When dealing with the issue of faith, it is important to understand the meaning of faith as it applies to the biblical usage of the word. Today, the word faith generally caries with it a more or less negative connotation. A person is said to have faith when they believe in something in spite of evidence to the contrary. In this modern usage, having faith means that a person is continuing to blindly believe something is true even though there is no proof. (Merrium-Webster Dictionary) This however is a very narrow definition of faith in the English langue and is not the biblical definition of faith at all.
The Greek word translated as faith in English versions of the New Testament is Pistis (πίστις), which is better translated as trust and stems from the Greek word Peithô (πείθω) which means to be persuaded or convinced. Therefore, in the biblical sense, a person has faith, not out of a misguided belief in spite of the evidence, but rather is convinced and persuaded to have confidence in their belief as a result of evidence. This idea of confident trust in God and His faithfulness is what the biblical authors were trying to convey to their original audience. It is only recently that faith has come to mean something less.
Understanding this subtle difference between having blind faith and a confident trust allows us to view faith-heavy passages like Hebrews 11 in an entirely different light. By simply replacing the word faith with trust the true meaning and intent of these passages becomes apparent. The great men of faith recorded in Hebrews 11 were not considered righteous because of a misguided superstition in an unseen reality for which there was no proof. They were counted as righteous because they believed God and trusted in Him and His faithfulness to keep His word.
In concision biblical faith and intellectual reason are not opposed to one another. On the contrary: they are intertwined! It is not irrational to have faith in God when He has given us proof of his faithfulness. Nor is it anti-intellectual to believe that something is true in light of affirming evidence.