156 SCIENTIFIC METHOD IN PHILOSOPHY 



" The world has no beginning and no limits in space, 

 but is infinite in respect of both time and space." Kant 

 professes to prove both these propositions, whereas, if 

 what we have said on modern logic has any truth, it must 

 be impossible to prove either. In order, however, to 

 rescue the world of sense, it is enough to destroy the 

 proof of one of the two. For our present purpose, it is 

 the proof that the world infinite that interests us. Kant's 

 argument as regards space here rests upon his argument 

 as regards time. We need therefore only examine the 

 argument as regards time. What he says is as follows : 



" For let us assume that the world has no beginning 

 as regards time, so that up to every given instant an 

 eternity has elapsed, and therefore an infinite series of 

 successive states of the things in the world has passed by. 

 But the infinity of a series consists just in this, that it can 

 never be completed by successive synthesis. Therefore 

 an infinite past world-series is impossible, and accordingly 

 a beginning of the world is a necessary condition of its 

 existence ; which was the first thing to be proved." 



Many different criticisms might be passed on this 

 argument, but we will content ourselves with a bare 

 minimum. To begin with, it is a mistake to define the 

 infinity of a series as " impossibility of completion by 

 successive synthesis." The notion of infinity, as we 

 shall see in the next lecture, is primarily a property of 

 classes^ and only derivatively applicable to series ; classes 

 which are infinite are given all at once by the defining 

 property of their members, so that there is no question 

 of "completion" or of " successive synthesis." And the 

 word " synthesis," by suggesting the mental activity of 

 synthesising, introduces, more or less surreptitiously, 

 that reference to mind by which all Kant's philosophy 

 was infected. In the second place, when Kant says that 



