The Architecture of Mathematics
Like Duolingo, but for The Architecture of Mathematics. Tomo turns the whole topic into a game you play five minutes a day, until it actually sticks.
For the part of you with thirty open tabs that never became anything.
79 levels across 8 sections, about 158 minutes end to end, roughly 32 days at five minutes a day. It moves through The Language of Rigor, Abstract Algebraic Landscapes, Analysis in the Infinite, The Shape of Space, The Secrets of Integers, Chance and Change, The Limits of Formalism, and Mathematics in Action.
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Key ideas in The Architecture of Mathematics
- The shift from algorithmic computation to structural verification
- Identifying unstated axioms in mathematical claims
- Contrapositive is preferred when the negation of the conclusion provides a more workable algebraic starting point than the original hypothesis
- The shift from algorithmic computation to structural verification in advanced mathematics
- Contradiction is essential for proving non-existence or infinitude where no direct construction is possible
- A counterexample must satisfy all hypotheses of the theorem (continuity) while violating the conclusion (differentiability)
- Identifying unstated axioms in seemingly intuitive mathematical claims
- The Weierstrass function or the absolute value function at the origin serve as boundary markers for the limits of differentiability
- Strategic selection between Contrapositive and Contradiction for non-constructive proofs
- Lemmas are 'helper' theorems that isolate specific technical hurdles
- Strategic selection between Contrapositive and Contradiction
- The main Theorem synthesizes multiple Lemmas into a final conclusion
- Using counterexamples to identify failure points
- Corollaries are immediate, low-effort consequences
- The modular function of Lemmas in constructing a 'Big Theorem'
- The Intermediate Value Theorem provides a non-constructive existence proof
You've tried the other tabs
Thirty open tabs. Four facts you actually kept.
You watched. You nodded. By Sunday it was gone.
One answer, then back to scrolling.
Eight weeks. You meant to finish. You didn't.
Tomo gives The Architecture of Mathematics the Duolingo treatment: levels, streaks, and quick quizzes that test what you just learned. That game loop is what the tabs above never had, so it's the one you actually finish.
Here's what playing it feels like
A real question from this course. Take your best guess.
When a mathematician looks at a new problem, what are they most excited to find?
Get it right to open this lesson and 78 more in the app.
Where The Architecture of Mathematics takes you
A comprehensive journey through the abstract structures and rigorous foundations of modern mathematics, from category theory to non-standard analysis.
- 1
The Language of Rigor
- The Architecture of Modern Proof
- Non-constructive Existence and Choice
- 2
Abstract Algebraic Landscapes
- Category Theory: The Universal Language
- Galois Theory and Symmetry
- Representation Theory
- 3
Analysis in the Infinite
- Measure Theory and Lebesgue Integration
- Functional Analysis
- Complex Analysis and Holomorphy
- 4
The Shape of Space
- Differential Geometry and Curvature
- Algebraic Topology
- Manifolds and Fiber Bundles
- 5
The Secrets of Integers
- Analytic Number Theory
- Algebraic Number Theory
- 6
Chance and Change
- Probability and Stochastic Calculus
- Ergodic Theory and Dynamics
- 7
The Limits of Formalism
- Logic and Incompleteness
- Model Theory
- 8
Mathematics in Action
- Information Theory and Entropy
- Cryptographic Hardness
- Numerical Stability and Complexity
8 sections · 20 units · 79 levels. Built to play, not to enroll.
You pick the voice
The Architecture of Mathematics is taught in the Explain Like I'm 5 style: no big words. promise.. Want a different feel? In the app you can spin up the same topic in any of Tomo's teaching styles. Same facts, totally different vibe.
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