on the midpoint of infinity

When I was about ten, visiting my mother for the summer in Bullhead City, AZ, I met a man by the name of Mike Anderson. He seemed to me to be rather intelligent, was undoubtedly quite an interesting fellow, and fueled my interest in a number of things that occupied my time throughout that summer and beyond. Most of the things he got me thinking about were “paranormal” – things like telekinesis, out-of-body experiences, and telepathy. In fact, I’m still interested in those things to a certain degree, as they appeal to my desire for an extraordinary existence, but I haven’t spent much time mulling them over lately. Instead, I’ve been thinking about discreet mathematics, which I know very little about, continuity, and the concept of infinity…

This goes back to Mike, because one of the tidbits he once left me to mull over was: “light is like a river, and nothing within the river can go faster than the river goes” – of course, he was trying to explain to a ten-year-old that the speed of light is a kind of universal speed-limit. It sounded neat, I didn’t really fully buy it then, and I’m still not sure if I do now. However, recently, I’ve been having the oddest thoughts about light-speed, midpoint paradoxes, and discreet mathematics. I’m basically under-qualified for discourse in all of the subjects, but let’s bundle them up for a bit and draw out what’s been bothering me.

The midpoint theorem is simple enough, to get from point A to point B on a continuous function you must pass through the points on the function between A and B. There are more rigorous definitions available, but that one should do for now, I hope. So, you walk in a straight line from point A to point B, and you must pass through the midpoint C. The paradox arises that you can never get to point B. There is always a point half-way between wherever you happen to be on the line and where you want to go; you must always get halfway before you can get where you want. You can always get to the midpoint, but you can never get to the end.

Now, here’s the catch, or so I think… for the paradox to hold, there must be a midpoint at every one of an infinite number of divisions. I do not believe that can happen. I’m highly suspicious of attempting to apply the conceptualization of infinity to the actual world. (Calculus is nifty and useful, right, I know… and I don’t think that I take issue with the use of infinity in that sense… as a symbol, or a designator of a mathematical process…) I’m thinking that the world does not have the kind of domain that permits of infinite divisions.

Naturally, things appear to have bounds… movement is bounded by the speed of light, the physical dimensions of objects by the size of atoms (or components thereof)… so that at some point it makes no sense to talk about dividing a step along a natural function. Maybe everything moves in discreet steps, with the number of possible divisions bound by the speed of light. When you try to divide time itself into a segment smaller than light can travel, maybe that just doesn’t make any sense… perhaps it’s an impossibility… and if it is – then maybe the paradox is misleading about the way the world is.

More than that, maybe the idea of infinity is misleading about the way the world is. Maybe the idea of continuity as applicable to the natural world is nothing more than a pleasantry…(though, would it make any practical difference if we changed our way of thinking about the number of possible midpoints on our walk  from our front door to the mailbox?) If we can’t divide time into infinity, then I don’t think we can divide anything else into infinity. It’s like time is the river, and everything that can happen can only happen as fast as time will permit.

Using Mike’s analogy: the speed of time can only bound by the speed of light (because, mustn’t time itself be in the river… or could it be the river?) – and then that’s our actual continuity stopper. We’re not moving continually, we’re taking a bunch of really, really small steps. Really small, but not infinitely so.

What happens at 299,792,459 meters per second? Nothing…? And light’s speed is constant… so we know where it must be at each time between any A and B. Take that with the limited dimensions of the light particle itself… and you have all the bounds you need to prevent the infinite division, or not? We can’t divide to any point that would make that little light particle move faster than it can move. Dammit, is time bound or not? I’m regressing into confusion…

What do you think? If you’ve read something somewhere that would help me think about the issue further, or have personal insight into what I’m confusing myself over, then please leave a comment and let me know.

good government, take one

My ideal government would be decentralized. The national government would be tiny, maintaining a national military and acting as a mediator between smaller governments. Local governments would hold a great deal of power and local citizens would control the means of production. There would be many powerful small governments, but no centralized big government. No big corporations.

The people would be taxed using a flat sales tax for necessary government services, but extra projects would be funded by inflation-indexed rate-capped government bonds. This way debt would be more fine-tuned by individual communities – and the nation would have less chance of overspending (especially on a national level).

Because communities would own patents collectively (granted by the national government), to foster innovation and productivity, large one-time cash awards and honors should be given to innovators. Say 10x the median income. This would ensure people were still excited about innovating, but prevent multi-billion dollar entities, groups, or people from concentrating power. Because local governments and people would benefit from innovators, they would be highly sought after. The local governments would set wages accordingly to keep and attract promising people. This would ensure that mediocrity didn’t run rampant.

Everyone would own arms, and participate in government/community at some level (even if it was just picking up trash in the park). This would make people feel connected with their community, and likely lead to more voluntary government involvement. Decisions at the local level would be made via direct democracy. State and national decisions would be made via representations. The overarching system would be a republic.

Governments would not be able to turn people away, but they could have policies in place to provide very low wages to new members of the community. Children would also become new members of the community when they were able to vote (which should require some type of national test, rather than an age requirement). This should lead to relatively normalized living conditions, and starting wages would not go too low (to deter new members) if people knew it would also affect their children.

I think that under a system like this, people would be guaranteed basic wages, but innovation would still be highly prized. Communities would become meaningful and cohesive, and people would not be making as many decisions while being removed from the effects of those decisions. Power would be with the people – political and economic power, both.

(A)I: On the Possibility of Separation between Hardware and Software

Whenever we start drawing parallels between men and computing machines we are bound to notice a particular incongruence rather quickly. Namely, men are apparently more indivisible than machines. That is to say, whereas we can talk of a computer requiring some hardware and some software to function, a man cannot be so easily disunited. A man has a brain that we may be tempted to associate with a processor and even memory (hardware), but it is not clear what part of a man we would want to label software. If we point to DNA or RNA, we do not ameliorate our difficulties. For one thing, that “software” creates its own hardware so that it is unintelligible to talk about a man without genetic code. There cannot be a human with “software” but no “hardware”. Of course, on machines today there certainly can be.

I’m not sure that this makes talking about artificial intelligence more difficult, but it may confuse the picture if not mentioned at the onset of a discussion. It can make the term “computer” somewhat ambiguous to the modern mind – and the object of artificial intelligence potentially elusive. If we inspect the hardware of a machine apart from the software, say, powered off – there would be very little of interest going on. If we took the software apart from the hardware, say, printed out – I think we’d have a hard time finding signs of intelligence then too. Only when the software is coupled with the hardware do interesting things become possible. Even when software can be embedded into hardware, it is easy for the concepts to admit separation. This may simply be due to the familiar organization of modern computers, but it may also be indicative of something more interesting – we should at least keep it in the back of our minds.

For now, at least to start, when discussing computers in relation to intelligence, it seems clear to me that we would do well to always discuss them as a bundle of software and hardware to avoid confusion. Despite the fact that one may install some “intelligent” program along many other programs, every program requires hardware to run. It is all too easy to think of the program itself as the sole cause of certain behavior – it should not be forgotten that the hardware is no less important in manifesting that behavior. So we are on the same page, in all that follows, unless I specify otherwise, when I talk of computers or computing machines, I am referring to a hardware-software couple. I am regarding the machine then, in that sense, as indivisible as a man.