Human pursuit of truth is essentially a process of extracting logic from concrete practice and approaching higher levels of cognition through infinite iteration. This is not only a philosophical speculation but also a core “craft” for solving complex real-world problems. The P vs. NP problem in computational complexity theory, along with the evolution of mathematical tools, provides a framework for understanding this process.
The “P problem” refers to problems that can be efficiently solved in polynomial time (i.e., for which an efficient algorithm exists); “NP problems” are those whose solutions can be verified in polynomial time. The classical “Zeno’s paradox” (such as Achilles and the tortoise), in the absence of rigorous mathematical tools in ancient times, can be seen as a kind of “NP problem”—logically incomplete within finite steps—since it involves the accumulation of infinitesimals, making it intuitively difficult to judge the final result quickly.
However, the birth of calculus, especially the introduction of the concept of limits, provided powerful analytical tools to deal with such infinite processes. It allowed problems that seemed infinitely divisible to be effectively converged and precisely calculated, thereby transforming Zeno’s paradox from a cognitive “NP problem” into a mathematical problem solvable in “P time.” This clearly demonstrates how human intelligence, through the creation of new abstract tools, can transform seemingly complex or unsolvable problems into manageable and understandable categories.
Nevertheless, whether P equals NP remains one of the central unsolved mysteries in computer science. This, in itself, constitutes a higher-order “transfinite iteration” problem. It concerns not only the limits of algorithmic efficiency but also symbolizes the continuous expansion of human cognitive boundaries.
Foundational paradoxes are often unified at higher levels of cognition, and the key to achieving this higher cognition lies precisely in a “craft” called “transfinite iteration.” This “craft” is not a simple repetition but a transcendent, progressively advancing infinite process involving thought and handling of infinitely layered structures.
Alan Turing, in his pioneering work Systems of Logic Based on Ordinals, profoundly elaborated on this kind of “craft.” He explored how to use ordinals—a mathematical concept used to measure the size of infinities—to construct more powerful logical systems capable of addressing undecidability issues revealed by Gödel’s incompleteness theorems. Turing’s theoretical attempt was precisely to introduce concepts beyond finite steps, seeking the possibility of unifying and resolving foundational paradoxes on a higher logical level.
Similarly, when Satoshi Nakamoto designed Bitcoin, he skillfully applied the “craft” of transfinite iteration to solve the trust paradox and double-spending issue inherent in traditional financial systems. Bitcoin’s “longest chain rule” is a classic example of this craft. It establishes and maintains consensus on value and transaction authenticity in a trustless environment.
Nature is the objective carrier of truth, and its complexity contains “infinite truths.” Humanity’s pursuit of truth is not merely the discovery of existing facts, but more a practice of creating a “real” that converges with nature through infinite transfinite iteration. While calculus can resolve first-order infinities, higher levels of infinity (such as transfinite numbers and different magnitudes of infinity in set theory) and deeper complexities still exist, continually driving humans to transcend current cognitive frameworks and, through endless exploration, to approach deeper layers of truth. This is an unending cognitive cycle, with each breakthrough representing a deeper iteration and understanding of truth.
