Why Mathematical Models Are the Backbone of Science: From Newton to Quantum Mechanics

Various Scientific and Mathematical Models

Scientific development relies on mathematics, and mathematical models play a crucial role. Whether in natural or social sciences, researchers study models—simplified representations that omit minor factors—rather than phenomena directly, using models to reveal objective laws of the real world.

Many fundamental theories across disciplines are expressed with mathematical (formula) models. Examples include Kirchhoff's laws in circuit theory and Marx's formula II describing the ratio of the first and second sectors in social reproduction, reflecting the proportion of concrete to abstract labor. Thus, mathematical modeling is a routine activity in scientific research, offering methodological value and acting as a bridge between mathematics and practical life.

Natural Sciences such as Physics and Their Mathematical Models

Mathematical models are vital for expressing concepts in natural sciences. In physics, many ideas are difficult to articulate verbally but become clear with mathematical expressions. For instance, instantaneous velocity was ambiguous before calculus; with calculus it is defined as the derivative of displacement with respect to time.

Similarly, the concept of work in physics is precisely defined as the line integral of force along a path, extending beyond the elementary product of force and distance. Numerous examples show how mathematical formulas clarify scientific concepts.

With the rise of electronic computers, computational branches of many fields have expanded, requiring mathematical models of systems. For example, theoretical and experimental physics together drive the discipline, yet many problems—such as solar evolution or fluid dynamics—are too complex for purely analytical or experimental approaches. Computational physics, enabled by mathematical modeling, now addresses astrophysical and fluid problems.

Historical giants like Newton, Einstein, and Heisenberg exemplify the power of mathematical models.

Newton expressed the laws of mechanics with simple equations, providing a unified picture of the physical world and enabling calculations from tides to planetary motion. His work illustrated the belief that science emerges from mathematically explaining nature.

Since Newton, a trend has emerged in natural sciences to represent laws with concise mathematical expressions, solve them, and compare results with experiments. Examples include Fermat’s principle of least time and Hamilton’s principle, showing that fields such as electromagnetism, optics, and thermodynamics rely on mathematical formulation.

Einstein’s theory of relativity was facilitated by mathematics: Euclidean geometry and Newtonian mechanics prompted him to question absolute space and time, leading to the special theory based on two postulates and a four‑dimensional tensor framework. Later, Riemannian geometry provided the tools for general relativity, demonstrating how non‑Euclidean geometry became essential for describing gravitation.

Einstein argued that theoretical physicists must achieve the highest standards of precision, attainable only through appropriate mathematical models.

Heisenberg’s development of quantum mechanics also illustrates this relationship. The uncertainty principle, expressed through non‑commuting matrix multiplication, was recognized as a mathematical structure already known decades earlier. By adopting matrix mechanics, Heisenberg clarified quantum behavior, establishing quantum mechanics.

Mathematical modeling thus updates physical concepts, guiding lengthy derivations toward clear physical insight.

Shen Wenxuan, Yang Qingtiao, Mathematical Modeling Attempts