Mathematical reasoning + proof¶

Weeks¶

Week 1¶

Spine: Book of Proof, 3.4

Reading: Ch. 1 Sets, pp. 3-33; Ch. 2 Logic, pp. 34-64

Know: Translate English claims into quantified statements; negate precisely; identify hidden assumptions and counterexamples.

Reconstruct: Rebuild De Morgan's laws for sets and logic; derive the contrapositive equivalence; write the negation of a nested quantified claim.

Do: Formalize a safety requirement for an autonomous lab and produce one implementation that satisfies the English wording but violates the intended requirement.

Defend: Why is testing many cases not a proof? When can a finite exhaustive test become one?

Gate: Closed-book 45 min: formalize 8 claims, negate 5, prove 2 equivalences with no notes.

Source: source

Week 2¶

Spine: Book of Proof, 3.4

Reading: Ch. 4 Direct Proof, pp. 113-127; Ch. 5 Contrapositive, pp. 128-136; Ch. 6 Contradiction, pp. 137-146

Know: Choose proof strategy from logical form; distinguish definition expansion from theorem invocation; write readable proofs.

Reconstruct: Regenerate the direct/contrapositive/contradiction templates and prove from definitions that the sum of two even integers is even.

Do: Prove or disprove three claims about graph connectivity, parity, and divisibility; for each explain why your proof strategy was appropriate.

Defend: What makes contradiction legitimate rather than rhetorical?

Gate: Three unseen proofs in 75 minutes; at least one must use a different valid route than the reference route.

Source: source

Week 3¶

Spine: Book of Proof, 3.4

Reading: Ch. 9 Disproof, pp. 172-179; Ch. 10 Induction, pp. 180-200; Ch. 12 Functions, pp. 223-243

Know: Use counterexamples, induction, functions/inverses, and invariant-style reasoning.

Reconstruct: Derive weak and strong induction from the well-ordering intuition; state injective/surjective/bijective conditions from memory.

Do: Specify and prove an invariant for a toy self-modifying program that is allowed to rewrite code but must preserve a safety property.

Defend: What is the difference between an invariant, an induction hypothesis, and an empirical regularity?

Gate: Oral defense: present a proof, then survive two adversarial counterexample attempts and revise if necessary.

Source: source

Exit gate¶

Closed-book: 90 min, no notes: formalize 10 claims; negate 5; prove 3; disprove 2; one induction.

Novel problem: Prove a safety invariant for a toy recursive/self-modifying process and state the exact assumptions.

Artifact: Write a one-page specification + proof artifact that another person can try to break.

Defend: Defend proof strategy; respond to counterexamples; distinguish definition/theorem/assumption/inference.

Pass criterion: Pass if all quantifiers are correct, no hidden assumption survives challenge, and at least 80% of proof steps are valid without prompting.

Transfer problems¶

Try these before consulting solutions or asking for the complete answer.

1. Logic: Translate: 'Every autonomous experiment that changes a safety-critical parameter must either be independently verified or automatically reverted.' Negate it exactly.

2. Proof: Prove from definitions that composition of two injective functions is injective.

3. Counterexample: Construct a plausible engineering claim of the form 'if P then Q' where Q is true in all your test cases but the claim is false.

4. Contrapositive: Prove: if n^2 is even then n is even.

5. Contradiction: Prove irrationality of sqrt(2) or another analogous statement without copying a memorized proof.

6. Induction: Prove a recurrence invariant for a simple repeated resource-allocation algorithm.

7. Functions: Give a real technical example of a many-to-one map and explain why inversion is ill-posed.

8. Specification: Formalize a requirement for a robot 'never entering an unsafe region' with explicit state and time quantifiers.

9. Assumptions: Take a short derivation and mark every step as definition, assumption, theorem, algebra, or empirical input.

10. Adversarial defense: Write a proof, then invent the strongest counterexample attempt you can and either refute it or revise the theorem.

Textbooks¶

See the five-book resource page.