Answer

Looks excellent to me (no surprise 🙂 ). Here are my comments in bold, with a few corrections of inaccuracies.
Best of luck,
Michi
 
What is the emptiness of the analytic, really? Let’s take a simple logical proof as an example:
 
All human beings are mortal
Aristotle is a human being
Conclusion: Aristotle is mortal
A logical proof brings to light every [MA1] conclusion that necessarily follows from the premises. That is, the truth of the conclusion is based on the truth of the premises: if the premises are true, the conclusion is necessarily [MA2] true. But how do we know whether the premises are true?
 
Let’s take gravity as an example. How do we know it exists?
True, scientists carried out experiments. But by virtue of what do we believe those experiments?
David Hume, an 18th-century British philosopher, asked this question and arrived at several conclusions.
We believe those experiments on the basis of several assumptions. One of them is the principle of causality, which Hume showed cannot be proven [MA3].
To make this clearer: the fact that every time I kick a ball, the ball flies away, does not mean that the reason the ball flies away is my kick. It could be that both happen one right after the other by chance, or because of a third cause that triggers them both. We assume that everything has a cause, and we reach the conclusion—with a few more assumptions added—that the kick is the cause of the ball’s flying away.
The second is the assumption of induction, which also cannot be proven. We assume that there are concise laws and rules [MA4] in our world, create such rules out of nothing [MA5], and believe in them so long as they have not been refuted [MA6].
That means that the theory of gravity—and really all of our science—is based on assumptions, and it is not clear how we arrived at them.
Of course I believe in gravity. But it is important to understand that the reliability of our scientific knowledge rests on assumptions that it is not clear how we arrived at, and the same degree of trust that we give to the scientific results we reach, we are really giving to those assumptions, which we arrived at in a way that is neither scientific nor mathematical.
(If someone finds some method or way to arrive at those assumptions, and claims that they are necessary, that method or way too rests on assumptions for which there is no formal basis; there is always a starting point to every thought process.)
And that means that we also give that same degree of trust to our intuition—to that sense that tells us what is right and what is not, a sense that does not rest on prior assumptions but rather creates them and assesses their correctness.
Science holds an honored place in our lives, and its importance and power in our society are enormous. So much so that we take the assumptions that we, humanity, created in order to advance it, and apply them to fields that are not necessarily related, such as the social sciences and the humanities. In philosophy, law, and psychology, the scientific approach, which is built mostly on quantification and precise definition, is often not suitable. And to the same extent that we learned to trust—or discovered in retrospect that we trust—our intuition, which gave us the basis for the science we developed, in my opinion it is worth giving it a chance to create new assumptions and rules in the various fields, in a way suited to each field.
 
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 [MA1] Why “every”? A proof yields a particular conclusion.
 
 
 [MA2] The conclusion is necessarily true, but not that it is true necessarily. This is a very important distinction in philosophy.
For example, in the argument (a substantive one, not a formal one): if Moshe is a bachelor, then he is unmarried. The fact that he is a bachelor is not a necessary truth (because he could have been married), and therefore the fact that he is unmarried is also true, but not necessarily so. It is necessarily true, but not true necessarily. In other words, what is necessary is the implication from his being a bachelor to his being unmarried.
 
 
 [MA3] Corroborated. Nothing scientific can be proven. In other words, this is not a scientific claim (it cannot be refuted or corroborated).
 
 
 [MA4] ???
 
 
 [MA5] Not out of nothing, but rather as a generalization from particular facts.
 
 
 [MA6] A Popperian formulation. I prefer a less analytical formulation: because they have been corroborated.