Mysticism and Logic and Other Essays

[11] This subject is due in the main to Mr. C.S. Peirce. [12] I ought to have added Frege, but his writings were unknown to me when this article was written. [Note added in 1917.] [13] Professor of Mathematics in the University of Berlin. He died in 1897. [14] [Note added in 1917.] Although some infinite numbers are greater than some others, it cannot be proved that of any two infinite numbers one must be the greater. [15] Cantor was not guilty of a fallacy on this point. His proof that there is no greatest number is valid. The solution of the puzzle is complicated and depends upon the theory of types, which is explained in Principia Mathematica, Vol. I (Camb. Univ. Press, 1910). [Note added in 1917.] [16] This must not be regarded as a historically correct account of what Zeno actually had in mind. It is a new argument for his conclusion, not the argument which influenced him. On this point, see e.g. C.D. Broad, "Note on Achilles and the Tortoise," Mind, N.S., Vol. XXII, pp. 318-19. Much valuable work on the interpretation of Zeno has been done since this article was written. [Note added in 1917.] [17] Since the above was written, he has ceased to be used as a textbook. But I fear many of the books now used are so bad that the change is no great improvement. [Note added in 1917.] [18] The greatest age of Greece was brought to an end by the Peloponnesian War. [Note added in 1917.] [97]