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.
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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
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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.
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When a mathematician looks at a new problem, what are they most excited to find?
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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.
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