B. Symbolic Systems

B.1 Mathematics

Phase: Phases 1–5
Study time: about 530 hours core, plus optional units
Prerequisites: secondary-school algebra
Leads to: C, B.2, B.3, E
Signature checkpoint: 18.01SC final at 80% closed-book

Course description

The units below run roughly in prerequisite order, and each is sized so you can see how much of the 530 hours it costs.

Algebra and trigonometry review (0–60 h). Diagnostic: take Khan Academy's Algebra 2 and Trigonometry course challenges cold; if you score 80% or better on both, skip this unit, and otherwise fill the gaps with Khan Academy. The Persian scholar al-Khwarizmi, writing in Arabic in Baghdad around 820, titled his treatise on solving equations the Compendious Book on Calculation by Completion and Balancing; al-jabr, the "completion" step, gave algebra its name. Two centuries earlier, in 628, the Indian astronomer Brahmagupta had already stated rules for arithmetic with zero and negative numbers
[1]
.
Proof (60 h, alongside A.2). Hammack's free Book of Proof
[2]
, solutions to the odd exercises included. Work a handful of propositions from Euclid's Elements, Book I, to see classical deduction in its original form. Pólya's How to Solve It
[3]
(referenced again from section IV) is the standing companion for every problem set from here on.
Single-variable calculus (150 h). MIT's 18.01SC
[4]
, built for independent study with graded problem sets and exams, or OpenStax Calculus Volumes 1–2
[5]
; 3Blue1Brown's Essence of Calculus
[6]
for geometric intuition alongside either. History carries the argument: Zeno's paradoxes and Archimedes' method of exhaustion pose the problem; the Kerala school worked out infinite series for sine and arctangent between the 14th and 16th centuries
[1]
; Newton and Leibniz independently built the general method between the 1660s and 1680s. Newton published his in a geometric idiom: read Book I, Section 1 of the Principia, the lemmas on "first and last ratios"
[7]
, and don't look for fluxions there, because he kept that notation out of the book. Leibniz's 1684 Nova methodus introduced the d notation still used today
[8]
, and the ∫ sign followed in 1686. Bishop Berkeley then attacked the whole apparatus as reasoning about "ghosts of departed quantities," an objection that stood for over a century until the 19th-century epsilon-delta definition of the limit answered it. That sequence (problem, method, objection, rigorous repair) is the cycle's skepticism stage inside one subject.
Multivariable basics (60 h, before C.1's electromagnetism). OpenStax Calculus Volume 3, the vector calculus chapters.
Linear algebra (120 h). Strang's course via MIT's 18.06SC
[9]
, 3Blue1Brown's Essence of Linear Algebra
[6]
for the geometry, then Axler's Linear Algebra Done Right, now open access
[10]
, for a proof-based second pass. The Chinese Nine Chapters on the Mathematical Art solved systems of linear equations by an elimination method equivalent to Gaussian elimination, some seventeen centuries before Gauss
[11]
.
Differential equations (80 h). MIT's 18.03SC
[12]
. This unit feeds directly into C.1, E.1, and E.3.
Optional: abstract algebra, real analysis, topology, or graph theory. Pick at least one to feel the abstraction stage properly, where content strips away and only structure remains. For topology, Richeson's Euler's Gem: The Polyhedron Formula and the Birth of Topology tells the story through one theorem. For group theory, Nathan Carter's Visual Group Theory builds the subject from pictures and puzzles such as the Rubik's cube, and Judson's free Abstract Algebra: Theory and Applications covers the same ground rigorously if you want no cost attached.

History and philosophy run as a thread through units 3–7 rather than a separate unit. Read Lakatos's Proofs and Refutations

[13]

once the proof unit is behind you: it reconstructs Euler's polyhedron formula being attacked by counterexamples and patched by "monster-barring," and it is itself an argument that mathematics goes through something like this document's whole cycle. The foundations crisis of the early 20th century, logicism against formalism against intuitionism

[14]

, and its resolution (or non-resolution) in Hilbert's program and Gödel's incompleteness theorems, is best read through Nagel and Newman's short Gödel's Proof

[15]

. Stillwell's Mathematics and Its History

[16]

is the companion volume for units 3 through 7 if you want more connective narrative than a textbook gives, and the MacTutor archive

[17]

is the place to check any mathematician's dates or a claim about who did what first. Probability and statistics are in B.2.

Checkpoints

Each checkpoint is tagged with the stage of the cycle it tests.

Stage	Checkpoint	Attempts
formalism	Without notes, prove that √2 is irrational, that there are infinitely many primes, and one further statement by induction.	Post the first attempt
formalism	Sit the 18.01SC final exam closed-book and score at least 80% against the published solutions.	Post the first attempt
skepticism	State Berkeley's objection to Newton's fluxions in your own words, then explain how the epsilon-delta definition of the limit answers it.	Post the first attempt
abstraction	Explain eigenvectors geometrically, then say why the same idea governs both a vibrating string and the long-run behavior of a Markov chain.	Post the first attempt
abstraction	Reconstruct Lakatos's first counterexample to Euler's polyhedron formula and the monster-barring response it provoked.	Post the first attempt
intuition	Before solving a damped oscillator for three different damping values, sketch what you expect the three solution curves to look like, then check yourself against the actual solutions.	Post the first attempt

Discussion

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