A mathematical model represents selected features of a system with mathematical objects and relations so that its behavior can be analyzed or predicted. Choosing the representation is part of the work: specify the quantities tracked, the equations or probability distributions connecting them, and the conditions under which those connections are expected to apply. A mathematical proof can establish a consequence of the model; comparing that consequence with observations tests its suitability for the intended use.
For a simple pendulum, the angle and gravitational acceleration give the ideal equation . Its assumptions include a point mass, fixed length , uniform gravitational acceleration, and no damping. The small-angle approximation gives the simpler harmonic oscillator equation and oscillation period . A larger amplitude changes the full model's period; drag requires a damped pendulum. The simpler model remains useful when its omitted effects are small for the question being asked.
A model need not track every detail. A Bernoulli process models repeated success/failure observations using one probability and an independence assumption; a Gaussian process specifies a mean and a covariance function to model related measurements. Each representation enables calculations by retaining some distinctions and omitting others. Useful simplification therefore comes with a stated domain of use and quantities against which its predictions can be checked.
