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.
2 Replies to “on the midpoint of infinity”
2nd to last paragraph you stated, “We can’t divide to any point that would make that little light particle move faster than it can move.” I don’t believe dividing more points would change the velocity of the light particle.
I think you’re right, actually. That’s why I re-examined this whole train of thought just a day later. Considering that velocity is just distance divided by time, if we thought that time and distance had upper and lower bounds, then velocity would have bounds too, right?
I think that’s what I’m actually interested in, but this rant didn’t make that interest clear.