For many years the twenty third Psalm was the most popular among American Christians. It conveys the impression that religion can have a very calming and soothing effect on one's life. Some ministers would "sell religion" to the masses by claiming that one who is devoutly religious will not have any contradictions in his life and will always be at peace. Rav Soloveitchik never liked this approach.
Shlomo Hamelech commented (Koheles 7:23) that he thought he could understand everything but soon enough he realized that there is much that is simply beyond him. According to tradition (Yalkut Shimoni, Melachim, #178) he was referring to the mitzvah of Parah Adumah. The details of this "chok" are simply contradictory. The main purpose of the ashes of the parah is to help one purify himself from tumas hameis, but at the same time it causes one who handles those ashes to become tameh. How can it be that the same thing can be metaher the tehorim and metameh the temeim?
The truth of the matter is that the world is full of contradictions. According to quantum physics a particle can indeed in two places at once and can travel in an infinite number of directions at once. A cat may be thought of as dead and alive at the same time. It is absolutely impossible to explain quantum phenomena using traditional principles of physics. If someone says he can think about quantum physics without becoming dizzy, that only shows that he has not understood anything whatsoever about it (see "Quantum Physics, Jewish Law, and Kabbalah", by Aaron M. Schreiber, z"l, pp. 14-16.)
Whenever we recite a beracha, we always start off in second person ("Blessed are You...") and conclude in third person (e.g. - "Who has brought out bread from the ground.") We often have the sense that G-d is very close to us, and on other occasions feel as if he is very far away from us. Both impressions are simultaneously correct. We are expected to both love Hashem and fear Him at the same time. These two miztvos are clearly interconnected. To the extent that we love Him, we come close to Him; and to the extent that we fear Him we withdraw from Him. When we recite Ashrei we continually switch back and forth between second and third person.
In Sefer Tehillim (chapter 8) Dovid Hamelech describes the duality of man's nature. On the one hand (in possukim 4 and 5) when we observe the vastness of the universe man's role seems so insignificant and petty that it leads one to wonder why G-d should think about man at all! And on the other hand in the very next possuk (6) Dovid Hamelech states, "You have created him (man) only slightly lower (in stature) than the angels, and crowned him with honor and glory, and gave him control over the entire creation."
The gemarah (Yoma 19b) derives from a passuk that when one is in the middle of reciting Krias Shema he ought to interrupt in order to show respect for someone else (mipnei kavod barhiyos.) but when one is in the middle of Shemoneh Esrei we consider man as insignificant and ignore him.
According to Professor Schreiber z"l (ibid p. 31) the Avnei Nezer deals with the possibility of tartei d'sasrei in Halacha in over thirty of his responsa. Rabbi Soloveitchik z"l adopted this notion of the Avnei Nezer and this was a recurrent theme in many of his shiurim (see Sefer Mipeninei Horav, Yoreh Deah, the section about Talmud Torah). If in the natural world there seem to be glaring contradictions, we ought not to be surprised if in halacha as well there will be contradictions.
Religion will not resolve any contradictions at all! If anything it will make us aware of more contradictions. Parah Adumah is not the only instance of the contradictory nature of Halacha. The parsha opens with the statement, "This is the chukah of the Torah." All of the Torah is one big chukah; the entire creation is one vast chukah. The midrashim on the opening passuk in Breishis make the comment that Hashem looked into the Torah and used it as a blueprint for creation. All of nature is interconnected with Torah.
Just as the study of physics guides us to maneuver with all the contradictions in the natural realm, so too the halachah guides us in how we should deal with contradictions in the spiritual realm.