On digital twins, precision medicine, and what happens when
we apply machine mathematics to systems that are not machines
There is a quiet crisis at the center of two of the most ambitious scientific programs of the 21st century. The first is precision medicine — the project of tailoring treatment to the specific biology of the individual patient rather than to population-level averages. The second is planetary stewardship — the project of managing human interactions with the Earth's climate and ecosystems with the kind of precision that their fragility demands. Both programs are underway. Both are producing results. And both are running into the same wall, for the same reason.
The mathematics being used to build them was not designed for the systems they are trying to model.
The machine problem
When engineers build a digital twin of a jet engine, the mathematics is straightforward in principle: model the components, specify their interactions, couple them through well-defined interfaces, and simulate. The jet engine is a machine — closed, decomposable, purpose-built. The mathematics of machines works because machines are designed to be modeled.
A human body is not a jet engine.
Neither is the Earth. Both are what biologists and systems theorists call living
systems — open to their environments in ways that continuously alter not just
their state but their operational structure; defined by networks of
relationships rather than by the intrinsic properties of their components;
capable of generating qualitative novelty that was not present in their initial
conditions; and fundamentally constituted by context in ways that make the same
input produce entirely different outcomes depending on where and when it
arrives.
The digital twin programs for both the body and the Earth have
inherited the mathematics of machines. They decompose their subjects into
components, model each component with differential equations, and couple the
components through interface conditions negotiated by specialists from different
disciplines. This works well under calibration conditions — in regimes close to
those used to build the model. It fails systematically at the boundaries: at
tipping points, at disease bifurcations, at the windows of therapeutic or
ecological leverage where the most consequential decisions have to be made.
"The
failure is not computational. Both programs have access to enormous computing
power and richer data than ever before. The failure is mathematical — a mismatch
between the formalism and the system."
Three frameworks, three dimensions
My new
white paper, Towards a Mathematics of Living Systems, argues that three
mathematical traditions together begin to define what an adequate mathematics of
living systems must look like — each addressing a different dimension of the
problem that machine mathematics cannot handle.
Chaos theory provides the tools
to represent instability and qualitative transition. Its formal apparatus —
phase space analysis, Lyapunov exponents, bifurcation theory — tells us where a
complex system is most sensitive, where prediction breaks down irreversibly, and
where the thresholds lie between qualitatively different regimes of behavior.
For a body twin, this means formally representing the bifurcation dynamics of
cardiac arrhythmia, epileptic seizure, and cancer progression. For an Earth
twin, it means mapping the proximity of the current climate system to its
tipping points — the AMOC collapse threshold, the Amazon dieback transition, the
ice sheet destabilization boundary. Chaos theory is indispensable for
identifying where the dangers are. Its limitation is that it does not tell us
how to act in response.
Applied category theory provides the tools to represent
the objects and relationships of a complex system with enough precision that
models from different disciplines can be formally assembled without conceptual
error. In current multi-component models — physiological or Earth system —
sub-models developed by specialists in different fields are coupled through
informal conventions. This works near calibration conditions and fails under
extrapolation, because informal conventions carry no mathematical guarantees
about behavior in novel regimes. Category theory makes the interfaces formal,
the type constraints explicit, and the compositional rules auditable. A protein
concentration and a cell count are not the same kind of mathematical object,
even if both are dimensionless numbers — and treating them as interchangeable
introduces errors that propagate silently through the model and compound when
conditions change.
Generative dynamics addresses the question that the other two
frameworks leave unanswered: how do interventions actually propagate through
living systems, and why do some inputs cascade into transformative change while
others dissipate without effect? The framework identifies four mechanisms —
cross-domain feedback loops, temporal leverage, network effects, and contextual
amplification — that together explain the intervention propagation problem.
Contextual amplification is the most consequential for precision medicine: the
same drug molecule does not produce the same effect in different patients, not
because of differences in its mechanism, but because the genomic, metabolomic,
and immune context constitutes what the drug becomes. This is not a statistical
nuance — it is the central mathematical challenge of personalized therapeutics.
The cost of the gap
Roughly 85–90 percent of drug candidates that demonstrate
efficacy in preclinical models fail in clinical trials. A substantial proportion
of those failures trace to the same source: preclinical models treat drug
effects as context-independent, and real patients present a range of biological
contexts that constitute different transformation environments. This is not a
data problem — more patient data fed into the same mathematical framework will
not close the gap. It is a mathematics problem.
The same logic applies to
planetary stewardship. The Earth system contains leverage points — places and
moments at which interventions produce disproportionately large and lasting
effects — but current Earth system models have no formal framework for
identifying them. Conservation and climate policy is therefore conducted without
reliable guidance about which interventions compound and which merely buffer.
Temporal leverage and cross-domain feedback loop mathematics are the formal
basis for answering that question, and their absence from current Earth system
modeling is a direct limitation on the precision of everything we are trying to
do.
A research program, not a finished theory
The paper does not claim to have
solved these problems. What it argues is that the shape of the solution is
visible, that the components exist in fragments across three mathematical
traditions, and that integrating them is both conceptually motivated and
practically urgent. The synthesis requires, at minimum, a dynamical categorical
framework that allows formal structure to evolve over time; a rigorous theory of
temporal leverage that connects bifurcation geometry to developmental
irreversibility; a contextual type theory in which what a morphism does depends
on the global configuration of the system; and a compositional dynamics that
combines all three.
That is substantial open mathematical work. But the case for
doing it is the case for precision medicine and planetary stewardship themselves
— two projects that the 21st century cannot afford to get wrong, and that the
mathematics of machines cannot get right.
The full white paper — including a
systematic framework comparison, detailed treatment of each mechanism,
mathematical directions for future research, a 32-term glossary, and an
annotated bibliography of 43 works — is available here:
Towards a Mathematics of Living Systems. You can also listen a summary on my Spotify podcast.
