Dedekind Dedekinds language in introducing irrational numbers leaves

介绍Dedekind分割的原文 • Dedekind’s language in introducing irrational numbers leaves a little to be desired. He introduces the irrational α as corresponding to the cut and defined by the cut. But he is not too clear about where α comes from. He should say that the irrational number α is no more than the cut. In fact Heinrich Weber told Dedekind this, and in a letter of 1888 Dedekind replied that the irrational number α is not the cut itself but is something distinct, which corresponds to the cut and which brings about the cut. Likewise, while the rational numbers generate cuts, they are not the same as the cuts. He says we have the mental power to create such concepts. --From the page 986 of the book 《 Mathematical Thought From Ancient to Modern Times》 by Morris Kline, Oxford University Press, 1972 ) 4

《古今数学思想》 对上述 作的评价（原文） • The irrational number, logically defined, is an intellectual monster （智慧的怪物）, and we can see why the Greeks and so many later generations of mathematicians found such numbers difficult to grasp. ——From the page 987 of the book 《Mathematical Thought From Ancient to Modern Times》 (by Morris Kline, Oxford University Press, 1972 ) 7

极限的思想必不可少 • Terence Tao 在讲实数的表示时说： But to get the reals from the rationals is to pass from a “discrete” system to a “continuous” one, and requires the introduction of a somewhat different notion ---that of a limit. ---from Terence Tao’s book 《 Analysis I 》， Hindustan Book Agency (India), 2006 ，Page 108 14