A Four‑Layer Path from Observation to Abstract Mathematical Models

Mathematical models play a huge role in solving problems, but not every problem can immediately be captured by a widely applicable, elegant model. Model building is usually a gradual approximation process, extracting core simplified aspects from complex real‑world phenomena and progressively abstracting them.

To solve problems effectively, the process is divided into four layers: Phenomenon layer, Logical analysis layer, Concrete model layer, and Abstract model layer . These layers represent both a pathway from reality to modeling and a common way of thinking in mathematical modeling.

1. Phenomenon Layer

The starting point of every mathematical model is the observation of a real‑world phenomenon. This layer focuses on the surface and actual existence of the problem, exploring which aspects are core to study.

We usually begin with intuitive perception, inductively identifying possible problems from a complex phenomenon.

1.1 Observation and Problem Discovery

Real problems are complex and vary across fields; for example, price fluctuations in economics or population changes in biology. These phenomena are identified through observation but initially lack a direct mathematical model.

1.2 Extracting Feature of Phenomena

Transitioning from the phenomenon layer to the logical analysis layer requires extracting key features. For instance, market price fluctuations can be reduced to supply‑demand relations, market sentiment, policy interventions, etc. We identify dependent and independent variables to analyze the core.

Mathematical expression: let the independent variable be ___ and the dependent variable be ___; analysis extracts the independent variables influencing the dependent one.

2. Logical Analysis Layer

After extracting core variables, we enter the logical analysis layer, using logical reasoning to clarify relationships and possible causality, shifting focus from intuitive phenomena to internal structure.

2.1 Determining Key Variables and Relationships

Logical analysis clarifies how factors relate; e.g., price may be affected by supply‑demand in economics, or population by environmental factors in biology. By analyzing causal relations we can identify core and auxiliary variables.

This part can be illustrated with diagrams.

2.2 Theoretical Derivation and Assumptions

Beyond relationships, we hypothesize how they operate. For a market model we might assume a linear relation between price and supply‑demand, or discover more complex relations from data.

Mathematical expression: assume market price ___ is influenced by supply ___ and demand ___, then price change can be expressed as:

In this formula, ___ represents the functional relationship, which may be linear, nonlinear, etc.

3. Concrete Model Layer

Once logical analysis is clear, we build a concrete mathematical model that quantifies variables and their interactions, providing a basis for analysis and decision‑making.

3.1 Mathematical Formulation and Testing

In this layer we choose appropriate mathematical tools—differential equations, regression, optimization—to describe the problem. For example, using regression to fit price‑supply‑demand relationship.

Through least‑squares we estimate coefficients ___ and ___, obtaining a price model.

3.2 Model Validation and Adjustment

After constructing the model we validate its correctness by comparing with actual data. If performance is poor, we return to previous layers to adjust variables or assumptions. Validation often relies on error analysis, e.g., computing the discrepancy between predicted and observed values.

4. Abstract Model Layer

When a concrete model is validated, we can abstract it to a more concise, widely applicable form. Abstract models are not tied to a specific problem but can be reused in similar contexts.

4.1 Induction and Abstraction

We extract commonalities from multiple concrete models, removing redundant details to create a universal structure. Abstract models emphasize universal relationships between variables rather than specifics of a single case.

For example, optimization models apply across many fields; a linear programming formulation to maximize profit can be written as:

subject to … (constraints)

where ___ are decision variables, ___ are profit coefficients, and ___ are constraint coefficients, representing resource limits.

4.2 Simplicity and Generality

The goal is a simple yet general model. By stripping unnecessary parts, we obtain a model that captures the essence and can be applied to other scenarios, such as a simplified supply‑demand model in economics.

The four‑layer abstraction provides a systematic thinking framework for problem solving, not only in mathematical modeling but also in other complex domains, enabling more efficient solutions and scientific decision support. (Author: Wang Haihua)