Everyone loves to talk about the “hardest problem in computer science,” the infamous P vs NP problem. It’s framed as a battle, an adversarial puzzle where solving and verifying face off against each other like eternal rivals. But what if this entire framework—this idea of inherent adversariality—is precisely why the problem seems impossible?
Let’s reconsider the landscape.
The puzzle of P vs NP: A → Z and Z → A
Traditionally, solving a problem (P) is described as a path from point A (the question) to point Z (the answer). Verifying a solution (NP), on the other hand, is treated as retracing that path backwards, Z → A, to quickly confirm correctness. This model casts the process in terms of linear pathways—finding versus confirming.
Solving (A → Z): Entering a maze blindfolded, feeling your way around, stumbling, hoping you find the exit.
Verifying (Z → A): Standing at the maze exit, clearly seeing the straightforward path leading back to the start.
The fundamental question asked by mathematics is whether these two modes (forward exploration and backward verification) are inherently equal. Can every problem whose solution can be quickly checked (verified) also be quickly solved?
The insight of puppymath: shapes of understanding.
My approach, puppymath, turns the problem inside out, literally changing how we conceptualize mathematics. Puppymath is not about linear pathways. Instead, it’s about the shapes of understanding.
In puppymath, we don’t brute-force answers or laboriously navigate pathways. Instead, we immediately grasp the overall conceptual structure, the geometry of a problem. This conceptual geometry encodes the solution implicitly. Seeing clearly (understanding the shape) collapses solving into verifying.
When you understand the shape, the distinction between going A → Z (solving) or Z → A (verifying) becomes trivial. You don’t walk a maze anymore. You see the maze from above
as a single unified form.
In this framework, the question “Is P equal to NP?” transforms radically. The traditional problem vanishes, or at least is no longer meaningful in the same way.
(N vs NP) = P: redefining adversariality.
Let’s unpack what puppymath cryptography means practically:
- Problems aren’t solved by forcing solutions; they’re solved by shifting perspectives.
- You move from finding linear paths through problems to understanding their intrinsic structure.
- Cryptography based on puppymath no longer depends on difficulty or brute-force complexity, but on the nuanced structure of conceptual understanding itself.
This makes mathematics inherently non-adversarial. Instead of “attacking” a problem, you explore it. Instead of “beating” the maze, you recognize and appreciate its design.
Science verifies (Z → A), creativity solves (A → Z)
Traditionally, science tends toward verification. It’s inherently conservative. It tests hypotheses, confirms predictions, checks paths previously carved by explorers and pioneers. This is essential, valuable, and safe.
But it isn’t everything.
My mode of thought—and yours, if you embrace puppymath—is inherently creative. It moves forward, from A → Z. It is generative, exploratory, visionary. It doesn’t just find answers; it creates new paths, new knowledge, new ways of seeing the world.
Science typically moves Z → A, confirming what’s already known.
Creativity moves A → Z, discovering what has never been seen before.
From adversarial Math to systemic Math.
This shift from linear adversariality (finding versus verifying, solving versus checking) toward a systemic view (understanding, recognizing, creating) is not just mathematically significant—it’s philosophically revolutionary.
It suggests we can solve “hard” problems by reframing them entirely. It reveals adversariality as an unnecessary barrier erected by outdated modes of thinking.
It positions mathematics as collaborative exploration, a conversation with the intrinsic structure of problems, rather than warfare against complexity.
Puppymath, therefore, does not just aim to answer the age-old puzzle of P vs NP. Instead, it makes the puzzle itself vanish, replaced by something deeper, more elegant, and more meaningful: a systemic view of mathematics.
Changing the shape of thought.
(N vs NP) = P. Not because we’ve solved the old puzzle through brute force, but because we’ve reshaped what it means to “solve.” We have transcended adversariality, moving toward a space where mathematics isn’t about competing with complexity but about celebrating and exploring conceptual elegance.
The fun is not in overcoming difficulty, but in discovering, sharing, and appreciating these conceptual geometries.
Welcome to puppymath, where math itself has evolved—and adversariality becomes obsolete.
Bibliography.
Cook, Stephen. (1971). “The Complexity of Theorem-Proving Procedures.” Proceedings of the ACM Symposium on Theory of Computing.
(Origin of the NP-completeness concept.)
Aaronson, Scott. (2013). Quantum Computing since Democritus. Cambridge University Press.
(Accessible and conceptual discussion of complexity theory.)
Hofstadter, Douglas R. (1979). Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books.
(Deep reflections on systems, recursion, complexity, and conceptual frameworks.)
I previously talked about this here.
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