Answer

I don’t think the cogito is necessary for understanding the book. It is only an introduction for the sake of illustration.
In any case, there is a detailed discussion of the cogito argument in the books of Abraham Tzvi Braun.
The claim “I think” is necessary because even if you think that you are not thinking, that itself is a thought. Either way, it turns out that you are thinking. In other words, when you say “I do not exist,” the subject of the sentence must necessarily exist (the one who does not exist, and of course the one who says it as well).
 
I did not understand the second part of what you wrote. Remember that Descartes begins his move by applying methodical doubt to everything. There is no body, and therefore there is no point in talking about coordination between it and experiences and thoughts.
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Questioner:
I am not thinking “I do not exist.”
With the appearance of the thought “I think,” there appears the thought “what is ‘I’?”, after that appear thoughts relating to all my self-schemas, and after them appears the thought (or doubt), “perhaps those schemas do not exist outside the thought itself?”
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Rabbi:
I don’t understand. Where do these thoughts “appear”? Who is the owner of the thought within which all this is taking place?
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Questioner:
 
Regarding the question — where do the thoughts appear? I would ask — what is place? (“Where” asks about place.)
And regarding the question — who is the owner of the thought? I would ask — what is ownership?
I did not speak about a thought within which everything takes place.
In other words, you are imposing on me concepts of place and ownership, though I have never felt them. The assumption that there is place and ownership is based on matters that first need to be proven. Instead of answering you that “I do not think,” I am breaking the question into parts and trying to understand the concepts.
It seems to me that the physical concept of place is meant for calculations, but nobody experiences place. In spoken language, place is meant to convey a message like “go to the grocery store,” but here too we are relying on there being something (a grocery store) that we do not sense. If we sensed everything, spoken non-graphic language would not have invented the concept of “place.” Everything would be “here,” and consequently there would be no “there,” and no difference between one place and another.
The concept of “ownership” belongs to control and ethics, which certainly need to be proven, and one cannot rely on them to prove my existence after I have placed everything under a question mark.
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Rabbi:
Just as an example. Here is your first sentence:
Regarding the question — where do the thoughts appear? I would ask — what is place? (“Where” asks about place.)
I am now waiting for a definition of “regarding,” “question,” “where,” “appear,” “thoughts,” “I would ask,” “what is,” and “place.” In fact, let’s begin by defining what a “definition” is.
I assume you understand that this is ridiculous. There are concepts we use on the assumption that they are understood, and there is no need at all to define them. When I think, everyone understands that the thoughts are taking place within me. The term place (which by the way I did not use; I asked “where,” and you turned that into a discussion about place) here is obviously broader than its geographical meaning. So what? Why does the geographical meaning seem clearer to you? On the contrary, it is only a particular case of location in the broader sense.
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Questioner:
I challenged the terms place and ownership not because I want to redefine language. The reason is that they seem misleading to me.
 
Let me clarify even further:
I do not understand what meaning there is to saying that thoughts are “within me” or “mine” apart from the empirical coordination between them and the conduct of the body. I understand that there is an intuitive feeling that it is obvious that the thoughts are “mine,” but I analyze this experience of “mine” and understand that it has no reality in itself (a bit similar to Hume’s move in arguing that we do not sense causality itself). This experience of “mine,” that thoughts are “mine” or “within me,” is created out of habit from the experience of coordination between the inner world and the outer world, or the organism as a whole. Therefore one can say “the thought appears” and “there is thought,” and there is no necessity to say “I think.” Only after applying induction can we say that I (the whole organism) am not a sensory illusion, and then say that the thought is “mine” in coordination with the whole organism, whose existence I have already gained confidence in through induction.
 
I invite you to define what the “I” is that we are talking about. Maybe that would clarify a lot. In the book you wrote that it is the “thinking I” or “the spirit.” Needless to say, “the spirit” is a very vague definition. “The thinking I” apparently assumes that there is an act of thinking, and that there is a difference between the I and the thought (such that the act of thought stands ontologically between them); these are entities whose existence I do not see any reason to assume (the I and the act of thought).
 
In a certain sense, I also want to argue that proving the existence of things comes before giving them a name in language. What I mean is that in my opinion the cogito is a word game.
 
“Hey, are you thinking?”
“No, I’m not.”
“So you exist, because you think that you don’t.”
“No, I’m not thinking.”
“So how did you deny that you think if you’re not thinking?”
And so on — “gotcha questions.”
 
But the basic assumption of the one doing the “gotcha” is that a “you” already exists. To such a question one can either say, “There is no such thing as ‘I,’” or ignore it, or play the game. But clearly it is just a word game. If I want to remove the sting by means of a philosophical discussion, I would say, “There is no such thing as ‘I,’” or “Define ‘I.’” In any case, in the “gotcha” there is a subtext that an I already exists, and I do not want to enter that game. And if I do enter it, it is not because suddenly I exist a moment after not having existed before the game. It is only because from the outset I was not really doubting everything with all my heart, and from the outset I already had a vague concept of my “I” (in my opinion arising from induction, which we cannot free ourselves from, even if we deceive ourselves into thinking we are doubting everything).
Is the sentence “This sentence is false” true or false? When you think it is true, did it suddenly become true, and when you think it is false, did it become false? No. It is a word game. When I make a claim, the subtext is that I am asserting a true sentence. Therefore the meaning of the claim is: “It is true that this sentence is false.” Since something true that is false is a logical contradiction, the sentence has no meaning.
 
Therefore I am asking for a definition. I want to hear the subtext.
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Rabbi:
I repeat: you are asking for definitions of basic things that have no definition. We grasp our I directly, and there is neither any need nor any possibility to define it. The same is true of the fact that thought takes place within me or with me. By the same token, you will not be able to define concepts like existence, matter, definition, and the like. All of these are self-evident and do not require definition, nor can they be defined.
I also do not agree with your assumption that the feeling of I has no meaning and is misleading. In my view, it has nothing to do with coordination with the body.
 
The claim that the organism is a sensory illusion or not is irrelevant to us. The cogito does not deal with the existence of the organism but with the existence of the thinking spirit.
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Questioner:
You wrote: “We grasp our I directly, and there is neither any need nor any possibility to define it.” If so, then you do not need the cogito. Not even “I think.” By your account, simply “there is an I” (which is the thinking I).
I personally do not grasp anything that I could call “I” directly. Maybe I’m blind.
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Rabbi:
Indeed, that is the deep meaning of the cogito. The moment you begin to use the tools of thought, you cannot conclude that you do not exist.
If you do not experience an “I” (which I find very hard to believe), then in my opinion there really is some kind of blindness here (unless I am the one who is mistaken). But that is no longer a matter for philosophical discussion. I do not know how to transmit my experiences to others.