Domain interactions: mathematical biology

Physical + Vital + Abstract

I. Introduction: the triadic nature of the inquiry

Mathematical biology occupies a unique position in the landscape of human knowledge. It is neither purely abstract speculation about living systems nor merely the collection of biological data subjected to statistical analysis. At its best, it is something far more profound: the formal practice of translating the dynamic, self-sustaining conferences of difference we call 'life' into the stable, revelatory language of mathematics. It is where the Physical, Vital, and Abstract domains of the Conference of Difference converge.

To understand why this convergence is possible—and why it proves so extraordinarily fruitful—we must first recall what each domain contributes in terms of the conference of difference.

The Physical domain provides the substrate, the 'bare conference' of matter-energy from which all else emerges. It is the realm of quantum fields fluctuating, molecules colliding, thermodynamic gradients seeking equilibrium. This is the hardware of existence, the stuff that must be organized in specific ways for life to appear. Its laws are consistent, its constraints non-negotiable and its behaviors mathematically tractable.

The Vital domain introduces something genuinely new: reflexivity. Here, the conference of difference becomes organized around its own continuation. The system organizes itself around its own continuation through internal cycles (metabolism) and codes (the genome) that preserve information across generations. This is autopoiesis—the 'self-structuring structure' that distinguishes the living from the merely physical. The Vital domain does not suspend physical laws; it conferences them, toward new being.

The Abstract domain provides the maps. Numbers, geometric forms, logical systems, and mathematical structures are not existents that undergo transformation through the CoD but are revealed by it, permitting us to model and make sense of the transformations occurring among existents. As established in the Abstract domain, 'abstracta conceptualize constructs of existence, while existents transform existence itself'.

Mathematical biology, viewed through the CoD lens, is the practice of bringing these three domains into alignment. It uses the stable syntax of the Abstract domain to write equations that reveal the grammar of the Physical and the narrative of the Vital. It is where we write the equations that allow us to map the terrain of existence—not to mistake the map for the territory, but to navigate the territory more wisely.

II. The physical foundation: the 'hardware' of life

Before there can be life, there must be a physical system capable of hosting it. The Vital domain does not float free of material constraint; it emerges from and remains embedded in the Physical domain. Mathematical biology's first task, therefore, is to model the physical substrate—the necessary, though not sufficient, conditions for living organization.

Consider the phenomenon of pattern formation in chemical systems. When certain reactants diffuse through a medium at different rates, they can spontaneously form stable spatial patterns—stripes, spots, spirals—without any external template. This is the Belousov-Zhabotinsky reaction, and it is described mathematically by reaction-diffusion equations first analyzed by Alan Turing in his 1952 paper: The Chemical Basis of Morphogenesis.

Through the CoD lens, these equations map what the Physical domain document calls the 'bare conference of difference'. The variables represent concentrations of different chemicals—distinct existents whose differences are the raw material of the interaction. The equations describe how these differences bear together and bear apart: the term for diffusion captures the 'bearing apart' of molecules as their relations extend; the reaction terms capture their 'bearing together' as their relations contract or reconfigure. The stable patterns that emerge are not imposed from outside but are the natural expression of this ongoing conference of difference under specific conditions.

Turing's insight was that such purely physical conferences could serve as prepatterns for biological development—that the stripes on a zebrafish or the spacing of digits on a limb bud might begin as chemical gradients established by reaction-diffusion dynamics. Here, mathematical biology reveals how the Physical domain provides a landscape of possibility upon which the Vital domain will later build.

Similarly, statistical mechanics and molecular dynamics model the microscopic conferences that give rise to reliable biological structures. Protein folding—the process by which a linear chain of amino acids assumes its functional three-dimensional shape—involves countless atomic interactions: van der Waals forces, hydrogen bonds, hydrophobic effects. These are physical conferences of difference at the smallest scale. Mathematical models do not simulate every atom (that remains computationally prohibitive) but instead reveal the probabilistic landscape of folding pathways—the field of potential within which actual proteins find their native states.

These physical models are necessary, but they describe a corpse, not a life. A reaction-diffusion system, left to itself, will eventually reach equilibrium and stop. A folded protein, in isolation, performs no function. To capture the living state, we must introduce the organizing principle of the Vital domain: the system's orientation toward its own continuation.

III. The vital phenomenon: the 'software' of self-continuation

What transforms a collection of chemical reactions into a living system is organization—specifically, organization around the project of persistence. The Vital domain identifies this as inherent reflexivity: the presence of cyclic, self-referential processes where information is about the behavior of the molecules participating in it. Mathematical biology models this reflexivity across scales, from metabolic pathways to ecosystems.

Perhaps the most famous equations in all of mathematical biology are the Lotka-Volterra predator-prey equations:

$$\frac{dx}{dt} = αx - βxy$$ $$\frac{dy}{dt} = δxy - γy$$

Here, x represents prey population, y predator population, and the parameters capture birth, death, and interaction rates. The equations produce characteristic oscillations: prey increase, predators increase in response, prey decline, predators decline, prey increase again.

Through the CoD lens, these equations are a formal expression of co-petition—the 'process of petitioning together' that the Gospel of Being distinguishes from competition (Koan 20.6). The predator and prey are not merely opponents in a zero-sum game. Their differences bear together in a reciprocal dance that regulates both populations, increasing the resilience and complexity of the ecosystem as a whole. The wolf that culls the weak deer strengthens the herd; the deer that shapes the forest through grazing creates niches for other species. The Lotka-Volterra oscillations are a mathematical portrait of this stable, dynamic conference of difference.

Note what the equations reveal that a purely descriptive account cannot: the existence of an equilibrium point around which the populations cycle, the dependence of stability on parameters, the possibility of extinction if perturbations push the system too far. These are not properties of predator or prey alone but of their ongoing conference of difference. The abstracta disclose the relational structure that constitutes the ecosystem's vitality.

At a smaller scale, systems biology models the internal conference of difference within cells. Metabolic Control Analysis, for example, uses differential equations to trace the flux of metabolites through enzymatic pathways. Here, the 'work' (Koan 70.1) of each enzyme transforms substrates into products, and feedback loops regulate the entire system to maintain homeostasis. The equations reveal how the cell's autopoietic organization—its being: 'action to be' emerges from the coordinated activity of thousands of molecular conferences of differences.

Consider the classic example of glycolysis, the metabolic pathway that breaks down glucose to produce energy. Mathematical models of glycolysis capture not just the sequence of reactions but the regulatory architecture: allosteric inhibition, product feedback, ATP sensing. These are mechanisms of reciprocity (Koan 80.1)—the 'like forward, like back' that maintains the system's internal equilibrium. When glucose is abundant, the pathway hums along; when scarce, the system adjusts. The equations show how the cell remains itself through ceaseless transformation.

At the highest scale of biological organization within an organism, mathematical neuroscience models the conference of difference that generates consciousness itself. Coupled differential equations describe populations of neurons, their firing rates and their synaptic connections. These models capture how billions of individual cells, each with its own internal dynamics, bear together into coherent patterns of activity—the neural correlates of perception, memory, and thought.

The Gospel of Being defines consciousness as a 'measure of knowing together' (Koan 50.5). Where mainstream neuroscience and psychology study intra-cognizance—the condition of knowing together within a single nervous system (synonymous with the Latin concius sibi: 'knowing together within')—the CoD framework recognizes this as one configuration of cognizance, bounded by biological limogenesis (the nervous system's functional boundary). Inter-cognizance and inter-consciousness (the condition and measure of knowing together across beings) are bounded by relational limogenesis—dyads, groups, institutions—and are the proper subjects of sociology, anthropology, and collective intelligence research. Mathematical neuroscience models intra-cognizance. When a visual cortex model reproduces the orientation tuning of neurons, or a thalamocortical model generates sleep spindle oscillations, we are witnessing the abstract domain revealing how the vital domain achieves one of its most sophisticated expressions of bounded knowing.

IV. The abstract machinery: the language of revelation

The success of mathematical biology depends on a remarkable fact: the abstracta we have developed for other purposes—calculus for physics, information theory for communications, graph theory for networks—turn out to be extraordinarily effective for modeling living systems. This is not coincidence but revelation. The Abstract domain provides the machinery for making the conference of difference of existence intelligible.

Calculus, particularly differential equations, is the mathematics of transformation. It captures rates of change, flows, accumulations, and equilibria—precisely the features that characterize biological processes. The Gospel of Being declares that 'all transformation is a conference of difference' (Koan 100.7). Differential equations are how we write that conference of difference in symbolic form. They reveal what remains invariant through change: the relationship between variables that defines the system's identity.

Information theory, developed by Claude Shannon for communication engineering, has proven indispensable for understanding genetics. The genetic code is a literal example of meaning in the Gospel of Being's sense: 'intending' sent and 'sense' received (Koan 60.4). Nucleotide bases are physical differences that, through the cellular machinery of transcription and translation, are transduced into functional proteins. Information theory allows us to measure this process: the information content of a gene, the channel capacity of the genetic code, the noise in replication. These are quantitative revelations of how the Vital domain stores and transmits its organizing principles.

Graph theory models the relational structure of biological systems directly. Protein interaction networks, metabolic pathways, food webs, neural connectomes—all can be represented as graphs: nodes connected by edges. Through the CoD lens, graphs are pure relation. They abstract away the physical substrate to show the structure of the conference of difference itself: who is conferencing with whom, how information or energy flows through the system, which nodes are hubs, where the system is vulnerable to disruption. A food web graph reveals the co-petitive structure of an ecosystem; a protein interaction graph reveals the functional architecture of a cell.

Category theory shifts focus from the internal structure of objects to the relations between them (morphisms) and their composition. A category is defined by both objects and morphisms, but the morphisms carry the explanatory weight: objects are characterized by their relationships to other objects (e.g., via universal properties). This aligns with the CoD's process-primitive view of existence, where relations precede relata. A biological system, from this perspective, is not a collection of entities defined by their intrinsic properties but a network of transformations. Enzymes are morphisms from substrates to products; signaling pathways are compositions of such morphisms; development is a sequence of transformations from zygote to adult. Category theory provides an abstract language for describing these transformations and their relationships.

What makes these abstracta so effective is their closure and consistency. As the Abstract domain notes, mathematical systems 'constitute complete possibility spaces—closed systems of relation where all potential configurations already implicitly exist'. When we model a biological process with differential equations, we are not inventing new mathematics but revealing which pre-existing relationships in the abstract space correspond to the vital dynamics we observe. The equations work because the conference of difference in the Vital domain shares the same relational logic as the abstract system—not because biology follows mathematics, but because mathematics reveals the patterns that biology embodies.

V. Synthesis: the dance of the three domains

To appreciate how the three domains interact in mathematical biology, consider a single, unified example: morphogenesis, the development of form in multicellular organisms.

The three domains are co-present from the outset. A developing embryo is already a Vital domain phenomenon: cells—autopoietic agents organized around their own continuation—secrete signaling molecules, maintain boundaries, and respond to their environment. But these Vital acts are only possible because the Physical domain provides the substrate: molecules diffuse, react, and bind according to physical laws. The Physical does not precede the Vital; it is concurrent with it, as the material condition of its expression. (A crucial clarification: the Abstract domain does not act upon the limb; it makes the limb's relational logic legible to us—a map, not a mover.)

Consider the morphogen gradients that pattern a developing limb:

• the Physical domain contributes the substrate: diffusion coefficients, degradation rates, binding affinities—the parameters that determine how morphogen concentration varies;
• the Vital domain contributes the agents: cells that synthesize and secrete morphogens, cells that express receptors to detect them, cells that interpret gradient concentrations as signals for differentiation, migration, or death; and
• the Abstract domain provides the maps by which we understand the process: reaction-diffusion equations (Turing's model) that reveal the possibility space of stable patterns—stripes, spots, spirals—latent within the physical parameters.

The three domains do not act in sequence; they conference together continuously. A cell produces a morphogen (Vital act) by synthesizing a protein according to a genetic program (Vital information storage). That morphogen is a physical molecule (Physical substrate) that diffuses through extracellular space (Physical process). The diffusion is described by equations (Abstract revelation) that predict where concentration peaks will form. A target cell senses the local concentration (Vital interpretation) and activates or represses genes accordingly (Vital response), changing its behavior and thus the physical configuration of the tissue.

The Turing instability analysis—a mathematical discovery—reveals something profound: through the lens of the Abstract domain, we see possibilities that neither the Physical nor the Vital alone would manifest without the specific coupling conditions. The equations show that under certain parameters (specific ratios of diffusion coefficients, particular reaction kinetics), a homogeneous field of morphogens will spontaneously break symmetry, forming regularly spaced peaks. This pattern formation is not caused by the Vital domain; it is a latent property of the Physical substrate, made legible by the mathematics. But which pattern emerges—whether a limb forms a digit or a zebrafish forms a stripe—depends on how the Vital domain selects from and shapes these latent possibilities.

The cell's interpretation of the morphogen gradient is not a physical necessity. Given a particular concentration, the cell could differentiate into neuron, skin, or undergo apoptosis—the outcome depends on its internal state (gene expression history, epigenetic marks, metabolic status). This is the Vital domain's contribution: intention—not conscious purpose, but the system's orientation toward its own continuation, encoded by evolution into the regulatory circuitry. The cell 'senses' the morphogen concentration (detects a physical difference) and 'responds' (acts to preserve or elaborate its role in the developing organism). This is the meaning of Koan 60.4: 'meaning sent' (the morphogen concentration as signal) becomes 'sense received' (the cell's interpreted response) within the conference of difference that is the developing tissue.

The resulting form—a properly structured limb, a correctly patterned neural tube—emerges from the conference of all three domains, not as a sum of separate contributions but as a simultaneous, interpenetrating process:

• The Physical provides constraint (what is physically possible) and substrate (what is physically present)
• The Vital provides agency (the autopoietic capacity to interpret and respond) and memory (genetic programs encoding evolutionary history)
• The Abstract provides revelation (the possibility space latent within the physical, made legible through mathematics—not as a cause, but as a map that makes the territory navigable)

Without the Physical, nothing happens—there is no substrate for life. Without the Vital, there is no agent to interpret or respond—only chemistry, not biology. Without the Abstract, we could not see the possibilities—we could not predict which patterns are stable, nor design interventions to guide development.

This synthesis illuminates why mathematical biology is so powerful. The models work not because one domain precedes another, but because the three domains are ontologically aligned from the start. The abstracta provide the map of possibility; the physical provides the territory of constraint; the vital provides the explorer who navigates both to persist. But the map, territory, and explorer are co-present: the explorer is already shaping the territory while reading the map; the map is being revised by what the explorer discovers; the territory is being transformed by the explorer's passage. The map itself, however, does not move the explorer or reshape the territory—it guides by revealing what is possible.

Mathematical biology, at its best, captures this dance. The equations do not describe a sequence of domain-specific events. They describe a single conference of difference in which the physical, vital, and abstract are inseparable aspects of a unified process—with the abstract serving as the legible grammar of that conference, not a separate actor within it. This is why the CoD framework is not merely a helpful heuristic but an ontological necessity: because existence, at every scale and in every domain, is the conference of difference. And mathematical biology gives us a language to listen to that conference, to write down its grammar, and to marvel at the coherence that emerges when we finally understand the language that life has been speaking all along.

VI. Conclusion: The Equation as Ontology

Mathematical biology, viewed through the Conference of Difference, reveals itself as something more profound than applied mathematics or quantitative biology. It is the practice of making the conference of difference legible—translating the silent 'bearing together' of atoms in the Physical domain and the reflexive 'will to power' of organisms in the Vital domain into the symbolic language of the Abstract domain.

An equation like Lotka-Volterra is not merely a predictive tool. It is a formal revelation of the co-petitive structure of reality that the Gospel of Being affirms: the 'process of petitioning together' rather than 'petitioning against' (Koan 20.6). It shows that stability in living systems is not stasis but dynamic reciprocity—the 'like forward, like back' (Koan 80.1) that maintains equilibrium through continuous transformation. The oscillations of predator and prey populations are not noise to be eliminated but the very music of the ecosystem's conference of difference.

In this sense, mathematical biology is ontology made quantitative. It demonstrates, with the precision of numbers and the rigor of proof, that 'all transformation is a conference of difference' (Koan 100.7). When we write equations for metabolic flux, population dynamics, or neural firing, we are not imposing mathematics on biology but revealing the mathematical structure that biology has embodied all along.

The maps we create are not the territory. The equations are not the organism. But they are faithful maps, reliable guides to the terrain of existence. And in their fidelity, they confirm the foundational insight of the CoD: that existence, at every level and in every domain, is relational. The physical, the vital, and the abstract are not separate realms but different registers of the same cosmic conference of difference—a conversation that has been underway since the first quantum fluctuation and will continue as long as difference bears together into existence.

Mathematical biology gives us a way to listen in on that conversation, to write down its grammar, and to marvel at the coherence that emerges when we finally understand the language that life has been speaking all along.

The Gospel of Being

by John Mackay

Discover the first principle of existence in 30 seconds.

Discover the book