The basis of the cosmological argument is that the universe cannot account for its own existence.
Plato argued that everything must be have been created by some cause. Aristotle argued that behind the series of cause and effect in the world there must be an Unmoved Moved. The Kalam arguement in Islam is an attempt to show that the universe must have a cause and is not the result of an infinite regress.
Aquinas and the cosmological arguement in Five Ways:
The Five Ways are ways of demostrating the existence of God through inductive argument, based on observation and evidence. Of Aquinas' Five Ways, the first three are different variants of the cosmological argument. Aquinas based his arguement on two assumptions:
a) the universe exists
b) there must be a reason why
People such as Bertrand Russel and Richard Dawkins would disagree with b), as they are happy to acept that universe just is, without moving to the conclusion that there should be some reason for it.
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