Beyond Techniques: The Philosophical Foundations of Mathematical Modeling

Mathematical modeling is widely recognized as a practical technique applied across natural sciences, social sciences, engineering, and other fields.

Is mathematical modeling merely a technique or tool?

It is not only a technique; learning it involves more than just acquiring a skill.

As the saying goes, “A gentleman is not a utensil”; focusing only on modeling techniques limits one’s perspective.

This article introduces the “Dao, Fa, Shu, Qi, Shi, Li, Yong, Shen, Xing” of mathematical modeling from a systematic viewpoint, offering new perspectives.

Dao

In ancient Chinese philosophy, “Dao” refers to the origin and fundamental law of the universe, emphasizing natural laws and harmony.

The “Dao” of mathematical modeling is its fundamental principles and philosophical foundation, the soul of modeling activities, involving the relationship between mathematics and the real world, how mathematics reflects reality, and the role of modeling in scientific exploration.

Li

“Li” denotes the internal laws and logic of things, the basis of rational analysis and reasoning.

It represents the scientific nature of modeling, the embodiment of “Dao” in concrete mathematical practice, ensuring scientific rigor, e.g., Newton’s laws as the “Li” for motion models.

Fa

“Fa” usually means rules or methods.

In modeling, “Fa” manifests as specific rules and methods for model construction and solution, including theory application, algorithm selection, data analysis, model selection, data collection, building, solving, and validation.

Shu

“Shu” refers to techniques, technologies, or strategies.

It includes concrete skills such as programming, software operation, experimental design, and specific modeling techniques, e.g., using MATLAB or Python for numerical computation, or R for statistical analysis.

Qi

“Qi” means tools or instruments, the material basis for achieving goals.

In modeling, “Qi” refers to tools like computer software, databases, statistical tools, etc.

Examples include SPSS for data analysis or Excel for preliminary data organization and model verification.

The “Dao” and “Li” form the philosophical foundation, the highest guiding principles; “Fa” provides the methodological steps to realize them; “Shu” and “Qi” focus on turning “Fa” into practice.

When learning and mastering modeling, one should aim for higher levels, moving from “Shu” to “Li”, but not neglect “Qi” and “Shu”, otherwise one may understand the theory yet be unable to apply it.

Shi

“Shi” involves power, energy, or situation, describing the state and trend of development.

In modeling, “Shi” reflects the model’s problem‑solving potential and impact; a model with greater “Shi” has higher theoretical and practical value, e.g., climate‑change models predicting global warming trends and informing policy.

Yong

“Yong” focuses on practicality and application value.

It concerns the usefulness of a model in solving real problems and optimizing decisions.

For example, a financial risk‑assessment model helps institutions evaluate and manage risk.

Shen

“Shen” can be understood as spirit, essence, or creativity.

In modeling, “Shen” is manifested as creative thinking and insight.

It involves intuition, insight, and innovative ability, the artistic component that drives breakthroughs, e.g., devising a novel algorithm for a complex optimization problem.

Xing

“Xing” refers to the form or appearance of a model, including graphs, charts, and visual tools.

These make results more intuitive and understandable, representing the “Shen” of modeling.