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The Limits of Comprehension
This is related to my concepts of "Superclasses", specifically Superclass 5. Someone on StackExchange Philosophy asked
If we begin with a notion of number N that we denote F(N) as a function of time, can a decidable procedure exist on definability of the growth of numbers? Inspired by Tipler's Omega point and Thomson's lamp, what would be the bound when definability cease to have meaning? [...]
Read the rest of the question for more of a context, but the important of the question, in my opinion, was this:
Suppose I have a library of books, and a card catalogue where each card describes the book. How many books can the library have so that each description is different?
This is somewhat ill-defined because we haven't bothered to define how many words or letters are allowed, whether special symbols or multiple languages are permissible, etc. but the answer will always come out to be something like 10M where M is the maximum number of letters allowed on a card.
The questioner's thought process eventually led to the idea that each book could be its own description, and then the question becomes:
Suppose I have a library of books, and people are allowed to read as many as they wish. How many books can the library have, so that if two books are selected at random and a person reads both, their recollection of the two books will be different?
Here the limit is something like 10B, where B is the amount of information one can assimilate into their memory/mind/brain.
I conversed with the questioner, and eventually wrote what follows:
I am going to state a more precise form of this question after corresponding off-line with the OP. I hope this still captures the intent of the question:
In a perfect world, where I will never grow old or go hungry, I am watching a giant computer display screen with enough space to show trillions, or quadrillions, or even centillions of characters or digits.
I have set up the computer to show "1" for a moment, then "10" for a moment, and then "100", and then "1000", and so on. Every moment (perhaps once per second) another 0 appears. Every time a 0 appears, there is a new number being shown.
Can I watch this "forever", and perceive a new number every time a 0 is added? Or is there a limit to my ability to perceive, comprehend, or remember what I am seeing? To what extent does this limit how we as humans can understand numbers and number systems?
I believe that there is a limit to the ability of human beings to perceive, comprehend what they are seeing, and remember what they have seen.
Every time a new "0" appears, it is clearly different from what was there a moment ago. I also know that every number I am seeing is different from each of the numbers I saw before. But as time goes on, I will repeatedly experience the feeling of "what I am seeing is very large, and I have been watching a very very long time". That feeling will be more and more common as time goes on, and eventually I'll be in exactly the same mental state that I was in at some earlier time.
Suppose I try to keep count of how many zeros there are? I can train myself to remember lots of facts, things that can be written out in letters and words.
The mind can remember a lot of information. Perhaps you have enough space in your mind, that if it were all written out it would take a billion = 10^9 letters. That means you can have about 26^(10^9) distinct mental states, because there are that many different combinations of a billion letters with a 26-letter alphabet.
With my mental capacity of 26^(10^9), I am "counting" the 0's as they get displayed, and I keep track of it with my mental state. When there are 876 zeros, I have the number "876" in my mind. There are about 10^3 zeros on the huge computer screen, and 3 digits in my mind. Since I can hold "about a billion letters" in my mind, that means I can "count" the 0's until there are about 26^(10^9) zeros on the screen. Then, because of the limited capacity of my mind, I must lose count. Beyond that, any perception of exactly how big the number is, must be subjective. I will eventually have the same "really big" mental state twice. The largest number I can comprehend, without being confused that it was some other number, is less than 10^(26^(10^9)).
This is like the "Poincare Recurrence Theorem" that the OP linked to, applied to minds. It is one of the natural limits that affect how well we can think about large numbers. I am not speaking of the literal Poincare theorem, which is very precise and mathematical. I merely use it as a metaphor: if a field is of limited finite size, and you walk around in the field indefinitely, you will eventually step on a spot where you have stepped before.
In our off-line discussion, the OP suggested that we can get larger numbers by programming the computer to display 2, then 2^2, then 2^2^2, then 2^2^2^2, or (using words) it could display "zwei", then "zweizenzic", then "zweizenzizenzic", and so on (see http://en.wikipedia.org/wiki/Zenzizenzizenzic ). The screen fills up with 2's, or with the letters "zenzi". Once again, there will come a point where I am no longer able to see anything changing, or perhaps I'll see it changing but my state of mind will eventually wander back to a point where it was at some time earlier. I know it is getting bigger every moment, but even that state of knowledge will eventually recur in exactly the same form.
We can so the same thing with any mathematical notation, like g(1), g(g(1)), g(g(g(1))), ... where g(N) is the "g function" described on wikipedia's "Graham's_number" page. This time instead of squaring each time, the numbers are getting bigger in a much faster way. Perhaps I have trained myself to understand what this means. If so, I could then watch the computer screen display "g(1)", then "g(g(1))", then "g(g(g(1)))", and so on... but again, eventually my mind would reach its "Poincare recurrence".
No amount of effort using more sophisticated or elaborate notation, or methods of abstraction and understanding, will overcome the finite limit of the human mind to perceive, comprehend, and remember.
This is all very similar to what my "Superclass 6" is about, near the end of my Large Numbers discussion: www.mrob.com/pub/math/largenum-4.html#superclass
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EDIT: I added a simple analogy for the "Poincare" reference, and pointed out that the mathematical Poincare theorem is not relevant. This is about the concept of re-visiting the same spot in a finite space.
I added the "To what extent..." bit at the end of the restatement to try to encompass more of the original question.
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