This article by Brian Miller is reprinted with permission from Evolution News.

The three iconoclast scholars Seymour Garte, Perry Marshall, and Stuart Kauffman have published an article in the journal Entropy titled “The Reasonable Ineffectiveness of Mathematics in the Biological Sciences.” Seymour Garte is a biochemist at Rutgers University who has written widely on the intersection of faith and science. Perry Marshall founded Evolution 2.0, and Stuart Kauffman is a leading theorist studying self-organization. Their article argues that simple natural processes cannot explain embryological development. Instead, biology is directed by cognition — the capacity to make genuine choices and act creatively in responding to environmental cues to pursue goals.

Demise of Reductionism

The authors contend that mathematical relationships cannot capture the complexity of biological processes. This limitation poses a challenge to reductionism — the idea that understanding the physical and chemical properties of biological molecules will ultimately explain life’s higher-level organization. Reductionism asserts that all biological phenomena are reducible to physics, and if physical processes are mathematically describable, then biology should, in principle, be largely explainable through mathematical models. 

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This assumption entails biological processes almost entirely governed by rule-based dynamics — much like the laws of physics. That in turn leads to expecting these processes to be directed by algorithms since algorithms are defined as step-by-step procedures that transform inputs into outputs based on defined rules. Yet, the authors argue, many biological operations are not computable (aka nonalgorithmic):

Marshall uses Turing mathematics to prove (and not merely hypothesize) that predictions about the future, assigning meaning to symbols, inductive reasoning, axioms in mathematics, negentropy, measurement, and perception are all undecidable propositions, equivalent to Turing’s Halting Problem. All require choices that are not computable from prior states; thus, biology transcends the limits of computation. 

Mathematics still provides a powerful tool for representing the abstract relationships between biological functions, but only if detached from their physical components:

Louie, drawing on Rashevsky, uses relational biology to demonstrate mathematically that some biological properties emerge from relationships that cannot be reduced to their physical components. Relational descriptions can apply to a large class of functionally identical but physically quite distinct systems.

Rashevsky demonstrated this concept through what he called the “principle of biological epimorphism.” One of his most famous examples involved comparing the digestive systems of different organisms. He showed mathematically that despite vast differences in physical structure — from the simple gut of a hydra to the complex digestive tract of mammals — these systems could be mapped to the same relational model describing the basic functions of nutrient absorption and waste removal. The physical implementations varied dramatically, but the fundamental relationships between inputs, transformations, and outputs remained invariant. This demonstrated that biological functions could be understood at an abstract relational level, separate from their specific physical manifestation. 

In simple terms, processes in biology can be mapped similarly to process flow diagrams in engineering. The higher-level organization of diverse biological systems often maps to the same design pattern. The design pattern is more fundamental than how it is implemented in a specific organism in the same way the general design of a car is more fundamental than the physical properties of the metal, glass, and steel used in a specific car model. The success of relational biology overturns the reductionist assumption that the fundamental essence of life resides in the chemistry and physics of its physical components.  

Cognition Precedes Chemistry

The authors push the argument further by denying that chemistry generated the genetic code and that the genetic code eventually gained the required information to manifest cognition. They instead argue that cognition generated the genetic code, which enables life to control chemistry: 

Marshall argues that causation in biology is cognition -> codes -> chemicals, running in the opposite direction of the standard reductionist model, which is chemicals -> codes -> cognition. A single empirical example of a chemical process producing coded information would falsify this paper’s thesis. 

Cognitive agency differs from standard algorithms in terms of the problems they can solve. Algorithms rely on mathematical models applied to highly constrained problems, such as identifying the most efficient route to visit multiple cities. In contrast, cognitive agents can solve open-ended problems that cannot be solved computationally due to their complexity and lack of clear constraints. Cognitive agents can develop creative solutions that entail choosing a mathematical framework and constraints out of vast possibilities:

Biology, on the other hand, performs induction which by definition creates mathematics. Organisms make inferences where exact answers cannot be precisely known. This means biology creates in ways that nonliving matter does not.

The thesis of this paper is that biological organisms really do create mathematics and, in effect, choose axioms, both implicitly and explicitly. Humans explicitly chose a base-10 number system. Many other numbering systems are possible.

Cognition and Developmental Biology

The authors illustrate cognition in embryology by describing two model organisms’ responses to environmental stresses, as reported by esteemed developmental and synthetic biologist Michael Levin. In the first example, cognition is demonstrated by newts altering their development in response to increased kidney cell size: 

In normal newts, kidney tubules are typically formed by the interaction of 8 to 10 small cells in a cross-section. These cells communicate and coordinate with each other to create a tubule with a lumen of a specific size. This process relies on cell-to-cell communication mechanisms….

In cases where the cells are made extremely large, the system adapts even further. Instead of relying on cell-to-cell communication, a single large cell wraps itself into a “C” shape to form the tubule. This process uses a completely different molecular mechanism — cytoskeletal bending — rather than the usual cell-to-cell constructions.

This highlights the plasticity and adaptability of the biological system, as it can creatively deploy different mechanisms to achieve the same structural goal (a kidney tubule with the correct lumen diameter) despite drastic changes in cell size.

This is an affordance [possible actions allowed by the environment], an example of a biological system choosing an alternate set of axioms and using them to compute a new result in the face of a threat. We use the term “compute” here deliberately: yes, the organism is computing, but this comes after a choice has been made to take advantage of an affordance.

In the second example, cognition is demonstrated by planarians responding to the toxin barium:

Similarly, when planarian flatworms are exposed to barium, a non-specific blocker of potassium channels, their heads undergo a process of degeneration due to the inability of neural tissues to maintain normal ionic balance. This results in the heads not developing or functioning correctly, and in some cases, the heads even “explode” or degrade entirely.

When these planarians are kept in a barium solution, their remaining tails regenerate new heads that are completely resistant to barium. This occurs immediately. This adaptation is particularly fascinating because barium is not a substance that planarians encounter in the wild, meaning that there has been no evolutionary pressure to develop a specific response to it. This is a novel response to an unforeseeable event. 

Particularly striking is that the solution to the toxin challenge appeared to modify the minimum number of genes possible. A simple algorithm could not have stumbled upon such an optimal response so quickly; only a cognitive agent can immediately devise a creative and optimal solution.

Expanding the Scientific Worldview

The authors do not attempt to explain the source of cognition or how it originated. Instead, they argue for the “necessity of finding new formalisms to describe biological reality.” They also invite the scientific community to expand their philosophical horizons to allow their understanding of biology to be defined by evidence instead of long-standing philosophical commitments: 

The ineffectiveness of mathematics in biology represents a fork in the road in the history of science. We are at the threshold of a “third transition,” where the Newtonian clockwork paradigm that was overturned by quantum mechanics is again transformed by the unruly creativity of life. Whereas classical physics shoehorned the world into tidy equations and phase spaces, biology beyond mathematics will have to grapple with fuzziness, context-dependency, and open-ended emergence. This shift will require not just new mathematical tools but a new scientific epistemology that can accommodate life’s creative freedom….

Ultimately, dethroning mathematics as the infallible language of biology allows us to step back in humility and wonder. By recognizing the ineffable creativity of life, we move closer to appreciating biological systems as they are, not as we want them to be for the sake of conceptual convenience….

The reasonable ineffectiveness of mathematics in biology, then, is not a failure but an invitation: an invitation to expand our scientific worldview, embrace the unknown, and learn anew to be astonished by the fecund creativity of the living world.