A formal system of equations, functions and constraints that represents selected features of a real or abstract system to enable analysis, prediction, or explanation.

Principle

Abstract phenomena are encoded in a formal language so that logical and quantitative manipulation yields testable consequences.

Demonstration

The Lotka–Volterra predator–prey equations use two coupled ordinary differential equations to represent population dynamics and predict oscillatory behavior under specified interaction rates.

Misapplication

Treating the model's outputs as literal truth without checking domain validity, parameter uncertainty, or overfitting to limited data.

Consequence

When well-specified, the model permits simulation, parameter estimation, hypothesis testing, and quantitative forecasts within its validity domain.

Reversal

A raw empirical description without formal relations or explicit assumptions; data without an abstract formalization.

Boundary

Includes explicitly formalized mathematical representations; excludes purely narrative descriptions, unverifiable metaphysical accounts, or unformalized software artifacts lacking stated assumptions.

Semantic Tension

Distinguished from 'theory' (broader explanatory framework) and 'simulation' (numerical execution); models can be parts of theories and implemented as simulations.

Synthesis

A mathematical model is a deliberately simplified, formal representation that encodes assumptions about a target system so that reasoning and computation yield interpretable, testable implications.