Modern Warfare and Mathematics
Throughout the centuries, the art of war has been at the forefront of technological innovation and has anticipated profound trends in economics and society. The development of conflicts, the definition of strategies, the construction of new instruments of war have often gone hand in hand with the evolution of thought and scientific conquests.
A very curious angle of observation today is represented by the philosophical reflection on mathematics as a discipline, observing with the macroscope its evolution in the 20th century one can extrapolate some tendencies that have also manifested themselves in the practice of contemporary warfare. According to Sir Michael Atiyah, a great mathematician of the last century and winner of the Fields Medal, the mathematics of the 20th century has been characterized by some fundamental tendencies that have changed its nature in a qualitative way. It is interesting to note how the same tendencies have now become evident also in the practice of the art of war.
Here the three fundamental areas.
Increase in Dimensions
Modern mathematics, within geometry but also in other branches, moved from 2D surfaces and 3D spaces to n-dimensional manifolds. In warfare, this parallels the expansion into multi-domain operations across physical, cyber, and cognitive spaces. Physical: Traditional “3D” battlefields involving land, sea and air now include space, especially in low Earth orbit, and cyber domains. Cyber attacks on critical infrastructure, companies and government institutions are now commonplace, as well as are information warfare operations, election interference and the production of fake news. Higher-dimensional warfare demands synchronized command structures to manage interconnected effects, and a more complex degree of awareness among multiple dimensions of activity.
Local to Global
In mathematics, this shift involved moving from local coordinate systems to global topological properties, increasing the degree of abstraction and generality. In warfare, it reflects the transition from geographically confined battles to multi-theater, interconnected conflicts with global repercussions. Modern warfare integrates local tactical actions, such as drone strikes and cyberattacks, into global strategic frameworks. For example, hybrid warfare tactics, such disinformation campaigns and asymmetric warfare techniques, exploit localized instability to destabilize broader geopolitical alliances. Just as topology studies global invariants, such as holes in manifolds, modern military activities leverage global supply chains, transnational terrorism, and climate-driven conflicts that transcend borders.
Linear to Non-Linear
The mathematical shift from the focus on linear systems and solutions to non-linear systems and chaotic dynamics reflects warfare’s evolution from frontal assaults to decentralized, adaptive strategies. Guerrilla tactics and non-state actors use decentralized networks to offset conventional force superiority. Autonomous drone swarms exemplify nonlinear dynamics, overwhelming adversaries through distributed coordination. In urban warfare cities become fractal-like battlegrounds where micro-tactical actions, like tunnel warfare, aggregate into strategic outcomes. Non-linearity demands new forms of planning and organization.
A final note: War and AI
A common denominator for the future of warfare could be the impact of artificial intelligence technologies. Abstraction, global complex planning, non-linearity and the combination of different dimensions of activity are a natural breeding ground for the application of intelligent automated systems. The art of war will tendentially increase its non-human dimension, with hyper-intelligent autonomous agents and forms of AI-driven planning that will accompany an hybrid, multi-theater, interconnected warfare scenario.