Has it not crossed Jeffrey Shallit's mind that Phillip Johnson is aware of those examples but doesn't find the case for the intermediate or transitional nature of those fossils very convincing? -- Jonathan M. on Shallit's claim that Phillip Johnson lied about the fossil recordSome notions of information are more closely tied to the Shannon notion of information (e.g. Thomas Schneider and Richard Dawkins) and tend not to wander much further. Much more closely related to some notion of meaning/function has been the Kolmogorov notion of information, also known as Kolmogorov complexity. Complexity and information tend to be closely related notions; both are related to probability, and probability has a combinatorial nature. The idea of mutual information is closely related to that of conditional probability. A complex thing generally needs more information to describe it (or information and time, but that will have to wait), while a much simpler thing generally required much less information. Highly improbable things tend to have many levels of contingencies, and convoluted histories/developments. Optimal codes match less probable events to more bits of information. An optimal code will tend to not only maximize the Shannon complexity of the messages but to also make the messages more compressed (maximizing K complexity per message length).
Jeffrey Shallit has become something of a public figure, not for math or computer science per se, but for being a "science defender" of the quixotic "anticreationist" variety. (Note: I have noticed Shallit also uses the term "anticreationist," though not in the ironic sense that I do.) He has made a name for himself ridiculing ideas and the people who promote them. He has complained in the past that 'specified complexity' would be better off called 'specified improbability.' His and Elsberry's long critique of the idea of 'complex specified information' scoffs at the difficulty in reducing CSI neatly to one of the two kinds of information . (Everything else is "creationist information.") Winston Ewert's notion of algorithmic specified complexity is very closely related to Dembski's notions of CSI and is itself a specie of randomness deficiency, combining notions of K-complexity and probability.
Something that is ubiquitous among complexity theorists and attempts to quantify 'biological information' is notably absent from Shallit's writings on the topic of complexity. It's what Jim Crutchfield calls 'humpology'. Most of the attempts over the last 20 years have tried to distinguish 'useful' information/complexity from both highly ordered and highly disordered configurations. Kolmogorov information considers highly disordered configurations to be provide more information, but most complexity theory tries to distinguish between useful information (what engineers typically mean by 'information': semantically "rich" information that carries some deep patterns) and the information maximized by total randomness. This fundamental concern in complexity theory doesn't seem to concern Shallit at all (which is odd given reasons that we'll get to).

