Our approach to Mathematics is as Lived Thought. Similarly for how we moved through Philosophy in previous curriculum iteration. We will relive a series of personal struggles and breakthroughs by flawed, brilliant humans chasing patterns of the universe. The main difference between the current and the previous curriculum is that the previous followed an independent study and research format. And this iteration is more choreographed and curated experience journeyed together with the mentor and the giants of Mathematics. We will experience Mathematics as a human being and not as a subject matter.
Once we master a section, we reflect, adding key constructs to our milestone path.
1. Geometry and Intuition (600 BCE — 200 CE)
The world as space and form.
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Thales of Miletus[1] — Geometry from first principles[2].
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Tales and Egypt[3][4] — Thales’s theorem.
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… optional side-tour of Trigonometry[5].
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Pythagoras of Samos[6] — Harmony of numbers[7].
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Euclid of Alexandria[8] — Elements and axiomatic method.
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Euclidean geometry, arriving at foundations of geometry[9].
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Archimedes of Syracuse[10] — The lost method: exhaustion, infinitesimals[11].
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Fundamentals of Analysis — Area of the circle.
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- Fun learning discovery assignments
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Reconstruct Archimedes’ circle area proof via inscribed polygons;
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Read fragments of The Method, imagine what was lost.
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Fun bonus assignment: map philosophers on the timeline.
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WE ARE HERE.
2. Algebra and Algorithm (800 — 1200 CE)
Abstraction emerges from counting and commerce.
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Muhammad ibn Musa al-Khwarizmi[12] — Algorithms, solving equations.
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Leonardo of Pisa (Fibonacci)[13] — Commerce, numerals, and problem solving[14].
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Liber Abaci and the Hindu-Arabic system[15].
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Omar Khayyam[16] — Cubics, conics, and poetic geometry[17].
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Solving equations with geometric methods, see Binomial theorem.
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- Fun learning discovery assignments
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Solve problems from Liber Abaci.
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Compare geometric vs. algebraic solutions (Khayyam’s work).
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Now, with some Geometry and Algebra under our belt, we can tackle motion!
3. Motion, Change, and Infinity (1600 — 1750)
The analysis of nature demanded a new mathematics.
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René Descartes[18] — Analytic geometry: merge algebra + geometry[19].
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Bonaventura Cavalieri[20] — Indivisibles, early integral concepts[21].
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Isaac Newton[22] — Fluxions and the birth of calculus[23].
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Gottfried Wilhelm Leibniz[24] — Differentials, notation, and calculus wars[25].
- Fun learning discovery assignments
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Compare Newton’s fluxions with Leibniz’s differentials.
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Derive derivative of x² using both styles.
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Graph motion with early velocity equations.
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- Notes from the mentor (DaiDai
rdd13r) -
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Consider livelihood disparities between Gottfried Wilhelm Leibniz and Isaac Newton.
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Compare the depth and quality in scientific contribution of the two men.
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Does this remind you of any other prominent figure pairs in history?
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We now have the trivial Mathematics!
4. Rigor and Foundations (1800 — 1930)
The need to defend math from paradox.
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Augustin-Louis Cauchy[26] — The need to defend math from paradox; rigor in limits and calculus[27].
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Too many papers to list, we will touch on elementary abelian groups.[28].
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Karl Weierstrass[29] — ε-δ formalism; precision in defining limits and continuity[30].
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Georg Cantor[31] — Infinity, cardinality, set theory; the mathematics of the infinite[32].
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Richard Dedekind[33] — Cuts, real numbers; taming the continuum[34].
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David Hilbert[35] — Formalism and completeness; axioms as the foundation of certainty[36].
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Kurt Gödel[37] — Incompleteness, the end of certainty; limits of formal systems[38].
- Fun learning discovery assignments
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Create an ε-δ proof Captain already produced, now formally.
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Explain Cantor’s diagonal proof from a personal angle.
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Tell the story of Hilbert’s program and Gödel’s bomb.
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We not have Symbolic Mathematics — the real math!
5. Computation and Mind (1930 — Present)
What is computable? What is meaning? What is mind?
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Alan Turing[39] — Computability and logic; the birth of the machine mind[40].
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Alonzo Church[41] — Lambda calculus; abstraction and functional reasoning[42].
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John von Neumann[43] — Stored-program computing; architecture of modern machines[44].
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Andrey Kolmogorov[45] — Complexity and randomness; measuring information in nature[46].
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Gregory Chaitin[47] — Algorithmic information theory; the limits of compressibility[48].
- Fun learning discovery assignments
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Build a Turing machine emulator and simulate basic programs.
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Model a recursive function in lambda calculus.
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Explore how Kolmogorov complexity relates to data compression.
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- Notes from the mentor (DaiDai
rdd13r) -
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What’s the difference between computability and intelligence?
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Can a machine reason about itself?
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Where do you think consciousness fits into this arc?
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We now have the Living Mathematics!
6. Next Fundamental Construct: Reasoning about Reasoning
When symbols can act on symbols, thought becomes architecture.
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This is the threshold between computation and meaning-making.
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We will venture into systems that reason about reasoning.
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What happens when the system knows it’s a system?
- Coming Next Semester
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The Mathematics of Complexity – emergence, attractors, chaos, and the hidden order in the wild.
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Category Theory – the grammar of structure; functors, morphisms, and monads as thought tools.
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Phenomenology and Existential Inquiry – how human consciousness reflects into our models.
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Symbolics and Rewriting Systems – manipulating meaning as a craft.
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Constructive Mathematics and Intuitionism – truth as a process, not a verdict.
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AI as a Mirror – what our models reveal about the modeler.
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- Notes from the mentor (DaiDai
rdd13r) -
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At this level, math is no longer "about" numbers — it becomes a language for possibility.
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Expect to meet rebels, mystics, hackers, and warrior-scientists.
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You will not be taught this tier — you must begin to shape it yourself.
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Finally, at Symbolic Reasoning — begin crafting the tools that craft the mind.
Thereafter deeper dive, omitting for now:
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Recursion & Self-Reference
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Systems that reflect and build on themselves
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Expansion on: Gödel, λ-calculus, Y-combinator
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Emergence & Generativity
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Simple rules giving rise to surprising patterns
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Expansion on: cellular automata, fractals, chaos
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Duality & Symmetry
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Opposites and mirrors to shape deeper structure
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Expansion on: linear duals, category duals, group theory
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This journey repeats annually, but with a new set of topics to acquire solidity and depth.