LECTURE VII 



THE POSITIVE THEORY OF INFINITY 



The positive theory of infinity, and the general theory 

 of number to which it has given rise, are among the 

 triumphs of scientific method in philosophy, and are 

 therefore specially suitable for illustrating the logical- 

 analytic character of that method. The work in this 

 subject has been done by mathematicians, and its results 

 can be expressed in mathematical symbolism. Why, 

 then, it may be said, should the subject be regarded 

 as philosophy rather than as mathematics ? This raises 

 a difficult question, partly concerned with the use of 

 words, but partly also of real importance in understand- 

 ing the function of philosophy. Every subject-matter, it 

 would seem, can give rise to philosophical investigations 

 as well as to the appropriate science, the difference 

 between the two treatments being in the direction of 

 movement and in the kind of truths which it is sought 

 to establish. In the special sciences, when they have 

 become fully developed, the movement is forward and 

 synthetic, from the simpler to the more complex. But 

 in philosophy we follow the inverse direction : from the 

 complex and relatively concrete we proceed towards the 

 simple and abstract by means of analysis, seeking, in the 

 process, to eliminate the particularity of the original 

 subject-matter, and to confine our attention entirely to 

 the logical form of the facts concerned. 



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