***To this, using Harv's words, the kid might offer a sort of compromise: "If you can agree that [the number of sand grains] is not fully communicable then I can agree that we shouldn't bother or even worry about fully comprehending [the number of them].... I agree, the important point is focusing on the communicable aspects of [the quantity of sand]."***
***Following this, you might take the approach I would call 'communicability in the current state of affairs' as follows. You might say to the kid, Yes, the number of grains is beyond your comprehension, but we can get some idea of how much sand is there. You could then proceed to relate the quantity to concepts already familiar to the kid, such as the number of buckets of sand, or the weight of it or a number of other things.***
Correct. A similar situation would apply if we picked up an ET signal and began interstellar communication. You naturally look for common reference points (e.g., laws of physics, mathematics, etc) that you think you would share with ET. We shouldn't expect to communicate at the same level of understanding with an ET culture, but having some common reference points allows further communication.
***Or, following 2), you might take the approach I would call 'communicability in principle' as follows: You might say to the kid, Yes, the number of grains is beyond your comprehension at the moment, but, trust me, with a little education you are capable of learning how to comprehend the number. So I can tell you that there is a numeral that identifies the number of grains of sand in that box, and it happens to be 84,372,996,186. So even though you don't comprehend the size of that number, you can learn to say it and write it, and you can trust me that it is correct, and you can expect that some day, when you are better educated, you will come to comprehend its size.***
Here I think your analogy breaks down from what I am saying. The analogy with ET might better illustrate what I'm saying. With an ET civilization you cannot expect to understand their thoughts for a number of reasons (advanced civilization, evolutionary differences, etc). Our goal is to understand them as much as possible while realizing that it is quite impossible to fully understand them. To understand them would require us to be them, which we aren't, and can never be. There is no means by which to converge our thoughts with their thoughts. We can only communicate using common reference points (laws of physics, mathematics, chemistry, biology, etc) and hope that these equations will have some type of meaning to them. If we are wrong then we couldn't communicate with them. If we are right, we would attempt to communicate beyond our sciences and mathematics maybe to philosophy, economics, etc once we have enough common references to make these transitions. No matter how successful we were at these attempts, we cannot expect their mental model of the world (which they communicate to each other) to be communicable to us humans.
***5) Or, following 2) you could attempt to convert a situation that is 'communicable only in principle' to one that is 'communicable in the current state of affairs' thus: You might say to the kid, Yes, the number of grains is beyond your comprehension now, but we have a lot of time so let's get to work. I can teach you about numbers, numerals, and numeracy so that you will be able to not only comprehend the number of grains, but you will be able to determine and verify the actual number.***
As you can see, this is the only means of communication that I see as feasible with ET. We cannot communicate abstract concepts until we build a basis for that communication. Abstract concepts have meaning only in that they reduce to tangible concepts. Without knowing that we communicate at a tangible level there can be no communication at an abstract level.
***Harv, I think you are holding back by adopting that "current state of affairs" connotation. You are desperately hanging on to what you think you know about the universe (all those herbs and spices that make it taste so good) and letting that limit what you can deduce about a state of affairs without those limitations.***
I have no problem of eventually shedding the tangible stuff for the abstract stuff once we have laid the basis of communication using the tangible things. However, every abstract concept has roots in the tangible, otherwise the abstract would be gibberish and non-sensible (non-communicable).
***Dick, I think you are assuming that people can easily ponder pure abstract concepts that are completely devoid of any connection with any universe or any notion of any universe. It is not easy to do so. In my training in math, that ability was what I think separated those who could proceed and those who could not. As I tried in vain to explain to Harv, mathematicians strive to do this, and in the best mathematical work, they have done so. There is absolutely no flavoring from "reality" spicing up their definitions, axioms, or theorems.***
I think anyone who tries to totally divorce themselves from the tangible world is deluding themselves. Abstract concepts only have meaning because there is a whole history of transition between the tangible to the abstract. At each point in the transition the tangible era is there as a supporting network providing the meaning behind the abstract. I'm not denying that you can move so far away from the tangible that you no longer think in terms of tangible objects, but the meaning of the abstract concepts has been acquired only because of this transition. At no time does the abstract become fully divorced from the tangible. Rather, the abstract concepts become meaningful enough so that we don't need to think in terms of the tangible.
For example, the concept of time for most people is based on the digital clock. Most of us don't even pay much attention to the sun, rotation of the earth, etc when considering the time of day. The digitial clock provides a much more abstract concept of time that our ancestors might have had trouble relating without the tangible position of the sun. Nevertheless, if we began to explore the meaning of time as it is displayed on a digital clock, then if we ask enough questions we eventually need to reference the tangible objects once again. We do this because ultimately all meaning of time is derived from the tangible objects.
***Harv, if I had continued the discussion with you on that topic, I would have taken up your challenge to produce theorems from your gibberish definitions and axioms. Of course, I wouldn't be able to do it literally because your gibberish was not "fruitful". It isn't easy to come up with gibberish that leads to interesting theorems. But, that's what mathematicians do. The way I would have responded to that challenge would have been to consult Van Der Waerden's treatment of Galois Theory and I am sure I could find some definitions, axioms, and theorems that would look every bit as gibberishtic as the challenge you posed.***
Paul, when I posted that gibberish I didn't intend any meaning behind the gibber. Therefore, there is no meaning in the gibber. If there happens to be a code whereby you could construct any kind of meaningful math theorems, then it would be entirely coincidental. This is what numerologists try to do. They try to find meaning when there is no meaning to find. When there is meaning intended (e.g., a code), then you can deduce theorems and code-breaking algorithms that can produce meaningful statements in one's own language, but that's only because the meaning is there to find. Language is ultimately based on the tangible things and from there we can abstract from those things.
***The point is, that you have to mentally let go of any and all pre-conceived notions when you consider Dick's "set of numbers". Only then can you see that it is exactly equivalent to any and all conceivable communicable-in-principle concepts.***
It's fine to mentally let go and consider abstractions such as a set of numbers. However, what you cannot do is divorce the original meaning that imputes those numbers with human meaning. It might look that way to someone who knows for a fact that they are thinking abstractly, but this is only fooling oneself. Your knowledge is only knowledge because it can be reduced to tangible objects and the original assumptions you made of those tangible things.
Hence, I am willing to consider abstract concepts if you and Dick are willing to concede that all abstract concepts require some epistemological basis for those concepts. This epistemological basis is the world of sense impressions which includes the untold number of inferences we make of those sense impressions. By making enough inferences we are able to form concepts which are have very little to do with the original sense impressions. These concepts are abstracts of those original sense impressions. They are still there in our mindset, but as we become so familiar with the abstract concept itself, we no longer need to think in terms of the original sense impressions and the basic inferences we made along the way.
Warm regards, Harv