finitism

简明释义

英[/ˈfɪnɪˌtɪzəm/]美[/ˈfɪnɪˌtɪzəm/]

n. 有限论；有极限论

英英释义

单词用法

同义词

反义词

例句

1.What is cosmological finitism?

宇宙有限论是甚么？

2.What is cosmological finitism?

宇宙有限论是甚么？

3.In mathematics, finitism refers to a philosophy that only considers finite mathematical objects.

在数学中，finitism 指的是一种哲学，只考虑有限的数学对象。

4.The debate between finitism and infinitism has been a central theme in the foundations of mathematics.

在数学基础的讨论中，finitism 和无穷主义之间的辩论一直是一个核心主题。

5.A finitism approach is often used in computer science to ensure that algorithms terminate.

在计算机科学中，finitism 方法常用于确保算法终止。

6.Philosophers who advocate for finitism argue that infinite sets cannot be fully comprehended.

提倡 finitism 的哲学家认为，无限集合无法被完全理解。

7.In a finitism framework, all mathematical proofs must ultimately rely on concrete examples.

在 finitism 框架下，所有数学证明必须最终依赖于具体示例。

作文

In the realm of mathematics and philosophy, the concept of finitism plays a crucial role in understanding the nature of infinity and the foundations of mathematical truth. Finitism is a philosophical viewpoint that asserts only finite mathematical objects and processes are meaningful or acceptable. This perspective stands in contrast to other philosophical approaches, such as classical mathematics, which often deals with infinite sets and concepts. The implications of finitism extend beyond mere academic discussions; they influence how we approach problems in mathematics, computer science, and even logic. The essence of finitism can be traced back to the early 20th century when mathematicians like David Hilbert and L.E.J. Brouwer began to explore the foundations of mathematics. Hilbert's program aimed to establish a secure foundation for all of mathematics, while Brouwer advocated for intuitionism, which emphasized the mental construction of mathematical objects. Within this context, finitism emerged as a distinct philosophy that rejected the existence of actual infinities. One of the significant contributions of finitism is its emphasis on constructive mathematics. In constructive mathematics, an object is considered to exist only if it can be explicitly constructed. This contrasts sharply with classical mathematics, where existence can be established through non-constructive proofs. For example, in classical mathematics, one might prove the existence of a solution to an equation without necessarily providing a method to find that solution. However, under finitism, such proofs are insufficient; one must demonstrate a concrete example or method. The implications of adopting a finitist viewpoint are profound. In computer science, for instance, finitism aligns closely with the principles of algorithmic computability. Algorithms, by their very nature, operate within finite time and space constraints. Thus, the finitist perspective provides a solid foundation for understanding computational limits and capabilities. It encourages mathematicians and computer scientists to focus on what can be practically achieved rather than what exists in an abstract sense. Furthermore, finitism has implications for the philosophy of mathematics. It raises questions about the nature of mathematical truth and existence. If one accepts finitism, then one must grapple with the idea that many classical results may not hold in a finitist framework. For instance, the law of excluded middle, a principle in classical logic stating that every proposition is either true or false, may not apply in a finitist context where only finite constructs are acknowledged. Critics of finitism argue that it is overly restrictive and dismisses the richness of mathematical exploration that includes infinitary concepts. They contend that mathematics has evolved precisely because it allows for the consideration of infinite sets and processes. Nevertheless, proponents of finitism argue that grounding mathematics in the finite provides clarity and prevents paradoxes that arise from dealing with the infinite. In conclusion, finitism offers a unique lens through which to view mathematics and its foundations. By prioritizing finite constructs and processes, it challenges us to rethink our assumptions about mathematical existence and truth. Whether in pure mathematics or applied fields like computer science, the principles of finitism encourage a more grounded and practical approach to understanding the universe of numbers and their relationships. As we continue to explore the depths of mathematical thought, the insights gained from finitism will undoubtedly play a pivotal role in shaping future discussions and discoveries.

有限主义在数学和哲学领域中扮演着至关重要的角色，帮助我们理解无限的本质以及数学真理的基础。有限主义是一种哲学观点，主张只有有限的数学对象和过程才是有意义或可接受的。这一观点与其他哲学方法形成鲜明对比，例如经典数学，后者通常涉及无限集合和概念。有限主义的影响超越了单纯的学术讨论，它影响着我们在数学、计算机科学甚至逻辑中的问题处理方式。 有限主义的本质可以追溯到20世纪初，当时数学家如大卫·希尔伯特和L.E.J.布劳威尔开始探索数学的基础。希尔伯特的计划旨在为所有数学建立一个安全的基础，而布劳威尔则提倡直觉主义，强调数学对象的心理构建。在这种背景下，有限主义作为一种独特的哲学观点出现，拒绝承认实际的无穷大。 有限主义的重要贡献之一是它对构造性数学的强调。在构造性数学中，只有当一个对象可以被明确构造时，才被认为是存在的。这与经典数学形成鲜明对比，后者可以通过非构造性证明来建立存在。例如，在经典数学中，人们可能证明某个方程的解是存在的，但并不一定提供找到该解的方法。然而，在有限主义下，这样的证明是不够的；必须展示一个具体的例子或方法。 采用有限主义观点的影响深远。在计算机科学中，有限主义与算法可计算性的原则密切相关。算法本质上在有限的时间和空间限制内运行。因此，有限主义的观点为理解计算限制和能力提供了坚实的基础。它鼓励数学家和计算机科学家关注可以实际实现的内容，而不是抽象意义上的存在。 此外，有限主义对数学哲学也有影响。它提出了关于数学真理和存在性质的问题。如果接受有限主义，那么必须面对许多经典结果在有限主义框架下可能不成立的想法。例如，排中律，这是一条经典逻辑中的原则，规定每个命题要么为真，要么为假，在承认仅有限构造的有限主义背景下可能不适用。 有限主义的批评者认为，它过于限制，并且忽视了包括无限概念在内的数学探索的丰富性。他们争辩说，数学之所以发展，正是因为它允许考虑无限集合和过程。然而，有限主义的支持者则认为，将数学建立在有限的基础上提供了清晰性，并防止了处理无限时出现的悖论。 总之，有限主义为我们提供了一种独特的视角，帮助我们看待数学及其基础。通过优先考虑有限的构造和过程，它挑战我们重新思考有关数学存在和真理的假设。无论是在纯数学还是应用领域，如计算机科学，有限主义的原则都鼓励我们以更扎实和实用的方法来理解数字及其关系的宇宙。随着我们继续探索数学思想的深度，从有限主义中获得的见解无疑将在塑造未来讨论和发现中发挥关键作用。