A mathematical proof and the verification of a hypothesis are quite different in process and outcome from the support of a theory with evidence that we understand how nature is really operating.

一项“ 数学证明”和“ 验证一项假设” 有很大的不同，无论过程及其结果，还是以证据支持一项理论与理解自然如何确切运行。

More and more attention has been received on the research of the training of students' ability in junior mathematical proof along with the development of junior mathematical course reform.

随着数学课程改革的深入，关于初中生数学证明能力的培养问题研究已经引起了越来越广泛的关注。

Graham’s number is the largest number that has been used in a serious mathematical proof.

格雷厄姆系数是一个在严谨的数学证明中被用到的最大的数值。

A computer can never check every number. Instead, what's needed is a mathematical proof.

计算机从来不能把每个数字都试一遍。相反，所需的是数学证明。

On one hand, as a typical content of mathematical subject, reasoning and proof embodies the mathematical rational spirit;

一方面，推理与证明作为数学学科的典型内容，体现了数学的理性精神；