VIII.] PRI^XIPLES OF NUMBER. l.^r, 



the two great branches of ab-stract science, interlacing and 

 co-operating in every discourse, should rest upon totally 

 distinct i'oundations. I as.sume that a connection exists, 

 and care only to incpire, What is its nature ? Does the 

 science of quantity rest upon that of quality ; or, vice 

 versa, does the science of quality rest upon that of 

 quantity ? Tiiere might conceivably be a third view, 

 that they both rest upon some still deeper set of prin- 

 ciples. 



It is generally supposed that Boole adopted the second 

 view, and treated logic as an application of algebra, a 

 special case of analytical reasoning which admits only two 

 quantities, unity and zero. It is not easy to ascertain 

 clearly which of these views really was accepted by Boole. 

 In his interesting biographical sketch of Boole, ^ the Rev. 

 R. Harley protests against the statement that Boole's 

 logical calculus imported the conditions of number and 

 quantity into logic. He says : " Logic is never identified 

 or confounded with mathematics ; the two systems of 

 thought are kept perfectly distinct, each being subject to 

 its own laws and conditions. The symbols are the same 

 for bo^.h systems, but they have not the same interpre- 

 tation." The Eev. J. Venn, again, in his review of Boole's 

 logical system,^ holds that Boole's processes are at bottom 

 logical, not mathematical, though stated in a highly gener- 

 alized form and witli a mathematical dress. But it is 

 quite likely that readers of Boole should be misled. Not 

 only have his logical works an entirely mathematical 

 appearance, but I tind on p. 12 of his Laus of Thoufjht 

 the following unequivocal statement: "That logic, as a 

 science, is susceptible of very wide applications is 

 admitted ; but it is equally certain that its ultimate 

 forms and processes are mathematical." A few lines 

 below he adds, " It is not of the essence of mathematics 

 to be conversant with the ideas of number and quantity." 



The solution of the difiiculty is that Boole used the 

 term mathematics in a wider sense than that usually 

 attributed to it. He proliably adopted the third view, so 

 that his mathematical Lavjs of Thouyht are the commou 



' British Quarterly Review, No. Ixxxvii, .July 1S66. 

 '^ Mind ; October 1 876, vol. i. p. 4^4, 



