Differential Equations and Historiography: A Hidden Mathematical Language of Historical Change

In the modern academic world, the divide between the “hard” sciences and the “soft” humanities has often been taken for granted. Mathematics occupies one domain, history another. Yet recent scholarly advances – and philosophical traditions stretching back to figures like Ibn Khaldun and Oswald Spengler – hint at an unexpected convergence: the use of differential equations as a model for understanding historiography itself. By translating the dynamics of historical change into systems of continuous variation, scholars have begun to treat history not merely as narrative, but as a dynamic system governed by the evolution of variables over time.

This article explores the profound implications of differential equations for historiography, reviewing actual research initiatives that blend mathematics with the philosophy of history, and proposing new paths forward for a more formal, predictive historiographical science.

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Differential Equations: A Primer

At its core, a differential equation describes how a quantity changes over time or space in relation to its current state. In physics, for instance, Newton’s second law of motion F = ma can be reformulated as a second-order differential equation relating acceleration to force. In epidemiology, the SIR model uses differential equations to model the spread of infectious diseases.