What can be known for certain from a few agreed starting points, and what must simply be granted?
Level 1
Learn it
5 ideas. Read the hook, pick an answer, see the picture.
Geometry, Euclid and Trigonometry
Geometry, Euclid and Trigonometry
Shrink a right triangle to napkin size or stretch it across a football pitch. Keep one angle at 30 degrees, and the side opposite it stays exactly half the hypotenuse. That ratio is the sine of 30 degrees.
Pick one
With no interval stated, how many solutions does sin t = 1/2 have?
One turn holds two, 30 and 150 degrees (pi/6 and 5 pi/6). Adding or removing whole turns gives infinitely many more. In general, t = pi/6 + 2 pi k or 5 pi/6 + 2 pi k, for any integer k.
From geometry to precalculus
1SimilarityEqual angles give proportional sides; this defines sine.
2Pythagorasa^2 + b^2 = c^2, plus its converse and coordinate distance.
3Unit circlePoint at angle t is (cos t, sin t).
4Laws, identitiesLaw of sines, law of cosines, identities and equations.
5PrecalculusInverse functions, conics, vectors, series; then calculus.
Trigonometry grows from similarity, so learn proportion before treating sine as a button.
Proof and Precise Reasoning: From Arguments to Theorems
Euclid's proof that the primes never run out is often presented as a proof by contradiction. His own version is a recipe. Give it any finite list of primes, and it produces a prime not on the list.
Pick one
Take 'If it rained, the road is wet'. Its truth table has four rows: rain or not, wet or dry. In how many rows is it false?
Only rain with a dry road makes it false. Without rain it promised nothing, so it counts as true, wet road or dry.
What Counts as Proof
Example
Shows a claim holds once
Can fail at the next case
Computation
Checks particular cases
Refutes; proves only if every case is checked
Diagram or simulation
Suggests a truth
Can hide assumptions
Evidence about a model only
Proof
Each step follows from definitions
Covers every case in the domain
Only a valid proof establishes a universal claim (checking every case of a finite domain is one); the rest suggest or refute.
The Seven Liberal Arts, Hands-On: Logic, Proof and the Quadrivium
Euler's polynomial, n squared plus n plus 41, gives a prime for every n from 0 to 39. At n = 40 it gives 1681, which is 41 squared. Testing refutes a rule quickly; it cannot prove it for every n.
Pick one
You want a new square with exactly twice the area of an old one. What should its side be?
Doubling the side quadruples the area; one and a half times the side gives 2.25 times. Cut the old square along its diagonal: the square on that diagonal holds four such halves, the old square two. Both wrong options are the slave boy's guesses in Plato's Meno (82e, 83e); Socrates leads him to the diagonal (85a-b).
How to Read a Great Book: Levels, Modes and the Completion Standard
Adler and Van Doren's Rule 9: you may not agree with a book, or disagree, until you can honestly say "I understand." So how do you know you understand? This page's test: you have made seven things from it.
Pick one
Adler and Van Doren teach slow, active reading of hard books. What do they have you do before the slow reading?
Inspectional reading: skim the title, preface, contents and index, then read the book through once without stopping to look things up. Slow analytical reading is the next level up.
Four Levels of Reading
4SyntopicalRead several books on one subject and build the conversation
3AnalyticalSlow, active reading: state its arguments, then judge them
2InspectionalSkim systematically to learn what kind of book it is
1ElementaryReading the words; assumed here
↑ higher level
Understand a book before you agree or disagree with it, and build up through the levels.
A Map of Mathematics: Twenty-Five Areas and What Depends on What
Euler proved no walk crosses Konigsberg's seven bridges once each, and graph theory began. Routers now run graph algorithms. Shannon showed Boole's algebra of logic describes switching circuits. Some regions of mathematics must be crossed before others.
Pick one
Which of these can you start with algebra and careful reading, before any calculus?
Introductory linear algebra needs algebra and proof familiarity, not calculus. Real analysis, differential equations and mathematical statistics all need calculus first.
Mathematics in Six Regions
FoundationsAlgebra, geometry, trigonometry, then precalculus and calculus.
TrunkProof, discrete mathematics, algorithms and probability.
DataStatistics, linear algebra and data modelling.
NetworksGraph theory, optimization, information theory, queueing and time series.
ChangeNumerical methods, differential equations, control and dynamics.
JudgmentBayesian decisions, causal inference and game theory.
Six regions, then three advanced bridges: computability, cryptography and spectral methods. Dependencies matter more than numbers; many areas can be studied in parallel.
8 questions from the Geometry curriculum, Round 1.
Question 1 of 8
Which of these is one of Euclid's postulates, that is, a request that something be granted?
Postulate 3 asks that a circle may be described with any centre and distance. The first two options are Common Notions 1 and 5, and the last is the content of Definition 15.
Question 2 of 8
Why is CA equal to AB in the construction?
Definition 15 gives the equality of radii. Common Notion 1 is used one step later, and the look of the figure is never a reason.
Question 3 of 8
The boy in the Meno first says that the side of the double square should be twice as long. Why is that wrong?
A square of side 4 contains four squares each equal to the 2-foot square, so its area is 16. Doubling the side quadruples the area.
Question 4 of 8
'If equals be added to equals, the wholes are equal.' Which kind of starting point is this?
It is Common Notion 2. Common notions (axioms) are shared by all reasoning, while postulates are requests local to one system, here geometry.
Question 5 of 8
Which step of the proof uses Common Notion 1?
Common Notion 1 says that things equal to the same thing are equal to one another. Both CA and CB equal AB, so they equal each other.
Question 6 of 8
Why does the square enclosed by the four diagonals in Socrates' figure have twice the area of the original square?
Each small square is cut by its diagonal into two equal triangles. The enclosed square holds four of them and the original square holds two, so the areas are in the ratio 4 : 2.
Question 7 of 8
In the proof of I.1, the step 'CA equals AB, because A is the centre of the circle' rests on which starting point?
Definition 15 says that all straight lines from a circle's centre to the circle are equal. Postulate 3 only lets you draw the circle, and Common Notion 1 is used one step later.
Question 8 of 8
What does Euclid assume in I.1 without proof?
Postulate 3 and Definition 15 cover the first two, and AB is given. No postulate, definition or common notion says that the two circles meet.