The point is a trap: Mathematics of Elastic States in Complex Multiscale Systems
Abstract
Observables in complex biological systems are often reduced in their inherent complexity, resulting in abstract, a priori mathematical representations based on simple concepts such as points in spaces or interconnected nodes in lattices or networks. This reduction is relevant not only for the formulation of multiscale mathematical models and field theories but also for experimental planning and data collection to gain systematic evidence for the nature of such systems, thus extracting robust causal relationships that would lead to better control of the system. However, mathematical abstraction remains challenging and problematic, especially given the tightly intertwined elements between objects and their environment, the system's affordance, and the fact that all these aspects limit fundamental mathematical concepts such as set theory, thus hindering the definition of phase spaces in extreme cases. In this work, I introduce a potential axiomatic representing a nonreducible representation of high order structures representing entanglement between observables in complex systems and their environment, considering not only their coupling but also their affordance in relation to the overall system and context, and thus intend to formalize concepts already used in complex systems such as Kantian wholes. In light of this axiomatic, I identify states that are rather less elastic with respect to their environment, discuss how low elasticity enables the application of set theory, while high elasticity limits the mathematical representation of complex systems, and present an exemplary Ising model with a network of elastic states with respect to different contexts as a limiting case for mathematical modelling in this context.
Related
- Quantum Multiscale Modeling: A Hierarchy of Algorithms for Complex Chemical Systems
- Multiscale evaluation of the elastic response of polycrystalline NiAl using molecular statics, EBSD-based finite element and analytical homogenization methods
- Multiscale Decoupling in Peptide-Membrane Interactions: From Elastic Softening to Domain-Mediated Stiffening
- Anderson localization in periodic elastic systems with random perturbations
- Frank Elastic Constants of Associating Rigid-Rodlike Systems
BibTeX
@misc{ochoa2026point,
title = {{The point is a trap: Mathematics of Elastic States in Complex Multiscale Systems}},
author = {Juan G. Diaz Ochoa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16994},
eprint = {2610.16994},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16994},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The point is a trap: Mathematics of Elastic States in Complex Multiscale Systems
AU - Juan G. Diaz Ochoa
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16994
UR - https://arxiv.org/abs/2610.16994
AB - Observables in complex biological systems are often reduced in their inherent complexity, resulting in abstract, a priori mathematical representations based on simple concepts such as points in spaces or interconnected nodes in lattices or networks. This reduction is relevant not only for the formulation of multiscale mathematical models and field theories but also for experimental planning and data collection to gain systematic evidence for the nature of such systems, thus extracting robust causal relationships that would lead to better control of the system. However, mathematical abstraction remains challenging and problematic, especially given the tightly intertwined elements between objects and their environment, the system's affordance, and the fact that all these aspects limit fundamental mathematical concepts such as set theory, thus hindering the definition of phase spaces in extreme cases. In this work, I introduce a potential axiomatic representing a nonreducible representation of high order structures representing entanglement between observables in complex systems and their environment, considering not only their coupling but also their affordance in relation to the overall system and context, and thus intend to formalize concepts already used in complex systems such as Kantian wholes. In light of this axiomatic, I identify states that are rather less elastic with respect to their environment, discuss how low elasticity enables the application of set theory, while high elasticity limits the mathematical representation of complex systems, and present an exemplary Ising model with a network of elastic states with respect to different contexts as a limiting case for mathematical modelling in this context.
ER -