​Axioms and Shared Premises

In Year 1, the Sudoku task and the axioms were moved to the start of Unit 3 for scheduling reasons — a grading benchmark closed before we could finish Unit 2. This is the kind of decision teachers sometimes have to make, and it is worth noting honestly: it cost something real. The axioms felt arbitrary arriving before students understood why a deductive system needs them. Here, after nine lessons of reasoning, they feel like the answer to a question students have actually been asking.

 

The Sudoku task is the first half of the lesson. Its purpose is not to solve the puzzle. It is to experience what it means to reason from shared rules. Critically, the puzzle does not need to be finished. Once groups have written enough justified placements to demonstrate that they understand what it means to convince someone a number belongs somewhere — typically five to eight well-written reasons — let them just work on the puzzle without the writing requirement. The reasoning was the lesson, not the completed grid. For students who found the writing tedious, make the connection explicit: writing ‘because the row already has 1, 2, 3, 4, 5, 6, 8, 9 and by Rule 1 each row contains each digit exactly once’ is exhausting. But this is why we develop shorthand. Eventually we will cite postulate numbers and theorem names instead of re-explaining the rules each time. These are the geometry equivalents of Sudoku’s rules — and they are worth knowing by name for exactly this reason.

 

Two notation tools introduced before the Sudoku task significantly improve the quality of written reasoning: a row:column cell reference system (top-left cell is 1:1, center cell is 5:5) and numbered boxes (the nine 3×3 boxes numbered 1–9, left to right, top to bottom). One group in the first year numbered the rules of Sudoku and cited them by number for every cell placement — exactly as they would later cite a postulate number in a geometric proof.

 

The disagreement question is the pivot of the lesson. Ask it in two stages. First: why do people disagree about things? The expected answer is that they don’t think carefully or don’t follow logic. Accept this honestly. Then ask the harder version: among people who are careful, who know the rules of logic, and who apply them correctly — why do some still reach very different conclusions about important things? The silence before someone says some version of ‘they started from different premises’ is the intellectual turning point of the unit. Stop and mark it.

 

Franklin, Jefferson, and Euclid:

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Jefferson drafted the Declaration of Independence with the phrase ‘sacred and undeniable’ truths. It was Benjamin Franklin who changed it to ‘self-evident.’ The edit matters: sacred and undeniable appeals to authority, to something external that declares the truths valid. Self-evident requires no such appeal. The truths are accepted because they are accepted — because they are too basic to derive from anything simpler. That is exactly what an axiom is.

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Euclid’s genius was not discovering geometric facts. Many were already known. His genius was showing that accepting just 10 self-evident premises makes all the others undeniable through chains of deductive reasoning. Some estimates put the number of theorems derivable from those 10 premises at around 465.

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Aristotle formalized this concept two generations before Euclid as Demonstrative Science: true knowledge must be built by deductive reasoning from first principles — things so basic they need no proof. The same idea appears in law, in mathematics, in ethics, in science. Before you can deduce anything, you need shared starting points.

 

Abraham Lincoln and the meaning of ‘demonstrate’:

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In 1864, the Reverend J. P. Gulliver of Norwich, Connecticut recalled a conversation with President Lincoln about how he had acquired his famously persuasive rhetorical skill. Lincoln’s answer was geometry.

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As a young lawyer, Lincoln had kept running into the word ‘demonstrate’ in his reading and could not satisfy himself that he understood it. He looked it up in Webster’s Dictionary, which defined it as ‘certain proof, proof beyond the possibility of doubt’ — but he could form no idea what sort of proof that was. He said to himself: ‘Lincoln, you can never make a lawyer if you do not understand what demonstrate means.’ He left his office in Springfield, went home to his father’s house, and worked through the first six books of Euclid until he could give any proposition at sight. His law partner William Herndon later described Lincoln staying up late into the night by candlelight, working through the Elements while fellow lawyers snored.

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Lincoln then knew what demonstrate meant. And he used that standard throughout his career. In the Lincoln-Douglas debates, he challenged his opponent: ‘If Judge Douglas will demonstrate somehow that this is popular sovereignty — the right of one man to make a slave of another, without any right in that other, or anyone else to object — demonstrate it as Euclid demonstrated propositions, there is no objection.’

Lincoln’s Cooper Union speech of 1860, which effectively launched his presidential campaign, has been analyzed as a Euclidean geometric proof: it opens with a postulate all parties could accept, develops each step as a logical consequence of the previous one, and arrives at a conclusion that — given the premises — cannot be denied.

The story is told in Jordan Ellenberg’s Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else (Penguin Press, 2021), and documented in Lincoln’s own words, recorded by Gulliver. It is one of the most powerful illustrations available of what it means to understand geometry — not as a collection of facts about shapes, but as a method for establishing what must be true.

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