论数学的不确定性：数学的三次危机及其未来
On Mathematical Uncertainty: The Three Crises of Mathematics and Their Future

从诠释学的观点看，不确定性乃是知识与生俱来的本质特性，这一点即便在数学中也不例外。在人们的习惯性认知中，数学和逻辑是最具确定性的学科，然而实际上并不是这样。数学并不是一个亘古不变的、适用于一切时间空间的学科，数学本身就体现了不确定性。在数学史上，数学不断地在经历着危机并解决危机，就说明了数学本身不过是人类对世界的认知方式而已。迄今为止，数学一共经历了三次危机。第一次数学危机是自然数的危机，危机的结果是严格的实数理论的建立。第二次数学危机是连续性的危机，危机的结果是微积分的严密基础的建立。第三次数学危机是集合论的危机，涉及了数学的基础。迄今为止，这一危机并没有得到完美的解决。从数学危机的产生和解决过程来看，数学正是由于危机的存在才得以不断前行，而数学知识也正是在一次次的危机中而得以构造。
From the point of view of Hermeneutics, uncertainty is the inherent nature of knowledge, even in mathematics. In people’s habitual perception, mathematics and logic are the most deterministic subjects, but in reality, they are not. Mathematics is not an immutable subject, applicable to all time and space, and mathematics itself reflects uncertainty. In the history of mathematics, mathematics continues to experience and solve crises, which shows that mathematics itself is only a way of human cognition of the world. So far, mathematics has experienced a total of three crises. The first mathematical crisis was the crisis of natural numbers, and the result of which was the establishment of a rigorous theory of real numbers. The second mathematical crisis is the crisis of continuity, and the result of which is the establishment of a rigorous foundation for calculus. The third mathematical crisis was the crisis of set theory, which dealt with the foundations of mathematics. So far, the crisis has not been solved perfectly. From the perspective of the generation and resolution of mathematical crisis, it is precisely because of the existence of crisis that mathematics can continue to move forward, and mathematical knowledge is also constructed in crisis after crisis.