I just listened to a talk given by Virginia Chaitin that can be found on academia.edu. The title of the talk is *A philosophical perspective on a metatheory of biological evolution*. In it she outlines Gregory Chaitin’s work on metabiology, which has been the subject of some of my previous posts – here, here, and here. But, since the emphasis of the talk is on the philosophical implications of the theory, I became particularly aware of what metabiology may be saying about mathematics. It is, after all, the mathematics that effects the paradigm shifts that bring about alternative philosophies.

Metabiology develops as a way to answer the question of whether or not one can prove mathematically that evolution, through random mutations and natural selection, is capable of producing the diversity of life forms that exist today. This, in itself, shines the light on mathematics. Proof is a mathematical idea. And in the Preface of Gregory Chaitin’s *Proving Darwin* he articulates the bigger idea:

The purpose of this book is to lay bare the deep inner mathematical structure of biology,

to show life’s hidden mathematical core. (emphasis my own)

And so it would seem clear that metabiology is as much about mathematics as it is about biology, perhaps more so.

Almost every discussion of metabiology addresses some of its philosophical implications, but there are points made in this talk that speak more directly to mathematics’ role in metabiology’s paradigm shifts. For example, Chaitin (Virginia) begins by stressing that metabiology makes *different use* of mathematics. By this she means that mathematics is not being used to model evolution, but to explore it, crack it open. Results in mathematics suggest a view that removes some of the habitual thoughts associated with what we think we see. She explains that this is possible because metabiology takes advantage of aspects of mathematics *that are not widely known or taught* – like its logical irreducibility and quasi-experimental nature. She also explains that exploratory strategies can combine or interweave computable and uncomputable steps. And so mathematics here is not being used ‘instrumentally,’ but as a way to express the creativity of evolution by way of its own creative nature. These strategies are some of the consequences of Gödel’s and Turing’s insights.

The fact that metabiology relies so heavily on a post-Gödel and post-Turing understanding of mathematics and computability, puts a spotlight on the depth and significance of these insights, and perhaps points to some yet to be discovered implications of incompleteness. I continue to find it particularly interesting that while both Gödel’s incompleteness theorems and Turing’s identification of the halting problem look like they are pointing to limitations within their respective disciplines, in metabiology, they each clearly inspire new biological paradigms, that could very well lead to new science. Metabiology affirms that our ideas concerning incompleteness, and uncomputability provide insights into nature as well as mathematics and computation. And so these important results from Gödel and Turing describe, not the limitations of mathematics or computers, but the limitations of a perspective, the limitations of a mechanistic point of view. Proofs, Chaitin tells us, are used in metabiology to generate and express novelty. And this is what nature does.

The talk makes the necessary alignment of biology with metabiology and, for the sake of thoroughness, I’ll repeat them here:

- Biology deals in natural software (DNA and RNA) while metabiological software is a computer program.
- In biology, organisms result from processes involving DNA, RNA and the environment while in metabiology the organism is the software itself.
- In biology evolution increases the sophistication of biological lifeforms, while in metabiology evolution is the increase in the information content of algorithmic life forms.
- The challenge to an organism in nature is to survive and adapt, while the challenge in metabiology is to solve a mathematical problem that requires creative uncomputable steps.
- In metabiology evolution is defined by an increase in the information content of an algorithmic life form, and fitness is understood as the growth of conceptual complexity. These are mathematical ideas, and together they create a lense through which we can view evolution.

In a recent paper on conceptual complexity and algorithmic information theory, Gregory Chaitin defines conceptual complexity in this way:

In this essay we define the conceptual complexity of an object X to be the size in bits of the most compact program for calculating X, presupposing that we have picked as our complexity standard a particular fixed, maximally compact, concise universal programming language

U. This is technically known as the algorithmic information content of the object X, denoted H_{u}(X) or simply H(X) sinceUis assumed fixed. In medieval terms, H(X) is the minimum number of yes/no decisions that God would have to make to create X.

Biological creativity, here, becomes associated with mathematical creativity and is understood as the generation of novelty, which is further understood as the generation of new information content. Virginia Chaitin also tells us that metabiology proposes a hybrid theory that relies on computability (something that can be understood mechanically) and uncomputability (something that cannot).

It seems to me that the application of incompleteness, uncomputability, and undecidability, in any context, serves to prune the mechanistic habits that have grown over the centuries in the sciences, as well as the habits of logic that are thought to lead to true things.

The need to address these issues can be seen even in economics, as explored in a 2008 paper I happened upon by K. Vela Velupillai presented at an International Conference on Unconventional Computation. The abstract of the paper says this:

Economic theory, game theory and mathematical statistics have all increasingly become algorithmic sciences. Computable Economics, Algorithmic Game Theory [Noam Nisan, Tim Roiughgarden, Éva Tardos, Vijay V. Vazirani (Eds.), Algorithmic Game Theory, Cambridge University Press, Cambridge, 2007] and Algorithmic Statistics [Péter Gács, John T. Tromp, Paul M.B. Vitányi, Algorithmic statistics, IEEE Transactions on Information Theory 47 (6) (2001) 2443–2463] are frontier research subjects. All of them, each in its own way, are underpinned by (classical) recursion theory – and its applied branches, say computational complexity theory or algorithmic information theory – and, occasionally, proof theory. These research paradigms have posed new mathematical and metamathematical questions and, inadvertently, undermined the traditional mathematical foundations of economic theory. A concise, but partial, pathway into these new frontiers is the subject matter of this paper.

Mathematical physicist John Baez writes about computability, uncomputability, logic, probability, and truth in a series of posts found here. They’re worth a look.

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