Why doesn't Zeno's paradox work?

The sum of an infinite series can be finite, as long as the terms get infinitesimally small. So even though the number of steps to reach a place can be thought of as being infinite, the time it takes to reach the place (sum of the times taken at each step) will still be finite.

Let us suppose, the total distance to cover is 1m. So breaking it down for Zeno's Paradox will give us distances - 1/2m, 1/4m, 1/8m... On adding all those distances, [math]S = 1/2 + 1/4 + 1/8 + 1/16 + ............. (i)[/math] Therefore, on multiplying by 2, [math](i)*2 =>[/math] [math]2S = 1 + 1/2 + 1/4 + 1/8 + .................(ii)[/math] Now, subtracting (i) from (ii) [math](ii) - (i) =>[/math] [math]2S - S = 1 + 1/2 - 1/2 + 1/4 - 1/4 + ...... [/math] [math]=> S = 1[/math] The sum of an infinite series might just be finite. The sum of an infinite number of distances might be a mundane finite distance... :(

Mathmatically it's not a paradox because it can be explained by the fact that time, just like position follows the real number system so a sequence of times each of which is later than the previous time doesn't necessarily mean a time later than all times in that sequence doesn't exist. However, in real life I don't believe in supertasks without a first step. For instance, if you drop a piece of salt onto ice, why does it even begin melting the ice? For it to melt a certain amount of ice, there has to first be salt water to depress the freezing point, but for there to be salt water, it first has to melt a smaller amount of ice. There must be a first step in order for that to happen. Indeed, there is one despite the fact that we don't observe one. The initial drop converts kinetic energy into a very small amount of heat which melts a microscopically small area of ice at the point of collision and then that sets off the chain reaction of more ice melting. In fact, I don't believe in supertasks without a last step either. I define any process without a first step or a last step to be a supertask including ordinary movement in a universe where time follows the real number system. Therefore, I believe time is quantized. Although I don't believe in the existence of an infinite descending sequence of time, I do believe in the existence of an infinite ascending sequence of time and just don't believe in the existence of a time later than all times in that sequence. That is, I believe that time has a beginning but not an end. I believe our universe is actually something like a Conway's Game of Life simulation that started with an initial state with only finitely many black squares. I don't see a reason to be so sure that's not the case. We can't make arbitrarily precise measurements according to quantum mechanics. We can never measure an amount of time smaller than the planch time. Although I believe time is quantized at the fundamental level, that doesn't mean in our physical universe, measured times will always take on integral multiples of a specific length of time. Rather, there's no direct correspondence between positions and times in the Conway's Game of Life simulation and those in our physical universe. Rather, it's a complex series of interactions in the simulation that give rise to a space time measurement in our physical universe. Following the time reversible laws of physics at the molecular level, it's been suggested by some people that there was a backwards arrow of time before the big bang but they defined it as a second future because cause and effect goes in the other direction there. However, in the Conway Game of Life simulation, it really is a future and not the past. Even if we lived in a universe where our measurement of where an object was at a certain time will become arbitrarily precise as more and more time goes by, that still doesn't necessairly mean it's not a Conway's Game of Life simulation. That means it wasn't really in that exact position and we just get tricked later into inferring from future observations and the observed laws that it was very close to that position. For instance, given an intial state and velocity of each particle in a 3 body system, there exists a turing machine that for any input time given, will keep on computing in terms of that time more digits of the position of each particle without bound.

Zeno's paradox suggests splitting a finite distance in to  smaller and smaller pieces without limit.  The distance is still finite, as is the travel time, even though it too is being split in to smaller and smaller intervals.If anyone actually claims to believe Zeno's conclusion, ask him to stand up in place of the wall while you shoot the arrow to see if it actually works.  This will dispose of him ... one way or another.

The question is clever posed from the wrong end of the telescope.  Instead of stepping off distance in fixed amounts as a function of time,  it keeps dividing the time intervals smaller and smaller, for no particularly good reason.   Sure, if you define the intervals that way, you never quite get there.

Let there be a line A-B-C-D where each dash is the same length. Define A as the starting point and C as the destination. First Zeno moves from A to B since B is half the distance to C. Then redefine B as the starting point (as in the original problem) and D as the destination. Zeno then moves from B to C since C is half the distance to D. Thus Zeno has reached C in a finite number of moves (and time). This 'solution' relies on two things: the arbitrary nature of start and end points in space and exploiting the fallacy of 'moving goalposts'. The nature of space is that start and end points are relative to the subject. They are arbitrarily defined by the participants. Thus question of what is the start and end point should therefore not be a limiting factor. Amber Giuliani details the 'moving goalposts' fallacy in her answer to this problem. She accuses Zeno of doing this by repeatedly changing the starting point to the landing point e.g after Zeno travels half the distance AC to B, B is the new starting point from which he must travel half the distance BC to C. Bringing these two together provides the solution as it reduces the restrictions in the problem. However only the nature of time and space as they exist in our reality should provide restrictions (See the 'physicist' answers to this question.) Therefore this solution should be allowed.

Suppose you are going to walk a certain distance in one minute.  You walk half the distance in half a minute, then a quarter of the distance in a quarter of a minute, an eighth of the distance in an eighth of a minute, and so on until the minute is up and you have walked the entire distance.  No problem; no paradox.  The only way you get a problem is by contriving to somehow forget that each time you cut the distance in half, you have to cut the time in half too.  So, if you walked halfway in half a minute, then a quarter of the way in another half minute, then an eighth of the way in yet another half minute, and so on, then it's true that you would never get there.  But it's also true that nobody ever walks in that manner.

One reason is Zeno didn't take into account velocity.  Each step in the process is completed twice as fast as the previous. The more complete, mathematical resolution is that the series 1/2+1/4+1/8...   converges to 1.

The "Dichotomy Paradox", in my opinion, is not a paradox at all. Here is how.The "Paradox" states that to get from point A to point B, the distance between which is say, x, you need to travel one half the distance, x/2. Then half of the half, x/4. Then x/8, ... and so on. This fraction will never be zero because no matter how large the denominator, the fraction will still be positive. One might argue that as the denominator gets to infinity, the fraction will be zero, and thus we reach B. But to make infinite moves we need infinite time. So we will never get to B.Now, here is how to resolve the "Paradox."Locate a point C that is twice as far from A as B, so that the distance from A to C is 2x. B lies in the middle between A and C. To reach C we need to travel half the distance from A, which is x. But then we are exactly at B!The "Paradox" is framed in a manner that leads to a false conclusion. When an object travels from A towards B, the object does not calculate the total distance between A and B and moves accordingly. Physical motion does not depend on the distance to destination. It depends (mainly) on mass and force. Travelled distance is determined by these variables, not the other way around.