200 SCIENTIFIC METHOD IN PHILOSOPHY 



Untersuchung uber den Begriff der Zahl} It is with this 

 book that the logical theory of arithmetic begins, arid it 

 will repay us to consider Frege's analysis in some detail. 



Frege begins by noting the increased desire for logical 

 strictness in mathematical demonstrations which distin- 

 guishes modern mathematicians from their predecessors, 

 and points out that this must lead to a critical investiga- 

 tion of the definition of number. He proceeds to show 

 the inadequacy of previous philosophical theories, especially 

 of the " synthetic a priori ' theory of Kant and the 

 empirical theory of Mill. This brings him to the ques- 

 tion : What kind of object is it that number can properly 

 be ascribed to ? He points out that physical things may 

 be regarded as one or many : for example, if a tree has 

 a thousand leaves, they may be taken altogether as con- 

 stituting its foliage, which would count as one, not as a 

 thousand ; and one pair of boots is the same object as 

 two boots. It follows that physical things are not the 

 subjects of which number is properly predicated ; for 

 when we have discovered the proper subjects, the number 

 to be ascribed must be unambiguous. This leads to a 

 discussion of the very prevalent view that number is 

 really something psychological and subjective, a view 

 which Frege emphatically rejects. " Number," he says, 

 " is as little an object of psychology or an outcome of 

 pscyhical processes as the North Sea. . . . The botanist 

 wishes to state something which is just as much a fact 

 when he gives the number of petals in a flower as when 

 he gives its colour. The one depends as little as the 

 other upon our caprice. There is therefore a certain 



1 The definition of number contained in this book, and elaborated in 

 the Grundgesetze der Arithmetik (vol. i., 1893 ; vol. ii., 1903), was re- 

 discovered by me in ignorance of Frege's work. I wish to state as 

 emphatically as possible what seems still often ignored that his dis- 

 covery antedated mine by eighteen years. 



