Two meanings of cause

I attended a Phil 101 class of a friend to see what his teacher was all about. I brought my Lectures on the Foundations of Mathematics (which had arrived in the mail that day) in case I got bored. He stared down my book and asked if it were Wittgenstein, to which I affirmed. He then said, What are you doing in an intro class if you are reading Wittgenstein?

Anyhoo, he decided to make some remarks on Wittgenstein and try a little bit of Wittgenstein philosophy- maybe to impress me since I did come to see his teaching abilities.

And he hit upon something found in the Lectures on Philosophy: namely the different responses that satisfy the question of CAUSE.

Wittgenstein's Lectures on Philosophy

His explanation failed the one I had offered my friend but a few days before, so I hope it is suitable to you.

Namely, when someone asks why it is raining (and thereby seeking a CAUSE) there are two different ways of answering. One way answers by giving some scientific hypothesis on precipitation. The other could simply respond with a description of the sky, or stating that it is springtime (and thus imply that springtime is the time for showers), etc.

From the end of item 4 in that lecture:
We are talking here of the grammar of the words "reason" and "cause": in what cases do we say we have given a reason for doing a certain thing, and in what cases, a cause? If one answers the question "Why did you move your arm?" by giving a behaviouristic explanation, one has specified a cause. Causes may be discovered by experiments, but experiments do not produce reasons. The word "reason" is not used in connection with experimentation. It is senseless to say a reason is found by experiment. The alternative, "mathematical argument or experiential evidence?" corresponds to "reason or cause?"

mathematical argument is what I wish to highlight. There seems to be much to say about this, and I wonder how it relates to Wittgenstein's proposition in the TLP that all the propositions of mathematics are tautological.

I remember that it infuriated my father when, as a young child, I answered questions as such- saying things like I am watching television because it is on, or (to use W again) that I have come to the classroom because there is a lecture.

One asnwer is open for dispute, the other, not so much. Thoughts?