 2 (2007): 15, doi:10.1186/1745-6150-2-15). The first sentence of the argument reads as follows: "Let us assume that, for the onset of biological evolution, a unique n-mer is required." Koonin conclude that the odds against abiogenesis are so long that we need to invoke a multiverse, in order to make it likely to happen somewhere. Dr. Axe finds the multiverse explanation wanting, and argues for a Creator instead.

In his response to Koonin's argument, Professor A noted that Biology Direct is not a peer-reviewed journal in the usual sense. Basically, an author can publish almost anything, so long as he/she selects a list of reviewers from the journal's Board, and the reviewers' comments are published along with the paper. Professor A readily agreed that the origin of life was an unsolved problem. But he didn't think Koonin had made any new criticisms of abiogenesis in his paper: instead, he had merely rehashed common objections to current RNA world models, noting that if they were valid, then the emergence of a system capable of further evolution was astronomically improbable. This, said Professor A, is not news.

Professor B commented that it was pure speculation on Dr. Koonin's part to assert that a unique n-mer was required in order to produce a coupled replication-translation system, and that this claim was almost certainly false. Most probably, he wrote, many different n-mers would do the job equally well. He added that given what we know of RNA structure, there is a combinatorially large number of RNA sequences that adopt the same structure, and that there was no reason why this magic sequence should be any different. Professor B concluded that Koonin's calculation could not be taken as a serious estimate, and expressed his astonishment that Koonin wasn't impaled by the peer reviewers for making such a bad case.

Professor C responded that he was not qualified to comment on Dr. Koonin's argument.

Summing up: the Achilles' heel of Dr. Koonin's argument is its initial assumption that, for the onset of biological evolution, a unique n-mer is required.

I might add that Axe's argument against the multiverse (which Dr. Koonin proposes as a way of beating the odds against abiogenesis) is far from compelling. Axe contends that if the multiverse hypothesis were true, odds are we'd be living on a planet that was barely rich enough to support intelligent life; but instead, we find ourselves on a planet with an abundance of ecological riches: "five-star accommodation" is how he describes Earth. This argument not only assumes that intelligent life would still exist if there were far fewer species of organisms on Earth, but also assumes that intelligent life would still have originated if there were far fewer species. I'm not so sure about that.

The odds against building a 150-amino-acid protein by chance

So much for the first argument. Now, let's examine Dr. Axe's second argument, that functional proteins consisting of 150-odd amino acids (which are required by all living things) are extremely isolated in sequence space (1 in 1074 is the figure he quotes in his book).

The importance of this particular argument to the case for Intelligent Design cannot be over-emphasized. Putting it succinctly: if it fails, then we're back at square one, in terms of building a mathematical case for ID. Dr. Stephen Meyer's two Intelligent Design best-sellers, Signature in the Cell and Darwin's Doubt, are built on the bedrock foundation of this argument: their whole case would collapse without it. The same goes for The Design Inference, by Dr. William Dembski and Dr. Jonathan Wells. Speaking for myself, I can't count how many times I've cited Dr. Axe's argument in my posts on Uncommon Descent, and referred readers to his paper, The Case Against a Darwinian Origin of Protein Folds, as well as his follow-up article, Correcting Four Misconceptions about my 2004 Article in JMB. Whenever I've had doubts about Intelligent Design, this argument has always been my shining star.

So, what did the three professors whom I contacted think of Axe's argument? In a nutshell: no-one had nice things to say about it. Professor A thought it ignored the possibility that the first life relied on much shorter proteins; Professor B queried the 1 in 1074 figure and said there was no way we could be sure about that; and Professor C said that he was neither qualified to comment on Axe's argument nor able to comment on it, as he hadn't read Axe's book, "Undeniable."

Professor A cited what an unnamed colleague of his, who is an expert in protein structure, had written to him about Dr. Axe's argument:

I think it's fair to argue that modern proteins (50 to 1000's of amino acids in length) probably didn't come into existence in one fell swoop by selection from huge sequence pools, as the probability of success would be vanishingly small. Nevertheless, there is no reason that I can see that primitive proteins had to be very large. For example, protein monomers of 10-15 residues could assemble into four-helix bundles or higher oligomers. The sequence information required for stable four-helix bundles is pretty minimal, largely having hydrophobic residues at buried positions. Michael Hecht's work at Princeton has shown that randomized sequences designed to adopt a four-helix bundle topology (albeit in this case, all of the helices were covalent connected) frequently have primitive enzymatic activity. I assume that once a useful cellular activity arose by chance, then mutagenesis, recombination to fuse protein segments, and selection would provide a path to the proteins and enzymes we now find.

So I think the counterargument to the ID folks is not that sequence populations of 10E80 needed to be searched to find a 100-mer with robust enzyme activity, but rather that random populations of a few million relatively small proteins could contain a few molecules from which to start the evolutionary process.

Professor A also mentioned that other work from Professor Hecht's lab had shown that short proteins that fold into 4-helix bundles have unexpectedly specific ligand binding properties. Professor A regarded Dr. Axe's work as highly biased, because he had based his studies and calculations on very large sequences of amino acids, even though much shorter sequences (such as polypeptides) are known to have biological functions.

When I contacted Professor B and asked for his opinion, he replied that Dr. Axe's estimate of 1 in 1074 was almost certainly wrong, and that in any case, it was a figure based on a very small sample. When I pressed him on this point, saying that several studies of proteins that were only slightly more complex than the shortest ones had suggested that around 1060 to 1080 attempts would be required to make one of the longer proteins, which are currently used by all living things, Professor B responded that a very large number of different amino acid sequences were capable of performing the same biological function. Hence the actual number of attempts that would be required to make a molecule with the same function as one of these proteins was likely to be much lower than 1060. Professor B added that it was actually very common for totally different proteins to have the same function, but acknowledged that we currently do not know how many 'clusters' of proteins there are that share any given function. He also pointed out that the probabilistic resources available to evolution have been estimated at 1042, but that the probabilistic resources available to man-made experiments are much smaller: about 1012, on an optimistic estimate, which meant that any protein which was too difficult for human experiments to generate (probability less than 10-12), could still be well within evolution's ability to locate. [Incidentally, the 1042 figure cited by Professor B seems to check out: apparently Dryden, Thomson & White estimate that up to 4×1043 different amino acid sequences could have been explored since the origin of life, which would imply that if life originated within the first 100 million years in the Earth's four-billion-year history, as is now believed, then about 1042 sequences would have been explored. In Dr. Axe's book, Michael Denton is quoted as estimating the number of possible protein sequences that could have existed on Earth as 1040 (pp. 30-31).]

Professor B added that that he was very sympathetic towards arguments against the natural origins of the first cell, and that of all the arguments raised by evolution critics, this argument had the most likelihood of being correct. He concluded by saying that he did not think that Dr. Axe was necessarily wrong about the unlikelihood of abiogenesis, but that Axe seemed to be trying to calculate the probability of an unknown process, and was therefore overstating his case.

When I asked Professor C about whether he thinks that Axe over-interprets his own data, he replied, "I am neither qualified to comment nor able to comment as I never read it."

Regarding Professor B's criticisms of Dr. Axe’s 1 in 10^74 figure, I asked Professor C if he thought Professor B's criticisms were legitimate? Professor C replied, "I have no idea."

I then asked Professor C, "Given what we know of RNA structure, there is a combinatorially large number of RNA sequences that adopt the same structure, and there is no reason this magic sequence should be different. How many solutions are there? We do not know.” Would you agree with this observation? Professor C replied, "I am further unqualified."

Finally, I asked Professor C, "As you are probably well-aware, Professor B is (like yourself) highly skeptical of abiogenesis, but he also feels that Dr. Axe has pushed his arguments too far, in his new book, 'Undeniable.' Do you share that view?" Professor C replied, "No."

The take-home message from all this is that Dr. Axe is trying to put a full stop where science leaves a comma. We don't really know how rare functional 150-amino-acid proteins are in sequence space, and we don't know that they couldn't have been derived from shorter proteins.

Getting from protein A to protein B

Dr. Axe's third mathematical argument against our being products of a blind evolutionary process is that if evolution is not even capable of bridging the gap between two highly similar enzymes like Kbl2 and BioF2, how much less capable is it of bringing about major evolutionary changes, such as the transition from microbe to man.

Professor A's response to this argument was very brief: as an experimental scientist, he said he was more impressed by empirical results than by theoretical calculations purporting to show what was possible and what was not. He referred me to work done in Dr. Lenski's lab at MSU and Dr. Thornton's lab at Chicago, which (he said) revealed the actual evolutionary pathways taken by the ancestors of living organisms, and which clearly showed that the evolutionary processes that Axe's calculations had suggested were impossible could be observed directly, in the laboratory.

Professor B, however, was harshly critical of Axe's argument. All Axe had done, he said, was to take two related proteins (A and B), and tweak them very slightly, in an attempt to change A into B. That didn't work, so Axe had leapt to the unwarranted conclusion that there was no good path from A to B, and that all proteins are isolated in function, when in reality, he had not proven this at all. Professor B argued that this inference was clearly mistaken, because Axe had not even tried a moderate number of paths from protein A to protein B - let alone all possible paths. Professor B added that most scientists believe that there was an ancestral protein C from which A and B could be easily derived, and that this protein would have performed the functions of both protein A and protein B. He faulted Axe for making no attempt to identify the ancestral protein C, and change C into A and into B.

Professor B thought Dr. Axe was extending his work far beyond what the data currently indicates. He contended that Dr. Axe was over-interpreting his data, and that he wasn't test the true evolutionary model in the correct fashion, but was attacking a straw man instead. Tellingly, Professor B endorsed Dr. Larry Moran's critiques of Axe's work (see here and here).

When I contacted Professor C, he responded that he was neither able nor qualified to comment on Dr. Axe's argument, as he had never read Axe's book. However, when I asked Professor C if he shared Professor B's view that Dr. Axe had pushed his arguments too far in his new book, "Undeniable," he answered that he did not.

Who is Professor C?

I mentioned above that Professors A and B would remain anonymous. Professor C made no request for anonymity, so I will now reveal his name: Professor James Tour. Readers of this blog will be familiar with Professor Tour's forthright criticisms of abiogenesis, and his frank admission that he could not see how macroevolution would work, on a biochemical level. It is worth noting, however, that Professor Tour refuses to describe himself as an Intelligent Design advocate, because he believes the evidence isn't strong enough. As he puts it:

I have been labeled as an Intelligent Design (ID) proponent. I am not. I do not know how to use science to prove intelligent design although some others might. I am sympathetic to the arguments on the matter and I find some of them intriguing, but the scientific proof is not there, in my opinion…

Recently, Dr. James Tour gave a talk titled The Origin of Life: An Inside Story:

Professor Tour's expose of theories of abiogenesis was nothing short of devastating. (I've blogged about it here.) However, he did acknowledge, in response to a question from the audience, that scientists might one day figure out how life arose [1:20:55], and he pointedly refused to say that the answer to this problem was too hard for humans to find. When pressed again, Tour accepted that there was indeed a scientific answer to the problem of life's origin [1:21:10].

Right near the end [1:23:00], when asked whether billions of years would suffice for the evolution of life, Professor Tour responded that many mathematicians would disagree with that claim, and that figures of 10-90 are commonly bandied around. So it seems that he may be inclined to agree with Dr. Axe's 1 in 1074 figure for functional proteins. What I find more telling, however, is that Professor Tour says he doesn't know if Dr. Axe's estimate is correct. He may well think it's true, but he isn't sure. And if he's not, then how can I be?

So, what remains of Axe's mathematical arguments?

Reviewing the comments by Professors A, B and C, I was struck by the fact that not one of them was willing to endorse Dr. Axe's mathematical arguments against abiogenesis and unguided evolution, even though all of them are Christians, two of them are skeptical of abiogenesis, and one of them is sympathetic towards Intelligent Design. So where does that leave me?

Like the guy in the Sam Cooke song, Wonderful World, I don't know much biology, and viewing the debate between Dr. Axe and the three professors whom I cited above makes me feel like an observer at a ping-pong game, as Admiral James Stockdale memorably put it. But it does seem to me that Dr. Eugene Koonin's argument makes an unwarranted assumption from the get-go, that Dr. Axe's argument that unguided processes are incapable of forming a 150-amino-acid protein is decidedly premature, and that his argument regarding the impossibility of getting from protein A to protein B overlooks the highly significant fact that despite their similar appearance, the two proteins last shared a common ancestor several hundred million years ago. I now think that the gene duplication hypothesis put forward by Dr. Larry Moran in his 2012 article, The Evolution of Enzymes from Promiscuous Precursors is a plausible one, especially when one takes into consideration his point that "most modern enzymes catalyze a variety of similar reactions even though the rate for one of them may be several orders of magnitude greater than that of the side reactions." At best, Dr. Axe's arguments fall far short of being proofs; at worst, they are fallacious.

Dr. Axe is well aware that critics have proposed a broad-specificity ancestral enzyme in order to explain why we can't get from protein A to protein B today, but he isn't impressed. "To put it bluntly," he writes, "evolutionary theory has become immune to refutation in the way that the stump of a tree has become immune to further pruning" (p. 227). He also points out, in response to biologist Dan Tawfik, who advances this idea, that we still need to explain how broad-specificity enzymes arose in the first place (p. 85). He even quotes Tawfik as acknowledging this difficulty: "Evolution has this catch-22: Nothing evolves unless it already exists." But the very article he quotes from cites research by Michael Hecht of Princeton University that may answer his own question. Some polypeptides, it seems, are capable of carrying out biologically useful functions. The article also quotes researcher Jack Szostak as saying that there's "pretty good evidence" that not all of the 20 amino acids found in proteins today were there at the beginning: in other words, the original set of amino acids was a subset of today's set. Of course, none of this adds up to a fully-fledged hypothesis. But it does discredit Dr. Axe's claim to have built a solid mathematical case for design, which is meant to run parallel to the popular argument for design which he puts forward in his book.

On a final note: I believe that when you present a scientific argument to the general public, you have a responsibility to draw notify your audience about the uncertainties attending that argument. It seems to me that Dr. Axe has not done that. Each of the three mathematical arguments for intelligent design in his book is highly questionable. For that reason, I shall refrain from citing them as arguments in support of ID, in future.

Are there any other good mathematical arguments against abiogenesis or evolution?

In a 2012 ENV post discussing the alleged transition from ape-like creatures to humans, Axe cites a 2008 paper by Durrett and Schmidt, and comments: "Any aspect of the transition that requires two or more mutations to act in combination in order to increase fitness would take way too long (>100 million years)." Here's my question for Dr. Axe: can you identify a single case in the line leading to human beings, where two or more mutations had to act in combination, in order to confer an increase in fitness in our ancestors? No? So much for Durrett and Schmidt, then.

Haldane's dilemma, which I blogged about here, is sometimes cited by Intelligent Design proponents. Personally, I think Ian Musgrave's take-down of Haldane's dilemma here (and here) looks pretty convincing.

Professor Michael Behe's argument that a "double CCC" event (with a probability of 10-40) constitutes the edge of evolution deserves more serious attention. Professor Larry Moran has put up a post with lots of handy links to his previous posts (for Behe's posts, see here, here, here, here and here). Trying to make sense of it has been very difficult for a layperson like myself. However, I came across a passage in a 2014 post by Moran which shed some helpful light on the evolution of chloroquine resistance in malaria, and which exposed what I consider to be a mathematical flaw in Behe's logic (bolding is mine - VJT):

One of the points of contention is whether the pathway to chloroquine resistance must involve a single mutation that is deleterious. If so, this would lower the probability of a resistant strain arising in the first place because the parasite with the first single mutation will likely die before the second mutation occurs.

This is where the data in the Summers et al. (2014) paper becomes relevant. They have a nice figure showing how the chloroquine resistant strains arose from a series of strains that acquired different mutations. The seven resistant strains are underlined (e.g. GB4). If you look at the group of strains on the left, you will see that there are two possible mutational routes to strain D32…

In one pathway, the mutation N75E occurs first giving rise to strain D39 and then the K76T mutation occurs in that strain creating D32. The order of the mutations is reversed in the other pathway. In either case, the first mutation has no effect on the chloroquine uptake while the addition of the second mutation produces a significant effect.

It is important for Behe's argument that the "first" mutation is deleterious and he claims that the K76T mutations is, in fact, deleterious on its own...

Assuming that Behe is correct about the K76T strain (this is not certain), then the pathway N75E → K76T → strain D32 does not have an intermediate that is "rate-limiting" because the K76T mutation is never present on its own. It seems to me that ALL the pathways have to have a deleterious intermediate in order for Behe's scenario to make sense. In other words, both N75E and K76T have to be deleterious.

This neutral pathway to resistance has been observed.

I can't comment on the biology, but Professor Moran's mathematical reasoning seems to be sound here.

Finally, there's an anti-evolutionary argument that I blogged about a few years ago, in a post titled, At last, a Darwinist mathematician tells the truth about evolution, which was based on a lecture given by Dr. Gregory Chaitin in Brazil in 2011, called "Life as Evolving Software." In his talk, Dr. Chaitin described a toy model he had created, mimicking three competing processes: an exhaustive search (where, for the next mutation, you pick an organism at random, without even looking at the state of the current organism), cumulative random evolution (i.e. natural selection) and intelligently guided evolution. Each of these processes was tested to see how long they took to solve a problem known to mathematicians as the Busy Beaver function. Of the three kinds of evolution examined by Dr. Chaitin, Intelligent Design was the only one guaranteed to get the job done on time. It took time N to calculate Busy Beaver function N. The exhaustive search procedure was the worst, taking time 2N. Darwinian evolution turned out to be much better than performing an exhaustive search of all possibilities: the time it took to reach Busy Beaver function N was, at worst, somewhere between the order of N2 and N3. Nevertheless, it was much slower than Intelligent Design (time N). And it needed an oracle to work. Not good news for Darwinism. But before we get too excited, let's remember that Chaitin's "toy model" of evolution is extremely unrealistic - for example, life itself isn't even embodied (it's purely software), there's no population, there's only one organism and there's no sex! That doesn't sound much like evolution, does it? The obvious thing to do would be to improve the model. However, Chaitin believes that if you try to make his toy model much more realistic and true to life, you won't be able to prove anything with it, mathematically, so there's a trade-off. In short: while Chaitin's argument raises some troubling questions for Darwinists, it doesn't establish the truth of Intelligent Design. When all is said and done, it's only a toy model.

The conclusion I have come to reach in the past couple of weeks is that there are no really strong mathematical arguments against unguided evolution or abiogenesis. Of course, there are no good mathematical arguments for these fanciful hypotheses, either: recent attempts to prove that there's plenty of time for evolution have been convincingly demolished, and my long-standing challenge to evolutionists to demonstrate that the chance of life originating via unguided natural processes on the primordial Earth was greater than 10-120 remains unanswered. On top of that, Professor James Tour has exposed the pretensions of scientists who claim to understand macroevolution, as discussed in my posts, A world-famous chemist tells the truth: there's no scientist alive today who understands macroevolution and Macroevolution, microevolution and chemistry: the devil is in the details. So I think a high degree of skepticism towards abiogenesis and large-scale evolutionary changes (e.g. the origin of animal body plans) is certainly warranted.

Finally, I would warmly endorse what Dr. Axe says in chapter 12 of his book: "The claim that evolution did invent proteins, cell types, organs and life-forms is scientifically legitimate only if we know evolution can invent these things" (p. 226).

Nevertheless, it seems that we have arrived at an odd epistemological impasse. The fact that emerges from the foregoing discussion, after the dust has cleared, is that neither the theory of Intelligent Design, nor the hypotheses of abiogenesis and unguided macroevolution, can be mathematically demonstrated to be true, or even probable. What's a person to do, when faced with a situation like that? For my part, I find the STOMPS principle which I discussed above to be a sensible one. Dr. Axe does a brilliant job of showing, in chapter 10 of his book, how the design of life is far, far superior, technologically speaking, to anything that our top scientists can create. The prudent conclusion to reach is that life was designed, after all. And that should be enough for us.

Origami cranes, recipe books and life: the importance of getting the analogy right

Another major problem with Axe's argument against the possibility of living things originating via a blind process is that the analogs for living things which feature in his book are all chemically unreactive and incapable of self-movement: cook books, digital photographs, origami cranes and mechanical contraptions. Three of these things don't do anything at all, and while things in the fourth category (mechanical contraptions) are capable of performing useful work, they are utterly incapable of harnessing energy on their own: they need to be hooked up in the right way by an intelligent agent before they can function at all. Living things, by contrast, are perfectly capable of chemically harnessing energy without the assistance of an intelligent agent, and of using that energy to power their metabolic reactions and make replicas of themselves. In short: life is chemically dynamic, whereas Axe's analogs for living things are all chemically inert. Axe has deliberately ignored the most salient feature of life: life is an exergonic chemical reaction, in which there is a positive flow of energy from the system to its surroundings. As biologist Bill Martin succinctly expressed it in a recent interview: "It's the energy releasing redox reaction at the core of metabolism that makes life run, and throughout all of life's history it is one and the same reaction that has been running in uninterrupted continuity from life's onset. Everything else is secondary, manifestations of what is possible when the energy is harnessed to make genes that pass the torch." That's what makes life special. Yes, we all know that recipes don't write themselves, that pixels don't arrange themselves into photos, that origami cranes don't fold themselves, and that mechanical contraptions require a maker. But so what? The real question we need to answer is whether the chemistry that powers life could have arisen via an accidental process. And to answer that question, we need to perform some chemical calculations. There are precious few of these in Axe's book, and as we'll see, the calculations that Axe does provide are highly questionable.

So it is very sad to see Axe urging his readers: "Forget the old textbook definition of life - something to the effect of life being a self-perpetuating, nonequilibrium process based on carbon chemistry and driven by the influx of solar energy" (p. 76). Evidently Axe prefers to write about omelet recipes and NASA's instructions for making a CO2-removing device. Apparently he really thinks organisms are like texts. But if he views his beloved salmon and orcas through the lens of a text, then perhaps he is viewing them the wrong way. They are what they are: material beings, living in a material world.

Funnily enough, Axe gets even the textbook definition of life wrong: his definition would exclude chemoautotrophs, which are bacteria that live in deep sea vents, that don't need to use solar energy, and that don't require any organic molecules as food sources. Instead, they obtain their energy entirely from inorganic sources. Many scientists (including Professor Larry Moran, who wrote about chemoautotrophs recently) believe that the first living things must have been like that.

Flawed rhetorical arguments against evolution

Of trampolines and orcas

Two of Axe's rhetorical arguments against evolution are also badly flawed. For instance, on page 18 of his book, Axe scoffs at the notion that "a pool of mineral water" - his derogatory term for primordial soup - set a motion in process that ultimately led to "actual working wonders, like brains and compound eyes and adaptive immune systems and submicroscopic molecular machines." And on page 253, he declares that if even "mundane things like bricks and shoes" are too good to happen by accident, "how can exquisite things like spiders and orcas be exceptions?"

The fallacy Axe commits in these passages is that of estimating the probability of reaching an end-point without looking at the intermediate points. This is particularly misleading when one of the intermediate points plays a critical role in boosting the probability of reaching the target. A simple example will suffice. Sitting in my upstairs study, I can see the fence separating my house from my neighbor's, about ten meters away. It would be nice if I could jump over that fence from my study, but I can't. Even with a long run-up, an assisting tail-wind and an open window that extends from the ceiling to the floor (which mine doesn't), there is no realistic chance of my clearing that fence and landing in my neighbor's garden. But now let us suppose that I place a large trampoline in my garden, between my house and my neighbor's fence. All I have to do now, when I take a running jump, is land on the trampoline very hard, and bounce off it at a sufficient velocity to clear the neighbor's fence. That's difficult, but doable. I doubt whether I could do it myself, but a younger and more athletic man might manage it.

Why, then, are blind processes capable of generating orcas, but utterly incapable of making bricks, which are far less complex? The answer is simple: orcas are targets that were reached by a trampolining process, which dramatically boosted the probability of their attainment. That process is Darwinian evolution, which automatically came into effect once the first Very Simple Replicator arose on the primordial Earth. (I am of course well aware that the vast majority of mutations in living organisms are neutral or near-neutral, but nearly all biologists agree that Darwinian evolution is the only natural process that can account for the evolution of complex structures such as the vertebrate eye, or the human brain.) Bricks, on the other hand, have no such trampolining process. And, as I've pointed out above, bricks are chemically inert, whereas life is chemically dynamic - which means that whereas life might arise in an environment where there is energy allowing its constituents to assemble, bricks don't have a hope in Hades of forming in this way. Simple as that.

Getting from A to B: examine the path, not the destination!

A second rhetorical argument employed by Axe involves an illustration (on page 225) showing two similar-looking bacterial enzymes (which he calls A and B), some complex life-forms (a human being, a fox, a dolphin, a fish and a dragon-fly) and a simple bacterium that supposedly corresponds to the ancestor of all living things. The caption to the illustration reads as follows: "If accidental causes can't invent this [enzyme B] from this [enzyme A], then how can they invent these [complex life-forms] starting from this [simple bacterium]?" This is the kind of rhetorical point which I would excuse, coming from a preacher or an orator, but never from a trained scientist. The fallacy Axe commits here is that of attempting to gauge the difficulty of a path by comparing the similarities between the starting point and end point: if the two look alike, then the end-point is deemed to be reachable, but if the end-point looks very different from the starting point, then Axe thinks we should be skeptical of the possibility of reaching the end-point. The illogicality in this argument should be obvious: one might as well argue that if people can't even walk from the southern tip of India to the island of Sri Lanka, which is but a short distance away and has a similar climate, similar terrain and similar wildlife, how much less possible should it be for people to walk the 5,000-odd kilometers from Mt. Everest to the southern tip of India, whose environment is totally different from Mt. Everest's. And yet they can, as any Sherpa will tell you. When considering the feasibility of a journey, what we really need to focus on is the traversability of the path, not the distance to the end-point. As we'll see below, there may be perfectly good reasons why it is practically impossible for one enzyme to evolve into another; but these reasons don't apply to the possibility of a one-celled primordial bacterium evolving into a complex life-form. The pathway is totally different in the latter case, and that's why it's reasonable to suppose that transitional forms linking the first cell to complex animals might exist.

Getting natural selection wrong, in so many ways

The importance of not straying too far from the path

Additionally, Axe's argument for the fantastic improbability of living organisms is both mathematically and biologically flawed. Axe confuses the improbability of a state with the improbability of a path, when he argues that because only a tiny proportion of all possible configurations of matter correspond to living things, the origin of these living things by blind processes is fantastically improbable. On page 192 of his book, Axe approvingly quotes a pithy aphorism from Dr. Richard Dawkins' best-seller, The Blind Watchmaker: "However many ways there may be of being alive, it is certain that there are vastly more ways of being dead, or rather not alive." Axe comments: "Coherent skeletons are impossibly rare among random arrangements of bones, as are coherent body plans among random arrangements of organs, and molecular machines among random arrangements of amino acids," before concluding that "none of these inventions had any prospect of coming together by accident" (p. 192).

What Axe implicitly assumes here is that the probability of a blind process hitting on these evolutionary "targets" is equivalent to the probability of a random configuration of atoms being in a state that happens to match one of the targets. This assumption would only be true if the process used to search for these targets were an exhaustive search of all possibilities - which natural selection is not. In reality, natural selection does not, and cannot, explore every possible molecular state corresponding to some way of being dead: it can only explore those ways which are but a short but fatal step away from being alive. (All it takes to kill an organism is one fatal mutation; evolution never explores what happens when an individual accumulates ten of these mutations, because it never gets the opportunity to do so: dead organisms don't reproduce, so fatal errors can never accumulate from one generation to the next.) Once the first living organism arose on Earth, the process of Darwinian evolution would have immediately come into effect, and the beauty of this process is that it does not have to search the entire space of all possible configurations: it sticks to a narrow path winding through this space, consisting of all viable arrangements of matter, and it selects those arrangements which prove to be more viable, and pitilessly winnows out those arrangements that turn out to be less so.

The photos below of Mt. Hua, in China (shown on the left), and of a perilous path leading up to the summit (showing on the right), illustrate my point about the narrow path that natural selection traverses. An unwary traveler might take one fatal step off this path; but he can never take ten.

Natural selection is not a random walk

While I'm on the topic of Darwinian evolution, I'd like to point out another fallacy Axe commits: the fallacy of overlooking (or at least blurring) the distinction between a blind search and a random walk. In chapter 7 of his book, he acknowledges that "natural selection tends to shift the genetic make-up of a species toward the highest fitness as judged from its present members" (p. 92), which implies that it has short-range targets, even if it has no long-range ones. However, it turns out that Axe has a very poor opinion of natural selection: he states that natural selection is very much like homing in on a target, but argues that because anything with a "homing ability" is liable to home in on the wrong signal, a homing device's successes in reaching its target are due entirely to repetition, and he even suggests that the device's homing ability serves no useful purpose: "just how significant was the homing ability in the first place?" (p. 92) In the following chapter, Axe then proceeds to castigate natural selection as a process of "aimless wandering," and he declares that "repetition is the only factor that can conceivably offset the improbability of stumbling upon biological inventions by accident" (p. 113). Reading these passages, a layperson could easily get the impression that natural selection is no better than a hit-and-miss random walk through evolutionary space. The only place in his book where Axe even hints that he thinks otherwise is a cryptic footnote (n. 3) to chapter 9 on page 283 (which very few people will take the trouble to read), where Axe acknowledges that blind attempts "need not be random." And that's it, folks.

The funny thing is that Axe provides a perfect illustration of the superiority of natural selection to a random walk on page 104 of his book, where he writes: "Small adjustments can sometimes mean the difference between working poorly and working well, and selection seems to have a knack for finding adjustments of that kind." He then goes on to describe an experiment in which, after just six rounds of mutation and natural selection, a beaten-up, artificially degraded enzyme, which was barely functional, managed to boost its functionality by a factor of 500, to the point where it worked better than the original, healthy enzyme! If natural selection proceeded by a random walk, then this sequence of changes could not possibly have happened so rapidly: unlike natural selection, random walks have no tendency to go uphill, even locally.

Axe's misrepresentation of natural selection gets worse when one examines his footnotes to chapter 7, where Axe informs his readers that "the likelihood of the first possessor of a new beneficial mutation passing that mutation to the entire species can easily be less than one in a million," and adds that this improbability "is further compounded by the rarity of beneficial mutations in the first place" (p. 281). In fact, recent research indicates that in E. coli bacteria and yeast, "almost 1 per cent of all mutations in experimental populations of these organisms are beneficial." (Another study yields a similar figure: "our estimate ... implies that 1 in 150 newly arising mutations is beneficial and that 1 in 10 fitness-affecting mutations increases the fitness of the individual carrying it.") Axe correctly notes that the likelihood of the first possessor of a new beneficial mutation passing that mutation to the entire species is equal to "twice the fractional fitness advantage multiplied by the ratio of the effective population size to the actual population size" (p. 281), but seems to think that the latter ratio is very low. There are a few cases in the literature where it has been found to be very low: Hauser (2002) discusses an overexploited species of fish, the New Zealand snapper, in which effective population sizes turned out to be five orders of magnitude smaller than census population sizes, and he speculates that similar ratios may occur in many other marine species. However, in a recent survey of the literature, Palstra and Fraser estimate that the median value of the ratio of the effective population size to the actual (census) population size to be 0.23, with a range from 0.01 to 0.95. (The authors acknowledge, however, that their survey is limited to fish, amphibia and insects.) What about the fractional fitness advantage, or selection coefficient, for a beneficial mutation? Studies appear to show that as the fractional fitness advantage increases, the probability of a mutation having that advantage declines exponentially (see Figure 5 for more details). For the purposes of illustration, I'm going to pick a fairly modest selection coefficient of 0.05: a mutation that makes an organism 5% fitter. If the ratio of the effective population size to the actual population size is 0.23, then the probability that the first possessor of a new beneficial mutation will pass it to the entire species is equal to about 0.01 - which is four orders of magnitude higher than the one in a million figure Axe quotes!

Another point which Axe omits to mention is that even when a beneficial mutation doesn't get fixed, often the reason is because it is competing with other beneficial mutations which get fixed instead. I wouldn't call that a failure of natural selection: would you?

A bogus argument for the intelligent design of each species of organism

While we're on the subject of faulty numbers, I was appalled to find Axe asserting, on page 183 of his book, that "every taxonomic group so far studied contains 10-20% of genes that lack recognizable homologs in other species," and that about one-third of the genes initially identified as one-offs in every species turn out to be "genuinely new." On the face of it, that would seem to imply that a large percentage of the genome of each and every species was intelligently crafted by God. Which is odd, because not even the creationists believe that: they tend to assert that the Biblical kinds created by God roughly correspond to the taxon of the family, rather than the species.

When I consulted Dr. Axe's footnotes on page 284 to discover the source of his astonishing claims, I found two papers dating from 2009 - which is virtually back in the Stone Age, as far as orphan gene research is concerned. Since then, a lot of genes that were initially identified as "orphan genes" have since been shown to have homologs in other species.

I would invite readers (including Dr. Axe) to have a look at my October 2015 post, Double debunking: Glenn Williamson on human-chimp DNA similarity and genes unique to human beings. In that post, I described how computer programmer Glenn Williamson had demonstrated that 60 de novo protein coding genes said to be unique to human beings turn out to have very similar counterparts in apes. How similar? Over 98% similar, in some cases. At the end of my post, I quoted from a 2014 wager by Glenn Williamson:

I'd happily take bets on the majority of these de novo genes having an evolutionary history (chimpanzee > 95% and/or gorilla > 90%).

Any takers?

Well? Is anyone feeling lucky?

While we're on the subject of orphan genes and de novo genes, here's an up-to-date paper that presents several hypotheses regarding their origin (h/t Professor Larry Moran): Schlotterer, C. (2015) Genes from scratch - the evolutionary fate of de novo genes (TRENDS in Genetics, 31:215-219. [doi: 10.1016/j.tig.2015.02.007]). The author makes the telling point that genuinely de novo genes are quite short. What that means, in practical terms, is that they are not beyond the reach of chance.

The silliest argument against natural selection

But without a doubt, Axe's silliest argument against natural selection is that it cannot invent. He calls this the Gaping Hole in Evolutionary Theory (p. 97). I would refer Dr. Axe to a recent post by University of Chicago Professor Jerry Coyne, in which he endeavors to summarize the entire theory of evolution in a single paragraph. His concluding point is as follows:

Finally, the "designoid" features of organisms - the features that make them look so well adapted to their environments and lifestyles - are the product of natural selection: the combination of a random process, mutation, that generates genetic variation without regard to whether it's "useful" or not, and a deterministic process, selection, that winnows the variation by retaining those mutations that are better able to make copies of themselves and eliminating the worse copiers. There are other important processes of evolutionary change, like random genetic drift, but only selection can produce the design-like features that so excite our wonder.

There you have it. Natural selection doesn't generate "inventions" (or biological innovations), because that's not its job: that's the job of mutation. It may be objected that Coyne is not typical of all evolutionary biologists; however, the respected evolutionary geneticist Allen MacNeill of Cornell University, whose perspective on evolution is very different from that of Coyne, agrees with him on this point. In a 2009 blog article titled, "Can Natural Selection Produce New Information?", MacNeill points out that natural selection has three prerequisites, one of which is Variety, generated by the "engines of variation." He continues:

...[T]he real dispute between evolutionary biologists and "intelligent design" supporters is not over natural selection per se, but rather the properties and capabilities of the "engines of variation". I have written extensively about these here and here.

Yes, natural selection … is conservative not creative. It produces no new genetic nor phenotypic information, which is why Darwin eventually came to prefer the term "natural preservation" rather than "natural selection". However, it is also clear that the "engines of variation" - that is, the processes the produce phenotypic variation among the members of populations of living organisms - are both extraordinarily creative and extraordinarily fecund. The real problem in biology is therefore not producing new variation, but rather limiting the production of new variation to the point that the "engines of variation" do not cause the inevitable disintegration of living systems.

How a blind search can invent new functions

Dr. Axe believes that a blind search is incapable of inventing anything: "Invention can't happen by accident," he writes (p. 160). How, then, does he account for the sudden appearance of bacteria with the ability to degrade waste chemicals derived from nylon (a substance not invented until 1935)? If that's not an invention, then I don't know what is. A recent article in New Scientist describes how this evolutionary invention arose:

Nylon was first made in 1935. Just 40 years later, in 1975, a bacterium was discovered that is able to digest and live off not nylon itself, but waste chemicals from its manufacture - chemicals that had not existed before nylon production began.

It was later shown this bacterium, now known as Arthrobacter KI72, has evolved several types of enzymes capable of utilising these waste products. One type, 6-aminohexanoic acid hydrolase, encoded by genes called nylBs, has become known popularly as "nylonase".

So, how did nylonase appear? A team led by Seiji Negoro of the University of Hyogo, Japan, finally uncovered the answer in 2005:

His team's study of the protein structure show that nylonase is very similar to a common type of enzyme that breaks down beta-lactamases - natural antibiotics produced by many organisms. Just two amino-acid changes - two mutations, in other words - are required to change the beta-lactamase binding site to one capable of binding the nylon by-product.

In other words, nylonase resulted from two point mutations in an existing esterase enzyme, without significantly reducing the original esterase activity. This enzyme enables a bacterium to digest a new source of food, which its ancestors were unable to digest 50 years ago. No two ways about it: that's a new invention. A prudent person, presented with this evidence, would modify his thesis that invention can't happen by accident. I don't know how Axe would respond to this objection, but the online responses of creationists (who deny that evolution can create new information) display a degree of mental gymnastics worthy of a contortionist. Over at Answers in Genesis, Dr. Georgia Purdom writes that since the mutations slightly alter the enzyme "so it can break down nylon that is very similar to the substance it normally breaks down," there is no gain of information here, but rather an alteration of currently existing information. Dr. Purdom gets her facts wrong here: the enzyme doesn't actually break down nylon, but a waste product derived from its manufacture. But let that pass. The argument that an alteration of currently existing information does not constitute new information simply won't hold water: it would imply, absurdly, that an intelligent Creator producing modern life-forms by a process of modifying their ancestors' genomes one step at a time would not have created any new information in the process. Dr. Purdom also argues that the mutations are degenerative to the enzyme, because they reduce its specificity: the bacteria can now "eat" the normal product and the nylon by-product. But the point here, surely, is that the bacteria are able to do something new that their ancestors couldn't do, and what's more, this new ability doesn't interfere with any existing ability they possess. Whichever way you slice it and dice it, that's an invention.

The New Scientist article describes four additional cases of gene evolution. One of these cases relates to color vision. In the ancestors of apes and some monkeys, the gene that codes for one of the two color-sensitive retina pigments (or opsins) found in most mammal eyes was duplicated, and mutations in one copy resulted in a pigment that could detect a new color, enabling these primates to distinguish red light (which has a long wavelength) from medium-wavelength green, and thereby pick out ripe yellow-orange fruit from a background of green leaves more easily. It turns out that the original gene which codes for long-wavelength opsins and the duplicate gene which codes for medium-wavelength opsins are highly similar, and when scientists examined the opsins themselves, they discovered that just three amino acid changes were sufficient to account for the primates' ability to detect medium-wavelength light. Curious readers can find out more by James Bowmaker's highly illuminating article, Evolution of color vision in vertebrates.

If the ability to distinguish red from green doesn't qualify as a new invention, then I ask: what does?

The second invention described in the New Scientist article was the crystalline proteins in the eye, which enable the lens to focus light on the retina. However, this invention didn't require the evolution of a new gene encoding a new protein: instead, evolution merely took an already existing heat-shock protein, alpha-crystallin, and used it to serve an optical function in the eye. A single gene, HspB5, codes for this protein in the body. Obviously, evolution didn't need to do much here, but it clearly invented something: it gave us the ability to focus, and see things clearly.

Bottom line: evolution can invent. The real question is: how much? That is the issue that Axe needs to address. In chapter 7 of his book, he draws a distinction between fiddling and inventing: evolution, he says, is very good at the former, but utterly incapable of the latter. For instance, Axe thinks that his experiment (discussed above) in which a barely functional beaten-up enzyme managed to improve its performance 500-fold after six rounds of mutation and natural selection, is a case of mere fiddling: selection had to be given a gene that codes for an enzyme in the first place (p. 104). But if he wishes to adopt that line of defense, then what Axe is conceding is that ordinary people can't identify new inventions, since they lack the scientific expertise to discern whether a new function (such as being able to degrade a nylon by-product, or distinguish red from green) arose from the appearance of a new gene or because of modifications in existing genes. And what about gene duplication? Which category does that fall into? Axe does not say. In short: if Axe wants to cast aside the everyday definition of "invention" as anything that can do something new, and define in a more academic sense, then he is effectively taking the Argument from Design away from the common folk - which I don't think he would wish to do.

The evolution of complex innovations

Dr. Axe might object that perhaps evolution can produce simple innovations, but not complex ones. In that case, he might be interested to read a recent article in Nature Communications, titledAdaptive evolution of complex innovations through stepwise metabolic niche expansion by Balazs Szappanos et al. Allow me to quote a brief excerpt:

Specifically, we employ detailed simulations on a pan-genome scale to demonstrate that complex metabolic innovations can evolve via the successive acquisition of single biochemical reactions that each confers a benefit to utilize specific nutrients. Thus, temporal changes in nutrient availability or complex environments (where multiple nutrients are available) can facilitate adaptive evolution of metabolic pathways through the step-by-step expansion of metabolic niches. Gene acquisition patterns across bacterial genomes and de novo laboratory evolution of nutrient utilizations in Escherichia coli (E. coli) provide clear support for the hypothesis.

The inadequacy of Axe's homing robot metaphor for natural selection

The final point I'd like to make about Axe's critique of natural selection is that its central metaphor is badly flawed: he writes that natural selection is like a homing robot that's designed to home in on the loudest sound in its vicinity. Axe then invites his readers to consider the likelihood of such a robot managing to find its way to the loudest location in a city: the local baseball stadium. In reality, the robot is liable to be waylaid long before it reaches the baseball stadium, by background noises such as the sound of traffic. The problem with homing, argues Axe, is that anything with a "homing ability" is liable to home in on the wrong signal, like the robot in his example. The robot will find the stadium only when it is in an ideal location, where the right source of noise (the stadium that it was intended to reach) drowns out other local sources of noise. Similarly, says Axe, natural selection will often fail to reach its biological target: success occurs only when the right source is located close enough in genetic space to outcompete other sources. What usually happens instead is that natural selection moves toward a "local maximum," where it gets trapped, rendering it incapable of reaching the global maximum where its biological target is located. However, what Axe overlooks is that the noise-seeking robot in his example (p. 89) has a unique, pre-defined target (the baseball stadium), whereas natural selection has no pre-defined target. What's more, Dr. Axe's homing robot is designed to home in on just one thing: the loudest local sound. But natural selection is not like that: it homes in on any biologically useful signal - and there may be thousands of these - that confers a reproductive advantage on an organism.

Axe's argument that natural selection tends toward local rather than global maxima (just as the homing robot will move toward nearby sources of noise instead of the baseball stadium) suffers from an additional mathematical flaw, as computer scientist Mark Chu-Carroll noted in a book review published several years ago:

The idea of local maxima and minima being common comes from thinking of things in terms of low-dimensional surfaces. A fitness landscape with two variables forms a three dimensional graph - and in three dimensions, we do frequently see things like hills and valleys. But that's because a local minimum is the result of an interaction between *only two* variables. In a landscape with 100 dimensions, you *don't* expect to see such uniformity. You may reach a local maximum in one dimension - but by switching direction, you can find another uphill slope to climb; and when that reaches a maximum, you can find an uphill slope in some *other* direction. High dimensionality means that there are *numerous* directions that you can move within the landscape; and a maximum means that there's no level or uphill slope in *any* direction.

As Chu-Carroll points out, the "fitness landscape" metaphor suffers from other major defects as well: first, it overlooks the fact that the landscape can suddenly alter (due to environmental changes); second, it assumes that evolution can only climb uphill, whereas in reality, small and temporary declines in the fitness of a population are possible, which means that it may sometimes be possible for evolution to travel from one local maximum to another; and finally, it assumes that small changes in an organism's genome (mutations) can only take it to locations in the immediate vicinity, whereas in fact, small genetic changes can sometimes give rise to huge effects (such as the panda's thumb) which land an organism in a completely different point on the landscape.

Let us hear no more talk of homing robots, then, when discussing the limitations of natural selection.

Chinese characters: a fairer test of the powers of natural selection and mutation?

In his book (pp. 210-212), Axe compares proteins to Chinese characters: both have a hierarchical structure and are composed of multiple parts, both can be put together in a variety of different ways, and both have functional forms numbering in the thousands. (As I've pointed out earlier, it's a mistake to say that a character of text has a function, as proteins do; rather, what it has is a meaning. And no, "being legible" is not a function, because a function is an activity, and the only activity performed in reading is done by the reader, not by the character. But let that pass.) Axe then describes a computer program called Stylus, which assesses how well a represents a set of strokes pointing in various directions (or a vector chain, as he calls it) resembles a particular Chinese character (such as the one depicted above, courtesy of Wikipedia and provides feedback, enabling a badly drawn character to be gradually corrected until it perfectly resembles the character - a process which Axe describes as selective homing. Some vector chains, however, may be so badly degraded that they can never be corrected in this stepwise fashion: they look nothing like the Chinese character, and they never will. Axe sees a parable here, of relevance to his thesis on natural selection: "As with the proteins in chapter 7, selective homing was of value only when functional coherence was already present in large measure." Or in other words: "Nothing evolves unless it already exists" (p. 214).

This attempt to improve Chinese characters on a stepwise basis is perhaps the most interesting analogy for natural selection in Axe's book. However, the problem with Axe's critique of natural selection, in this instance, is that Chinese characters have internal components of their own, and any fair test of natural selection would begin with those components being "functional" (or legible), because the task of natural selection is to take existing functional units and co-ordinate them to create something capable of performing a new function. Axe's test doesn't do that: it begins with a set of strokes which is so badly corrupted that it doesn't possess anything that's even vaguely recognizable as those components. For instance, the Chinese character depicted above has three separate components, but the vector chain that was incapable of being improved in Axe's experiment didn't even have three components, let alone three that resembled the ones in the character. How fair is it to apply natural selection to a mess like that? Putting it another way: how can you possibly make a jigsaw puzzle where there are no recognizable pieces of the puzzle, to begin with?

Harnessing the power of exchange and duplication: how Axe's Chinese character experiment could have been improved

Reading about Axe's experiment with the Chinese characters, though, I had an idea. (Full disclosure: I live in Japan, where similar characters are used - including the one depicted above, although its meaning is somewhat different: in Chinese it means section, whereas in Japanese it means step, rank or grade.) I thought to myself: what if Axe were to use the basic components of all the Chinese characters as his starting materials, instead of using badly degraded versions of existing characters, and randomly select these components, two or three at a time, and then (and only then) use a selective homing procedure (i.e. the Stylus program) to build them into a Chinese character that happens to contain exactly those components, if there is one? Of course, some of the components would have to be slightly modified in shape, and size, in order to fit together in the right way, and all of the components would have to be in the right relative positions to one another, in order to make a proper Chinese character.

Readers may be wondering about the combinations of components that didn't correspond to any Chinese character: what would happen to them? Here's where my proposal gets interesting. What if there were a computer program which allowed these combinations to randomly swap components with one another, until they achieved a combination which matched some existing character, and then the Stylus program would kick in, refining the combination until it closely resembled the character? And what if, additionally, "functional" (i.e. legible) combinations were allowed to make duplicates of themselves that could be mutated yet again, and/or merged with other combinations, in an attempt to make new and even more complex characters? (Some Chinese characters have dozens of strokes.) It seems to me that this would be a fairer test of the power of natural selection and random mutation than the one described by Axe in his book.

It should be obvious to readers with a biology background that the processes I'm proposing to simulate here are those of horizontal gene transfer (swapping) and gene duplication. And I'm sure readers can think of many other genetic processes that could also be simulated: Professor Allen MacNeill has identified no less than 47 different engines of variation. Without these processes, I don't think natural selection is going to take us very far. But since we know that these genetic processes are common in Nature, shouldn't we be trying to model them properly, in an attempt to find out exactly how much functionality unguided natural processes are capable of generating? That, I believe, is what Axe needs to do.

Natural selection: summing up

The list of errors and misrepresentations regarding natural selection in Axe's book is beyond a joke. As I have emphasized previously, the fault here is not Axe's: he solicited comments on his manuscript from no less than fifteen scholars and other eminent people, before publishing it as a book. What I cannot understand is why the biologists reviewing his manuscript didn't alert him to the holes in his argument. Guys, what were you doing?

(NOTE: In the interests of journalistic accuracy, I feel obliged to state that: (a) at least one reviewer has since contacted me to let me know that some of the changes he proposed were rejected by the author; and (b) not all of the reviewers were eminent people. I should note that this reviewer mentioned in point (a) was overall very positive about Axe's book and said "overall I strongly agree with Doug's thesis, and I am proud to be listed as a reviewer who would endorse the book.")

It must be fairly acknowledged, however, that we currently lack a Darwinian explanation for how functional coherence can arise on a massive scale. Had Axe contented himself with hammering home this point, he would have been on very strong ground. Where he went too far, in my opinion, was in arguing that Darwinism could not, in principle, explain the degree of functional coherence we find in living things. He hasn't demonstrated that. Nevertheless, Axe is fully entitled to ridicule the failure of current attempts by Darwinists to come to grips with this problem, because so far, they haven't.

Mind: mysticism versus materialism

Axe on the immateriality of consciousness

I now turn to Dr. Axe's views on the human mind. Axe forcefully argues against materialism throughout his book: for instance, on pages 239-240, he describes a hilarious thought experiment intended to discredit materialism, in which a brain scientist voices his frustration that he cannot find anything in his patient's brain which matches what the patient means by the words "two," "circle," "triangle," "line," "around," "between," "love," "hate," "true," "false," "one," "none" and "time." Funny as it is, I don't think Axe's thought experiment proves much: all it shows is that what we mean by these terms isn't something in the brain - but then, who said that meaning was in the brain? All the materialist claims is that that whereby I understand terms like "two" and "circle" is my brain. What I understand, however, is something outside my brain. If someone asked me what I mean by the number two, I'd point to material objects like pairs and number lines, and say, "That's what I mean by the number two." And if someone asked me what I meant by a circle, I'd point to something round. To be sure, some materialists (known as representationalists) maintain that when I entertain a concept of something like "two" or "circle," there has to be a representation of these things in my brain. But while the representationalist thesis may be true for our low-level concepts, there's no reason to suppose that it's true for all of them. So a materialist might shrug and say, "OK, so I don't have a representation of 'two' in my brain, let alone 'true.' But I still grasp these concepts with my brain, somehow, even if they're not in my brain. Perhaps my brain simply stores information that tells me how these various concepts hang together, rather than what they mean." That strikes me as a tenable position.

Axe makes a strong claim about consciousness on page 44 of his book: "It is intuitively obvious to me that a mere collection of atoms cannot attain consciousness. It can never become aware of its own existence." I'm not sure why Axe thinks it self-evident that atoms can't be conscious, but I would guess it's because he thinks they can be described in purely mechanical terms. What he needs to recognize is that not all metaphysical naturalists are mechanists. For instance, Bertrand Russell's neutral monism imputes both physical and mental properties to objects: neither is more fundamental than the other.

Axe tells his readers that he is a big fan of orcas, and I presume he would impute to them a consciousness of some sort, even if he denies them self-consciousness. At the same time, I would hazard a guess that Axe doesn't believe orcas have an immaterial soul. (Or does he?) That being the case, it seems clear that a collection of atoms - for that is what a living body is, even if it is a highly integrated collection - can attain consciousness. What Axe needs to explain, then, is why consciousness lies within the reach of mere matter, but self-consciousness does not. He might like to have a look at what philosophers Edward Feser and James Ross have written on the subject. And if he really wants to know how to discredit materialism, the numerous online articles published by Edward Feser on the mind-body question should set him straight. Several prominent Intelligent Design advocates are Thomists, and I'm sure that neurosurgeon Dr. Michael Egnor would be happy to assist him.

For my part, while I would wholeheartedly applaud Axe's human exceptionalism, I'm somewhat agnostic regarding the immateriality of animal consciousness, for reasons I've discussed here.

Axe on the uniqueness of insight

Throughout his book, Axe insists on the primacy of insight: "Insight," he writes, "is absolutely unique, without rival among the mindless causes to which the materialists limit themselves and, as we will later see, not reducible to those causes either… Parallels for insight ... are non-existent" (p. 100). He then proceeds to deduce that "any instance of mindless causes doing the work of insight would have to be a coincidence" (p. 101). Later on, Axe argues that "the amount of functional coherence produced by insight truly can't be produced by accident," because "accidental causes mimicking insight on this scale would be a fantastically improbable coincidence, which means a physically impossible coincidence" (p. 152).

Speaking personally, I wish that Axe had written more about the nature of insight in his book. Yes, we all know that insight is unique, but why? What makes it unique? Without a clear answer to these questions, hostile critics are liable to dismiss Axe's claims as mere anthropocentric assertions.

Human invention: a top-down, intuitive process?

Perhaps Axe's remarks on the process of human invention might shed light on what he is driving at. On pages 136-138 of his book, in a section called "How Humans Invent," he describes this process in some detail. In the first stage, the mental stage (or conception stage), the idea for the invention is developed into a detailed plan, by breaking down the big, top-level idea into smaller ideas, which are further broken down, and so on. In the second stage, the methodical stage (or construction stage), the aim is "to form a physical device in a bottom-up manner, going from the raw materials and supplies at hand to a working prototype." This building process "must take careful account of the properties of the physical objects and materials being manipulated." The third and final stage is the mechanical stage (or operation stage): a purely physical activity where humans can watch the device do what it was designed to do - in other words, test it. Hopefully, by now it will work properly, and if it does, spectators will be able to infer that because the device is a busy whole (an active thing that accomplishes a big result by bringing lots of smaller things together in just the right way), it must have been designed.

All well and good; but the process of human invention is usually not like that, and Axe cites no sources in his footnotes, to support his claims. In particular, Axe's claim that invention is a top-down process is a highly debatable one. Canadian biochemist and bioinformatician Chris Hogue thinks otherwise. He writes:

The individual steps in human design are small, slow and absolutely require the intellectual imprinting of lessons by trial and error. Students who are led to think falsely about human design, or any complexity as having mystical origins are harmed by the diminishment of their own aspirations of creativity. We all need to understand how small steps and tools lead to human creativity and any object of complexity...

Human creativity is always applied in small increments, as has been well stated by Thomas Edison and more recently by vacuum designer James Dyson. Complexity never gushes forth in a single setting. It accumulates, incrementally over time, and can be copied as a meme and reapplied. It is always more evolutionary than revolutionary.

Professor Hogue was refreshingly candid about the implications of his view for the Intelligent Design debate. ID concepts "force the gerrymandering of human design history, and surround it with mystical borders to make their claims," he writes. Intelligent Design is indistinguishable from the incremental process of evolution, which means that there can be no evidence in principle that would count in favor of Intelligent Design and against evolution:

Evolutionary processes have wonderful analogies in human design and I will go over many cases to show that complexity does in fact always arise from small steps. And when we take human design and shrink it down to the molecular level, as I myself have done, the human design process is indistinguishable from evolution.

I blogged about Professor Hogue's argument here, back in 2012. At the time, I thought I had a pretty good comeback. "It is quite true," I wrote, "that if we look at the history of the electric light bulb, for instance, we will find a pattern of incremental improvement. Tesla's inventions, however, were quite different." Tesla actually despised Edison's method of inventing. For his part, Tesla claimed that he never needed to design things via a trial-and-error, incremental process. In the first chapter of his autobiography, Tesla describes his own method of invention, which he first came upon at the age of seventeen, after discovering that he could "visualize with the greatest facility" and that he "needed no models, drawings or experiments":

My method is different. I do not rush into actual work. When I get an idea I start at once building it up in my imagination. I change the construction, make improvements and operate the device in my mind. It is absolutely immaterial to me whether I run my turbine in thought or test it in my shop. I even note if it is out of balance. There is no difference whatever, the results are the same. In this way I am able to rapidly develop and perfect a conception without touching anything. When I have gone so far as to embody in the invention every possible improvement I can think of and see no fault anywhere, I put into concrete form this final product of my brain. Invariably my device works as I conceived that it should, and the experiment comes out exactly as I planned it. In twenty years there has not been a single exception.

In chapter three of his autobiography, Tesla narrates that the idea for his induction motor "came like a flash of lightning and in an instant the truth was revealed." If ever an inventor could be said to exemplify Douglas Axe's intuitive, top-down conception of invention, it would surely be Nikola Tesla. (I should mention, however, that Tesla, who is believed to have suffered from synaesthesia, states in the first chapter of his autobiography that while still a young man, he came to believe that he was "merely an automaton endowed with power of movement, responding to the stimuli of the sense organs and thinking and acting accordingly" - a conclusion which hardly sits well with Axe's claims about the uniqueness of human insight.)

Human invention is always incremental, after all

Recently, however, I discovered that Professor Hogue had responded to my post some time ago, and refuted my example of Tesla as an intuitive inventor. The reality, contended Hogue, was quite different when Tesla was viewed in the context of his contemporaries:

Tesla's AC motor and radio inventions were incremental improvements and insights completely dependent on his experience with the designs and efforts of his predecessors. Changes Tesla made to existing designs of DC generators, which he did while under the employ of Thomas Edison, were incremental changes to an accumulated complex design...

...If all who contributed ideas to manufacture AC motors were alive today and assembled in one place, Tesla would be lost in a crowd.

Tesla's contributions to radio were also incremental and replaced by better ideas soon after. The critical thinker can find all the relevant information on Wikipedia and Google Patents to see that a body of precedent designs and training led Tesla's mind in the direction of invention. Precedent and existing information, far more than divine inspiration, was the key to his success…

Touche. In retrospect, I am now inclined to agree with Hogue's thesis that human invention never really advances by leaps and bounds, and that it is invariably an incremental process.

Why invention has to be bottom-up as well as top-down, from the get-go

In his book, Axe maintains that the first stage of human invention (the mental stage) is a purely top-down stage, which is followed by a bottom-up construction stage. I have to say that disagree with this picture. What it presupposes, in effect, is that you can reverse-engineer matter from form: in other words, that you can start off with the function you want to create, ask yourself what lower-level functions would be needed to support it, and what subsidiary functions would be required to support those functions, and so on, until you reach some "bedrock level" of basic functionality which can be easily imposed on the raw material you are using. On the contrary, I would contend that you need to be aware of the limitations of your raw material from the very beginning, otherwise you may end up trying to force it to do something which it is incapable of doing. Invention is a process in which human beings need to be simultaneously engaging in top-down and bottom-up thinking, in order to ensure that form and matter meld seamlessly with one another.

A Divine Inventor, creating ex nihilo, would not be faced with this problem. Nevertheless, there would be another reason why such an Inventor would still need to think bottom-up while engaging in top-down design: namely, the problem of under-determination. If you're going to make a man, whether from the dust of the ground, from the body of a hominid, or from nothing at all, you need to ask yourself questions like: how tall do I want him to be? How heavy? What color hair and eyes? What blood type? What DNA sequence? Not even God could command dust to become a man, for that leaves unanswered the question: which man? Design, I would suggest, is never a purely top-down process: it always incorporates the bottom-up, as well.

Why human innovation beats Darwinian evolution, even if it's incremental

There is, however, a flaw in Professor Hogue's own reasoning, which I pointed out in my 2012 post: "Even if human innovation is always incremental, it is nevertheless often directed at long-term goals (e.g. putting a man on Mars, or making a computer that can win at Jeopardy) while Darwinian evolution is never directed at anything long-term." For that very reason, it's inefficient.

Hogue might be tempted to reply by citing Orgel's Second Rule: "Evolution is smarter than you are." He could argue that evolutionary processes are capable of creating designs which no human designer could even conceive of, as ID critic Michael Rumraket Rasmussen has pointed out, and he might add that a computer can beat one of the world's top human players at the game of go - a game which supposedly requires insight (or intuition) in order to be played well. But as science journalists have pointed out, the computer has to play itself millions of times, in order to attain a level of proficiency which a human being can acquire after a few thousand games. So I don't think that a pre-programmed computer which is very slow on the uptake and which doesn't acquire insights as we do, presents much of a threat to human uniqueness.

The design paradox: a problem for Intelligent Design

The real problem, however, with Professor Hogue's thesis that human invention is invariably incremental is that it seems to force Axe to make a terrible choice between arguing for Intelligent Design and arguing for God. It appears that he cannot do both. Let me explain why.

Axe's case for Intelligent Design is built on the assumption that the Designer is very much like a human inventor, who goes through three stages (mental, methodical and mechanical) when searching for new ways of doing things (pp. 136-138). (Although Axe begins by saying he isn't presuming that all invention must proceed the way human invention does, it is hard to see what purpose his exposition of the three stages would serve, if invention could happen in some radically different fashion.) Axe also tells his readers that invention, unlike blind processes, requires knowledge: knowledge of what you are trying to make, and how the various components you are assembling will help you make it. The functional coherence of life explains why invention can't happen by accident (p. 160). "Each and every new form of life," Axe writes, "must therefore be a masterful invention in its own right, embodying its own version of functional coherence at the very highest level" (p. 184).

All well and good: but what if Professor Hogue is correct in arguing that invention is essentially an incremental process? Then we are forced to conclude that the Designer of the cosmos and of life also thinks incrementally, one step at a time. What's more, we'll have to jettison the notion that the Designer is purely a top-down thinker: invention, as we've seen, is both top-down and bottom-up. However, many Jews, Christians and Muslims would adamantly reject the idea that God can only invent incrementally and that He can't design in a purely top-down fashion, as being nothing short of blasphemous. The traditional view of classical theism is that God's knowledge is non-discursive, and that He does not need to reason His way from premises to conclusions. Nor does He need to invent one step at a time: He can see the goal from the very beginning. And because His ideas are said to be simple (i.e. devoid of parts), His idea of whatever He wants to invent is supposedly very simple, too, which means that He understands things in a purely top-down manner.

We might seek to avoid the conclusion that God thinks bottom-up and incrementally, by supposing that God is an Inventor like no other, and that His Mind is utterly unlike our own. Classical theists envisage God in precisely these terms. God, they say, does not belong to any genus: He is sui generis. But if we adopt this portrait of God, then there is no hope of our being able to reason our way up to God by drawing upon analogies with human inventors, as Axe does (pp. 136-138). Axe's entire argument for Design presupposes that the Designer is very like us, psychologically: He is "someone who invested [in the cosmos] not just intellectually but also emotionally, just as we invest in our creations" (p. 250), and He has a personality as well (p. 250). But if God's Mind is not at all like ours, then Axe's argument can never take us to the Mind of God. It will always fall short.

How could Axe resolve a dilemma like this? Since his portrait of God is very personalistic, I would recommend that he try embracing the first option, and openly acknowledge that God does design things incrementally, and that He does reason His way from premises to a conclusion. (I defended the view that God's knowledge is discursive, in a blog post back in 2014.) This may not be as shocking as it first seems: the fact that there is a logical progression in God's thoughts does not mean that there needs to be a temporal progression, or that God requires lots of time, in order to think things out. A God Who thinks step-by-step could still be outside time. Interestingly, if this picture of God is correct, it would explain why He had to resort to the process of (guided) evolution in order to invent us, instead of just making us from the dust of the ground (say). Maybe God can't make a man that way. Maybe the very notion of making any functionally coherent system that way is logically absurd. But if that's right, then God couldn't have made life that way either: He must have designed it by a stepwise process. And if that's correct, then Axe shouldn't be arguing against stepping stones: he should be arguing for them. There must have been a series of steps leading to the first living cell. In that case, the task of Intelligent Design would be to show that these were guided steps. How would one do that? Presumably, by showing that they were the fastest set of steps that would take us from simple chemicals to life itself.

I discussed the question of whether God has to think in a bottom-up as well as a top-down fashion, in a blog post two years ago. All I'll say here is that imputing complex ideas to God does not detract from the traditional doctrine that God's innermost Being (or essence) is absolutely simple, even if His thoughts are not.

I'd like to conclude by quoting from Professor James Tour's talk, "The Origin of Life: An Inside Story."

As Professor Tour pointed out, what makes the puzzle of life’s origin all the more baffling is that even if you had a "Dream Team" of brilliant chemists and gave them all the ingredients they wanted, they would still have no idea how to assemble a simple cell:

All right, now let's assemble the Dream Team. We've got good professors here, so let's assemble the Dream Team. Let's further assume that the world's top 100 synthetic chemists, top 100 biochemists and top 100 evolutionary biologists combined forces into a limitlessly funded Dream Team. The Dream Team has all the carbohydrates, lipids, amino acids and nucleic acids stored in freezers in their laboratories... All of them are in 100% enantiomer purity. [Let's] even give the team all the reagents they wish, the most advanced laboratories, and the analytical facilities, and complete scientific literature, and synthetic and natural non-living coupling agents. Mobilize the Dream Team to assemble the building blocks into a living system - nothing complex, just a single cell. The members scratch their heads and walk away, frustrated...

So let's help the Dream Team out by providing the polymerized forms: polypeptides, all the enzymes they desire, the polysaccharides, DNA and RNA in any sequence they desire, cleanly assembled. The level of sophistication in even the simplest of possible living cells is so chemically complex that we are even more clueless now than with anything discussed regarding prebiotic chemistry or macroevolution. The Dream Team will not know where to start. Moving all this off Earth does not solve the problem, because our physical laws are universal.

You see the problem for the chemists? Welcome to my world. This is what I’m confronted with, every day.

What the Dream Team story seems to suggest is that life built, even by a team of very intelligent agents, by a process of reasoning backwards from its higher-level functions to the lower-level components (or ingredients) required to support those functions, which would undercut the whole case for top-down Design. That leaves us with two choices. We could say that life was designed, but that the Designer used a stepwise process to create life, which is why a high-level, purely top-down approach by a team of chemists is doomed to failure.

Alternatively, we could say that God is not a Designer at all, but more of an Author, and that He somehow created life holus bolus - which means that Intelligent Design movement (most of whose members believe in God) has an identity problem: it would have to rechristen itself the Intelligent Author movement.

Which option is correct? Honestly, I don't know. But I do know that the field of Intelligent Design is in for an intellectual shake-up over the next few years. Somehow, it will have to wrestle with these questions and figure out its own approach. Perhaps it might be best if the Intelligent Design movement splits into two wings: an "incremental design" wing, which tries to build life in a manner that combines top-down and bottom-up thinking; and an "ideal" design wing, which envisages life in a more Platonic fashion, as an Idea in the Mind of an Author. Time will tell which approach turns out to be more fruitful. I would suspect that front-loaders and believers in common descent in the ID camp would tend to favor the "incremental design" approach, while Old Earth and Young Earth creationists would favor the "ideal" approach. But a little friendly competition might reinvigorate the movement.

Before I close, I'd like to give Dr. Axe a little tip which he might find useful in future: when writing a book, don't ask your friends or people who think like you to read the manuscript and offer helpful comments; instead, ask tough-minded but fair critics, whose viewpoint is opposed to your own, to preview your book, and comment as they see fit.

Let me conclude by wishing Dr. Axe every success with the revision of his book. He is a superb writer, and I hope he will incorporate some of my criticisms, and write a new version of his book that dazzles readers even more than the current one.

What do readers think? Over to you.