First PrinciplesStart anywhere. Prove it, then move on.

Rung 13

Deduction

Every step justified by a step already earned.

Best attempted after 1. Instructions a machine could follow and 5. Balancing equations. Nothing stops you trying this now — the gate will tell you if you were right.

The gate

Prove one of Euclid's early propositions with every line justified by a postulate, a definition, or a proposition already proved. Then hand it to someone instructed to challenge each line with "why is that allowed?" — and answer every challenge from your written proof, adding nothing new.

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.

Rung 1 asked you to write instructions with nothing left to judgement, and to hand them to someone who would follow them literally. This is that rung again, addressed to a reader who is not merely literal but hostile — someone who will grant nothing you have not earned.

Euclid built the whole of Book I from a handful of definitions and five postulates. Everything else is derived, in order, each proposition standing on ones already proved. Nothing is assumed because it looks obvious in the diagram.

Why this is on the ladder

Because this is the same skill as programming, viewed from the other side. A proof is a procedure a sceptic can execute. A program is a procedure a processor can execute. Both fail in the same way: a step that seemed too obvious to state.

It also inoculates you against the most expensive error ahead. At rung 45 your network will produce a number, and you will want to believe it. The habit of asking what licenses this step? is what stops you believing a result because it appeared, rather than because it follows.

Do this

Read the definitions and the five postulates first, slowly. They are short, and everything rests on them.

Then take Proposition 1 — constructing an equilateral triangle on a given segment — and write the proof out in your own hand, with a justification beside every line. Not "obviously"; name the postulate.

Then find the famous gap. Euclid asserts that the two circles intersect. Which postulate grants that? Work at it honestly before looking it up. The answer is that none of them do — it is an assumption smuggled in from the picture, and mathematicians took two thousand years to notice.

That discovery is the real gate. Once you have caught Euclid, you can catch yourself.

Where people get stuck

The diagram does the arguing. You look at the figure, see that two lines cross, and use it — but "I can see it" is not a justification, and the whole discipline is refusing that move.

The other difficulty is that the propositions must be taken in order. Proposition 7 may use 1 through 6 and nothing else. Reaching for a fact you know from elsewhere breaks the chain even when the fact is true.

Reading