Rationality

Despite the fact that philosophy is based on rationality, are there any philosophers who embraced the irrational side of man or irrationality in general, and how could they justify this except by contradicting themselves by using rational arguments?

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Good question and good suggestion! Beginning with the last point, there are philosophers who love self-refuting arguments, the most famous being the Cartesian (and Augustinian) proposal that claims such as "I do not exist" have a habit of self-destructing. But some philosophers (and not a few poets) have sometimes introduced a narrow conception of 'rationality' in contrast to the emotions or experiences that seem to defy easy rational analysis, e.g. experiences that are moral, aesthetic or religious. So, when Pascal claimed that "the heart has its reasons whereof reason knows nothing" he was still appealing to reason but in contrast with what you might call abstract, emotionless rationality. This is probably best seen, not as a philosopher recommending we be irrational, but that we not be restricted to a narrow concept of the rational. On this point, a follower of Pascal might be in the same company with romantic poets such as Wordsworth or Coleridge and Blake or even St. Thomas Aquinas. All these thinkers resist narrow notions of reason, and allow for faith or insights that go beyond scientific rationality or passionless reason.

Two philosophers or philosophical theologians who come close to recommending the irrational are Tertullian, a third century Christian who is often quoted as claiming he believes Christianity because it is absurd, and the 19th century Danish, Christian philosopher Soren Kierkegaard who is sometimes understood to hold that, from the stand point of reason, Christianity cannot be true. I suggest both cases turn out to be less exciting, and that Tertullian only meant that Christianity only appears absurd from a pagan point of view and Kierkegaard that Christianity is only impossible from a certain kind of Enlightenment use of reason, whereas a broader point of view would show faith to be more reasonable than doubt. Nietzsche may be your best bet for a thinker who seeks to throw off a concern for "objective truths" discernible by reason. In a very early essay on history, Nietzsche suggests (or he may be understood as suggesting) that truth itself (or at least the truths in history) need to be subordinate to the value of life itself.

One other modest point may be worth making: many philosophers recognize that occasions can arise when it is rational (or reasonable) to appear to be unreasonable or irrational. Think of the figure Hamlet in Shakespeare's famous play: he wisely pretended to be crazy in order to not arose suspicion that he suspected the King of killing his father. There is a good, recent book on such matters: Appearances of the Good; An Essay on the Nature of Practical Reason by Sergio Tenenbaum (Cambridge University Press), especially the last three chapters

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