Step 1: We sum n numbers.
Step 2: The average number is roughly n²/3
(Most squares are less than n²/2, pulling the average down)
Step 3: So the sum ≈ n × (n²/3) = n³/3
A visual synthesis of Terence Tao's three stages of mathematical development and Ben Orlin's distinction between proof and explanation.
"There's more to mathematics than rigour and proofs"
"Against Mathematical Proof"
"Mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. The emphasis is more on computation than on theory." — Terence Tao
This is where mathematics feels like play. Calculus means slopes and areas, not epsilon-delta. Numbers have personality. Shapes tell stories. Understanding emerges from doing, not from axioms.
The derivative is the slope.
Drag to see how steepness changes.
↑ Steeper line = bigger slope = faster change
The ε-δ definition of limit.
For all ε > 0, there exists δ > 0...
"One is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually 'mean'." — Terence Tao
The transition is "rather traumatic." Intuition gets deprecated as "non-rigorous." Mathematics becomes symbol manipulation—powerful but often disconnected from meaning. Many get stuck here, unable to return to intuition.
"The ideal state to reach is when every heuristic argument naturally suggests its rigorous counterpart, and vice versa." — Terence Tao
Here, intuition returns—transfigured. You see through formalism to meaning. You can think heuristically and know exactly how to make it rigorous. Both hemispheres working together. This is mathematical fluency.
Unified vision: The derivative as both slope and limit.
Same truth, two languages—perfectly aligned.
Ben Orlin identifies a crucial distinction that illuminates the three-stage journey.
Shows how a statement fits into the body of shared mathematical knowledge— axioms, definitions, earlier proofs. It's the castle walls of mathematics.
Shows how a statement fits into my personal knowledge— experiences, mental models, approximations, vivid images. It's psychological, not logical.
"Explanation is not just a fuzzy proof, and proof is not just a rigorous explanation. They are different species entirely: one logical, the other psychological."
1² + 2² + 3² + ... + n² = ?
Step 1: We sum n numbers.
Step 2: The average number is roughly n²/3
(Most squares are less than n²/2, pulling the average down)
Step 3: So the sum ≈ n × (n²/3) = n³/3
Approximation accuracy: 75.8%
"Have we proved the formula? Not at all. We've given a squishy argument for something vaguely similar to the formula. But that has its own benefits." — Ben Orlin
Intuition leads, proof absent
Proof dominates, intuition suppressed
Proof and intuition unified
The deep insight emerging from both Tao and Orlin is this: mathematical maturity is not the triumph of rigor over intuition, but their reunion.
The purpose of rigor is not to destroy intuition—it is to destroy bad intuition while clarifying and elevating good intuition. When you reach the post-rigorous stage, you don't abandon proof; you see through it to the meaning beneath.
The ideal state: every heuristic naturally suggests its rigorous counterpart. Every proof illuminates rather than obscures. Both hemispheres working together— the same way you already tackle problems in "real life."
"Proof is the architecture of mathematics. It's our castle walls... Don't abandon proof. Just acknowledge its true purpose."