Pure mathematics is the study of mathematics for its own sake.

纯粹的数学是为了研究数学而研究数学。

Bergson apparently holds that capacity for this kind of knowledge is less explicable by the struggle for existence than, for example, capacity for pure mathematics.

柏格森显然认为， 与纯数学的能力相比， 这种知识的能力更难通过生存斗争来解释。

Thus he was led to the conclusion that all pure mathematics, though a priori, is synthetic; and this conclusion raised a new problem of which he endeavoured to find the solution.

因此，他得出结论，即所有纯数学，尽管是先验的，都是综合的；这个结论提出了一个新问题，他努力寻找解决办法。

And I’m not being political here; it’s pure mathematics: If you have 99 people making $50,000 per year and 1 person making $50 million per year – what’s gonna happen to the mean income?

而且我在这里不是政治；这是纯数学：如果你有 99 个人每年赚 50,000 美元， 而 1 个人每年赚 5000 万美元——平均收入会发生什么变化？

The question which Kant put at the beginning of his philosophy, namely 'How is pure mathematics possible'? is an interesting and difficult one, to which every philosophy which is not purely sceptical must find some answer.

康德在其哲学开篇提出的问题，即“纯数学何以可能”？是一个有趣而困难的问题，每一种非纯粹怀疑的哲学都必须找到一些答案。

Many were engaged in writing manuscript books of various kinds; one (Chang said) had made valuable researches into pure mathematics; another was coordinating Gibbon and Spengler into a vast thesis on the history of European civilization.

许多人从事各种手稿书籍的写作；一个（Chang 说）对纯数学进行了有价值的研究；另一个是将吉本和斯宾格勒协调成一篇关于欧洲文明史的庞大论文。

All pure mathematics is a priori, like logic. This was strenuously denied by the empirical philosophers, who maintained that experience was as much the source of our knowledge of arithmetic as of our knowledge of geography.

所有纯数学都是先验的，就像逻辑一样。经验哲学家极力否认这一点，他们坚持认为经验是我们算术知识和地理知识的源泉。