Criticize Thomas Kuhn’s paradigm theory to explain the continuity and interdependence of scientific knowledge. Dr. Perelman’s solution of the Poincaré conjecture highlights the close connectivity of scientific knowledge.
In The Structure of Scientific Revolutions, Thomas Kuhn proposes that scientific theories are not continuous and do not develop gradually, nor is their direction the pursuit of truth. This had a profound impact on many scientists. Before Thomas Kuhn’s paradigmatic theory, scientists believed in and respected the incremental and cumulative development of scientific knowledge to date, epitomized by Newton’s phrase “the shoulders of giants.” They also believed that the ultimate goal of science was the pursuit of truth, and that scientific progress was a movement toward truth with a clear direction. However, Thomas Kuhn denied this in a major way in his book, which has led many scientists to reflect on the question: Is science really just a mere expedient of knowledge without a “long lineage” that changes its identity with paradigm shifts? The answer is a resounding no.
In 2002, while the country was buzzing with excitement over the World Cup in South Korea, the math world was buzzing for a completely different reason. The Poincaré conjecture, named one of the seven problems of the millennium by the Clay Mathematics Institute, had been solved for the first time by Russian mathematician Dr. Grigory Perelman, who had withstood an onslaught of proofs from leading mathematicians for more than 100 years since it was first posed in 1904. The world’s attention was focused on how Dr. Perelman solved the conundrum of the century, and the journey he took to solve it is relatively well known. I will use this process of solving the Poincaré conjecture to show that scientific knowledge is by no means discrete or independent, but rather very closely connected, with the implication that we should take theories in the philosophy of science very critically.
The struggle to resolve Poincaré’s conjecture has been a long one, and one of the most pivotal parts of the process has been the field of topology. Topology is a geometry that looks at shapes in a completely different way than traditional Euclidean geometry, treating shapes as the same if they can be deformed into different shapes without cutting them. For example, a square can be made into a circle by rounding its corners. In topology, these two shapes are the same shape. Topology has given us a logical and intuitive way of looking at a lot of knowledge that was once strictly geometry, and it is in topology that Poincaré’s conjecture was born. Poincaré’s conjecture states that ‘if all closed curves in a single, closed, three-dimensional space can be contracted into a point without creating a knot, then the space can be deformed into a sphere’, and it explicitly uses topology in that it introduces the concept of deformation of space and shape. According to the paradigm theory proposed by Thomas Kuhn, topology exists as the core of a new paradigm, and Poincaré’s conjecture, knot theory, graph theory, etc. derived from topology are theories on the fringes of paradigm theory. In addition, prominent scholars studying topology have been immersed in research and solving problems within the framework of ‘topology’, and revolutionary scientific advances should have been made from the results of such research. But that’s not how Dr. Perelman solved his problem, which was to solve a century-old mathematical conundrum and make a revolutionary scientific breakthrough.
Attempts to use topology to solve the Geometrization Conjecture and Poincaré’s Conjecture have been around since the 1960s. After repeated attempts to solve the problem in three dimensions were thwarted by the creation of knots, Dr. Steven Smail from the United States tried to solve the problem in higher dimensions. Consider the rails of a roller coaster. The shadow of the roller coaster rails on the ground is tangled and intertwined, creating many knots, but the actual roller coaster rails are knotless. If we try to solve the problem at this higher level, we can easily solve the knot problem. In fact, it worked, and Poincaré’s conjecture was solved in five and more dimensions and in four dimensions using pure topology alone, and the mathematicians who solved it, Dr. Smail and Dr. Friedman, both won the Fields Prize, the Nobel Prize of mathematics. Dr. Thurston won the Fields Prize again for his work on the classification of 3-dimensional manifolds (the geometrization conjecture).
So far, this is remarkably consistent with Thomas Kuhn’s paradigm theory. A new paradigm shift (from Euclidean geometry to topological geometry) has taken place, and within this new paradigm, prominent researchers have formed their own nuclei and fringes, and have made a remarkable expansion of scientific knowledge. However, proving the conjecture in three dimensions proved to be difficult, and no progress was made for more than two decades. It was Dr. Perelman who made the breakthrough. The problem is that Dr. Perelman’s method for solving the conjecture is rooted in classical physics, not topology. Classical physics has nothing to do with the new paradigm of topology. There is not even a common denominator of geometry. However, conjectures were solved by classical physics. The Ricci flow equation, a variant of the heat diffusion equation in thermodynamics, provided an important clue to the solution, which eventually led to the proof. It also required a lot of prior knowledge to solve Poincaré’s conjecture. The Ricci flow equation, Den’s auxiliary theorem, solving the Poincaré conjecture in four dimensions and beyond, collapse theory, the soul conjecture, and more. The Ricci flow equation is rooted in thermodynamics in space, which in turn is rooted in other physical theories. This is yet another argument against Thomas Kuhn’s claim that scientific knowledge is not continuous and does not accumulate.
The problem with this process is that it is a clear refutation of the “science is not unified, it is independent” characterization that Thomas Kuhn gave in his argument for paradigm theory. Clearly, topology has provided many clues to solving the conundrum of Poincaré’s conjecture. Dr. Thurston’s geometrization conjecture and the solution of Poincaré’s conjecture in multidimensions are both interspersed with concepts of topology, and can be said to be part of the paradigm of topology in Thomas Kuhn’s sense of the term. However, the ‘giant’s shoulder’ of the topology world that helped to solve Poincaré’s conjecture was the Ricci flow equation, which is rooted in the heat diffusion equation of physics, not topology, so Thomas Kuhn’s theory must be denied. The singularity of the spatial function caused by the contraction of a closed curve in three-dimensional space was simplified by the Ricci flow equation, which allowed him to solve the knot problem. Concepts from classical physics, seemingly unrelated to topology, played a crucial role in solving topology-derived challenges, thus negating one of the main thrusts of Thomas Kuhn’s argument that paradigms in science are neither unified nor mutually interfering. This demonstrates that scientific knowledge does not change its identity as scientists shift paradigms. It makes it clear that the paradigm theory, which has been instrumental in explaining the structure of the scientific revolution, has blind spots.
Thomas Kuhn’s theory of paradigms and the structure of scientific revolutions has revolutionized the way people perceive science. The questioning of positivist science by Thomas Kuhn has allowed people to think critically about the validity of positivist science once again. It shook up the notion, deeply ingrained in people’s consciousness, that empirical science was a fact to be trusted, and that the accumulation of facts was a powerful step toward ultimate truth. Paradigm theories are to be commended for their ability to make us think a little deeper and look at science in a more critical light. It also sparked discussions about the philosophy and ethics of science, providing clues that scientists’ ideas might be involved in the pursuit of scientific knowledge.
However, as the examples above show, there are many weaknesses and blind spots in his theory. The independence of scientific knowledge and the mutual inviolability of paradigms cannot be guaranteed, as we have seen. In addition, the theory has been criticized for its vagueness in defining paradigms. Kuhn’s definition of paradigm is so vague that it has been criticized as a conveniently loose definition. Although Thomas Kuhn made great advances in the philosophy of science, the independence of scientific bodies of knowledge that he claimed is hard to convince in light of the example of solving the Poincaré conjecture. Also, the example of Thomas Kuhn claiming that scientific knowledge is not continuous or accumulative is hard to convince. He gives the example of how new scientific theories are often forgotten or made obsolete as new ones are developed, but I would argue that we don’t “lose” anything as we move forward by embracing new disciplines. For example, imagine that we know addition, and then we learn the concept of multiplication. The multiplication formula 7 × 3 actually comes from 7+7+7. Just because we accept the more advanced expression 7 × 3 doesn’t mean that the conceptual principles of 7+7+7 become obsolete.
In the process of resolving the Poincaré conjecture, we have seen that some of the arguments underlying Thomas Kuhn’s paradigmatic theory are negated. Clearly, Thomas Kuhn has made a significant contribution to the history of the philosophy of science, and his theories have had a profound impact on the scientific community. However, we have seen that some of his arguments are either irrefutable or poorly supported by examples. As such, I believe that theories and explanations related to science contain a mixture of some convincing universal truths and others that are not, and we should be able to accept them with a critical and cautious attitude.
