Mathematical Proof Construction
8 weeks · 0 milestones
Construct a complete formal mathematical proof in continuous mathematics — real analysis (limits, continuity, differentiation, integration), linear algebra (eigenvalues, vector spaces, matrix decomposition), differential equations (existence, uniqueness, stability), or probability theory (convergence, distributions, central limit theorem). The proof may be original or a reconstruction of a named non-trivial result with all steps made explicit and no reasoning omitted. Mathematical proof construction is an applied intellectual skill — the same reasoning that places legal argumentation in SKILLS rather than LEARNING. The skill is not knowing what a theorem says but demonstrating you can derive it from first principles with all logical steps documented. Mathematical proofs require no equipment — paper, pen, and primary mathematical texts are sufficient. Reviewed by a mathematician or physicist who confirms the proof is logically valid and complete; the reviewer presents an unseen claim during the review session for the student to work through live, confirming that the student can construct proofs rather than reproduce memorised ones.
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3 milestones
Choose a specific mathematical claim you will prove and determine the proof strategy. The claim must be one where the proof is not trivially a reproduction of a standard textbook proof — an extension of a known result, a novel application of a standard technique to a less-studied case, or a claim you have formulated yourself. Research the existing proof landscape for related results before committing to a strategy, so that the approach you choose is informed by what is already known.
Proof required
Submit the precise statement of the mathematical claim you intend to prove (using standard mathematical notation with all terms defined), a description of the existing proof techniques for related results you have reviewed, your chosen proof strategy (direct proof, proof by contradiction, mathematical induction, construction, or other), and a justification for why that strategy is appropriate for this particular claim.
What gets checked
- Claim is stated precisely in formal mathematical notation — every variable is quantified, every set is defined, and the statement is unambiguous
- Proof strategy justification connects the chosen approach to the specific structure of the claim — not 'I will use induction' but 'I will use strong induction on n because the step from case n-1 to n requires knowing the result holds for all smaller values'
- Evidence of research into related proofs — at least one reference to a related result whose proof technique informed the strategy choice