Arguments
One of the arguments the contributed to Flew's change of mind about God was the cosmological argument. It goes something like the following:
1. Everything that exists had a cause.
2. The universe exists.
3. Therefore, the universe had a cause.
4. That cause was God.
Now, this argument is very old, and Flew surely knew about it in his days as an atheist. But, due to the advances in science, e.g. the big bang theory, certain premises of the cosmological argument began to receive scientific support. For example, the big bang theory states that there once was a beginning of time. This is a crucial discovery because previously, the prevailing theory was that the universe was eternal (and therefore had no cause. This used to be the main objection to the cosmological argument). Therefore, the big bang theory is cited as scientific support for points 1-3, above.
Limits
(I'm always pleased when the limits of an argument are openly acknowledged)
Point 4 above is controversial because it doesn't necessarily follow from the previous points. The cause of the universe could be something other than God. For example, it might have been an alien from another universe. The argument doesn't necessarily say anything about the nature of the cause. Ultimately, in order to establish that God was the cause of the universe, one would need more corroborating evidence.
Another objection is that point 1 is true for causes and effects within this universe. Therefore, it's not necessarily true for causes and effects outside of this universe. So, some say that it doesn't necessarily follow to conclude that there was a cause of the universe that came from outside the universe, e.g. God.
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