Can Human Emotions Belong in Mathematical Models? Exploring the Essence of Modeling

The Essence of Mathematical Modeling

The core mission of mathematical modeling is to translate complex real‑world problems into solvable mathematical structures, typically expressed as a mapping from a problem space P to a model space M via a relationship f constructed by the modeler.

Traditional views stress two qualities of a good model: objectivity and generality . For example, the classic logistic population growth equation uses population size N, intrinsic growth rate r, and carrying capacity K. While elegant, it reduces each living individual to a statistical unit.

The question arises: does such simplification overlook essential aspects of reality?

Mathematical Interpretation of “Human Touch”

In modeling, “human touch” means incorporating individual heterogeneity , emotional factors , and ethical considerations . This is not a betrayal of mathematical rigor but a pursuit of model completeness .

Introducing Individual Heterogeneity

Traditional models often assume homogeneous agents, yet real populations differ markedly. Consider an epidemic model with heterogeneity:

For each sub‑population i, let β_i denote susceptibility, C_{ij} the contact matrix between groups, and γ_i the recovery rate.

This model acknowledges that elderly and young people face different disease vulnerabilities and that healthcare workers have distinct exposure risks—an embodiment of “human touch” by recognizing each group’s special needs.

Emotional Dimension in Utility Functions

Economic utility functions can be extended to include emotional satisfaction:

Traditional consumption quantity x is supplemented by an emotional satisfaction term e. An altruistic utility component α measures concern for others’ welfare. When α=0, the model reverts to a purely self‑interested agent; when α>0, the model gains a human‑centric dimension, acknowledging that people are not just rational calculators but also social beings with feelings.

Embedding Fairness Constraints

In resource allocation problems, fairness can be introduced as an explicit constraint. Let F be an unfairness metric such as the Gini coefficient. By setting a tolerance level τ, the model balances efficiency with equity, mathematically encoding ethical considerations.

Unifying Human Touch and Mathematical Rigor

Some fear that adding “human touch” undermines scientific modeling, mistakenly equating objectivity with coldness and rigor with oversimplification. In fact, true rigor demands an accurate depiction of reality; ignoring emotions, morals, and social dimensions makes a model less rigorous because it systematically omits key factors.

From a mathematical perspective, incorporating human factors involves:

Expanding the state space : augmenting the traditional space S with an emotional/ethical dimension E.

Enriching constraints : adding ethical and fairness constraints alongside technical ones.

Multi‑objective optimization : shifting from a single objective function f to a vector‑valued objective that captures multiple goals.

These extensions do not violate mathematical principles; they make models more complete and realistic .

Case Study: Human‑Centric Modeling in Pandemic Control

COVID‑19 provides a vivid example. A purely optimal control model might recommend strict lockdowns at any cost. A human‑centric model assigns non‑zero weight to psychological health loss, reflecting the value judgment that people need not only to survive but also to live with dignity, freedom, and social connection.

Conclusion

The answer to the opening question is affirmative: models that incorporate human touch are still mathematical models. Their inclusion expands the boundary of modeling, reminding us that mathematics is a tool serving humanity, not a cold authority imposing itself on people.

Mathematics is a tool, not an end.

Models serve people, not the other way around.

Rigor does not equal coldness; scientific inquiry can be warm.

“Mathematics is a cultural phenomenon.” By embedding human considerations, we return mathematics to its original purpose: a wise instrument for human well‑being.