∎ Mathematical Proof & Logic

What you’ll learn

1. The Discipline That Doesn't Take EvidenceExplain why mathematics refuses evidence as a standard of truth, using conjectures that survived centuries of confirming cases and then failed — and state the standard that replaces evidence: an argument that survives a hostile reader.Euler's sum-of-powers conjecture went unrefuted for 197 years before a 1966 computer search produced 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵, and Pólya's conjecture holds for every number up to 906,150,257 before failing. In every other discipline that much confirming evidence would settle the question; in mathematics it settles nothing, because a claim about infinitely many cases can never be checked case by case. What replaces evidence is proof: an argument constructed to survive a reader who is actively trying to make it fail — the standard this whole course is about meeting.
2. Definitions Are Where Proofs Are WonRecognise the mathematical definition as the machinery that makes proof possible — a conversion of a vague word into a checkable condition — and execute the unfold–work–refold pattern that underlies almost every direct proof.A mathematical definition is not a description of a word's meaning but a replacement of it: 'even' stops meaning 'divisible into two equal halves' and starts meaning 'equal to 2k for some integer k', a form you can do algebra to. Almost every elementary proof follows one pattern — unfold the definitions into their algebraic form, work in that form, then refold the result back into the defined word. Because a definition is a stipulation rather than a claim, it is the one thing a hostile reader cannot argue with, which is why an unclear definition is a proof's most common point of failure.
3. Direct Proof and the Price of 'Therefore'Execute a direct proof of a conditional statement, and audit an argument for the two failure modes that hide inside the word 'therefore': the unjustified leap and the assumed conclusion.A direct proof of 'if P then Q' assumes P and derives Q, and the whole art is that every 'therefore' must be cashable — each one is a promise that the previous line forces the next. Two failures hide there: the leap, where a step is true but unjustified, and circularity, where the conclusion is quietly used to prove itself. The reliable audit is to read a proof backwards, asking of each line what earlier line compels it.
4. The Contrapositive GambitUse the contrapositive as a strategic choice — proving ¬Q ⇒ ¬P in place of P ⇒ Q when the negated hypothesis is the more workable one — and distinguish it sharply from the converse and inverse.The contrapositive ¬Q ⇒ ¬P says exactly what P ⇒ Q says, since forbidding 'P true with Q false' is a single prohibition that can be read from either end. That makes it a free strategic move: 'n² is even' is a hypothesis you cannot unfold usefully, while its contrapositive hands you 'n is odd', which unfolds immediately. The converse and inverse are different statements with different truth values, and confusing them with the contrapositive is the most consequential error in elementary proof.
5. Contradiction: Renting the Enemy's HouseConstruct a proof by contradiction, understand why deriving a single impossibility from ¬P establishes P, and recognise the technique's characteristic overuse.Proof by contradiction assumes the negation of the claim, reasons until something impossible appears, and concludes that the assumption was the only thing that could have been wrong — the classic instance being the irrationality of √2, where 'in lowest terms' is loaded like a trap and springs when both numerator and denominator turn out even. Its power is that it converts a claim about nothing existing into a concrete object you can manipulate, but its cost is a proof that certifies without explaining, which is why an argument that never actually uses its contradictory assumption is a direct proof in disguise.
6. Induction: The Ladder, and How It BreaksProve statements about all natural numbers by induction, justify why the technique is legitimate rather than circular, and diagnose the two ways a real induction fails — using the 'all horses are the same colour' argument as the diagnostic instrument.Induction proves infinitely many statements with two finite acts — a base case and an inductive step that manufactures each rung from the one below — and it is legitimate because the alternative would require a smallest counterexample that the step forbids. Its failures are almost never in the algebra: the base case can be true yet unreachable from the step, as in 'all horses are the same colour', where the inductive step secretly assumes a group large enough to overlap and so cannot bridge n = 1 to n = 2. Strong induction extends the technique to cases where the previous rung alone is not enough, as in the proof that every integer above 1 has a prime factorisation.
7. Proving Something Exists Without Finding ItDistinguish constructive from non-constructive existence proofs, and understand what the non-constructive proof does and does not deliver — using the √2^√2 argument, which proves a number exists while leaving its identity undetermined.An existence claim can be settled by exhibiting an example or by showing that non-existence is impossible, and the two are not equally informative: the √2^√2 argument proves that irrational a, b exist with a^b rational by presenting two candidates and proving one of them works — without determining which. Constructive proofs hand you the object and are what engineering and computation need; non-constructive proofs are often shorter and sometimes the only available route, and the distinction matters enough that a school of mathematics rejects the non-constructive existence claim outright.
8. The Counterexample Is a Proof TooDisprove a universal claim by exhibiting a single counterexample, negate quantified statements correctly to know what a disproof owes, and calibrate against patterns that hold for many cases and then fail.Disproving 'for all x, P(x)' requires only one x where P fails, because the negation of a universal claim is an existence claim — which is why Euler needed one equation to be refuted and Fermat's conjecture that 2^(2^n)+1 is always prime died when Euler factored 4,294,967,297 = 641 × 6,700,417. The asymmetry is total: universal claims are expensive to prove and cheap to destroy, existence claims the reverse. Patterns like n² + n + 41, prime for n = 0 through 39 and composite at 40, exist to calibrate how little a run of confirming cases is worth.
9. Proofs Are Read by PeopleUnderstand proof as a social and communicative act as well as a formal object — why the most scrutinised proof of the century still contained a gap, why computer-assisted proofs unsettled the field, and what formal verification changes about the hostile reader.Wiles's 1993 proof of Fermat's Last Theorem was found to have a gap in its Euler system argument and took until 1994 to repair, which is not an embarrassment but a demonstration that proofs are validated by readers rather than possessing validity intrinsically. Appel and Haken's 1976 Four Colour Theorem forced the question of what a proof is for when no human can read the case analysis, and formal verification — Hales's Flyspeck project, completed 2014 — answers it by making the hostile reader a machine that accepts nothing. What the machine cannot supply is the judgement that the theorem was worth proving.

Questions this course answers

Grounded in trusted sources

• L. J. Lander & T. R. Parkin, 'Counterexample to Euler's conjecture on sums of like powers' (Bull. Amer. Math. Soc. 72:6, 1966) — https://www.ams.org/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf
• Pólya conjecture — counterexamples of Haselgrove (1958), Lehman (1960) and Tanaka (1980): https://en.wikipedia.org/wiki/P%C3%B3lya_conjecture
• G. Pólya, How to Solve It (Princeton University Press, 1945)
• Daniel J. Velleman, How to Prove It: A Structured Approach (3rd ed., Cambridge University Press, 2019)
• Kenneth Rosen, Discrete Mathematics and Its Applications (8th ed., McGraw-Hill)
• Andrew Wiles, 'Modular elliptic curves and Fermat's Last Theorem' (Annals of Mathematics 141, 1995, 443–551) — https://jontalle.web.engr.illinois.edu/Public/Wiles-Fermat.95.pdf
• Wiles's proof of Fermat's Last Theorem — the 1993 gap and its 1994 repair with Richard Taylor: https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem
• Kenneth Appel & Wolfgang Haken, 'Every planar map is four colorable' (Illinois J. Math. 21, 1977)

Every Wunder lesson is built from real, reputable sources — never invented.

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