Quadrivium · the arts of number · art 4 of 7
Arithmetic
Arithmetica
Number and its properties
- Classical definition
- The science of multitude, or discrete quantity, considered in itself. (Nicomachus, Introduction to Arithmetic I.3; Boethius, De institutione arithmetica I.1.)
- Proper object
- Number as such: unity and plurality, even and odd, perfect numbers.
- Place in the order
- Quadrivium: multitude in itself · see the whole order
Arithmetic is the study of number in itself, as music is the study of number in relation (Nicomachus, Introduction to Arithmetic I.3). It covers whole numbers, what divides them, what measure two numbers share and what patterns they follow. It holds some of the oldest surviving proofs: Euclid's argument that no finite list of primes is complete, and the proof, already familiar to Aristotle, that a square's diagonal has no common measure with its side. The same ideas now sit under computing, cryptography and any careful use of units, rates and growth.
The guiding questionWhat can be proved about whole numbers and their ratios, and where do they stop being enough?
The spiral
Three strands, three rounds
Each strand comes back in every round, a level deeper. Finish Round 1 across the arts before Round 2: breadth first, then depth.
Round I Foundations
- Primes and divisibilityPrimes and Euclid's proofI can reconstruct Euclid's proof that no finite list of primes is complete, say exactly what it claims and what it does not, and use a counterexample to test a claim about primes.
- Measure and ratioRatio and common measureI can find the greatest common measure of two whole-number lengths by repeated subtraction, write their ratio in lowest terms, and reconstruct the proof that a square's diagonal and side have no common measure.
- Pattern and inductionPatterns, recurrence and inductionI can compute terms from a recurrence or a formula, and prove a formula for every n by induction with the base case and the step written out.
Round II Practice
- Primes and divisibilityFactoring, sieving and testingI can factor whole numbers up to 1,000 into primes by hand, list the primes up to 100 with the sieve of Eratosthenes, and write a short program that tests a conjecture about primes and say what its output does and does not show.
- Measure and ratioThe Euclidean algorithmI can run the Euclidean algorithm with division and remainder, prove that it returns the greatest common divisor, and use it to reduce ratios and compare them.
- Pattern and inductionInduction in practiceI can prove claims about sums, divisibility and recurrences by induction, and say why each step is allowed.
Round III Mastery
- Primes and divisibilityPrimes at scaleI can state what is proved, what is only conjectured and what is assumed about the distribution and use of primes, and explain why large primes matter in public-key cryptography without claiming more than is known.
- Measure and ratioMagnitudes without fractionsI can state Eudoxus's definition of equal ratios (Elements V, Definition 5), say why it handles incommensurable magnitudes, and explain how Dedekind's cuts define irrational numbers from the rationals.
- Pattern and inductionInduction and the foundations of numberI can explain why induction is justified from Dedekind's definition of the natural numbers, state Goedel's first incompleteness theorem for arithmetic with its conditions, and model a real process with a recurrence whose limits I state.
Labs
Core reading in the Library
- LibraryThe Seven Liberal Arts, Hands-On: Logic, Proof and the QuadriviumA practical guide to the trivium and quadrivium, with a practice and a made thing for each art, plus a path through Socratic questioning, Aristotle's logic and Euclid's method.
- LibraryProof and Precise Reasoning: From Arguments to TheoremsThe grammar of valid inference: validity and soundness, truth tables, quantifiers, the main proof techniques and counterexamples, framed by the trivium habit of defining, testing and explaining.
- LibraryDiscrete Mathematics and Graphs: Counting, Relations and NetworksSets, relations, counting, induction, recurrences, graphs and trees, data modelling, and the discipline of separating an algorithm's result from a proof that it works.
- LibraryAlgebra: The Skills That Carry EverythingThe dependency groups of school algebra and the reasons behind each rule, so that errors become diagnosable instead of mysterious.
- LibraryA Map of Mathematics: Twenty-Five Areas and What Depends on WhatA guided tour of the branches of mathematics in learning order, with prerequisites and the philosophical and engineering parallels of each.
- LibraryMathematics for Better DecisionsA tiered curriculum of the practical mathematics that protects you from being fooled by numbers, risk and scale.
Where it is used
- LibraryQuantitative Reasoning with Stated Assumptions · Recovery durationA rate is a ratio with units: 500,000 MB at 100 MB/s is 5,000 s, and reading 500 GB as 500 GiB adds about six minutes.2 min glance · 6 min
- LibraryMathematics for Better Decisions · Tier 2: compoundingLinear growth adds, compounding multiplies, and the Rule of 72 is a quick check on a doubling time.2 min glance · 5 min
- LibraryThe Practical Kitchen: Staples, Meal Prep and Food Safety · Buying rulesCost per usable pound is a ratio: paying 3.00 a pound and eating half of what you buy costs about 6.00 per usable pound.2 min glance · 5 min
- LibraryProtein for Muscle on a Plant-Based Diet · Estimating a targetA per-kilogram range times a stated reference weight gives a daily range: 80 kg at 1.4 to 2.0 g per kg is 112 to 160 g a day.2 min glance · 6 min
Seminars
Read the sources
One primary text for each round, with a question to think through. All 21 seminars →
Arithmetic · Round 1
- Nicomachus of Gerasa, Introduction to Arithmetic, translated by Martin Luther D'Ooge (Macmillan, 1926), Book I, chapters 7 to 13Chapter 7 divides number into even and odd. Chapters 8 to 10 divide the even. Chapters 11 to 13 divide the odd into the prime and incomposite, the secondary and composite, and the kind that is composite in itself but prime and incomposite relative to another. Chapter 13 also describes the sieve of Eratosthenes and a test by repeated subtraction for whether two odd numbers have a common measure. D'Ooge's translation was published in 1926, so it entered the public domain in the United States on 1 January 2022. This Zenodo record holds a full PDF scan of the 1926 volume. Cite by book and chapter, not by page.
- Euclid, Elements, Book VII (David Joyce's online edition, Clark University), Definitions 1 to 16; Propositions 1 and 2Definitions 11 to 14 define prime, relatively prime, composite and relatively composite numbers by what measures them. Propositions 1 and 2 are linked from the same page. Proposition 2 finds the greatest common measure of two numbers by repeated subtraction; Elements X.2 applies the same idea to magnitudes.
Euclid calls a prime a number 'measured by a unit alone' (Book VII, Definition 11), while Nicomachus calls it 'prime and incomposite' and files it among the odd numbers (I.11). What does each way of defining a prime make you notice, and what does each hide?
- On Euclid's Definition 11, does the number 2 count as prime? Where would Nicomachus put it, given that he sorts primes among the odd numbers?
- Nicomachus describes a sieve (I.13) that separates primes from composites by a procedure. Does knowing how to find primes tell you what a prime is?
- In VII.1 and VII.2 Euclid subtracts the smaller number from the larger again and again. What does he assume about what it means for one number to 'measure' another?
Arithmetic · Round 2
- Euclid, Elements, Book IX (David Joyce's online edition, Clark University), Book IX, Proposition 20Read Euclid's own proof, not a summary. It takes the least number measured by the assigned primes (their product), adds a unit, and considers two cases: the result is prime, or it is measured by some prime that is not among those assigned (VII.31). The statement does not use the words 'infinitely many'.
- Euclid, Elements, Book IX (David Joyce's online edition, Clark University), Book IX, Proposition 36, with Book VII, Definition 22Definition 22 of Book VII says a perfect number is equal to the sum of its own parts. The first four perfect numbers, 6, 28, 496 and 8128, all have the form Euclid builds. Joyce's page links each step of the proof to the earlier propositions and definitions it uses.
IX.20 shows that no assigned list of primes is complete. IX.36 shows how a prime of one particular form yields a perfect number. What does each proposition let us say about how many primes and how many perfect numbers there are, and where does each stop?
- Euclid starts from a unit and doubles: 1, 2, 4, 8, 16, 32. Find the running sums, say where the sum is prime, and compute the perfect numbers that IX.36 then gives.
- IX.20 gives a way to produce a prime that is missing from any list. Does IX.36 give an equally good way to produce a new perfect number? What must be true before it applies?
- IX.36 says that numbers of this form are perfect. Does it say that every perfect number has this form?
Arithmetic · Round 3
- Richard Dedekind, Continuity and Irrational Numbers (Stetigkeit und irrationale Zahlen, 1872), translated by W. W. Beman in Essays on the Theory of Numbers (Open Court, 1901), Preface and Sections III and IVBeman's 1901 translation is in the public domain, and Project Gutenberg ebook 21016 (PDF and TeX source) prints both of Dedekind's essays. In Section III Dedekind recalls that the ancient Greeks had shown that some lengths are incommensurable with a unit, demands that arithmetic be developed out of itself, and asks what continuity is. Section IV defines the cut and gives an indirect proof that a positive integer that is not the square of an integer has no rational square root. Cite by section, not by page.
- Richard Dedekind, The Nature and Meaning of Numbers (Was sind und was sollen die Zahlen?, 1888), translated by W. W. Beman in Essays on the Theory of Numbers (Open Court, 1901), Preface to the first edition; Sections 71 and 80This is the second essay in the same Gutenberg ebook; the numbered sections are Dedekind's own paragraphs. The Preface to the first edition opens with the demand that nothing capable of proof be accepted without proof. Section 71 defines a simply infinite system. Section 80 states the theorem of complete induction (inference from n to n') and derives it from his general theorem on chains (Section 59).
Dedekind says that arithmetic shall be 'developed out of itself' and that irrational numbers must be defined 'by means of the rational numbers alone' (Section III). When he defines the square root of 2 by a cut in the rational numbers, has he found a number or made one?
- Dedekind writes that he cannot prove the principle of continuity of the straight line and that it is 'nothing else than an axiom'. What does he say would follow if space were not continuous, and what does that suggest about the link between number and space?
- In the Preface of 1888 he writes both that 'nothing capable of proof ought to be accepted without proof' and that numbers are 'free creations of the human mind'. Can both stand? Which parts of arithmetic are chosen, and which are proved?
- Section IV gives its own indirect proof that no rational number squares to D. Compare it with the proof you reconstructed for the diagonal of a square. What do the two proofs share, and what differs?