[51] In his Grundlagen einer allgemeinen Mannichfaltigkeitslehre and in articles in Acta Mathematica, vol. ii.

[52] The definition of number contained in this book, and elaborated in the Grundgesetze der Arithmetik (vol. i., 1893; vol. ii., 1903), was rediscovered by me in ignorance of Frege's work. I wish to state as emphatically as possible—what seems still often ignored—that his discovery antedated mine by eighteen years.

[53] Giles, The Civilisation of China (Home University Library), p. 147.

[54] Cf. Principia Mathematica, § 20, and Introduction, chapter iii.

[55] In the above remarks I am making use of unpublished work by my friend Ludwig Wittgenstein.

[56] Thus we are not using “thing” here in the sense of a class of correlated “aspects,” as we did in [Lecture III]. Each “aspect” will count separately in stating causal laws.

[57] The above remarks, for purposes of illustration, adopt one of several possible opinions on each of several disputed points.

[Transcriber's Note:]

The following is a list of corrections made to the original. The first passage is the original passage, the second the corrected one.