Editor’s note: We are delighted to present a new series by William Dembski, adapted from his Substack. This is the 18th post. Find the full series so far here, “My Personal History with Information.”
In 2023, when the second edition of The Design Inference was published, I would have said that all the important results about conservation of information were in place and that I would have been in a position to write a book on the topic simply on the basis of existing work that my colleagues and I had done. And I would have been wrong.
Conservation of information as I had developed it was a mathematical result that applied to search but where the actual mathematics had to be adapted to how the search itself was represented mathematically. There were thus different conservation of information theorems. There was a measure-theoretic version, as in my searching large spaces paper from 2005. There were also fitness- and function-theoretic versions, as stated and proved in a paper I wrote with Bob Marks titled “Life’s Conservation Law: Why Darwinian Evolution Cannot Create Biological Information,” available as a pdf here and published as a chapter in Bruce L. Gordon and William A. Dembski, eds., The Nature of Nature: Examining the Role of Naturalism in Science (Wilmington, DE: ISI Books, 2011). It may be my favorite information-theoretic paper from that period and, in my view, it is still worth reading.
A Meta-Mathematical Result
Because there were so many conservation of information theorems, what I called the Law of Conservation of Information ended up not being a strictly mathematical result but more a meta-mathematical result, in the spirit of the Church-Turing Thesis. As explained in “Life’s Conservation Law”:
The Law of Conservation of Information [LCI], in both scope and function, parallels the Church-Turing Thesis of theoretical computer science. The Church-Turing Thesis is a deep claim in mathematical logic about the nature of computation. LCI functions within the theory of intelligent design as the Church-Turing Thesis functions within the theory of computation. According to this thesis, given a procedure that is informally computable (i.e., that is finitely specifiable and rule-based), it can be coded as an algorithm running on a Turing machine. According to LCI, given an alternative search that does better than blind search, the alternative search was purchased at an information cost no less than I+. The task of the design theorist in that case is to “follow the information trail” and show where the information that this search outputs in locating a target was first inputted (much as the task of the computer scientist is to show how some procedure that is informally computable can be explicitly formulated as an algorithm capable of being run on a Turing machine).
It follows that there is no — and indeed can be no — strict mathematical proof of either the Church-Turing Thesis or LCI. Nonetheless, the two are subject to independent verification. With the Church-Turing Thesis, the challenge is to show that any operation that is informally computable is also, upon a close mathematical analysis, formally computable (to date, this has always been confirmed). Likewise with LCI, the challenge is to find where the information that improves an alternative search over a blind search was inputted and then to calculate the amount of this information, showing that it does not fall below I+ = log(q/p) (to date, this has also always been confirmed).
Until November 2024, that was my position. Come up with any reasonable mathematical representation of a search, and a (indefinite article) conservation of information theorem can be proven for it and shown to apply to it. But then in November 2024, I was in San Diego at an Evangelical Philosophical Society symposium on intelligent design moderated by John Bloom of Biola University. There both Doug Axe and I gave presentations.
An Arrow and an Archer
At a moment in his talk that proved critical for me, Doug gave an example in which an event E denotes an arrow shot by an archer hitting a target and an event F denotes a wind blowing the arrow to the target. The events E and F don’t belong to the same probability spaces, E to a space characterizing an archer’s shooting, F to a space characterizing wind blowing. And yet, by thinking of F as a search for a search, I suddenly realized that the Law of Conservation of Information was not meta-mathematics in the sense of Church-Turing, but a direct consequence of a very simple probabilistic relationship, one I’ve dubbed the conservation of information inequality:
This inequality is a very elementary probabilistic result. And yet, because E and F belong to different probability spaces, this inequality needs to be interpreted in a way that allows probabilities to be applied to E and F in concert. As it is, Bayesian probabilities do just this in case of E varying over events and F varying over hypotheses. But if we think of events as hitting targets and hypotheses as characterizing search for a search, then this inequality captures the fundamental claim of conservation of information.
Accordingly, if we think of E as hitting a given target and having small probability, and if we think of F as a search for a search that makes it highly probable that the target will be found, then F must itself be highly improbable. All of this is captured in this inequality, which then ensures that conservation of information is a result of pure mathematics and not merely a type of mathematical result that has been widely verified in specific cases and has to date escaped all counterexamples. It is thus a law in the strongest possible sense.
A Nontrivial Result
There are a lot of details here, and the result is nontrivial even though the conservation of information inequality looks utterly elementary and straightforward. I wrote all of this up in a monograph published in BIO-Complexity that appeared in August 2025. Here is the abstract for that monograph. I include this abstract here because it recaps much of our discussion about conservation of information in this history.
Conservation of information sparked scientific interest once a recurring pattern was noticed in the evolutionary computing literature. In grappling with the creation of information through evolutionary algorithms, this literature consistently revealed that the information outputted by such algorithms always needed first to be programmed into them. Thus, the primary goal of this literature — to uncover how information could be created from scratch or de novo — was shown to be misconceived: the information was not created but instead shuffled around or smuggled in, implying that it already existed in some form or other. Information output in these situations therefore always presupposed a counterbalancing input of prior information.
Once this pattern was seen, the next logical step was to quantify the amount of information inputted and outputted, demonstrating a consistent mathematical relation between the two. This led to the proof of a number of theorems about search. In these theorems, a baseline search with probability p of success gave way to an improved search with probability q of success. Typically p would be very small and close to zero, implying a practically impossible search (like searching for a needle in a haystack). By contrast, q would be much larger and close to one, implying an eminently doable search. The punchline of these theorems was that, as the improved search became itself the subject of a search (a search for a search, or S4S), the probability of finding it could not exceed p/q, rendering success of the improved search no more probable than success of the original baseline search, in effect filling one hole by digging another.
Such conservation-of-information theorems, as they came to be called, were search-space specific, adapted to different kinds of search across a range of search spaces. There was a measure-theoretic theorem in which probability measures guided search. There were also function-theoretic and fitness-theoretic theorems where mappings into the search space as well as fitness functions on the search space respectively guided search. The key insight of this paper is that all these conservation-of-information theorems are special cases of a simple probabilistic relation based on elementary probability theory. This paper identifies the underlying rationale that makes all the previous conservation-of-information theorems work. In so doing, it provides a straightforward proof and general formulation of what may rightly be called the Law of Conservation of Information.
Next, “Recent Work: A Theory of Intelligence Metrics.”