Can science account for everything?

The answer is NO! Science cannot account for everything!
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There are a good number of things that cannot be scientifically proven but are all rational to accept.

1) Logical and mathematical truths cannot be proven using science. Science presupposes logical and maths. Using science to prove logic and maths would cause circular reasoning.

2) Metaphysical truths such as:
a. The acceptance of minds other than our own.
b. The acceptance of the external world or that past is not recent, with the appearance of age.
c. Rationality cannot be scientifically proven

3) Ethical beliefs about statements of value cannot be accounted for by scientific methods. You cannot tell whether the Nazi scientists in the concentration camps did anything evil as opposed to the scientists in the western democracies.

4) Aesthetic judgments cannot be accessed by scientific methods because beautiful life cannot be scientifically proven.
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5) Science itself cannot be justified by the scientific methods.
Science is filled with scientific assumptions that cannot be proven. None of these beliefs can be scientifically proven yet they are accepted by all of us. Therefore, there are many truths that science cannot account for.