Mathematics: Start Here

Most people meet mathematics as a long list of topics to survive. This wing treats it differently: as a small set of connected abilities that you can build in a sensible order and prove to yourself that you have. Start here to see the whole shape, then follow the links to the page that fits your need.

The thesis: a spine, not a survey

A survey tries to touch every branch. A spine picks the load-bearing ones and builds them in order. The spine proposed here runs:

1. Logic and proof, for knowing when a claim is actually established.
2. Discrete mathematics, probability and algorithms, for the structures and uncertainties of technical work.
3. Linear algebra and optimization, for transformations and for "best under constraints".
4. Then feedback, information, networks and dynamics, for reasoning about whole systems that change.

The reasoning is practical. Each later area leans on earlier ones, so skipping the trunk makes the branches harder, not faster. And a learner who goes deep on a few connected areas can usually pick up a neighbouring field later, while a learner who has sampled twenty topics shallowly usually cannot. Provisional This is a design argument about how to order study, not a theorem. A Map of Mathematics: Twenty-Five Areas and What Depends on What lays out the full order and the dependencies.

Six abilities mathematics trains

"Knowing maths" bundles several separable skills. This wing tracks six:

They overlap with what researchers call the strands of mathematical proficiency (conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, productive disposition) in the National Research Council report Adding It Up. Established as a useful framing; the six-way split here is a teaching convenience, not a measured psychological structure. Learning Mathematics: Procedures and the Six Abilities gives a practice for each.

The core idea: readiness is demonstrated

Hours spent, videos watched and chapters finished are measures of effort. They are not measures of skill. A platform's progress meter can read high while a whole domain is weak, because it samples some skills and not others. The rule of this wing is simple: you are ready for the next subject when you can solve fresh problems on its prerequisites without help, explain why the methods work, and do it again a week or more later. That standard is stated with working numbers in Placement, Repair and Retention: Keeping Mathematics Alive.

Operating rules

• Two streams at a time. One foundational stream, one applied stream. More than that and nothing compounds.
• Problems before summaries. A book read without retrieval, derivation or implementation is exposure, not mastery. Dunlosky and colleagues (2013) rated practice testing and distributed practice as the most broadly useful techniques they reviewed, and rereading and highlighting as low in utility. Established for those techniques; effect sizes vary by material.
• Every area ends in one artifact. A page of proofs, a small program, a query, a diagram, a simulation or a short memo. See Twelve-Week Arcs and the Capstone Project Method.
• Practice with observable targets. Vague goals cannot be tested. Deliberate Practice: Choose Targets You Can Observe shows how to turn them into things you can check.

Three entry doors

Pick the door that matches where you are, not where you wish you were.

1. Repair gaps. You studied maths once and it did not stick. Start by testing arithmetic and algebra, repair only what fails, and move on. Go to Placement, Repair and Retention: Keeping Mathematics Alive, then Algebra: The Skills That Carry Everything.
2. Learn forward. The foundations are solid and you want new territory. Use A Map of Mathematics: Twenty-Five Areas and What Depends on What to choose the next area, and Calculus and Beyond: Change, Structure and Uncertainty for the continuous branch.
3. Learn to reason. You care less about computation than about thinking clearly. Begin with Proof and Precise Reasoning: From Arguments to Theorems and Discrete Mathematics and Graphs: Counting, Relations and Networks; proof needs little more than careful algebra.

The trivium applied to mathematics

The classical trivium of grammar, logic and rhetoric, introduced in The Seven Liberal Arts, Hands-On: Logic, Proof and the Quadrivium, translates cleanly:

• Grammar: define terms, fix notation, state units and assumptions. Rewrite each key definition in your own words and give one example and one non-example.
• Logic: test inferences. Before accepting a claim or a chart, ask what would falsify it.
• Rhetoric: explain to a real audience. A one-page memo with the claim, the model, the evidence, the uncertainty and a recommendation shows whether you understand. For the writing side, see English Expression Mastery: Words, Sentences, Paragraphs and Speech.

Anti-drift rules

• A model is not metaphysics. Systems language, cybernetics and simulations are models with assumptions and limits. A simulation is evidence about a specified model, not proof of a universal claim and not evidence about the nature of reality. This is the same discipline as separating fact from analogy elsewhere in the library; see Liberal Arts and Reasoning: Start Here.
• Do not collect courses. Do not take a course or buy a book until it has a named place in an active plan. Depth compounds; novelty restarts the clock.
• Do not advance because material feels familiar. Advance when you can solve, explain and critique.

Free courses and textbooks, used responsibly

Excellent open material exists: Hammack's Book of Proof and the Open Logic Project's forall x texts are free, Mathematics for Computer Science is on MIT OpenCourseWare, and OpenStax publishes free calculus and precalculus books. Use each for a named purpose, check its stated prerequisites, and do its exercises. A free resource you finish in part with exercises beats an expensive one you only skim. Check licences and editions before relying on a link, since editions change.

Map of the wing

Music as applied number

The old quadrivium paired arithmetic with music because pitch intervals correspond to whole-number ratios. On a stretched string, halving the length raises the pitch an octave (2:1); two-thirds of the length gives a fifth (3:2). You can hear this on the monochord. Stack twelve fifths and seven octaves and they almost meet but not quite: the gap is the Pythagorean comma, a ratio of 531441 to 524288, about 23 cents. Demonstrated as arithmetic. That Pythagoras himself discovered the ratios is a tradition recorded centuries later, not a verified fact. Plato's Republic (Book VII) presents arithmetic as drawing the soul toward being; read that as the philosopher's claim about education, not a result.

Try this

1. Write down the six abilities and rate yourself honestly on each with one piece of evidence, such as a fresh problem you solved without help last week. Where you have no evidence, write "untested".
2. Choose one door above and one stream to start. Write the first artifact you will make and a date to check it.
3. Take one definition you rely on (for example, "function") and write it in your own words with an example and a non-example.

Further reading

• Kilpatrick, Swafford and Findell, Adding It Up (National Academies Press, 2001), free to read online.
• Daniel Velleman, How to Prove It (Cambridge University Press, 3rd ed., 2019).
• Richard Hammack, Book of Proof (3rd ed., free from the author's site).
• Eric Lehman, F. Thomson Leighton and Albert Meyer, Mathematics for Computer Science (MIT OpenCourseWare 6.042J).
• Donella Meadows, Thinking in Systems (Chelsea Green, 2008).
• Barak Rosenshine, "Principles of Instruction", American Educator, 2012.