I disagree with your entire premise. I think we should pin down this concept of "levels of perspective" with some good jargon at some point, but regardless...
You can look at a computer from the level of perspective of "there are windows on the screen and I can move the mouse around. I can manipulate files on the hard drive with the mouse and the keyboard, and those changes will be reflected inside information boxes in the windows." This is the perspective most people see a computer from, but it is not a complete description of a computer (i.e. if someone unfamiliar with the concept of computers heard this description, they could not build a computer from base materials.)
You might also see the perspective, "There are many tiny dots of light on a flat surface, lit up in various patterns. Those patterns are caused by electricity moving in certain ways through silica wires arranged in certain ways." This is, I think, one level lower, but an unfamiliar person could not build a computer from scratch from this description.
Another level down, the description might be: "There is a CPU, which is composed of hundreds of thousands of transistors, arranged into logic gates such that when electricity is sent through them you can perform meaningful calculations. These calculations are written in files using a specific instruction set ("assembly language"). The files are stored on a disk in binary, with the disk containing many cesium atoms arranged in a certain order, which have either an extra electron or do not, representing 1 and 0 respectively. When the CPU needs to temporarily store a value useful in its calculations, it does so in the RAM, which is like the disk except much faster and smaller. Some of the calculations are used to turn certain square-shaped lights on a large flat surface blink in certain ways, which provides arbitrary information to the user". We are getting to the point where an unfamiliar human might be able to recreate a computer from scratch, and therefore can be said to actually "understand" the system.
But still yet there are lower levels. Describing the actual logic gate organization in the CPU, the system used by RAM to store variables, how the magnetic needle accesses a specific bit on the hard drive by spinning it... All of these things must be known and understood in order to rebuild a computer from scratch.
Humans designed the computer at the level of "logic gates", "bits on a hard drive", "registries", etc, and so it is not necessary to go deeper than this to understand the entire system (just as you don't have to go deeper than "gears and cogs" to understand how a clock works, or how you don't have to go deeper than "classical physics (billiards balls bouncing into each other)" to understand how a brain works.
But I hope that it's clear that the mechanisms at the lower levels of a system completely contain within them the behavior of the higher levels of the system. There are no new behaviors which you can only learn about by studying the system from a higher level of perspective; those complicated upper-level behaviors are entirely formed by the simple lower-level mechanisms, all the way down to the wave function describing the entire universe.
That is what reductionism means. If you know the state of the entire wavefunction describing the universe, you know everything there is to know about the universe. You could use it to predict that, in some everette branches, the assassination of Franz Ferdinand on the third planet from the star Sol in the milky way galaxy would cause a large war on that planet. You could use it to predict the exact moment at which any particular "slice" of the wavefunction (representing a particular possible universe) will enter its maximum entropy state. You could use it to predict any possible behavior of anything and you will never be surprised. That is what it means to say that all of reality reduces down to the base-level physics. That is what it means to posit reductionism; that from an information theoretical standpoint, you can make entirely accurate predictions about a system with only knowledge about its most basic level of perspective.
If you can demonstrate to me that there is some organizational structure of matter which causes that matter to behave differently from what would be predicted by just looking at the matter in question without considering its organization (which would require, by the way, all of reality to keep track not only of mass and of velocity but also of its organizational structure relative to nearby reality), then I will accept such a demonstration as being a complete and utter refutation of reductionism. But there is no such behavior.
You are right; my example was a bad one, and it does not support the point that I thought it supported. The mere fact that something takes unreasonably long to calculate does not mean that it is not an informative endeavour. (I may have been working from a bad definition of reductionism).
If you can demonstrate to me that there is some organizational structure of matter which causes that matter to behave differently from what would be predicted by just looking at the matter in question without considering its organization
Um. I suspect that this may have...
(At this point, I fear that I must recurse into a subsequence; but if all goes as planned, it really will be short.)
I once lent Xiaoguang "Mike" Li my copy of "Probability Theory: The Logic of Science". Mike Li read some of it, and then came back and said:
Then Mike said, "No, wait, let me explain that—" and I said, "No, I know exactly what you mean." It's a convention in fantasy literature that the older a vampire gets, the more powerful they become.
I'd enjoyed math proofs before I encountered Jaynes. But E.T. Jaynes was the first time I picked up a sense of formidability from mathematical arguments. Maybe because Jaynes was lining up "paradoxes" that had been used to object to Bayesianism, and then blasting them to pieces with overwhelming firepower—power being used to overcome others. Or maybe the sense of formidability came from Jaynes not treating his math as a game of aesthetics; Jaynes cared about probability theory, it was bound up with other considerations that mattered, to him and to me too.
For whatever reason, the sense I get of Jaynes is one of terrifying swift perfection—something that would arrive at the correct answer by the shortest possible route, tearing all surrounding mistakes to shreds in the same motion. Of course, when you write a book, you get a chance to show only your best side. But still.
It spoke well of Mike Li that he was able to sense the aura of formidability surrounding Jaynes. It's a general rule, I've observed, that you can't discriminate between levels too far above your own. E.g., someone once earnestly told me that I was really bright, and "ought to go to college". Maybe anything more than around one standard deviation above you starts to blur together, though that's just a cool-sounding wild guess.
So, having heard Mike Li compare Jaynes to a thousand-year-old vampire, one question immediately popped into my mind:
"Do you get the same sense off me?" I asked.
Mike shook his head. "Sorry," he said, sounding somewhat awkward, "it's just that Jaynes is..."
"No, I know," I said. I hadn't thought I'd reached Jaynes's level. I'd only been curious about how I came across to other people.
I aspire to Jaynes's level. I aspire to become as much the master of Artificial Intelligence / reflectivity, as Jaynes was master of Bayesian probability theory. I can even plead that the art I'm trying to master is more difficult than Jaynes's, making a mockery of deference. Even so, and embarrassingly, there is no art of which I am as much the master now, as Jaynes was of probability theory.
This is not, necessarily, to place myself beneath Jaynes as a person—to say that Jaynes had a magical aura of destiny, and I don't.
Rather I recognize in Jaynes a level of expertise, of sheer formidability, which I have not yet achieved. I can argue forcefully in my chosen subject, but that is not the same as writing out the equations and saying: DONE.
For so long as I have not yet achieved that level, I must acknowledge the possibility that I can never achieve it, that my native talent is not sufficient. When Marcello Herreshoff had known me for long enough, I asked him if he knew of anyone who struck him as substantially more natively intelligent than myself. Marcello thought for a moment and said "John Conway—I met him at a summer math camp." Darn, I thought, he thought of someone, and worse, it's some ultra-famous old guy I can't grab. I inquired how Marcello had arrived at the judgment. Marcello said, "He just struck me as having a tremendous amount of mental horsepower," and started to explain a math problem he'd had a chance to work on with Conway.
Not what I wanted to hear.
Perhaps, relative to Marcello's experience of Conway and his experience of me, I haven't had a chance to show off on any subject that I've mastered as thoroughly as Conway had mastered his many fields of mathematics.
Or it might be that Conway's brain is specialized off in a different direction from mine, and that I could never approach Conway's level on math, yet Conway wouldn't do so well on AI research.
Or...
...or I'm strictly dumber than Conway, dominated by him along all dimensions. Maybe, if I could find a young proto-Conway and tell them the basics, they would blaze right past me, solve the problems that have weighed on me for years, and zip off to places I can't follow.
Is it damaging to my ego to confess that last possibility? Yes. It would be futile to deny that.
Have I really accepted that awful possibility, or am I only pretending to myself to have accepted it? Here I will say: "No, I think I have accepted it." Why do I dare give myself so much credit? Because I've invested specific effort into that awful possibility. I am blogging here for many reasons, but a major one is the vision of some younger mind reading these words and zipping off past me. It might happen, it might not.
Or sadder: Maybe I just wasted too much time on setting up the resources to support me, instead of studying math full-time through my whole youth; or I wasted too much youth on non-mathy ideas. And this choice, my past, is irrevocable. I'll hit a brick wall at 40, and there won't be anything left but to pass on the resources to another mind with the potential I wasted, still young enough to learn. So to save them time, I should leave a trail to my successes, and post warning signs on my mistakes.
Such specific efforts predicated on an ego-damaging possibility—that's the only kind of humility that seems real enough for me to dare credit myself. Or giving up my precious theories, when I realized that they didn't meet the standard Jaynes had shown me—that was hard, and it was real. Modest demeanors are cheap. Humble admissions of doubt are cheap. I've known too many people who, presented with a counterargument, say "I am but a fallible mortal, of course I could be wrong" and then go on to do exactly what they planned to do previously.
You'll note that I don't try to modestly say anything like, "Well, I may not be as brilliant as Jaynes or Conway, but that doesn't mean I can't do important things in my chosen field."
Because I do know... that's not how it works.