DEFINITION. 91 



brings to light what the thing is ; thus mathematics commences 

 from the exposition of what unity is, what an uneven number is, 

 and so on .&quot; 



As an illustration of the work of analysis and synthesis involved in form 

 ing definitions, the following few examples may prove instructive. In de 

 ductive sciences like mathematics, whose principles are got by direct intellec 

 tual intuition (5), the process of definition is mainly synthetic, an accumulating 

 of the simple attributes so obtained into more and more complex wholes. 

 Each apart, being simpler than the complex whole, is also wider in extent than 

 the latter : the extent is gradually limited (opos, opio-pos) by definition, 



Suppose we are in possession of the simple arithmetical notions of unity, 

 number, prime number, odd number, even number ; of the series, one, two, 

 three, four, etc. How should we define the number three, for example ? 

 Which of our stock of notions will help us ? 



Three is an odd number, not divisible by two. Three is a prime number, in 

 the twofold sense that it is neither a multiple of any other number, nor re 

 solvable into other numbers. 1 Three is therefore defined as an odd prime 

 number. Each of these notions apart is found in other numbers ; all to 

 gether are found only in the number three. They define it, therefore. The 

 notion &quot;number&quot; is true of odd and even alike. The notion &quot;odd &quot; is true 

 rffive, seven, nine, etc. The notion &quot; prime,&quot; in the sense of not being a 

 multiple, is true of five, seven, eleven, thirteen, etc. The notion &quot; prime &quot; in 

 the sense of &quot;not resolvable into other numbers&quot; is also true of the number 

 two. But the combination &quot; odd, prime number&quot; is true only of the simgle 

 number three? 



In the inductive sciences, synthesis is preceded by a long and often 

 laborious work of analysis. Let us take an example from psychology, the 

 science of life. To define life we begin by observing &quot; living &quot; things. Here 

 is a rose-tree in full bloom : it is alive. Here is a puppy, barking and frisk 

 ing : full of life. Here is a man, hard at work : he is alive. Why do we 

 say of all these that they live ? Why ascribe an identical attribute to them ? 

 The rose-tree assimilates nourishment, grows, and perpetuates its kind by 

 seeding ; and so of all vegetables. The puppy is endowed, moreover, with 

 powers of sensation and locomotion ; and so are animals generally. Man, 

 furthermore, has intelligence and free will. Have all these various activities 

 nutrition, growth, reproduction ; sensation, locomotion ; thought and voli 

 tion anything in common ? If not, each group should have a name of its 

 own, nor should we be justified in giving all a common name. But they have 

 a common characteristic, which we may detect by abstracting from their differ 

 ences ; and this common feature is that all these activities have their term 

 within the agent, which they perfect : they are not transitive but immanent ; 

 the definition of life is, therefore, immanent activity* 



The elimination of the differentiating attributes from the classes com 

 pared the rose-tree, the puppy, the man was simultaneously a process of 



1 Unity is not regarded as a &quot; number &quot; but as the &quot; principle &quot; of number, the 

 latter notion implying plurality. 



ARISTOTLE, Anal. Post., ii., 8. MERCIER, Psychologic, Pt. i., ch. i. 



