Basis Lab

The mathematics underneath

Say exactly what a claim asserts and establish that it holds in every case without checking every case, using the handful of argument shapes that every proof, every invariant and every correctness argument in this library is built from

Arguing That It Always Holds

Claims, their negations, and the ladder argument that covers infinitely many cases at once. The reasoning that every later derivation assumes you already accept.

8 lessons, written and corrected before you arrived. Reading them here needs no account. The first reads the whole way through; the others open and then stop, because a page nobody owns cannot tell who is reading it. Starting the course gives you your own copy, where every idea has problems standing under it and you can ask about any sentence.

Start reading

This is course 5 of 8 in a reading track: read any page here without stopping at the symbols. The track is an order, not a gate, and the rest of it is on one page.

  1. 01A Claim You Can TestBefore anything can be proved it has to be said precisely enough to be wrong. Most disagreements that look mathematical are really two people arguing about an unsharpened claim.
  2. 02And, Or, and Notopening onlyThree ways of joining claims, two of which behave as expected and one of which does not. The rules for negating a joined claim are the ones people get wrong in writing and in code.
  3. 03What an Implication Promisesopening onlyAn if-then claim rules out exactly one combination and says nothing about the rest. Reading it as though it ran backwards is the most expensive mistake in the subject.
  4. 04Every, and At Least Oneopening onlyClaims about all cases and claims about some case are opposites of each other, and negating one turns it into the other. Getting the order of two of them wrong changes the claim entirely.
  5. 05Collections and What Is In Themopening onlyA collection described by a rule rather than a list, and the four ways of combining two of them. The notation is light and it is what every later claim about all cases is quantified over.
  6. 06Breaking a Claim With One Caseopening onlyOne case is enough to destroy a claim about everything, which makes disproof cheap. The skill is constructing awkward cases on purpose rather than waiting to be surprised by one.
  7. 07Proving It for Every Numberopening onlyInfinitely many cases proved by two finite checks: that the first case holds, and that each case forces the next. The second check is where nearly every broken argument of this kind breaks.
  8. 08What This Buys You in Practiceopening onlyA loop is correct after any number of passes for the same reason a ladder argument works. Writing down what stays true on every pass is the working form of the whole course.