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How do mathematicians go from conjecture to theorem? How difficult is the process?
Mathematics is a subject that explores truth, and conjecture, as an important part of this process, often triggers countless research and discussions. Conjectures in mathematics are unproven conclusions or propositions. These conjectures are like guiding lights, guiding mathematicians to sail in the infinite ocean of mathematics. From ancient times to the present, there have been many famous conjectures, such as Riemann's hypothesis and Fermat's last theorem. The challenges brought by these conjectures have not only inspired the development of new mathematical fields, but also deepened people's understanding of the nature of mathematics.
The core of mathematics lies in provable truth. Any universal conjecture supported by big data cannot establish its authenticity, because a counterexample may shake its foundation.
In the world of mathematics, proof is a difficult road. For a conjecture, mathematicians need to go through repeated testing and reasoning until they finally establish that its logic cannot be false. Various evidence supporting the conjecture, including verification of its derived results and close connection with existing theories, are all casting the cornerstone of these theories. At the same time, if there are limited cases that may lead to counterexamples, mathematicians will also use the "violent proof" method to safely check all possible situations. For example, the four-color theorem was verified through computer algorithms, and its proof method using digital technology for the first time also aroused heated discussions.
The four-color theorem marked an advancement in mathematics because it was the first major theorem to be proved with computer assistance.
In the field of mathematics, the failure of conjecture is equally eye-catching. For example, certain conjectures that have been proven by counterexamples, such as the Praiat conjecture and Euler's sum of powers conjecture, have become one of the counterexamples known as pseudoconjectures. These situations raise deep thoughts about the boundaries of mathematics, especially the circumstances under which a conjecture might be completely overturned.
The world of mathematics is complex and diverse, and not all conjectures will be proven correctly. For example, the existence of the continuum hypothesis shows that there are some independent propositions in the generally accepted axioms of set theory. This means that one can adopt the proposition or its negation as a new axiom in a consistent way. This situation has triggered more in-depth thinking and discussion on the stability of axiomatic systems in the mathematical community.
Sometimes people discover that the assumptions they relied on are simply unreliable, challenging the entire mathematical system.
In the process of mathematics, many famous theorems were once conjectures, such as the geometrization theorem, Fermat's last theorem, etc. Their establishment has gone through a long and arduous process. Fermat's Last Theorem was first proposed by Pierre de Fermat in 1637, and was not successfully proved until 1994 by Andrew Wiles. For a full 358 years, its journey has condensed the efforts of several generations of mathematicians.
Another important example is Poincaré's conjecture. Although it was proven nearly a century ago, its significance has not diminished at all. Before Grigory Perelman published his proof in 2003, this problem attracted countless mathematicians to study, and it was called the "Holy Grail" of mathematics.
The journey of exploration in mathematics is difficult, and every theorem successfully proved is a testimony to the perseverance and wisdom of mathematicians.
Whether it is a mathematical problem closely integrated with practical applications or a theory deeply related to philosophy, the solution of conjectures allows us to see the power of mathematics. In the process of conjecture, mathematicians go from doubt to belief, from exploration to confirmation. The difficulties and twists and turns of this road reflect the beauty of mathematics. In the future, how many unanswered questions and unproven conjectures will await us to explore?
