Deep Roots

The (Un)reasonable effectiveness lemma

Hungary is a curious nation. Steeped in history, rich in culture, and the womb of Martians on Earth. Not the cruel Martians that have been popularized by pulp fiction and b-movies, but the actual historically relevant conception of what Martians could represent: Superhuman intelligence. In the last century, if you could pin-up all major paradigm shifts and revolutions in science and engineering on a board and run a tracer algorithm that maps out the nations where the ideas were birthed from, you'd land in Hungary on more occasions than imagined. Watch:

Why Hungary became the epicenter of this "Martian revolution" needs a lot more study and will reveal amazing connections between sociology, genealogy, politics and distant branches of reasoning and logic, so we'll steer clear of that in this tiny blog. Instead, we turn our gaze to something that Eugene P. Wigner once famously wrote in an essay that has time and again revealed its prophetic qualities:

"It is important to point out that the mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena."

This is true not only for physics, but for all branches of inference. The ability to infer truth from experience, rests on a rich base of mathematics that spreads beyond context. How else, could nonlinear dynamics find itself rooted in the neural networks that power your favorite chatbot? Or how could space exploration be hinged on the accelerated radiation off electrons in an electromagnetic field?

All of these, and more, are deducible from first principles, in principle. The real muck is in the abstraction. To prevent our heads from exploding under the barrage of nested integrals and analytic procedures, we choose to abstract some details away from the calculation at hand and thus build a meta-language for inference on the go. And the common thread in every single procedure, is our trusted friend & guide: mathematics.

Mathematics is invented, that much is true. But the juice that lies underneath the patchwork of symbols and computational arguments, is a fundamental lemma, or maybe a governing body of lemmas, which steers all physical phenomena towards an explainable truth. All inference is derived through the thread of math. Which is why a matrix multiplication procedure invented centuries ago powers the massively parallel GPU clusters behind AlphaFold and Claude. Or the elegance of Lie groups drives our understanding of the fundamental forces and matter in the universe. At each step of human inference, math holds the candle and guides us farther into the horizon than foreseen.

So this unreasonable effectiveness of math at guiding our inference almost presciently, is not a mere observation in passing, but rather a fundamental lemma. If our observed truths are theorems, then almost every single one is derivable from this lemma. Why almost? In science, there is an unspoken rule of leaving a bit of room for the incomprehensible. Not all of the world can be accurately inferred from data alone. Gödel's incompleteness theorems enforce this rule across the entire field of logic. There indeed might be truths far more profound than this lemma, which might turn out to be the progenitor of this lemma indeed.

But all of that lies beyond the horizon of inference. We, mere mortals, find ourselves on the ledge, staring ahead at the vast, imperceptible horizon, and carry but a tiny stick to prod at the skin of reality. This unreasonably effective stick has served us beyond expectations and shall continue to serve us far beyond the trodden path. It is our innate drive, that shall guide us to find the tree from whence this stick emerged. That, shall be our singularity.