Maths vs. Darwin.

 Marcel-Paul Schützenberger

Introduction

Until his death, the mathematician and doctor of medicine Marcel-Paul Schützenberger (1920-1996) was Professor of the Faculty of Sciences at the University of Paris and a member of the Academy of Sciences. [See "From the Editors" for additional biographical information.] In 1966, Schützenberger participated in the Wistar Symposium on mathematical objections to neo-Darwinism. His arguments were subtle and often misunderstood by biologists. Darwin's theory, he observed, and the interpretation of biological systems as formal objects, were at odds insofar as randomness is known to degrade meaning in formal contexts. But Schützenberger also argued that Darwin's theory logically required some active principle of coordination between the typographic space of the informational macromolecules (DNA and RNA) and the organic space of living creatures themselves -- which Darwin's theory does not provide. In this January 1996 interview with the French science monthly La Recherche, here published in English for the first time, he pursued these themes anew, finding inspiration for his ideas both in the mathematical ideas that he had pioneered and in the speculative tradition of French biological thought that stretched from Georges Cuvier to Lucien Cuenot. M.P. Schützenberger was a man of universal curiosity and great wit; throughout his life, he was both joyful and unafraid. The culture that he so brilliantly represented disappears with him, of course. It was his finest invention and it now belongs to the inventory of remembered things.

---

Q: What is your definition of Darwinism?

S: The most current, of course, a position generically embodied, for example, by Richard Dawkins. The essential idea is well-known. Evolution, Darwinists argue, is explained by the double action of chance mutations and natural selection. The general doctrine embodies two mutually contradictory schools -- gradualists, on the one hand, saltationists, on the other. Gradualists insist that evolution proceeds by means of small successive changes; saltationists that it proceeds by jumps. Richard Dawkins has come to champion radical gradualism; Stephen Jay Gould, a no less radical version of saltationism.

Q: You are known as a mathematician rather than a specialist in evolutionary biology...

S: Biology is, of course, not my specialty. The participation of mathemeticians in the overall assessment of evolutionary thought has been encouraged by the biologists themselves, if only because they presented such an irresistible target. Richard Dawkins, for example, has been fatally attracted to arguments that would appear to hinge on concepts drawn from mathematics and from the computer sciences, the technical stuff imposed on innocent readers with all of his comic authority. Mathematicians are, in any case, epistemological zealots. It is normal for them to bring their critical scruples to the foundations of other disciplines. And finally, it is worth observing that the great turbid wave of cybernetics has carried mathematicians from their normal mid-ocean haunts to the far shores of evolutionary biology. There up ahead, Rene Thom and Ilya Prigogine may be observed paddling sedately toward dry land, members of the Santa Fe Institute thrashing in their wake. Stuart Kauffman is among them. An interesting case, a physician half in love with mathematical logic, burdened now and forever by having received a Papal Kiss from Murray Gell-Mann. This ecumenical movement has endeavored to apply the concepts of mathematics to the fundamental problems of evolution -- the interpretation of functional complexity, for example.

Q: What do you mean by functional complexity?

S: It is impossible to grasp the phenomenon of life without that concept, the two words each expressing a crucial and essential idea. The laboratory biologists' normal and unforced vernacular is almost always couched in functional terms: the function of an eye, the function of an enzyme, or a ribosome, or the fruit fly's antennae -- their function; the concept by which such language is animated is one perfectly adapted to reality. Physiologists see this better than anyone else. Within their world, everything is a matter of function, the various systems that they study -- circulatory, digestive, excretory, and the like -- all characterized in simple, ineliminable functional terms. At the level of molecular biology, functionality may seem to pose certain conceptual problems, perhaps because the very notion of an organ has disappeared when biological relationships are specified in biochemical terms; but appearances are misleading, certain functions remaining even in the absence of an organ or organ systems. Complexity is also a crucial concept. Even among unicellular organisms, the mechanisms involved in the separation and fusion of chromosomes during mitosis and meiosis are processes of unbelieveable complexity and subtlety. Organisms present themselves to us as a complex ensemble of functional interrelationships. If one is going to explain their evolution, one must at the same time explain their functionality and their complexity.

Q: What is it that makes functional complexity so difficult to comprehend?

S: The evolution of living creatures appears to require an essential ingredient, a specific form of organization. Whatever it is, it lies beyond anything that our present knowledge of physics or chemistry might suggest; it is a property upon which formal logic sheds absolutely no light. Whether gradualists or saltationists, Darwinians have too simple a conception of biology, rather like a locksmith improbably convinced that his handful of keys will open any lock. Darwinians, for example, tend to think of the gene rather as if it were the expression of a simple command: do this, get that done, drop that side chain. Walter Gehring's work on the regulatory genes controlling the development of the insect eye reflects this conception. The relevant genes may well function this way, but the story on this level is surely incomplete, and Darwinian theory is not apt to fill in the pieces.

Q: You claim that biologists think of a gene as a command. Could you be more specific?

S: Schematically, a gene is like a unit of information. It has simple binary properties. When active, it is an elementary information-theoretic unit, the cascade of gene instructions resembling the cascade involved in specifying a recipe. Now let us return to the example of the eye. Darwinists imagine that it requires what? A thousand or two thousand genes to assemble an eye, the specification of the organ thus requiring one or two thousand units of information? This is absurd! Suppose that a European firm proposes to manufacture an entirely new household appliance in a Southeast Asian factory. And suppose that for commercial reasons, the firm does not wish to communicate to the factory any details of the appliance's function -- how it works, what purposes it will serve. With only a few thousand bits of information, the factory is not going to proceed very far or very fast. A few thousand bits of information, after all, yields only a single paragraph of text. The appliance in question is bound to be vastly simpler than the eye; charged with its manufacture, the factory will yet need to know the significance of the operations to which they have committed themselves in engaging their machinery. This can be achieved only if they already have some sense of the object's nature before they undertake to manufacture it. A considerable body of knowledge, held in common between the European firm and its Asian factory, is necessary before manufacturing instructions may be executed.

Q: Would you argue that the genome does not contain the requisite information for explaining organisms?

S:Not according to the understanding of the genome we now possess. The biological properties invoked by biologists are in this respect quite insufficient; while biologists may understand that a gene triggers the production of a particular protein, that knowledge -- that kind of knowledge -- does not allow them to comprehend how one or two thousand genes suffice to direct the course of embryonic development.

Q: You are going to be accused of preformationism...

S: And of many other crimes. My position is nevertheless strictly a rational one. I've formulated a problem that appears significant to me: how is it that with so few elementary instructions, the materials of life can fabricate objects that are so marvelously complicated and efficient? This property with which they are endowed -- just what is its nature? Nothing within our actual knowledge of physics and chemistry allows us intellectually to grasp it. If one starts from an evolutionary point of view, it must be acknowledged that in one manner or another, the earliest fish contained the capacity, and the appropriate neural wiring, to bring into existence organs which they did not possess or even need, but which would be the common property of their successors when they left the water for the firm ground, or for the air.

Q: You assert that, in fact, Darwinism doesn't explain much.

S: It seems to me that the union of chance mutation and selection has a certain descriptive value; in no case does the description count as an explanation. Darwinism relates ecological data to the relative abundance of species and environments. In any case, the descriptive value of Darwinian models is pretty limited. Besides, as saltationists have indicated, the gradualist thesis seems completely demented in light of the growth of paleontological knowledge. The miracles of saltationism, on the other hand, cannot discharge the mystery I have described.

Q: Let's return to natural selection. Isn't it the case that despite everything the idea has a certain explanatory value?

S: No one could possibly deny the general thesis that stability is a necessary condition for existence -- the real content of the doctrine of natural selection. The outstanding application of this general principle is Berthollet's laws in elementary chemistry. In a desert, the species that die rapidly are those that require water the most; yet that does not explain the appearance among the survivors of those structures whose particular features permits them to resist aridity. The thesis of natural selection is not very powerful. Except for certain artificial cases, we are yet unable to predict whether this or that species or this or that variety will be favored or not as the result of changes in the environment. What we can do is establish after the fact the effects of natural selection -- to show, for, example that certain birds are disposed to eat this species of snails less often than other species, perhaps because their shell is not as visible. That's ecology: very interesting. To put it another way, natural selection is a weak instrument of proof because the phenomena subsumed by natural selection are obvious and yet they establish nothing from the point of view of the theory.

Q: Isn't the significant explanatory feature of Darwinian theory the connection established between chance mutations and natural selection?

S:With the discovery of coding, we have come to understand that a gene is like a word composed in the DNA alphabet; such words form the genomic text. It is that word that tells the cell to make this or that protein. Either a given protein is structural, or a protein itself works in combination with other signals given by the genome to fabricate yet another protein. All the experimental results we know fall within this scheme. The following scenario then becomes standard. A gene undergoes a mutation, one that may facilitate the reproduction of those individuals carrying it; over time, and with respect to a specific environment, mutants come to be statistically favored, replacing individuals lacking the requisite mutation. Evolution could not be an accumulation of such typographical errors. Population geneticists can study the speed with which a favorable mutation propagates itself under these circumstances. They do this with a lot of skill, but these are academic exercises if only because none of the parameters that they use can be empirically determined. In addition, there are the obstacles I have already mentioned. We know the number of genes in an organism. There are about one hundred thousand for a higher vertebrate. This we know fairly well. But this seems grossly insufficient to explain the incredible quantity of information needed to accomplish evolution within a given line of species.

Q: A concrete example?

S: Darwinists say that horses, which were once mammals as large as rabbits, increased their size to escape more quickly from predators. Within the gradualist model, one might isolate a specific trait -- increase in body size -- and consider it to be the result of a series of typographic changes. The explanatory effect achieved is rhetorical, imposed entirely by trick of insisting that what counts for a herbivore is the speed of its flight when faced by a predator. Now this may even be partially true, but there are no biological grounds that permit us to determine that this is in fact the decisive consideration. After all, increase in body size may well have a negative effect. Darwinists seem to me to have preserved a mechanic vision of evolution, one that prompts them to observe merely a linear succession of causes and effects. The idea that causes may interact with one another is now standard in mathematical physics; it is a point that has had difficulty in penetrating the carapace of biological thought. In fact, within the quasi-totality of observable phenomena, local changes interact in a dramatic fashion; after all, there is hardly an issue of La Recherche that does not contain an allusion to the Butterfly Effect. Information theory is precisely the domain that sharpens our intuitions about these phenomena. A typographical change in a computer program does not change it just a little. It wipes the program out, purely and simply. It is the same with a telephone number. If I intend to call a correspondent by telephone, it doesn't much matter if I am fooled by one, two, three or eight figures in his number.

Q: You accept the idea that biological mutations genuinely have the character of typographical errors?

S: Yes, in the sense that one base is a template for another, one codon for another, but at the level of biochemical activity, one is no longer able properly to speak of typography. There is an entire grammar for the formation of proteins in three dimensions, one that we understand poorly. We do not have at our disposal physical or chemical rules permitting us to construct a mapping from typographical mutations or modifications to biologically effective structures. To return to the example of the eye: a few thousand genes are needed for its fabrication, but each in isolation signifies nothing. What is significant is the combination of their interactions. These cascading interactions, with their feedback loops, express an organization whose complexity we do not know how to analyze (See Figure 1). It is possible we may be able to do so in the future, but there is no doubt that we are unable to do so now. Gehring has recently discovered a segment of DNA which is both involved in the development of the vertebrate eye and which can induce the development of an eye in the wing of a butterfly. His work comprises a demonstration of something utterly astonishing, but not an explanation.

Q:But Dawkins, for example, believes in the possibility of a cumulative process.

S: Dawkins believes in an effect that he calls "the cumulative selection of beneficial mutations." To support his thesis, he resorts to a metaphor introduced by the mathematician Emile Borel -- that of a monkey typing by chance and in the end producing a work of literature. It is a metaphor, I regret to say, embraced by Francis Crick, the co-discoverer of the double helix. Dawkins has his computer write a series of thirty letters, these corresponding to the number of letters in a verse by Shakespeare. He then proceeds to simulate the Darwinian mechanism of chance mutations and selection. His imaginary monkey types and retypes the same letters, the computer successively choosing the phrase that most resembles the target verse. By means of cumulative selection, the monkey reaches its target in forty or sixty generations.

Q: But you don't believe that a monkey typing on a typewriter, even aided by a computer...

S:This demonstration is a trompe-l'oeil, and what is more, Dawkins doesn't describe precisely how it proceeds. At the beginning of the exercise, randomly generated phrases appear rapidly to approach the target; the closer the approach, the more the process begins to slow. It is the action of mutations in the wrong direction that pulls things backward. In fact, a simple argument shows that unless the numerical parameters are chosen deliberately, the progression begins to bog down completely.

Q:You would say that the model of cumulative selection, imagined by Dawkins, is out of touch with palpable biological realities?

S: Exactly. Dawkins's model lays entirely to the side the triple problems of complexity, functionality, and their interaction.

Q: You are a mathematician. Suppose that you try, despite your reservations, to formalize the concept of functional complexity...

S: I would appeal to a notion banned by the scientific community, but one understood perfectly by everyone else -- that of a goal. As a computer scientist, I could express this in the following way. One constructs a space within which one of the coordinates serves in effect as the thread of Ariane, guiding the trajectory toward the goal. Once the space is constructed, the system evolves in a mechanical way toward its goal. But look, the construction of the relevant space cannot proceed until a preliminary analysis has been carried out, one in which the set of all possible trajectories is assessed, this together with an estimation of their average distance from the specified goal. The preliminary analysis is beyond the reach of empirical study. It presupposes -- the same word that seems to recur in theoretical biology -- that the biologist (or computer scientist) know the totality of the situation, the properties of the ensemble of trajectories. In terms of mathematical logic, the nature of this space is entirely enigmatic. Nonetheless, it is important to remember that the conceptual problems we face, life has entirely solved; the systems embodied in living creatures are entirely successful in reaching their goals. The trick involved in Dawkin's somewhat sheepish example proceeds via the surreptitious introduction of a relevant space. His computer program calculates from a random phrase to a target, a calculation corresponding to nothing in biological reality. The function that he employs flatters the imagination, however, because it has that property of apparent simplicity that elicits naïve approval. In biological reality, the space of even the simplest function has a complexity that defies understanding, and indeed, defies any and all calculations.

Q: Even when they dissent from Darwin, the saltationists are more moderate: they don't pretend to hold the key that would permit them to explain evolution...

S: Before we discuss the saltationists, however, I must say a word about the Japanese biologist Mooto Kimura. He has shown that the majority of mutations are neutral, without any selective effect. For Darwinians upholding the central Darwinian thesis, this is embarrassing... The saltationist view, revived by Stephen Jay Gould, in the end represents an idea due to Richard Goldschmidt. In 1940 or so, he postulated the existence of very intense mutations, no doubt involving hundreds of genes, and taking place rapidly, in less than one thousand generations, thus below the threshold of resolution of paleontology. Curiously enough, Gould does not seem concerned to preserve the union of chance mutations and selection. The saltationists run afoul of two types of criticism. On the one hand, the functionality of their supposed macromutations is inexplicable within the framework of molecular biology. On the other hand, Gould ignores in silence the great trends in biology, such as the increasing complexity of the nervous system. He imagines that the success of new, more sophisticated species, such as the mammals, is a contingent phenomenon. He is not in a position to offer an account of the essential movement of evolution, or at the least, an account of its main trajectories. The saltationists are thus reduced to invoking two types of miracles: macromutations, and the great trajectories of evolution.

Q: In what sense are you employing the word 'miracle'?

S:A miracle is an event that should appear impossible to a Darwinian in view of its ultra-cosmological improbability within the framework of his own theory. Now speaking of macromutations, let me observe that to generate a proper elephant, it will not suffice suddenly to endow it with a full-grown trunk. As the trunk is being organized, a different but complementary system -- the cerebellum -- must be modified in order to establish a place for the ensemble of wiring that the elephant will require to use his trunk. These macromutations must be coordinated by a system of genes in embryogenesis. If one considers the history of evolution, we must postulate thousands of miracles; miracles, in fact, without end. No more than the gradualists, the saltationists are unable to provide an account of those miracles. The second category of miracles are directional, offering instruction to the great evolutionary progressions and trends -- the elaboration of the nervous system, of course, but the internalization of the reproductive process as well, and the appearance of bone, the emergence of ears, the enrichment of various functional relationships, and so on. Each is a series of miracles, whose accumulation has the effect of increasing the complexity and efficiency of various organisms. From this point of view, the notion of bricolage [tinkering], introduced by Francois Jacob, involves a fine turn of phrase, but one concealing an utter absence of explanation.

Q: The appearance of human beings -- is that a miracle, in the sense you mean?

S: Naturally. And here it does seem that there are voices among contemporary biologists -- I mean voices other than mine -- who might cast doubt on the Darwinian paradigm that has dominated discussion for the past twenty years. Gradualists and saltationists alike are completely incapable of giving a convincing explanation of the quasi-simultaneous emergence of a number of biological systems that distinguish human beings from the higher primates: bipedalism, with the concomitant modification of the pelvis, and, without a doubt, the cerebellum, a much more dexterous hand, with fingerprints conferring an especially fine tactile sense; the modifications of the pharynx which permits phonation; the modification of the central nervous system, notably at the level of the temporal lobes, permitting the specific recognition of speech. From the point of view of embryogenesis, these anatomical systems are completely different from one another. Each modification constitutes a gift, a bequest from a primate family to its descendants. It is astonishing that these gifts should have developed simultaneously. Some biologists speak of a predisposition of the genome. Can anyone actually recover the predisposition, supposing that it actually existed? Was it present in the first of the fish? The reality is that we are confronted with total conceptual bankruptcy.

Q:You mentioned the Santa Fe school earlier in our discussion. Do appeals to such notions as chaos...

S:I should have alluded to a succession of highly competent people who have discovered a number of poetic but essentially hollow forms of expression. I am referring here to the noisy crowd collected under the rubric of cybernetics; and beyond, there lie the dissipative structures of Prigogine, or the systems of Varela, or, moving to the present, Stuart Kauffman's edge of chaos -- an organized form of inanity that is certain soon to make its way to France. The Santa Fe school takes complexity to apply to absolutely everything. They draw their representative examples from certain chemical reactions, the pattern of the sea coast, atmosphere turbulence, or the structure of a chain of mountains. The complexity of these structures is certainly considerable, but in comparison with the living world, they exhibit in every case an impoverished form of organization, one that is strictly non-functional. No algorithm allows us to understand the complexity of living creatures, this despite these examples, which owe their initial plausibility to the assumption that the physico-chemical world exhibits functional properties that in reality it does not possess.

Q: Should one take your position as a statement of resignation, an appeal to have greater modesty, or something else altogether?

S: Speaking ironically, I might say that all we can hear at the present time is the great anthropic hymnal, with even a number of mathematically sophisticated scholars keeping time as the great hymn is intoned by tapping their feet. The rest of us should, of course, practice a certain suspension of judgment.