There are three things we might be talking about when we ask about reasons for a mitzvah:

1. (if it’s Torahitic:) the source in the pasuq directly or through derashah;
2. the halachic mechanism by which a given conclusion was reached;
3. the philosophical principle behind the mitzvah.

I’m speaking of the third. But the gemara’s usual question is the first: How do we know something is the law?

The two most cited sefarim aimed at discussing the function of mitzvos are the Chinuch and R’ SR Hirsch’s Horeb. An interesting (to me) difference between each of those books, and R’ JB Soloveitchik. All three agree that we can’t fully grasp the reason for a mitzvah. Torahitic mitzvos require knowing the Infinite “Mind” of G-d, and therefore are beyond our abilities to comprehend. Rabbinic ones, unless the reason is explicitely given as part of the legistlation, are products of cultures that are beyond our current abilities to recreate.

They insist, as does the Rambam before them, that while we can’t fully explain a mitzvah, we are also not freed from trying to explore reasons that we can grasp. What’s interesting is that each has a different description of what we’re trying to find:

1- The Chinuch often introduces the reason for a mitzvah with the words “mishorshei hamitzvah” — among the roots of the mitzvah. It would seem that he believes that we can find part of the telos that motivated G-d to give us a mitzvah. But never the whole.

2- R’ Samson Raphael Hirsch sees ritual as a means of communication, G-d transmitting truths to man by means of symbols. In order to fully integrate these symbols into ourselves, and fully explore their richness, they are presented as acts for us to perform.

Any aspect of the message that we understand fully justifies doing it. And every aspect we don’t yet understand, fully justifies doing it until we reach that comprehension. The mitzvah can never be fully comprehended because there is no limit to human growth — there are always new things to learn from it.

3- In R’ JB Soloveitchik’s writings, he calls his explorations into the purpose of mitzvos “halachic homiletics”. In other words, he sees these lessons as things gleaned from the mitzvah, and have value, and should be internalized — but do not necessarily have any connection to the “Idea” that motivated their legislation.

Notice that all three approaches fully conform to the idea of “na’aseh vinishmah”, where doing causes thinking. None of them would say that these ideas should have impact on behavior or legislation. R’ Soloveitchik entire position is based on the premise that there is no causal connection.

Rav Hirsch contrasts Geiger’s Wissenschaft des Judentums (the “science of Judaism”) with true science. Geiger changes Jewish practice to fit his understanding of what Torah is. Fitting experiment to theory is alchemy, not science. A scientific approach to Judaism is one that takes halakhah, the givens, and constructs theories to explain them.

The Chinuch, though, by saying that these are parts of “The Reason”, might support the creation of stringencies based upon these “shorashim”. However, he couldn’t justify a leniency that might run counter from one of the roots he didn’t uncover.