AS A MENTAL OPERATION 7 



from the concrete to the abstract, to the units of abstract 

 arithmetic, and the points of abstract geometry. The 

 Greeks achieved this analysis from concrete to abstract, 

 and thus converted mathematics from analysis to syn- 

 thesis, which begins with the abstract unit as origin of 

 number, and with the abstract point as simpler than the 

 line. But the order of discovery was from the concrete 

 and analytical, although afterwards the order of develop- 

 ment was from the abstract and synthetic. Nor did the 

 certainty of mathematical principles require an a priori 

 origin. They are equations, or rather identical judge- 

 ments ; and, after experience of their objects, the mind 

 has the power of perceiving their identity. Thus once 

 we have experienced two we perceive that it is the 

 same as one and one ; having experienced a triangle we 

 perceive that there is no difference between it and a 

 three-sided rectilineal figure ; after experience of wholes 

 we perceive that a whole must be greater than a part of 

 it, otherwise it would not be a whole. Nothing is abso- | 

 lutely necessary except that a thing is itself. * The 

 method then of mathematics consists of principles, dis- 

 covered from experience and apprehended as certain 

 by the power of perceiving identity, and of demonstra- 

 tions or deductions of consequences from these identical 

 and therefore necessary, principles. , 



Though we have the mathematical power of appre- 

 hending identical certainty, and demonstrating conse- 

 quent certainties, this power is limited. It requires! 

 simple objects, such as numbers and figures, which after/ 

 experience admit of great abstraction, and of such simpliJ 

 fication that we are able to grasp the whole abstract 

 object and so apprehend with certainty what it identically 

 is. We may indeed extend this power of abstract simpli- 

 fication and identification from mathematics to physics, 



