Ask any assembly language programmer. Is every file a single positive binary Integer? Yes, it most certainly is. Therefore all possible computer files will, proven beyond any shadow of a doubt, be created by a COUNTER.
Do you have a secret idea? If you will answer honestly YES or NO, I can find it out by asking 20 questions. This is a lot quicker than counting to your file number.
In few words I have evidence and final proof that nothing can be invented on a computer because everything already exists as a countable or calculable number.
And it is Finite. Every possible song can be played as an MP3file with a number Less Than 2^2^32. So can almost every fileyou will ever make. 2^2^32-1 is 4 megabytes full of "ones".
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Didn't Godel or somebody prove a theorem along those lines a long time ago? My cat typed this. Or maybe I'm a bot. You'll have to guess!
Woody, I believe you're thinking of mathematician Alan Turing's "Turing Machine", a simple, theortical computer that has been proven to run any program that any other computer can run. This is basically the result being referred to here. But it doesn't lead to "No More Secrets", to borrow from the movie Sneakers. On the other hand, Turing did do outstanding work during WWII breaking the German Enigma cryptographic machine.
It ALMOST leads to "No More Secrets". Without any musical skill, I algorithmically synthesized a recording of a song from one of my own dreams, using the 20-questions-like method. However being interested in privacy I keep secrets in very unusual ways. In this case, "imaginary numbers" are relevant.
The same thing can be said of the English language--don't all works of literature have a unique integer representation? Ditto with works of visual art, say painting--it's just color and position, both of which can be represented numerically. Sure the realm of expression is finitely-generated. But it's BIG, and people have historically recognized that it's big enough for original expressions of creativity.
If I pick a number between 1 and 2^2^32, we'd have to play 2^32 questions, not 20, for you to guess it using binary search.
Astute observations, but I don't think your conclusion ("nothing can be invented on a computer") follows from your premises (digitized expressions can be represented numerically).
It can be shown that many numbers produce nearly identical information content, and also that those numbers can be obtained with far fewer than 2^32 questions. For example, 1000 people scanning the same picture will certainly get 1000 different image files, and also, it takes a forensic artist not too many more than 20 questions to draw a picture of the perp. This is by no means an exhausted "theory", because there are many small algorithms capable of generating large numbers with specific properties. The COUNTER is not a theory but a fact. But I have a strong theory that I may be able to create a bootable floppy (algorithm of less than 1.44 megabytes) which is self contained and that can create recordings of Elvis singing your lyrics and playing your midi file. No bets, though, I don't have the interest or the time. I just happen to be experienced in voice synthesis enough to know how it can be done.
You are assuming binary encoding and vaguely defining "nearly identical information." I could use a system based on real numbers and have an infinite number of options, i.e. an infinite number of volumes between 0 and 10 (11 if you're Spinal Tap). As to whether anything can be "invented" on a computer, this is more of a metaphysical question.
I specified a finite range of binary integers which contain most interesting files, and suggested my methods of generating them. The binary numbers are all less than 2^2^32 and fit in 4 megabytes. All the files I find interesting, including "all music" files, can each fit in4 Megabytes, and again, counting to 2^2^32 will generate them all, and also, they can be generated much fasterusing other methods than counting, when I know what result I want. Please disregard the existence of files larger than 4 meg, I do not consider them relevant, nor is it practical to consider the infinite number of real numbers between 0 and 1 as relevant either.
As a child I experimented with testing algorithmically generated binary numbers, by loading them into a "shift register" and serially feeding them to a speaker and a video buffer at the same time. Uninteresting numbers produced "TV static". Algorithms that produced this effect were labeled random number generators. However, this experiment was designed to discover random number generating algorithms, and more often than not, FAILED to do so. In other words, my results were almost always audibly and/or visually Interesting. And this led me further into unconventional "synthesizer" experiments.
As the saying goes about an infinite number of monkeys with an infinite number of typewriters, random numbers or letters could eventually form the entire creative work of mankind. However, what tiny fraction is useful. Perhaps 10^-5 ?
In your example of MP3s, you said that they all fall in the range of 2^2^32 numbers. With an optomistic view of 1.5 million songs, with perhaps 1 thousand million (10^9) recognizable alternate versions apiece (a big allowance), you are still left with an infinitesimally small number. This is a curiosity but ultimately totally useless.
When I took a course in Critical Reasoning, I had a terrible time with the course the first couple of weeks. The point of the course was to figure out if something someone tells you has the possibility of being true or not. Everything the instructor said I could find examples to contradict him. So I stayed after one night and asked to speak with him. I proposed that anything he said that he thought was true, I could prove it was false. And anything he thought was false, I could prove it to be true. He took my challenge and simply stated that a square has four sides. I said no, that is not true. In the majority of cases a square will not have four sides. In multi-dimensional space a square will only have four sides in one out an infinite number of dimensions. So statistically that is the equivalent of a proof that a square does not have four sides. He lent back slightly and pondered. Then said to my surprise and respect "But what good is that? The best you can do is take the information you have and use it to your best benefit." I realized what he was saying, there are no absolute truths. We simply have to use our meager minds and limited sensory perceptions to predict events as best we can to attain a desired goal. Of what use is that?
Modern computers have clock frequencies of about 5 GHz. Assuming that a comparison could be made for desireable information on each clock cycle. How long would it take you to cycle through every possible combination of 128 bits to look for desireable information?
Answer: 1^111 years. And how many bits represent all that is known? Of what use is that?
Okay, maybe I'm being a bit pesimistic. If I were to approach this problem and be as realistic as I can to produce something useful. Hmmm.
Take a snapshot of all that we currently know and digitize it. Then periodically take a snapshot of all that we know and do a comparison. Mathematically would the parallel comparison show anything? Especially when the binary pattern is so erratic 0111 1000 only one digit appart, but geometrically so diverse. Sorry, nothing useful comes to mind.