Mathematical Models and the Newtonian Worldview

Further Reading: Differential Equations

• Recommended Textbooks: Blanchard, Devaney, and Hall ($2011$) is highlighted for its dynamical systems perspective and balance of qualitative, numerical, and analytic methods.

• Mathematical Biology Resources: Key texts include Britton ($2005$), Edelstein-Keshet ($2005$), Ellner and Guckenheimer ($2006$), Garfinkel et al. ($2017$), Mangel ($2006$), and Vandermeer and Goldberg ($2013$).

The Newtonian Mechanistic Worldview

• Newton's Principia ($1687$): Isaac Newton established a unified account of gravity and three universal laws of motion, framing the universe as a machine governed by unchanging mathematical rules.

• Core Assumptions: The traditional view assumes that simple mathematical laws lead to simple, predictable behavior. Complexity was historically thought to arise only from many interacting objects or non-simple laws.

Laplacian Determinism

• Laplace's Demon ($1814$): Pierre-Simon de Laplace posited that a "vast intelligence" knowing the position and forces of all particles could perfectly predict the past and future.

• Scientific Requirements for Prediction:     1. Knowledge of the universal laws of nature.     2. Accurate measurements of the current state.     3. Sufficient computing power.

• Challenges from Dynamical Systems: Modern study of chaos shows that even simple deterministic systems can be unpredictable due to sensitive dependence on initial conditions.

Styles and Levels of Mathematical Models

• Faithful vs. Caricature Models: Some models aim for exact quantitative prediction (like projectile motion), while others (caricatures) highlight essential features or qualitative traits (like an artist's sketch in a bird guide).

• First-Principles vs. Empirical Models:     * First-principles: Mechanistic models based on fundamental laws governing a system's constituents.     * Empirical: Descriptive models (e.g., linear regression) that capture trends in data without explaining the underlying mechanism.

• Abstraction Levels: Elements in models range from physical objects (point particles) to coarse-grained variables like temperature, population, and density.

• Agent-Based Models (ABMs): Instead of using population averages, ABMs simulate the movement and interaction of individual entities (agents) to see emergent behavior.

Pluralistic Modeling

• Contextual Effectiveness: The value of a model depends on its purpose. For example, a fetal pig and a mannequin are both valid models of a human body, but only for specific, non-interchangeable goals (physiology vs. clothing design).

• The Pluralistic View: No single model is "true." A full understanding of complex systems requires multiple models at different levels to illuminate various facets of the phenomenon.