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

語法和語義之間的相互作用是形式語言研究的核心。

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

形式系統的發展徹底改變了我們對數學的理解。

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

計算的研究對于理解可計算性的極限至關重要。

Logic is the foundation of all mathematical reasoning.

邏輯是所有數學推理的基礎。

The concept of infinity is both fascinating and perplexing.

無窮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.

數學的美在于它既是藝術又是科學。

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

證明是一種用于使他人相信數學陳述為真的工具。

The essence of mathematics lies in its freedom.

數學的本質在于其自由性。

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

在數學中，你并不理解事物。你只是習慣了它們。

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夠強大以包含算術的形式系統中都是不可定義的。