I read your answer with interest, Cecil, and you are correct.
However, in addition to your explanation that it is a fluke of the limitations of our decimal system, there is also a physical limitation. I believe that Max Planck conluded that time and space cannot be subdivided forever. The fundamental quantum of length is Planck length, and is the distance that light can travel in Planck time (10^-43 seconds). It is 10^-35 of a meter. In other words, one millimeter contains 100,000,000,000,000,000,000,000,000,000,000 Planck lengths. Thus, a measurement cannot be cut into thirds infinitely. Thus, at some point in repeating 0.999~ you will reach Planck length, and the answer becomes zero when all added up.
In regards to Zeno’s paradox, it was solved by Sir Isaac Newton when he developed calculus, originally used to calculate the moon’s orbit accurately. If the moon was stationary at the beginning of every second, adding up all of its tiny “falls” does not give the right answer.
The whole misunderstanding of the Planck scales is a minor pet peeve of mine. We don’t know that you can’t divide up space finer than the Planck length. In fact, we know precious little at all about what happens at the Planck length. That’s just what the Planck scale is: It’s a back-of-the-envelope estimate of the scale at which our current theories break down.
The problem, basically, is this. We have General Relativity, which involves the constants G (Newton’s gravitational constant) and c (the speed of light), and which works very well. We also have quantum mechanics, which involves the constants hbar (Planck’s quantum of action) and c, and which also works very well. So long as we’re in a situation where only one of those theories applies (which is everything we have any experimental experience with at all), we’re fine: We apply one theory, ignore the other, and get an answer.
But if we squish all three of those constants together in the right way, we get a unit of length (squish them slightly differently, and you can get a time, or mass, or energy, or most any other sort of unit). If we’re working with something with about that size, then both GR and QM will be significant simultaneously. Except that GR and QM don’t really seem to play well together. Surely, there’s some way of explaining what happens in such situations, but we have no idea what it is.
Still, the fact that spacetime cannot be subdivided is still relevant, right? Or is it simply the fact that our theories break down on the Planck scale that makes it seem so?
No, it is entirely irrelevant. Zeno’s paradox merely uses a physical metaphor to talk about infinite mathematical series. What does or doesn’t happen in the real world is of no matter, so to speak.
And if spacetime cannot be subdivided indefinitely, it it even less relevant, because then distance is merely the addition of a finite number of increments, so the whole problem of converging or diverging infinite sums doesn’t come into it.
It is almost always a mistake to put a physical structure onto purely mathematical problems. People who try to trisect an angle with straightedge and compass often make this same mistake.
Infinity is not a physical concept. Don’t think about distance here. Think about pure numbers.
This illustrates part of the difficulty of the Planck units. Definitely, the Planck Speed (the Planck Length divided by the Planck Time; just a fancy name for the Speed of Light) is very big. And the Planck Action is assuredly very small, since it’s impossible to get any action (or angular momentum, which has the same units) which is any smaller. And everyone will agree that the Planck Length and Time are very small.
But what about the Planck Mass? Is that big or small? Well, that depends on whom you ask. The Planck Mass is about a microgram, which is inconceivably small from the perspective of someone doing GR, but it’s inconceivably big to a particle physicist. And to someone working with Newtonian systems, it’s a perfectly reasonable mass (about the mass of a single bacterium). You can have things with half that mass, or twice that mass, and they behave exactly as you’d expect. And if you look at the Planck Momentum (mass times the speed of light), you’ll find that it’s about the same as the momentum of a running housecat. You don’t even need a microscope to study systems like that.
Ney, non, niet, the Planck length is not a quantized unit of space, infact observations by astronomers have proved that if space is quantized it must be on a much smaller scale than the Planck length. The Planck length is the length over which yet undiscoverd quantum gravitational effects should take over from classical gravitational effects.
I am no physicist. I wouldn’t even call myself a mathematician, but even I know there is no “physical limitation” in a purely mathematical statement such as 1=.99…
