H4 MYSTICISM AND LOGIC 



Kant's Transcendental ^Esthetic, and suppose we wish 

 to discover what are the elements of the problem and 

 what hope there is of obtaining a solution of them. It 

 will soon appear that three entirely distinct problems, 

 belonging to different studies, and requiring different 

 methods for their solution, have been confusedly combined 

 in the supposed single problem with which Kant is 

 concerned. There is a problem of logic, a problem of 

 physics, and a problem of theory of knowledge. Of 

 these three, the problem of logic can be solved exactly 

 and perfectly ; the problem of physics can probably be 

 solved with as great a degree of certainty and as great 

 an approach to exactness as can be hoped in an empirical 

 region ; the problem of theory of knowledge, however, 

 remains very obscure and very difficult to deal with. 

 Let us see how these three problems arise. 



(i) The logical problem has arisen through the 

 suggestions of non-Euclidean geometry. Given a body 

 of geometrical propositions, it is not difficult to find 

 a minimum statement of the axioms from which this 

 body of propositions can be deduced. It is also not 

 difficult, by dropping or altering some of these axioms, 

 to obtain a more general or a different geometry, having, 

 from the point of view of pure mathematics, the same 

 logical coherence and the same title to respect as the 

 more familiar Euclidean geometry. The Euclidean 

 geometry itself is true perhaps of actual space (though 

 this is doubtful), but certainly of an infinite number of 

 purely arithmetical systems, each of which, from the 

 point of view of abstract logic, has an equal and inde- 

 feasible right to be called a Euclidean space. Thus 

 space as an object of logical or mathematical study loses 

 its uniqueness ; not only are there many kinds of spaces, 

 but there are an infinity of examples of each kind, 



