ESSAY-REVIEWS 143 



Pythagoreanism to combine the more purely mathematical properties of numbers 

 with their mystical inheritance from the past and declare that all things are 

 numbers. 



Locke, in his chapter on number, drops a hint at another reason why number 

 was so prominent in early speculation. "Amongst all the ideas we have, as there 

 is none suggested to the mind by more ways, so there is none more simple than 

 that of unity, or one. It has no shadow of variety of composition in it ; every 

 object our senses are employed about, every idea in our understandings, every 

 thought of our minds, brings this idea along with it : and therefore it is the most 

 intimate to our thoughts, as well as it is, in its agreement to all other things, the 

 most universal idea we have. For number applies itself to men, angels, actions, 

 thoughts — everything that either doth exist or can be imagined." 



The subsidence of the first mental excitement roused by the discovery of 

 numbers and their properties has left a problem for solution. The most familiar 

 and universal concepts are the most mysterious. What is time or space or 

 causation ? St. Augustine said that he knew what time was so long as he was not 

 asked. We know what number is so long as no one asks us or we do not ask 

 ourselves. Is number a property of the objects to which it is attached ? Locke 

 apparently thought that it is, for he includes it among the " original or primary 

 qualities . . . such as are utterly inseparable from the body, in what state soever 

 it be. . . ." The particular number " of the parts of fire or snow are in them," 

 he adds, "whether any one's senses perceive them or no." If number is really a 

 property of objects it seems to be a peculiarly sliding property. A group of six 

 stones may be thought of as six individuals or as one heap. Number is a 

 collective property if it be one at all. Each member of the class of red objects is 

 red, but each member of a group of six is not six. A body may be in the classes 

 of round objects, of wooden objects, of blue objects, and of manufactured objects, 

 because it has the characteristic property of each class. Any object can be 

 included in any collection denoted by any number, and apparently it can be so 

 included because it has none of these numbers itself. Since any collection can be 

 regarded as one or many, it is tempting to suppose that number is a purely mental 

 thing. Some deference certainly seems due to the objects, for a group of five 

 stones must be regarded as either one or five, but number appears to be some- 

 thing to which the objects dispose the mind rather than a quality possessed by 

 them. Number has been described as strokes of attention— a definition that 

 transfers it from external to mental existence. Is number, then, a mental entity?! 

 '' The question, ' What is number ? ' is one which has often been asked," writes 

 Bertrand Russell in his recent Introduction to Mathematical Philosophy, "but 

 has only been correctly answered in our own time. The answer was given by 

 Frege in 1884, in his Grundlagen der Arithmetik. Although this book is quite 

 short, not difficult, and of the very highest importance, it attracted almost no 

 attention, and the definition of number which it contains remained practically 

 unknown until it was rediscovered by the present author in 1901." Towards the 

 end of the chapter he condenses this discovery into definitions running as follows : 

 ' The number of a class is the class of all those classes that are similar to it," and 

 "A number is anything which is the number of some class." Many may feel 

 inclined to leave Mr. Russell in peace with his definitions, to continue to count 

 their money without peering into the inward mystery of numbers, and to wait 

 for something simpler to remove their metaphysical itch. A little patience will, 

 however, dispel the apparent portentousness of this revelation of the nature of 

 number, even if it be decided that the whole mystery has not been disclosed in it. 



