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In the present note, we establish a finiteness theorem for $L^p$ harmonic 1-forms on hypersurfaces with finite index, which is an extension of the result of Choi and Seo (J. Geom. Phys.129 (2018), 125–132).
The authors run through the major arguments for the existence of God: Anselm’s ontological argument (and also Descartes’s version), arguing that the very notion of God a priori proves hs existence; Aquinas’s cosmological (or causal) argument, that God is needed to stop an infinite regression of causes from the present to the past; and the teleological argument or the argument from design, that the design-like natural objects of this world demand a designer. Then they raise the standard objections: Gaunilo’s criticism that the ontological argument proves the existence of perfect islands, which is ridiculous, and Kant’s objection that you cannot infer matters of fact by a priori reasoning; Dawkins’s criticism that the cosmological argument raises the unanswered question of what causes God; and Hume’s criticism of the design argument, and Darwin’s subsequent demonstration that natural selection can explain final causes naturalistically, and so there is no need to invoke a Designer God.
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