If you’re like me, you’ve spent countless hours in real analysis carefully constructing epsilon-delta arguments, in topology proving that certain sets are open, and in abstract algebra showing that certain maps are homomorphisms. You know the satisfaction of finally seeing how all the pieces of a proof fit together—and the frustration of getting stuck on a seemingly simple step.

But here’s something you might not have experienced: what if you could write your proofs in a language where the computer checks every single step, catches your mistakes immediately, and even helps you figure out what to do next?

That’s Lean.

What is Lean?

Lean is a proof assistant—software designed to verify mathematical proofs with absolute certainty. Unlike the proofs you’ve written on paper or typed in LaTeX, proofs in Lean are interactive dialogues with the computer. You state what you want to prove, and Lean shows you exactly what you need to establish. You apply a theorem or make a logical step, and Lean immediately updates to show you what remains. If you make an error—using the wrong theorem, forgetting a hypothesis, or making an invalid logical leap—Lean won’t let you continue until you fix it.

A Different Way of Thinking

At first, this might sound frustrating or overly restrictive. After all, you already know how to write proofs! But here’s what makes Lean fascinating: it forces you to confront the gap between understanding a proof conceptually and knowing it with absolute rigor. That step you thought was “obvious”? Lean will ask you to justify it. That theorem you remember from class? You’ll need to state it precisely. It’s mathematics at its most unforgiving—and, surprisingly, at its most rewarding.

What drew me in was discovering that Lean isn’t just about checking proofs; it’s about thinking differently about mathematics itself. In Lean, you work backwards from what you want to prove, strategically transforming your goal until it becomes something you can establish from your hypotheses. It’s proof as problem-solving, proof as puzzle, proof as conversation with the machine.

Building on a Foundation

And the best part? The mathematical community is building something remarkable. Thousands of theorems from undergraduate and graduate mathematics have already been formalized in Lean—from calculus and linear algebra to measure theory and beyond. You can actually use these theorems in your own proofs, building on a verified foundation of mathematical knowledge.

If you’ve ever wanted to be absolutely certain your proof is correct, if you’re curious about how computers understand mathematics, or if you simply want a new way to engage with the subject you love—Lean might just be worth exploring.

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