Before Proof

The emergence of mathematical thought

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Why Before Proof works this way

Textbooks run mathematics backwards — definition, theorem, proof, and an example if there is room. Excellent for storage, useless for learning. Ideas are actually made in the opposite order, and everything here follows that order instead.

Play

The idea is turned over for its own sake. Nothing is at stake, and nothing has to be justified yet.

Pressure

It meets a case it cannot survive. The comfortable version has to give way.

Necessity

One route remains open. The proof is not invented so much as forced.

New possibility

The result opens ground that could not be seen from where the question began.

So each topic is built in three parts

Not three kinds of content — one argument at three depths, each designed around a different part of the cycle above.

Play and pressure

The film

The introduction, and the narrative of discovery. It asks the question, plays with the obvious approach and breaks it, then stops at the point of maximum tension — before the proof.

Motivation before formalism. A definition met after the problem it solves is understood rather than memorised.
Necessity

The document

Much longer and completely technical, every proof given in full. What it does not do is change voice: each proof is oriented before it is stated and reflected on afterwards.

The same explanatory voice at higher resolution, so there is no handover to a cold reference.
New possibility

The problems

Statements first, so a problem is met before it is answered. Hints kept separate from solutions, written solutions as standard, and a walkthrough film on choosing an approach.

Retrieving an argument yourself builds something reading one never does. The struggle is preserved on purpose.
Read the philosophy The full argument: why the standard order fails, how each part is designed, and what this deliberately is not.

Explore the subjects

Thirteen subjects, ordered so that each assumes only the ones before it. Every subject holds films and documents on the same topics, plus problem sets.

Logic & Proof

What makes an argument binding?

4 films · 4 documents · 3 problem sets

  • Statements›
  • Quantifiers›
  • Methods of proof›
Enter Logic & Proof  →

Analysis

What does it mean to approach without arriving?

9 films · 9 documents · 7 problem sets

  • Sequences›
  • Continuity›
  • Derivative & integral›
Enter Analysis  →

Algebra

When does structure reveal more than quantity?

8 films · 8 documents · 6 problem sets

  • Groups›
  • Rings›
  • Fields›
Enter Algebra  →
All thirteen subjects Foundations, the continuous, structure, space, and the limits — a dependency order rather than a difficulty ranking.

Articles

Shorter pieces that sit outside the subjects: essays on what mathematics is and how it is taught, notes written while working through something, and occasional commentary on how a piece of mathematics came to be. Published as they are finished, and revised in the open.

Essay

The definition always comes last

By Joshua Allen

Every definition in mathematics is an answer to a question that was asked first. We teach them in the opposite order and then wonder why students experience the subject as a list of arbitrary rules to be endured. This is the long version of the argument the whole project rests on.

Read more  →
Notes

What I got wrong about compactness for two years

By Joshua Allen

I could state the definition, prove the standard theorems, and had no idea what the thing was for. These are the notes from the week that fixed it, including the wrong pictures I was carrying around and the example that finally dislodged them.

Read more  →
Essay

Rigour is a destination, not a filter

By Joshua Allen

There is a style of teaching that treats the difficulty of the presentation as a measure of seriousness. It selects for tolerance rather than ability, and it has cost mathematics more good people than any other habit in the discipline.

Read more  →
All articles Nothing is sorted into categories. Newest first, and the kind of piece is noted beside each one.

Get the latest

New films, documents and articles as they are finished. No more than one email a month, and nothing else.

Drafts go up before they are finished, and the revisions stay visible.