Like most students, I first encountered the Pythagorean theorem during my school years. At the most superficial level of engagement, one merely needed to memorize the formula a² + b² = c² in order to complete the required problem sets. Sadly, that’s what I did. A deeper engagement might involve studying the standard algebraic proof. For those with both curiosity and time, there existed an entire corpus of alternative demonstrations: proofs by rearrangement, proofs using similar triangles, Euclid’s geometric construction, dissection and area-preserving arguments, and even Einstein’s proof by dissection without rearrangement.

It was only a few months ago, while reading about the historical development of mathematical ideas, that I realized just how impoverished my own education had been. The true intellectual richness of the theorem lay not in the diversity of proofs, but in the historical and conceptual journey that produced it.

Far more powerful than rote application would have been an education that revealed how the Greeks arrived at such an idea in the first place. The Greeks conceived of numbers primarily through geometric representation. Their integration of arithmetic with spatial reasoning yielded not only enduring mathematical principles but also remarkable architectural achievements.

This framework was so foundational to their worldview that, when the concept of zero eventually traveled from India, they rejected it. Their reluctance stemmed not only from difficulty in comprehending “nothingness,” but also from the recognition that adopting such a notion would destabilize their entire mathematical edifice.

It is conceivable that, had they embraced this radical abstraction, they might have anticipated developments akin to calculus centuries earlier. Zeno’s paradox had already pushed them toward the conceptual boundary. However, it was not until Fibonacci reintroduced the Hindu–Arabic numeral system to Europe in 1202, emphasizing its practical utility in commerce, that zero and positional notation were broadly accepted in the West.

I wonder how much more beautiful the lesson would have been if it had not ended with the formula for finding the hypotenuse of a right-angled triangle. Imagine if it had shown instead how ideas emerge and evolve, how they move across cultures, and how they can grow so useful and rigid that they begin to limit growth and inquiry. Imagine if it had revealed how utility often overshadows truth, how economic forces can propel new ideas into motion even when most people reject them. Most of all, imagine if the real lesson had been that openness to new ideas and the willingness to reinvent oneself is not just acceptable but essential, even when it means dismantling everything one thought was certain. How much more curious, and how much braver, I might have become.

As we face a crisis of curiosity, or more precisely a crisis of misplaced curiosity that feeds on celebrity gossip or the Tesla dude’s latest beef, perhaps the remedy lies in a pedagogy rooted in storytelling. Maybe a pedagogy that highlights the continuity of intellectual history and the interconnectedness of ideas could help rekindle the right kind of curiosity. Maybe the biggest tragedy in education today is treating knowledge as fragmented pieces instead of one single, unfolding story. Or maybe I am wrong.

Math Without Meaning