Even better would have been "irreducible functionality", since the real issue that Behe fixates on is not "complexity", per se (since even something as simple as a mousetrap can be declared "IC"), but functionality itself (or the lack thereof).
Nonetheless, whatever he chooses to call it, the notion still suffers from the many flaws that have already been pointed out.
But if we insert the adjective “functional” – i.e. “functional complexity” - then we do have a new meaning, both in math and biology:
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...
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.