passing either through zero or through infinity. In the latter case there must be one intermediate value of the variable for which the variant is indifferently negative infinity and positive 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 principles appear otherwise than contradictory to our common notions of space.
The hypothesis that there is a Creator at once all-powerful 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 consciousness of the extreme finitude of our intellectual powers, there can arise only nobler and wider notions of the purpose 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 surpass 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 conceivable 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.