Theory of Functions
Theory of Functions
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DOI：
10.1038/059533d0
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发表时间：
1899-04
期刊：
影响因子：
64.8
通讯作者：
W. Burnside
中科院分区：
文献类型：
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作者：
W. Burnside
教授们对书中第3页所引段落的批评。Harkness和Morley （NATURE， 2月23日，第347页）揭示了这样一个事实，即在处理与测量分离的数时，作者使用了短语“无限的对象”，而没有明确说明其含义。我不确定我是否理解他们信中提到这一点的段落；但在我看来，这似乎是在暗示“有限”与“无限”之间的区别是不需要下定义的。这并不是唯一被接受的观点。例如，这不是戴德金先生的书中所持的观点，“我是一个人，我是一个人。”关于第十五章开头几句话。，作者显然误解了我反对的要点。根据通常得到的无穷积收敛的定义，Π（1-αn）如果收敛，则不等于零。根据引用的文章，Π（1-αn）可能是零；因此，如果假设通常的收敛定义，它是不收敛的。至于引用第232页的段落，我必须向作者表示遗憾，因为我忽略了这样一个事实，即在第八章中使用的特定重新排列是完全合理的。不管Logxis是不是，在第四章的开头，Logxis是用一根绳子和一个圆锥体来定义的，对于任何一个读过整篇文章（第46页，第16行，到第47页，第9行）的人来说，都是显而易见的。
THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Logxis or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.