I’m positive this has an easy answer, but it eludes me.
As I sit here typing, there’s an inviting glass of scotch/rocks on my desk about two feet from me. When I reach for it, my hand has to cover half the distance, which takes a certain amount of time. One foot to go, taking another half of the time remaining. Only 6" left, but that takes a certain amount of time to complete. 3" now, and my hand has yet to reach the glass. When the glass and my hand are only microns apart, the space between them and the time it takes crossing it can still be halved.
See where I’m going with this question?
If the distance between my hand and the glass, and the duration of the time involved in completing my goal can always be halved, how does my hand ever reach the glass? Each halving of distance takes measurable time, and each halving of time allows for an incompletion of my goal.
This question has vexed me since my teens, and I’ve gotten a few headaches thinking about it very long. In my defense, I never claimed to be smart, just easily perplexed.
That’s like wondering how, since there are infinite numbers between 1 and 2, how 2 can exist at all. Why stop at halving it? Why not take it into 1/10000s?
Imagine a graph with a line. The line intersects the X axis at 10. We’ll call the X axis “time” and the vertical axis “distance remaining to cup”. Your question is merely pointing out that 5 exists, as does 7.5, and 8.75, etc.
So to answer your question: So what? That doesn’t mean that the 10 second mark never arrives.
The effect you are describing is also known as Xeno’s Paradox, except in his case he described it as the continuing halving of distance an arrow experiences as it reaches its target.
mathematically it is x= 1/2+1/4+1/8+1/16… where x is the value of the entire distance.
obviously you will never reach x, but instead become infinitely close to it.
The paradox is “real” in a mathematical sense, but in the real world, it becomes irrelevant as very quickly the diminishing distance between the arrow and the target, or your hand and the scotch becomes less than the diameter of an electron.
One explanation would be that there are units of time/distance which are indivisible. As a result, you can’t cut in halves infinitely - at some point, you get down to being one indivisible-unit away, and after dividing that in half you have reached your precious scotch.
One explanation would be that there are units of time/distance which are indivisible. As a result, you can’t cut in halves infinitely - at some point, you get down to being one indivisible-unit away, and after dividing that in half you have reached your precious scotch.
Then to my math-challenged mind, this seems to make the concept of “infinity” moot. Not arguing with you, just sayin’.
The paradox is not real in any sense, mathematical or otherwise.
What Xeno didn’t understand, because the concepts had not been fully formed, is that an infinite series can add up to a finite sum. Xeno considered that an utter impossibility, hence the “paradox,” but the invention of the calculus and modern understandings of limits make it a commonplace.
All you are saying is that 1/2 + 1/4 + 1/8 + 1/16 and on indefinitely equal (or have the limit of) 1.
There are many infinite series that have a finite sum. (Also called converging series.) They can be expressed in either fractional or decimal form.
.3 + .03 + .003 + .0003 … = 1/3.
Search on Xeno’s paradox and you’ll find a zillion threads discussing this in every possible level of mathematical detail.
And while I’d say IntelSoldier’s first post really outlines the proper response to this observation (you don’t have to know how to sum an infinite series to realize there are unwarranted assumptions going on in finding this situation paradoxical), his last post makes me fearful this may devolve into yet another round in the unceasing game of “Does 0.9999… = 1?”.
(But perhaps each new round brings us asymptotically closer to the point where ignorance is fully eliminated…)
The fallacy in your question lies in the word “measurable”. No, as the distances get infinitesimally small, they’re no longer measurable in any practical sense. And practicality is an important part of this scenario.
Here’s another way of looking at it: Your problem would be real if each step took the same amount of time. One second for the first foot, another second for the next six inches, another second for the next three inches, and so on. If that was the setup, then indeed, you would never get all the way to the glass. But that’s not what’s happening. What is happening is that the distances get smaller, but the times get smaller too, and at the very same rate. Thus they cancel out, and you do reach the glass.
The critical factor in all this, is to understand that not all infinities are equal. There are twice as many points in a 12-inch line than in a six-inch line, despite that one could use “infinify” to describe both numbers. And ditto for the moments in a one-second interval and a half-second interval.
On the contrary - there are the same number of points in a 12 inch line as there are in a six inch line, since they can be mapped 1-1 to each other. Similarly, there are the same number of integers as there are even integers, the same number of integers as there are rational fractions (x/y where x and y are integers), but there are not the same number of integers as points on a line. Arithmetic with infinities is, shall we say, rather surprising at times.
No, practicality plays no part at all in scenario.
It is absolutely and completely a thought experiment. The physical steps are meaningless. It is about infinite sums, not about actual distances. Zeno [sorry about that] knew that at the time.
Canadjun already took care of the other and equally wrong half of your post. Not only are there the same number of points in 12 inches as six inches, there are the same number of points in a line as long as the universe, in a plane that bisects the universe, and in the whole 3-D universe as in a six-inch line. If we could see into 4, 5, or n dimensional universes, there would still be the same number of points, because they can all be put into a one-to-one correspondence.
That’s why proper understanding of infinite math had to wait until Georg Cantor in the second half of the 19th century. It is so counterintuitive that even now that it’s been fully explained, people keep getting it wrong.
There is also the idea that space-time id quantumized. You cannot divide it in half after a certain number of repitions. The concept of Planck length at time describe this concept.
The idea that space or matter cannot be infinitely subdivided in hardly new. Democritus, who lived at about the same time as Zeno, had an atomic theory in which particles were indivisible. Nor was he first.
As I say earlier, Zeno knew this. He didn’t care. Whether space or time could be physically divided indefinitely was beside the point. He was thinking in terms of perfect mathematical reasoning, the same as the diagonal of a unit square not being measurable but being perfectly representable.
The physical divisibility of space-time is a red herring. It has nothing to do with Zeno’s paradox, and bringing it up only confuses the matter by dragging in an irrelevancy.
Are you saying that taking a problem phrased as a real-world problem, then trying to use math to solve it, and then discovering there is a physical answer to the whole thing is irrelevant? Most people don’t care what the solution is or what discipline it comes from as long as it ultimately works.
If physical space is not continuous then it makes a huge difference. It would imply that the mathematical model of the problem is not even accurate, regardless of math with inifity.
If spacetime is discrete, then the sum only has a (large) finite number of terms, and there’s no paradox. I took Exapno’s comment to mean that the divisibility of spacetime is a red herring in that there’s a solution either way, so it’s not relevant to resolving the paradox.