Good to hear from you. As I have explained a few times before, I don't usually comment on your posts for several reasons. One is that I am usually too short of time to do them justice. Another is that I have already told you what I think of your posts in general. But right now I have a little time to respond so I'll tell you essentially the same thing again.
I think you have some special insight into some profound truths. I don't think you can articulate those insights very well in words -- not because of any inadequacy on your part but because of the inadequacy of language itself. And even though you do articulate those insights as well as language allows, I don't think the rest of us comprehend very well what you are trying to get across. This last, I think, is due for the most part to our own prejudices which get in the way of understanding your words.
Now, I think each of us has some special insight into some profound truths but I think you have pushed the envelope further than most of us.
For my part, I have the very vague feeling that I am in possession of some sort of faculty for discerning some profound truths but I don't know exactly how to recognize them or how to express them even if I recognized them. I get little "aha" glimmers now and then when I learn something that I think is profound. I get these same little glimmers when I read what you write. But alas, I am at a loss to get very specific without producing what appears to be obvious gibberish. But I am going to try to comment on your post nonetheless.
Just as I think you have, I have spent a lot of time during my lifetime pondering some of the profound questions about reality and in particular about consciousness. After weighing all the alternative candidate explanations I have heard, I have settled on those that seem to make the most sense to me. As a result, I have formed a sort of explanation of reality that I would bet on if I had to bet.
"May I suggest that there appears to be a lot of self-reference involved in the "converging series" concept; that the limit, the infinity, is consciousness?"
I suppose you could see it that way, but I don't see exactly like that. In thinking about your suggestion, it occurs to me that mathematics requires consciousness but that consciousness is outside of the mathematics.
Do this thought experiment: Suppose there were no such thing as consciousness but suppose there were a vast number (some might even say an infinite number) of material universes. And suppose that these universes took on all possible configurations of material particles. Somewhere among them, there would exist an arrangement of paper and ink that just happened to be a duplicate of "Euclid's Elements". Could we say that mathematics existed in that universe? That is, is it sufficient to have symbols arranged into axioms, theorems, proofs, etc. in order to have mathematics, or is consciousness required for mathematics to exist?
In my view, consciousness is essential for mathematics. Mathematics is a collection of specifically organized sets of thoughts in some consciousness. It is not some collection of symbols in a set of books. In particular, the "converging series" concept is nothing but a specific set of thoughts in some consciousness. Similarly, the concepts of limits and infinity are nothing but specific sets of thoughts in some consciousness.
Now, you mentioned self reference. I think we need to be careful here. What exactly is a reference? And then, what exactly is a self reference? Well, I'd say that a reference is a symbol which is understood to stand for something else. And I'd say that self reference is a symbol which is understood to stand for something else but where the symbol itself happens to be a part of the "something else".
So, for example, the statement, "This statement is true." is self-referential because the symbol 'This statement' is understood to stand for the statement, "This statement is true." (i.e. the "something else") and the symbol 'This statement' happens to be a part of the statement "This statement is true.".
But....Notice I didn't say that the symbol stands for something. I said the symbol is understood to stand for something. There is, in my mind (so to speak), a big difference. In our hypothetical non-conscious universe, there might be an appearance of the sentence "This statement is true." but I maintain it would not be self-referential. I maintain that it would have no meaning at all. I think it can have meaning, or be a reference, if and only if some consciousness understands the meaning or the reference. In other words, I think that consciousness is a necessary requirement for meaning or understanding. A corollary would be that consciousness is a necessary requirement for mathematics.
So I would suggest that no concept, whether of converging series, limits, or infinity, whether they are self-referential or not, is consciousness. I think consciousness is something that needs to exist before there can be any concepts at all.
But, there might be another way to look at self-reference. The word 'self' is the giveaway. Suppose some consciousness came to an understanding, say, that 1+1=2. Then we could say that that consciousness knows that 1+1=2. Now, could we say that that consciousness knows that it knows that 1+1=2? I don't think so until a process of reflection takes place. I think that reflection is a deliberate conscious act of examining the state of knowledge and concluding that one thing that is known is the fact that 1+1=2. Similarly, before the consciousness knows that it knows that it knows that 1+1=2 it must go through a similar process of reflection.
After thinking about this series of processes I am convinced that not only can it never become infinite, but that it doesn't, or can't, really progress very far at all. I think it might be limited to a very small number, maybe like 15, or 10^trillion^trillion (which compared to infinity is a tiny, miniscule number). The reason is that the process takes time (whatever that is) and it becomes more complex with each new level.
What I am trying to suggest with this alternative way of looking at self-reference is that it might be that the number of levels of knowing that you know that you know that you know... might be some kind of measure of the depth or power of a particular consciousness, or aspect of consciousness. I don't know. The brief little glimmer that was illuminating this idea for me has just winked out so I will have to leave it at this point. Sorry.
"This Zeno's Arrow "paradox" is resolved by seeing that "moment" was defined in terms of the halving distances; the moments were also being halved."
Yes, that is the mathematical trick of canceling out appearances of infinity. But I think Zeno's question is similar to my question of filling an infinite tube with a finite amount of paint. (See http://www.paulandellen.com/essays/essay121.htm ). The mathematical result is one thing, but it is quite another thing to ask how that result relates to reality -- in particular to physical reality. If we are talking about a real physical arrow, then there is a huge amount of ambiguity about the question. For example, what do we mean by the position of the arrow? Do we mean the atom at the very front of the leading edge of the tip? If so, we might have difficulty picking that atom out of the billion or so contenders all of which are vibrating and moving with respect to one another. But even if we could pick it out, where exactly would we say that atom is? Or where the leading edge of it is? Not only is the leading position continually being represented by a different electron as they spin around, but the location of any one of those electrons can't be pinned down exactly. So to talk about the arrow "moving half of the remaining distance" gets hopelessly confounded by the inability to precisely define where it is at any time. And even if we could somehow be precise about the location of the arrow, the definition of time at that scale would be equally ambiguous. I think there is quite a gulf between mathematics and reality.
"What is a "sum"? We say "1 = 1 = 2". But in reality, this "real sum" is a partial sum? Because there must be SOME aspect of each of the "ones" that does not equal (or you wouldn't have two ones to add, you would only have one!)
To see "two" items requires a SAME background surely; or you wouldn't be able to count the items as you would get them muddled with two different backgrounds?"
Here I think you are probing the same conundrum I wonder about. That is, if reality started with nothing but consciousness and no thoughts, what are the possibilities for the very first thoughts and the progression from there to more complex structures like concepts? For example, there couldn't be language yet, or perceptions, or qualia. It would have to be something like you suggest which is simply some sort of recognition of a difference of some kind. Like the difference between a bit and nothing. Then, as you point out, that recognition itself immediately becomes something real, so now we have two things: the bit and the recognition. I know I am not representing your ideas very well, but I am not representing my own very well either. It's just that I get one of those glimmers that suggests that you and I are thinking along the same lines. I also think that Chris Langan's ideas of how something arose from the nothing of his Unbounded Telesis (or whatever) are an attempt to explain the same thing. The difference between Chris and me is that I think he thinks consciousness itself was produced somehow this way, where I think consciousness is primary and preceded all thoughts and everything else. This has probably long since reached the state of "obvious gibberish" that I mentioned earlier so I think I'd better quit before it gets worse.
It's great fun for me to think about these things and it's fun to talk (write) about them too if I think anyone else might be interested. Thanks for your interest, Alan. It's fun talking with you.