First PrinciplesStart anywhere. Prove it, then move on.

Rung 14

Proof technique

Contradiction and induction, the two workhorses.

Best attempted after 13. Deduction. Nothing stops you trying this now — the gate will tell you if you were right.

The gate

Produce a correct proof by contradiction and a correct proof by induction, each on a claim you have not been shown worked. For the induction, state the base case and the inductive step separately and say precisely where the assumption is used — a proof that never uses it has proved nothing.

Nobody checks this but you. Do it honestly and the rungs above hold; do it loosely and they will not, somewhere further up where the cause is much harder to find.

Euclid gave you the discipline: nothing without justification. This rung gives you two machines for producing justifications when the direct route is blocked.

Contradiction: assume the opposite of what you want, follow it faithfully, and arrive somewhere impossible. Since the reasoning was valid, the assumption must have been false.

Induction: prove it for the first case, then prove that whenever it holds for one case it holds for the next. Two finite arguments, infinitely many cases.

Why this is on the ladder

Induction and recursion are the same idea in two costumes — which is why rung 25 will feel familiar rather than strange. A recursive function has a base case and a step that assumes the smaller call works; an induction has a base case and a step that assumes the smaller claim holds. Learning them separately and then noticing they are one thing is worth more than being told.

Contradiction, meanwhile, is how you establish that something is impossible — and knowing what cannot be done is what stops you spending a week attempting it.

Do this

For contradiction, prove √2 is irrational. Assume it equals a fraction in lowest terms, follow the consequences, and watch "lowest terms" destroy itself.

For induction, prove that the first n odd numbers sum to n². Base case: one odd number, 1 = 1². Step: assume it for n, add the next odd number 2n + 1, and show you land on (n + 1)².

Then apply the real test to your induction: find the line where you used the assumption. Point at it. If you cannot, you have written something that is not an induction, however true its conclusion.

Where people get stuck

Induction gets written as a restatement — a page that never actually invokes the n case to reach n + 1. It reads like a proof and proves nothing. The pointing test catches it every time.

With contradiction, the usual error is arriving at something merely surprising rather than genuinely impossible. Surprising is not a contradiction. You need an outright collision: a number both even and odd, a fraction in lowest terms that is not.

Reading