Beliefs that are contrary to the dictates of reason are irrational. Rational beliefs have also been contrasted with beliefs arrived at through emotion , faith, authority, or by an arbitrary choice. The point of each contrast is to capture a sense in which we believe a proposition either without carrying out an appropriate assessment or in spite of the results of such an assessment. For example, we determine the balance in a cheque-book rationally when we enter the correct credits and debits and do the arithmetic. Irrational ways of determining a balance include picking a number at random or choosing a number because we find it pleasant.
Rational belief is established in mathematics by providing a formal proof.
There has been an intense debate throughout the history of philosophy on the question whether matters of value are subject to rational assessment. It has long been held that rational assessment requires rigorous rules for deciding whether a proposition should be believed. Formal logic and mathematics provide the clearest examples of such rules. Science has also been considered a model of rationality because it was held to proceed in accordance with the scientific method which provides the rules for gathering evidence and evaluating hypotheses on the basis of this evidence. More recent discussions have proposed accounts of bounded rationality that pay closer attention to human cognitive limitations and recognize considerable scope for rational disagreement.