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Hold On; Harv May Have A Point.

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Posted by Paul R. Martin on February 26, 2002 01:20:41 UTC

Dick,

I think we need to more precisely define 'communicability'. As I see it, you and Harv each have a slightly different connotation in mind.

I think you mean communicable in principle, and I think Harv means communicable in the current state of affairs. Let me illustrate with an example.

Suppose you were in a room with a sandbox full of sand and a bright 10 year old kid. Let's say you want to discuss the sandbox and its contents with the kid. In particular, you want to discuss the concept of the cardinal number of grains of sand in the box.

Here are some scenarios which I hope illustrate the different connotations I mentioned:

1) Using Dick's words, it would be reasonable for the kid to take the following position: "As far as I am concerned, I have no interest whatsoever in any concept which cannot be communicated for I clearly have no chance of ever understanding such an idea[, i.e. the number of grains in the box]." He would go on to say that even if you could somehow count all those grains, I simply don't know enough numbers (he means numerals) to even say how many there are. And if you told me how many there are, I just would have no concept of that number. There are just too many for me to comprehend.

2) 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]."

3) 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.

4) 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.

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.

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.

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.

Harv is right, of course, that historically no concept was ever conceived without those spices, but what he doesn't realize is that it is possible to discuss and consider ideas that are strictly formal, meaningless, and having no connection with anything real.

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.

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.

WR

Paul

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