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Rashi 19:22 explains in the name of Rabbi Moshe Hadarshan, ויקחו אליך-משלהם כשם שהם פרקו נזמי הזהב לעגל משלהם כך יביאו זו לכפרה משלהם- And have them take for you: From their own possessions; just as they removed their own golden earrings for the golden calf, so shall they bring this cow from their own possessions in atonement. It is clear that Parah Adumah (the red heifer) is an atonement for the sin of eigel (the golden calf).
Another Rashi in 19:22 explains this concept further: פרה אדמה: משל לבן שפחה שטינף פלטין של מלך. אמרו תבא אמו ותקנח הצואה, כך תבא פרה ותכפר על העגל - A red cow: This can be compared to the son of a maidservant who soiled the king’s palace. They said, “Let his mother come and clean up the mess.” Similarly, let the cow come and atone for the calf.
The first answer seems to be clear and valid; why did Rashi feel the need to add the example as further elucidation?
The problem with the first explanation is that Parah Adumah was one of the few mitzvos that were given to us before the actual giving of the Torah. We were commanded in Moroh (as mentioned in tractate Sanhedrin), and the sin of the eigel occurred after the giving of the Torah.
One of the most famous philosophical questions in the Jewish religion is understanding bechira and yedia. In a nutshell: if God knows every move a human being is going to make, then how can we explain that every person has a choice between good and evil? If God knows in advance what’s going to happen, it would seem that the choice has been taken away from the person.
When addressing this query in his book, Moreh Nevuchim (guide for the perplexed), the Rambam says there definitely is an answer to this question. However, no paper in the world will suffice to adequately answer it. Therefore, we’ve got to have a blind faith that we still have a choice in choosing between right and wrong, despite God’s knowledge in advance of what we plan on doing.
The other Rishonim (e.g the Ramban) agree with the assessment of the Rambam and disagree on minor points concerning the explanation. (There is not enough space here to go into the explanation!) However, the common theme is that this is the most complex philosophical question concerning the Jewish religion.
Therefore, we can perhaps suggest that the question was how God could command us to do the mitzvah of Parah Adumah before we actually sinned with the golden calf. There should have been nothing to atone for yet, if we do indeed have our own choice of choosing not to sin. The answer is that this is a chok (a commandment which we don’t understand), since this truly is the most difficult concept in Judaism. Shlomo Hamelech (King Solomon) said, in regard to this mitzvah, ”Omarti echkama vehi rechoika mimeni“ (I wanted to understand but it was distant from me: I couldn’t understand).