From rr-dual sets to uniform contractions

Let $M^d$ denote the $d$-dimensional Euclidean, hyperbolic, or spherical space. The $r$-dual set of given set in $M^d$ is the intersection of balls of radii $r$ centered at the points of the given set. In this paper we prove that for any set of given volume in $M^d$ the volume of the $r$-dual set becomes maximal if the set is a ball. As an application we prove the following. The Kneser-Poulsen Conjecture states that if the centers of a family of $N$ congruent balls in Euclidean $d$-space is contracted, then the volume of the intersection does not decrease. A uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. We prove the Kneser-Poulsen conjecture for uniform contractions (with $N$ sufficiently large) in $M^d$.Comment: 8 page

An atheological argument from evil : a thesis presented in partial fulfillment of the requirements for the degree of Master of Arts in Philosophy at Massey University

Ever since ancient times it has occurred to many people, great and small, that the existence of evil constitutes evidence against that of God. The central claim of this thesis is that, contrary to theistic belief, this evidence is decisive. In the introduction it is argued that all previous attempts to show this fact have been unsuccessful. These attempts have been vitiated by the fallacy of supposing that God, as an all-powerful, all-knowing and all-good agent, is always required to do the best. He is capable of doing. Though other possibilities remain, this supposition usually manifested itself in arguments which claimed that a God of the above mentioned sort cannot exist, because if such a being existed, He would have created a much better world than the actual one. Besides an appeal to God's above mentioned qualities, the sole justification offered for this claim usually has been only to point out the fact that it was in God's power to actualise a better world than the actual one. But this argument is invalid. Given God's qualities, the mere fact that the creation of a better world was an option to God cannot constitute a sufficient reason for Him to take advantage of that option. For, given the fact that there is virtually no limit to what a being like God can do, it is true of any possible world which was in God's power to actualise that He could have created a better one than it. Consequently, if God decided to create, say value, He would have to be quite irrational to decide not to create some particular world just because it was in His power to create a better one than it. For, if He did that, He ultimately would altogether have to forego creating anything at all - which is absurd because it cannot be the case that a being like God is unable to perform His own will. And this is a problem for atheist endeavours because it shows that the claim that if God existed, He would have created a better world than the actual one inevitably remains unsupported if we proceed from this traditional approach. The chief novelty of this thesis lies in showing the way out of this particular difficulty. It is argued here that if God existed, He would have created a better world than the actual one not only because it was in His power to do so, but because the actual world fails to meet a certain adequacy threshold of being good enough for a product of the creative activity of a perfect being like God. The justification offered for this claim relies on a distinction between ends and means. It goes in two steps: Firstly, it is argued that in God's hands the actual world could only be a means to an end. And secondly, it is argued that whatever God's purpose with the actual world might have been, on account of His benevolence it would have to be a morally good one. Consequently, He could have achieved that purpose by creating a world without superfluous, unnecessary evils in it. Failure to do this conflicts with His benevolence. Further, in defence of this last claim it is argued that although God cannot be reasonably required to attempt realizing the best possible moral goal (for nothing qualifies as such), it conflicts with His benevolence if. He fails to employ the best possible moral means available to Him for realizing His goals. The rest of the thesis contains the details and defence of an argument from evil which is advanced within the framework of this new approach