Mathematical reasoning and proof¶

Open Frontier

When does an argument establish a claim, rather than merely make it plausible?

What you will develop¶

Own definitions, quantifiers, implication, counterexample, induction and proof structure.

Before you begin¶

None beyond algebra

First attempt: Negate 'Every tested component is safe.' Give an example where many successful tests fail to establish safety. Explain the missing inference.

Study and practice¶

Progress by the work you can do, not by a fixed calendar.

Practice sequence 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

Practice sequence 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

Practice sequence 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.

Reading options¶

Assigned readings, where available, appear in the practice sequence above. These additional recommendations are not all required reading.

• How to Prove It: A Structured Approach

Continue¶

Connections above reflect recorded prerequisites, not a compulsory calendar.

• Related inquiry: Argument, dialectic, debate, rhetoric & adversarial reasoning
• Related inquiry: Social Evidence, Interpretation & Comparative Method
• Related inquiry: Rhetoric, Argument, Audience & Occasion

Integrated practice | Choose a study route