You actually cannot divide a distance infinitely. There is a smallest unit of length, the Planck Length. You cannot subdivide the Planck Length. Thus there is a finite number of Planck Lengths between any two points. There is not an infinite subdivision of anything.
You cannot even subdivide time infinitely. There is a smallest unit of time that has any meaning, the Planck time. Anything smaller than that would be so unstable, so variable due to quantum fluctuations, as to make that 'thing' spontaneously change in position or time by an amplitude equal to the Planck length or Planck time (which is why these are the minimal units). That is, there is no absolute 'stillness' or 'present'. There is only position that varies at least by a variation equal to the Planck length, and only a moment in time that is at least as variable as one Planck unit of time. Zeno's paradox is a dead issue. 1 centimeter equals 6.1873559 × 10^32 Planck Lengths. One second equals 1.855094832e+43 Planck time units.
Infinity is itself just an abstract concept. No actual infinities can exist. And even the thought experiment of dividing distance over and over again by halving your distance to the wall, taking a step, halving that distance, taking a step, halving that step and so on... even that fails because you cannot keep this up forever. You eventually will be One Planck unit away from the wall and won't be able to get any closer because the wall itself varies in position by at least one planck unit constantly. Given repulsion effects of atoms dominate at many times larger than a Planck unit, you won't be able to get closer than the orbits of the electrons of the molecules of the wall and those of your body. Rather, the magnetic repulsion of your electron shells on the surface of your body and that of the wall will dominate well before you get to even that distance.
But as a though experiment, we need to recognize that the universe we live in does abide by the rules of quantum mechanics. Reality is the reference frame for even our thought experiments. As such, you cannot have a paradox of subdivision like the one called Zeno's paradox. Given there is a minimal time or distance that has any meaning, there can be no actual infinities of any property defined in space or time.
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It then a count of the number of steps to solve, which is infinite, not the addition of the fractions.
That's right, and Zeno's "paradox" arises because those infinitely many additions is conceived to somehow "add up" to an infinite amount of ... something ... an infinite amount of processing in one's mind, or something like that.
However, infinite sums can have a definite value.
1 + 1/2 + 1/4 + 1/8 + ... is one of the infinite sums that does have a finite value.
It seems the ancient Greeks didn't have a good handle on the concept of infinity. Their thinking was different from ours in many ways that are not obvious.
Mathematics has developed since then to handle infinities. There are even different sizes of infinity.
Can I just say how enjoyable it is to chat about quantum mechanics? Jeez. I have too much time on my hands. Sorry everybody. Not really a subject relevant on an atheism website. Not even sure why I brought it up, except that Zeno's paradox came up in a recent discussion with a rather ignorant theist.
I picture it this way. I draw a dot on the floor (with a very very fine tip marker) that is one Planck distance from another dot. If I look closely at the two dots, they randomly go from being one Planck distance apart to overlapping, in a fuzzy sort of blur. Trying then to add a dot in between the two is impossible (with my magic ultra-microscopic sub-atomic sharpie).
We are not permitted to "know" what happens below the Planck scale. The deterministic way we handle location, speed, and thus also time breaks down because quantum fluctuations dominate. It is the barrier beyond which classical notions of predictability break down. A position that is one Planck distance from another position, is for all intents and purposes the same position as that position varies due to quantum fluctuations by the Planck amount, whether that is a point in time or a point in space (Planck distance vs Planck time). We actually CANNOT know what happens below the Planck distance, and that is the point of defining the Planck distance in the first place. Determinism is simply no longer possible below the Planck scale. We 'know' that there is no WAY to 'know' or define any point relative to another in space time if those points occur below the Planck limits. That is what makes subdividing the Planck scale meaningless. Measurement is meaningless when the universe itself is working against you, swapping location points by a variable amount equal to the Planck scale.
Wait... I see the issue here. The math is not about adding up the various remaining halves. It is a matter of the infinite halves that one can generate by just dividing whatever is left in half. The fractional values of the original length are not the issue. It is that there is an infinite number of divisions possible unless the universe is indeed quantized as it appears to be in agreement with quantum mechanics.
It is dividing 1 by 2, dividing that by 2, dividing that by 2, and then into an infinite number of further operations. It would be listed not as a series of added values, but as a sigma function with the number under that sigma symbol as x=1 and the number above the sigma value as infinity. The function to the right of the sigma symbol would be x/2. The equation is unsolvable as one would be plugging x into the equation an infinite number of times.
It then a count of the number of steps to solve, which is infinite, not the addition of the fractions.
It's not yet known what happens to spacetime at the Planck distance.
But even if spacetime were infinitely divisible, a continuum, there still would be no paradox in Zeno's paradox.
1 + 1/2 + 1/4 + 1/8 + ... sums to 2, and there are numbers bigger than 2.
I looked into Zeno's various paradoxes a bit further and concluded he was making claims based on a primitive understanding of the universe.
For example, take the position that since objects occupy the same space when in motion as they do at rest means they are always at rest even if they are in motion. This presumes that the space an object occupies does not change when it is in motion. Einstein's theory of Relativity shows that this is actually not true. As you move at any velocity, your occupy less space than you do when you are at rest. If you travel at a significant fraction of the speed of light, you are actually flattened from the point of view of a stationary observer, taking up less apparent space. There is no absolute space, as Zeno would have assumed, so the apparent change in size is an actual change in size, if only from the point of view of a stationary (relatively) observer.
Then there is the distance paradox, where to get from point a to point b you must first halve the distance, then halve that remaining distance, then halve that remaining distance into infinity, never reaching your destination. This is invalid as space is again, not absolute, nor an infinite continuum that can be subdivided infinitely. There is indeed a finite number of points between two points, in a sort of grid of 1 Planck length per segment. If you halve the distance, you have half as many Planck lengths left to travel. You do in fact eventually get to the point of having only one Planck unit of distance between yourself and the destination. At that point you cannot halve the distance any more, as there is no such thing as half a Planck length. And as no two objects can occupy the same space, the closest two objects, the traveler and destination, can ever be is one Planck length apart. If you are that close, you have by definition 'arrived.'
Alternatively, you might want to be in the same place as the destination, or rather standing over a specific point. If the distance was marked out as dots on the floor, each one Planck length apart, you could then get to the destination position of being on top of that last remaining dot. However, once you are left with just one Planck length between your position and the last possible position, you indeed cannot subdivide the distance. You indeed cannot first travel half the distance before getting to that destination point since you cannot halve a Planck length.
If point a and point b are only one Planck length apart, the distance itself becomes unstable. Rather, your position, whether you are standing over point a or point b varies due to quantum fluctuations by exactly the amount of the Planck distance. That means that once you get to where you are only one Planck length away from your destination, the universe itself, operating in accordance to quantum mechanics, will itself move you back and forth between point a and point b automatically, as you will only ever be able to be in ANY position at rest with a margin of probability of at least one Planck length. This is what I meant by the statement that there is no such thing as absolute stillness. The universe operates in such a way that a probability wave makes positions separated by the Planck length unstable and undefinable.
Zeno's paradoxes are indeed invalid, largely solved by the invention of calculus. But the way our expanded understanding of our physical universe also destroys such classical-mechanical paradoxes is still quite intriguing.
Perhaps some day we will figure out that god concepts are also reliant on a classical-mechanical universe, and will likewise not only have justification to reject god claims due to a lack of evidence, but also be able to once and for all make the gnostic atheist claim that no gods CAN exist. There are already hints at this, whereby the Kalam and TAG fail, because they rely on neo-Lorentzian mechanics, when we KNOW the universe is Lorentz invariant. Apologists like Dr. W. L. Craig like to ignore that fact, that very nature of the universe is incompatible with his argument for his god's existence. But the truth of the matter is there for all to review, if they only obtain the necessary knowledge to understand the difference between a Lorentz invariant universe like our own and the neo-Lorentzian universe, which is absurdly different than ours, that theists require to exist.
OK, but the problem with Zeno's paradox isn't that time is quantized (very likely it is).
It's hard to see how anyone could think Zeno's paradox is a real paradox.
Time is not merely our invention. It is indeed an actual dimension, which along with those of space and the curled up hyper dimensions, is absolutely real. If it were just an invention, it would not be necessary to take it into account when doing any science. If it were just our invention, it would not be required to describe reality at all. It would merely be the consequence of our perception, which should be independent of the laws of physics. But it indeed MUST be real and independent of our perception, as the affects of the actual dimension of time are measurable and have to be compensated for in any calculation from quantum physics all the way up to cosmology.
There are several types of time. Stephen Hawking's famous book 'A Brief History of Time' goes into this quite well, and likely better than my reply here.
The second law of thermodynamics and the tendency of systems to evolve to higher entropy necessarily creates a direction for what is called cosmological time. The expansion of the universe also creates a direction. The causal chains of events that occur as the universe expands align to this same direction (except at the quantum level in some interactions). This sense of time is independent of human minds and would exist even if no contemplative entities existed. As we reverse the expansion mathematically, and thus get closer to the big bang singularity, we are necessarily moving along an axis only definable as time, in the opposite direction of its natural flow.
This cosmological arrow of time is true for all entities, independent of their personal frames of reference. It is not an absolute time, in that sense, but it is an independent axis, defined by the various stages of the expanding universe, and flowing towards increasing entropy.
Our local time is also true for everybody. It is not the SAME for anybody or any thing in motion, which is everything. That is, any two clocks in the universe cannot match exactly. The difference of these clocks is exaggerated when either clock is moving at a significant velocity relative to the other or when one clock is closer to a massive object than the other. This axis of time is still always pointing in the direction of increasing entropy, but it is a sort of warped and twisted axis, affected by mass and velocity.
Our psychological sense of time is indeed something that we as thinking entities possess ad hoc. But it is not really imaginary, but a necessary construct of sentient entities that have memory and thus require some means of organizing those memories. As our universe moves on its cosmological time axis to the future, we observe changes and processes that we alocate to memory in our brains. The cosmological axis is indeed real, and the processes that we observe are real. The brain is itself real, and the structure of the brain makes it efficient at capturing memories in parallel sequence to the cosmological axis. Our recollection of memory invokes our psychological sense of time, which can indeed be somewhat subjective, and maleable, even to the extent that it can be rendered inefficient (as we sometimes remember sequences of events inaccurately). But generally speaking, our psychological sense of time is a reliable parallel of the actual cosmological axis of time.
The quantum realm is what makes it impossible to infinitely subdivide anything in reality. Our reality necessarily conforms to the probabilities and rules of quantum mechanics. As such, there is a certain fuzziness about actuality when we get down to a certain ultra-tiny scale. This quantum foam makes it impossible to pinpoint any position absolutely, and renders it impossible to make any time stamp 100% accurate. The foam fluctuates at a very small scale, but it is a measurable and actual phenomenon. Thus we can only define things in classical terms of position and moment down to a certain level, the Planck scale. Any further subdivision, like saying 'half a Planck unit' has no meaning as there is nothing to divide INTO smaller than the Planck scale. You might still be able to form the THOUGHT experiment of infinitely subdividing time or position, which does lead to Zeno's paradox. But REALITY does not work that way, rendering the paradox moot.
Zeno's paradox wouldn't be a paradox even if time and space could be infinitely subdivided.
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