I’ve been diving deep into category theory lately and had a realization that’s honestly kind of infuriating: mathematics education has suffered a profound pedagogical inversion that has taken me significant, deliberate effort to recover from.

For eighty years, we’ve taught mathematics backwards, starting with disconnected procedures instead of the unifying categorical principles that generate them, obscuring the fact that mathematics is actually a beautiful, coherent theory of how reality optimally organizes itself. Most of us never escape this gauntlet of rote techniques to see the deeper picture.

Whether through institutional inertia or deliberate gatekeeping, this system creates mathematical “priests” who discover or are taught the deep structural connections at graduate level and mathematical “peasants” (most of us) who remain trapped in procedural thinking, never realizing that arithmetic, algebra, calculus, and beyond all emerge naturally from the same elegant principles that govern optimal relationships throughout reality itself.

Even worse, this isn’t just about what mathematical topics we teach first, it’s about the type of mathematical thinking we embed from the start. We train students in ‘Type I’ abstract universals (set theory’s non-participating collections) where mathematical objects are things that have properties independently, rather than ‘Type II’ concrete universals (category theory’s participating constructions) where mathematical structure flows through optimal relationships.

So even when students eventually encounter ‘advanced’ mathematics, they’re still thinking in terms of objects with properties being sorted into categories, rather than understanding that mathematical properties emerge from participation in universal constructions. Therefore, we have not just taught math backwards, but we have embedded a fundamentally limiting philosophical framework about what mathematical reality even is!

A category-theoretic pedagogical foundation doesn’t need to replace the classic pipeline of arithmetic -> algebra -> calculus -> …, but we should at least introduce and weave its structural insights in from the start. Doing so could help more students become ‘proto-priests’ rather than remain lifelong ‘peasants.’