The interplay between syntax and semantics is central to the study of formal languages.

语法和语义之间的相互作用是形式语言研究的核心。

The development of formal systems has revolutionized our understanding of mathematics.

形式系统的发展彻底改变了我们对数学的理解。

The study of computation is essential to understanding the limits of what can be computed.

计算的研究对于理解可计算性的极限至关重要。

Logic is the foundation of all mathematical reasoning.

逻辑是所有数学推理的基础。

The concept of infinity is both fascinating and perplexing.

无穷的概念既迷人又令人困惑。

The beauty of mathematics is that it is both an art and a science.

数学的美在于它既是艺术又是科学。

A proof is a device for convincing someone else that a mathematical statement is true.

证明是一种用于使他人相信数学陈述为真的工具。

The essence of mathematics lies in its freedom.

数学的本质在于其自由性。

In mathematics, you don't understand things. You just get used to them.

在数学中，你并不理解事物。你只是习惯了它们。

The notion of truth is not definable in any formal system that is strong enough to include arithmetic.

真理的概念在任何足够强大以包含算术的形式系统中都是不可定义的。