Okay, so basically everything depends on something. Something that does not change is called "not dependent on anything," and this still needs a bit of thought.

If you send your beloved a love letter, that is love that depends on the mailman.

Gal,
My emphasis is on "depends." If the thing does not change, then the love does not depend on it in any meaningful sense (the thing does not cease, and therefore the love does not cease).

Not Related,
About that it was said: "Mailman by mailman, in due form."
[Something that happened to a friend of mine who is a judge. A well-known lawyer appeared before him and used the Hebrew idiom "each word in its proper way," but pronounced it incorrectly. My friend, who is a language enthusiast, immediately corrected him and said it should be pronounced differently. The lawyer did not get flustered, and answered immediately: I meant to say "mailman by mailman on his bicycle"—and there the pronunciation really is as I said.]

Rabbi, I did not understand why, when the thing does not change, the love does not depend on it. On the contrary, exactly the opposite: the love does not change because it depends on it, and that is what does not change. No?

*"…the thing does not change, so the love does not depend on it"…

Of course there is dependence, but in such a case the dependence is meaningless. The meaning of dependence is that if the thing changes, what depends on it changes too. It is like saying to someone: I am willing to do X for you, provided that your name is Moshe. Does the action depend on his being called Moshe? Yes. Is that dependence meaningful? No, because obviously his name is Moshe and that is how it will always be.

Yes, I agree. And even so, it seems to me that the wording "does not depend on anything" still needs a bit of thought. Strange that the commentators there did not stop and explain something.

I interpret the concept "love that does not depend on anything" as "love for its own sake." To love the other not because of the functions he fills in my world, but to love him because there is value in love itself.

Obviously. But the question is what you love in him (on what basis is there value in relating lovingly to him). His being Jewish, human, a convert? All of these are reasons for love, and that is why he asked why this is not love that depends on something.

Gal,
Maimonides' version:

"Any love that depends on something that can cease—when the thing ceases, the love ceases. But if it does not depend on something that can cease—it never ceases."

The Mishnah brings examples: "the love of Amnon and Tamar," "the love of David and Jonathan."
The difference between them is that the love of Amnon and Tamar is a love that comes from need or desire; after Amnon committed the outrage, "he hated her."

In the love of David and Jonathan, this is not a love that comes from desire. And about that the Mishnah says that it does not depend on anything.

Dependent on something = arising from desire.

But on the face of it, it seems that two tannaim wrote the Mishnah. The one who wrote the first clause meant one thing, which is hard to understand and requires interpretation, and then the examples are brought in the latter clause.

Interesting comment. The dependence is on something in the lover, not in the beloved. But in practice there is not much difference. Clearly, in the end, what causes dependent love is some factor in the lover (which may perhaps be awakened by something in the beloved).

My suggestion says that there is a difference between "love for its own sake," meaning the desire to give to another without personal interest, and love that is interest-dependent. Love devoid of interests will never cease because it does not depend on anything external to the loving person.

Copenhagen Interpretation,

Thank you for your words! Indeed, Maimonides' version goes down more smoothly for me. Thanks.

If in your opinion love of Israel and love of the Creator are love that does not depend on anything,
why then has providence ceased?

I did not understand the question.

What Shalom Cohen must mean is:
A. One who loves necessarily watches over the beloved. (And if he has excuses for why he stops, then he does not love.)
B. Love that does not depend on anything never ceases.
C. God loves His people Israel with a love that does not depend on anything.
D. Therefore God's providence over His people Israel necessarily never ceases.

And I ask: is this what is called formalization?

Formalization is giving something mathematical form. What you did is only preparation for formalization.
As for the argument you put in his mouth, I do not accept assumptions A and C.

And were I not afraid, I would have wanted to say explicitly regarding the subject of the thread that love that does not depend on anything never ceases. What is a "thing"? What is the property called a "thing" that is not part of the object itself? The stimulus? The conversion? The Jewishness? But the object certainly has some specific property that arouses love. So what kinds of properties are merely a "thing," and what are the properties of the thing itself? About this Maimonides wrote that we regard properties as belonging to the thing itself when they do not cease. According to this, the understanding of Maimonides is completely different, since the Mishnah is not coming to tell me something that is not novel at all—that if the cause of the love never ceases, then the love never ceases either. Rather, perhaps one could say that the Mishnah is making a point about the power of the lover, that his love endures without needing novelty and change. And what Maimonides wrote is the reason. According to what I wrote in my comment, Maimonides is giving a sign, a marker: when does a property become the thing itself, and when not? When it does not cease, it is the thing itself. And the reason the love does not cease is because he loves the thing itself. Therefore even when the thing itself no longer exists in a certain sense, such as the death of a beloved family member, the love continues because he is still a beloved family member, and such a property remains even when life has been taken from the object. So until today he experienced the beloved concept in a certain way, whereas now in a different way. But the concept continues on forever, and that property does not cease eternally. Therefore the Mishnah wrote that if it does not depend on something but rather is the thing itself, then the love does not cease. And Maimonides brought an identifying sign for the properties of the thing itself. True, the reason the love does not cease is because the thing itself does not cease. But from the Mishnah's perspective there is a tremendous insight: if this is a property of the thing itself, then there is no cessation of love. And Maimonides explained what the sign is, and in fact that is also the reason. But the Mishnah formulated it this way, and did not write it as Maimonides did, apparently in order to teach the exclusivity of eternity only regarding "that which does not depend on anything"—there is only this and nothing else that does not cease.
Of course, my "thing itself" has nothing to do with the Kantian concept. I simply did not know what to call it, until I remembered the Aramaic word gufa, "the thing itself."
So much for my attempt to expand on the Rabbi's third comment above: when do we say that a certain dependence is meaningful, and when not. The Rabbi indeed meant meaningful with respect to the cessation of the love, but I emphasized more the side of significance for love that does not cease.
Does the Rabbi think it would be better for me to comment like this, or should I prepare a formalization?

This somewhat resembles the version of Maimonides brought above. But it seems forced to me. The fact that I have two legs does not cease. Is someone who loves that about me engaged in love that does not depend on something that can cease? Clearly, beyond non-cessation there must be some substantive characteristic of the beloved.
I also did not understand the "tremendous" novelty you found in the Mishnah. Your question still remains: if the thing that causes the love does not cease, then obviously the love will not cease. But your original question is not really a question. The claim is that love that depends on something may or is liable to cease. And there is a novelty in this even though it is a simple point (see the introduction to Mesillat Yesharim, and similar to what Malcolm once wrote: this is an illuminating tautology).
There is no need at all for formalization when it adds nothing. What I wrote above was not criticism, but a substantive answer to your question whether there is formalization here.