Algebra vitae: a manifesto for the new mathematics of biology - S. V. Kozyrev (Translational Medicine, 2026)
This paper formulates the problem of developing a mathematical language intrinsically suited to the historicity and organized complexity of living systems. The challenge lies not in the insufficient expressive power of modern mathematics, but in its underlying ontological assumptions. The mathematical framework developed for describing inert physical systems encounters fundamental limitations when applied to living systems. Models formulated within this framework cannot adequately describe systems whose state space expands throughout their history and whose dynamics are governed by memory, as well as by relational and functional organization. Following the historical pattern whereby new mathematical languages often prove productive before acquiring rigorous foundations, we propose, as a first step, a conceptual framework assembled from existing theoretical building blocks. These include Diekmann’s models of physiologically structured populations, Rosen’s relational biology, Kauffman’s concepts of the expanding state space and the adjacent possible, fractional and non-Markovian dynamics, the Doi-Peliti and Schwinger-Keldysh field-theoretic formalisms, and homotopy type theory, which reimagines equality as a structure of paths. Studies of chronic inflammation, tumor evolution, and carcino-evo-devo theory demonstrate that none of these approaches, taken individually, is sufficient to address the questions that biology poses to mathematics. However, each shifts the analytical focus, in its own way, from individual entities to relations, processes, and reproducible patterns of organization, thereby aligning with process ontology. The aim of this article is to identify existing conceptual gaps, argue for the methodological importance of the ontological shift from the classical “substantial” perspective to a process-oriented one, and formulate the requirements for a future “algebra of life”. From the perspective of translational medicine, a reformed mathematical paradigm for living systems is of critical importance, as it offers a means to overcome the looming deficit of theoretical tools for the conceptual compression and interpretation of exponentially growing volumes of multi-omics and clinical data.