This blog post explores whether scientific knowledge is closely interconnected by examining the process of solving the Poincaré Conjecture.
In ‘The Structure of Scientific Revolutions,’ author Thomas Kuhn argued that scientific theories are not continuous, do not develop gradually, and their direction is not the pursuit of truth. This caused significant ripples among many scientists. Before Thomas Kuhn proposed his paradigm theory, scientists held a belief and respect for the gradual, cumulative nature of scientific knowledge, epitomized by Newton’s phrase “standing on the shoulders of giants.” Furthermore, they firmly believed that the ultimate goal of science was the pursuit of truth, and that scientific progress advanced toward truth with a clear direction. However, Thomas Kuhn decisively refuted this in his work, providing many scientists with an opportunity to reconsider these assumptions. So, is science truly nothing more than fragmented knowledge, lacking a ‘long lineage’ and changing its identity according to paradigm shifts? My answer to that is emphatically ‘No’.
In 2002, while the entire nation was buzzing with excitement over the World Cup in Korea, the mathematical community was in a frenzy for entirely different reasons. The Poincaré Conjecture, selected by the Clay Mathematics Institute as one of the seven Millennium Problems and which had withstood over a century of relentless attempts at proof by outstanding mathematicians since its initial formulation in 1904, was solved for the first time by Russian mathematician Dr. Grigori Perelman. The world’s attention focused on how Dr. Perelman solved this century-old problem, and the journey he undertook to solve it is already relatively well-known. Through the process of solving the Poincaré Conjecture, I will demonstrate that scientific knowledge is never isolated or independent but rather deeply interconnected. From this, I will draw the implication that theories in the philosophy of science must be approached with a highly critical eye.
The struggle to solve Poincaré’s conjecture persisted for an exceptionally long time. Central to this process was the field of ‘topology’. Topology offers a perspective on shapes that is fundamentally different from traditional Euclidean geometry; it treats shapes as equivalent if they can be transformed into one another without cutting or joining. For instance, a square can be made into a circle by rounding its corners. In topology, these two shapes are considered the same. Topology provided a foundation for viewing much knowledge logically and intuitively, breaking away from the strictness of traditional geometry. It is also the theoretical background that allowed Poincaré’s conjecture to emerge. Poincaré’s conjecture states: ‘If every closed curve in a single, closed three-dimensional space can be shrunk into a single point without forming a knot, then that space must be transformable into a sphere.’ This explicitly employs topology because it introduces the concept of transformation of space and shapes. According to Thomas Kuhn’s paradigm theory, topology exists as the core of a new paradigm, while theories derived from topology—such as the Poincaré Conjecture, knot theory, and graph theory—would correspond to the periphery of that paradigm. Moreover, the prominent scholars researching these areas immersed themselves in their work within the vast framework of ‘topology’ and solved problems. Revolutionary scientific progress should have emerged from the results of this research. However, the manner in which Dr. Perelman solved the problem—achieving revolutionary scientific progress by resolving a mathematical conundrum that had remained unsolved for a century—was entirely different.
Attempts to use topology to solve the Geometrization Conjecture and Poincaré’s Conjecture have persisted steadily since the 1960s. When attempts to solve the problem in three dimensions were repeatedly thwarted by the creation of knots, Dr. Stephen Smale of the United States attempted to solve the problem in higher dimensions. Consider a roller coaster track.
The shadow of the roller coaster track cast on the ground is tangled and intertwined, creating many knots, but the actual roller coaster track has no knots. Similarly, attempting to solve the problem in higher dimensions allows the knot problem to be easily resolved. Indeed, this method proved effective: purely through topology, the Poincaré conjecture was solved for dimensions five and above, as well as for four dimensions. Both Dr. Smale and Dr. Friedman, the mathematicians who solved the problem, were awarded the Fields Medal, often called the Nobel Prize of mathematics. Later, Dr. Thurston won another Fields Medal for his research on classifying three-dimensional manifolds (the Geometrization Conjecture).
Up to this point, it aligns remarkably well with Thomas Kuhn’s paradigm theory. A shift to a new paradigm (from Euclidean geometry to topology) occurred, and within this new paradigm, prominent researchers formed their own cores and peripheries through their work. Moreover, they achieved remarkable expansion of scientific knowledge. However, proving the conjecture in three dimensions faced obstacle after obstacle, with no progress for over twenty years. The breakthrough came from Dr. Perelman. The problem is that the method Dr. Perelman used to solve the conjecture was rooted not in ‘topology’ but in ‘classical physics’. Classical physics has no connection whatsoever to the new paradigm of topology. They don’t even share a common denominator like geometry. Yet, the conjecture was solved by classical physics. A crucial clue to the solution was found through the Ricci flow equation, a modification of the heat diffusion equation from thermodynamics, which ultimately led to the proof. Moreover, solving the Poincaré conjecture required extensive prior knowledge: the Ricci flow equation, Dehn’s auxiliary theorem, the resolution of the Poincaré conjecture in dimensions higher than four, collapse theory, the Soul conjecture, and more. The Ricci flow equation has its roots in thermodynamics within space, and thermodynamics itself is grounded in other physical theories. This provides another basis for refuting Thomas Kuhn’s claim that scientific knowledge is neither continuous nor cumulative.
The problem with this process is that it clearly contradicts the characteristic Kuhn presented in his paradigm theory: that “science is not unified and is independent.” Clearly, topology provided many clues for solving the difficult problem of the Poincaré Conjecture. Concepts from topology are scattered throughout the resolution of Dr. Thurston’s Geometrization Conjecture and the Poincaré Conjecture in higher dimensions; from Thomas Kuhn’s epistemological perspective, this belongs to the paradigm of topology. However, Thomas Kuhn’s theory must be refuted because the decisive contribution to solving the Poincaré Conjecture—the ‘giant’s shoulders’ within this topological framework—was not topology itself, but rather the Ricci flow equation, rooted in physics’ heat diffusion equation. The singularities of the space function that appear when contracting a closed curve in three-dimensional space were simplified through the Ricci flow equation, enabling the resolution of the knot problem. The intervention of a concept from classical physics, seemingly unrelated to topology, to play a crucial role in solving a problem derived from topology negates one of the central tenets of Thomas Kuhn’s argument: that paradigms in science cannot unify or interfere with each other. This serves as an example demonstrating that scientific knowledge does not change its identity through paradigm shifts among scientists. This provides clear evidence that the paradigm theory, which played a crucial role in explaining the structure of scientific revolutions, has blind spots.
Thomas Kuhn’s proposed structure of scientific revolutions and paradigm theory brought about a major transformation in how people accept science. Those who had accepted positivist science without question were prompted by Thomas Kuhn’s challenges to positivist science to critically reconsider its validity. It shook the deeply rooted notion that verified science was an indisputable fact, and that the accumulation of such facts represented a powerful stride toward ultimate truth. The paradigm theory deserves high praise for providing the initial impetus to think more deeply about science and view it from a more critical perspective. Furthermore, by suggesting that scientists’ ideas can influence the pursuit of scientific knowledge, it also played a role in sparking discussions on the philosophy and ethics of science.
However, as seen in the earlier example, Thomas Kuhn’s theory contains many weaknesses and blind spots. As noted earlier, the independence of scientific knowledge and the mutual inviolability between paradigms, as claimed by Thomas Kuhn, could not be guaranteed. Furthermore, his theory faced criticism for the ambiguity in defining paradigms. Indeed, Kuhn’s paradigm was criticized as being so vaguely defined that it was conveniently ambiguous. While Thomas Kuhn achieved significant progress in the philosophy of science, the independence of scientific knowledge bodies he advocated is difficult to accept when viewed through the lens of the resolution of the Poincaré Conjecture. Furthermore, the examples Thomas Kuhn used to argue that scientific knowledge is not continuous or cumulative are also hard to accept. He cited examples where new scientific theories develop, causing existing theories to be forgotten or rendered useless. However, my view is that just because we progress by accepting new knowledge, it does not mean we ‘lose’ the old. Consider, for instance, already knowing addition and then newly accepting the concept of multiplication. The multiplication expression 7 × 3 actually originates from 7+7+7. Just because we accept the more advanced expression 7 × 3 doesn’t mean the conceptual principles embodied in 7+7+7 become completely useless.
Thus far, while examining the process of solving the Poincaré Conjecture, we have presented arguments refuting several claims that form the basis of Thomas Kuhn’s paradigm theory. Undoubtedly, Thomas Kuhn presented a theory that will endure in the history of the philosophy of science and caused significant resonance within the scientific community. However, the basis for some of his claims has been shown to be refutable through counterexamples or to lack sufficient justification and reference points. Thus, I believe theories and explanations related to science contain a mixture of universally acceptable truths and elements that are not. Therefore, we must cultivate the ability to approach these aspects with a critical and cautious attitude.
