768 THE PRINCIPLES OF SCIENCE [CHAP. 



infinity. Professor Clifford tells me that he has found a 

 mathematical function which approaches infinity as the 

 variable approaches a certain limit ; yet at the limit the 

 function is finite ! Mathematicians may shirk difficulties, 

 but they cannot make such results of mathematical prin- 

 ciples appear otherwise than contradictory to our common 

 notions of space. 



The hypothesis that there is a Creator at once all-power- 

 ful and all-benevolent is pressed, as it must seem to every 

 candid investigator, with difficulties verging closely upon 

 logical contradiction. The existence of the smallest amount 

 of pain and evil would seem to show that He is either not 

 perfectly benevolent, or not all-powerful. No one can 

 have lived long without experiencing sorrowful events 

 of which the significance is inexplicable. But if we 

 cannot succeed in avoiding contradiction in our notions of 

 elementary geometry, can we expect that the ultimate 

 purposes of existence shall present themselves to us with 

 perfect clearness ? I can see nothing to forbid the notion 

 that in a higher state of intelligence much that is now 

 obscure may become clear. We perpetually find ourselves 

 in the position of finite minds attempting infinite problems, 

 and can we be sure that where we see contradiction, an 

 infinite intelligence might not discover perfect logical 

 harmony ? 



From science, modestly pursued, with a due conscious- 

 ness of the extreme finitude of our intellectual powers, 

 there can arise only nobler and wider notions of the pur- 

 pose of Creation. Our philosophy will be an affirmative 

 one, not the false and negative dogmas of Auguste Comte, 

 which have usurped the name, and misrepresented the 

 tendencies of a true positive philosophy . True science will 

 not deny the existence of things because they cannot be 

 weighed and measured. It will rather lead us to believe 

 that the wonders and subtleties of possible existence sur- 

 pass all that our mental powers allow us clearly to perceive. 

 The study of logical and mathematical forms has convinced 

 me that even space itself is no requisite condition of con- 

 ceivable existence. Everything, we are told by materialists, 

 must be here or there, nearer or further, before or after. I 

 deny this, and point to logical relations as my proof. 



There formerly seemed to me to be something mysterious 



