THE PROBLEM OF INFINITY 165 



property of being unable to be counted is characteristic 

 of infinite collections, and is a source of many of their 

 paradoxical qualities. So paradoxical are these qualities 

 that until our own day they were thought to constitute 

 logical contradictions. A long line of philosophers, from 

 Zeno 1 toM. Bergson, have based much of their metaphysics 

 upon the supposed impossibility of infinite collections. 

 Broadly speaking, the difficulties were stated by Zeno, 

 and nothing material was added until we reach Bolzano's 

 Paradoxien des Unendlichen, a little work written in 1847-8, 

 and published posthumously in 1851. Intervening at- 

 tempts to deal with the problem are futile and negligible. 

 The definitive solution of the difficulties is due, not to 

 Bolzano, but to Georg Cantor, whose work on this subject 

 first appeared in 1882. 



In order to understand Zeno, and to realise how little 

 modern orthodox metaphysics has added to the achieve- 

 ments of the Greeks, we must consider for a moment his 

 master Parmenides, in whose interest the paradoxes were 

 invented. 2 Parmenides expounded his views in a poem 

 divided into two parts, called " the way of truth ' and 

 " the way of opinion " like Mr Bradley's " Appearance ' 

 and " Reality," except that Parmenides tells us first about 

 reality and then about appearance. " The way of opinion," 

 in his philosophy, is, broadly speaking, Pythagoreanism ; 

 it begins with a warning : " Here I shall close my trust- 

 worthy speech and thought about the truth. Hence- 

 forward learn the opinions of mortals, giving ear to the 

 deceptive ordering of my words." What has gone before 

 has been revealed by a goddess, who tells him what 



1 In regard to Zeno and the Pythagoreans, I have derived much valuable 

 information and criticism from Mr P. E. B. Jourdain. 



2 So Plato makes Zeno say in the Parmenides, apropos of his philosophy 

 as a whole ; and all internal and external evidence supports this view. 



