What I understand rationality to be. Rationality is that you reason in a manner that can be expressed using proof's in first order logic assuming certain facts. A fact is a statement whose truth value you assume to be true. Axiom is another name for statement that are assumed to be true. Generating proofs is not necessarily done but could be if desired.
Science combines rational reasoning with observations that occur in reality. Basically you make observations in reality that you assume as facts. Then rational reasoning is used to derive theorems from the facts. The hypothesis is a statement whose truth value is to be calculated. Science it a method to get starting facts for rational reasoning that prove or disprove the hypothesis.
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