This exploration weaves together two profound ideas. One comes from Thomas Hertog and Stephen Hawking’s early-universe cosmology: in the beginning, there was no single, sharp “first moment” with fixed rules, but a quantum haze of possibilities. As the universe unfolded, certain patterns became stable and self-replicating, leading to the effective laws of nature we observe today. The other idea comes from Gödel’s incompleteness theorems: as territory, mathematics is an inexhaustible landscape of truth, a realm so vast that no finite set of axioms can ever capture it all.
A central metaphor illuminates this connection. Imagine a timeless Library containing every consistent mathematical blueprint for a universe—all possible geometries, interaction rules, and even complete cosmic histories. This library is static. Now, imagine a single Path being traced through its shelves. That path is our world.
Early on, the path winds through foggy regions where countless alternatives overlap. Later, it enters chapters where stable, law-like behavior dominates—gravity holds, atoms persist, and chemistry works. Our present data acts like clues at the end of a mystery, allowing us to perform a top-down filtering of possibilities and deduce which earlier chapters could have led to our here-and-now.
Observers are not external to this story; they are the mechanism by which a branch of the Library becomes legible. This is the “worm’s-eye tube”-perspective, explored by thinkers like Hertog and Hawking, where origin, evolution, and observership are inextricably linked. We can visualize this as a ladder:
No contradiction: Gödel speaks about mathematics-as-territory (the Library). What follows speaks about mathematics-as-practice (our traversal on the Path). The territory is inexhaustible; therefore the practice is, by necessity, an infinite game.
This brings us to the nature of mathematics as practiced—not only a static collection of truths housed in the Library, but also the active process of navigating the Path. This practice is best understood as an infinite game.
First, we must distinguish between two kinds of games, as defined by James P. Carse.
When we reduce mathematics to exams, grades, and right-or-wrong answers, we treat it as a finite game. But the practice of mathematics—the act of discovering, creating, and connecting new concepts—is the ultimate infinite game.
Every theorem is not an ending, but a new beginning. Euclid organized a language for space; calculus opened a language for change; set theory and logic widened the frame of inquiry. Modern formalisms now reach toward the structures of quantum mechanics and information. None of this “finishes” mathematics; it extends observership, clarifying the Path as it moves forward.
The right attitude for an infinite game is humble wonder. A true mathematician knows that every answer is a temporary resting place. What seems settled today may be revealed tomorrow in a deeper, more encompassing light. This pursuit demands not only logic but also courage—the courage to say, “I don’t know yet.”
If mathematics is the practice of observership, then education is not about hoarding tricks but about learning to see. The goal is to ask better questions, to connect disparate concepts, and to treat errors not as failures but as informative steps. The classroom becomes a space where mistakes are not defeats, but new moves in the game. The question shifts from “What is the answer?” to “What makes this answer possible?” and “What happens if we change the rules?”
Finite goals demand that we “cover the syllabus.” The infinite goal is to keep understanding alive. The durable purpose—the “Just Cause”—to play the game is simple and sufficient: to sustain the search for pattern, clarity, and truth, and to pass on the capacity to continue it.
Gödel’s theorems provide a gentle horizon to our confidence. If mathematical truth will always outrun any fixed list of axioms, we should not expect a final, compact “theory of everything” that explains every fact once and for all.
Instead, we should expect what we have found: excellent, domain-true laws—theories good enough to land probes on comets and build microchips—that bend, break, or generalize at their extremes (the earliest moments, the strongest fields, the deepest scales). This horizon doesn’t paralyze us; it motivates our observership to keep refining its tools as our Path enters new chapters of the Library.
Time belongs to the Path: memories accrue, entropy grows, and causes precede effects. The Library, however, is timeless: its structures simply are. Both views are true because they describe different levels: the journey and the map. Mathematics bridges the two: as territory, an inexhaustible structure; as practice, the evolving craft by which observers extract stable order from that structure—and test whether yesterday’s models will still guide them tomorrow.
Mathematics (two senses):
In short: the Library holds more than any theory can exhaust. Our Path stabilizes into a world of law-like order. Observers arise within it and, through the infinite game of mathematics, keep the route intelligible. The point is not to finish the book, but to keep reading—and to keep the language clear for those who will read after us.