ideasthesia

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Ryan Collison

I write about cool science and share it here.

Set Theory Explained

Set Theory Explained

In 1901, Bertrand Russell found a paradox that threatened to unravel the logical foundations of all mathematics. The fix required building set theory from scratch — axiom by careful axiom. What emerged became the bedrock every other branch of math now stands on.

Synthesis: Algebra as the Grammar of Mathematics

Synthesis: Algebra as the Grammar of Mathematics

Arithmetic gives you numbers. Algebra gives you the rules those numbers follow—making it the grammar that every branch of higher mathematics is written in.

Graphing: Making Equations Visible

Graphing: Making Equations Visible

The coordinate plane was a philosophical breakthrough as much as a mathematical one. When Descartes unified algebra and geometry, equations became pictures and pictures became solvable. Here's why that translation works.

Functions: The Input-Output Machines of Mathematics

Functions: The Input-Output Machines of Mathematics

Functions are math's input-output machines: put something in, get exactly one thing out. That constraint—no ambiguity allowed—turns out to be the organizing principle behind nearly all of higher mathematics.

Polynomials: Expressions with Multiple Powers of x

Polynomials: Expressions with Multiple Powers of x

Every smooth curve you've ever seen in math can be approximated by a polynomial. They're the simplest tools that let you model complexity — which is why calculus, physics, and data science all start here.

Exponents and Powers: Repeated Multiplication Compressed

Exponents and Powers: Repeated Multiplication Compressed

2³ is just 2×2×2—but the logic extends further than most people realize. Negative exponents, fractional exponents, and zero powers all follow from the same simple idea, once you see the pattern.

Inequalities: When Equals Is Not Enough

Inequalities: When Equals Is Not Enough

Equations pin down a single point. Inequalities define a space of solutions — and most real-world constraints are inequalities, not equalities. Optimization, economics, and logic all run on this distinction.