Mathematical Reasoning, Proof and Modelling 0.2.0

Learning outcomes

1. Translate technical statements into precise definitions, predicates, quantifiers, sets, functions, and relations and negate or challenge them correctly.
2. Construct and critique direct, case, contrapositive, contradiction, and induction proofs with explicit hypotheses and domains.
3. Use invariants, equivalence, conjecture testing, and representation changes to expose mathematical structure.
4. Carry units, derive dimensional scalings, estimate orders of magnitude, calculate bounds and errors, and analyse sensitivity.
5. Translate system stories into variables, assumptions, balances, constraints, calibrated parameters, validation checks, and targeted revisions.
6. Write readable derivations and distinguish proof, calculation, finite computation, simulation evidence, and empirical model validation.
7. Create and defend an independent quantitative resource model with dimensional checks, calibration, holdout validation, sensitivity analysis, and explicit failure modes.