# What ramifications will my scientific breakthrough have?
**URL:** https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889
**Category:** Factual Questions
**Created:** [May 18, 2009, 7:24pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889 "2009-05-18T19:24:12Z")
**Posts on this page:** 20
**Page:** 2
### Author:  [@Lemur866](https://boards.straightdope.com/u/Lemur866)
#### Post date: [May 19, 2009, 12:53pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/21 "2009-05-19T12:53:04Z")
But if each increment only takes half the time of the previous increment to travel half the distance, then you can quite easily sum those infinite slices of movement into a finite movement. As Newton and Leibniz proved.
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### Author:  [@straggler](https://boards.straightdope.com/u/straggler)
#### Post date: [May 19, 2009, 12:59pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/22 "2009-05-19T12:59:41Z")
> [@Lemur866](#):
>
> But if each increment only takes half the time of the previous increment to travel half the distance, then you can quite easily sum those infinite slices of movement into a finite movement. As Newton and Leibniz proved.
You can only “sum infinity” in theory. In reality, you can not “sum infinity”.
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### Author:  [@Lemur866](https://boards.straightdope.com/u/Lemur866)
#### Post date: [May 19, 2009, 1:23pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/23 "2009-05-19T13:23:07Z")
Your theory is that in reality you cannot sum infinity. In reality, it turns out you can.
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### Author:  [@straggler](https://boards.straightdope.com/u/straggler)
#### Post date: [May 19, 2009, 1:26pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/24 "2009-05-19T13:26:17Z")
> [@Lemur866](#):
>
> Your theory is that in reality you cannot sum infinity. In reality, it turns out you can.
Can you elaborate on that? I hold, that in reality, you can’t.
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### Author:  [@Rolken](https://boards.straightdope.com/u/Rolken)
#### Post date: [May 19, 2009, 1:52pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/25 "2009-05-19T13:52:48Z")
> [@straggler](#):
>
> Can you elaborate on that? I hold, that in reality, you can’t.
It can be mathematically demonstrated that the sum of the infinite series of halves from your OP is double the original distance. In what sense is that not real?
Or to take another tack on it: I assume your point is that your hand is getting closer and closer to the apple but never reaching it, yes? Thus, your hand inches over a finite distance and gets closer and closer but never passes the apple.
But the flaw in your model there is that the time spent reaching for the apple _also_ is diminishing in each step. So if it takes infinite steps to reach that finite distance towards the apple, then it necessarily follows that those infinite steps are achieved in finite time as well, and there’s no paradox.
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### Author:  [@straggler](https://boards.straightdope.com/u/straggler)
#### Post date: [May 19, 2009, 1:55pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/26 "2009-05-19T13:55:46Z")
> [@Rolken](#):
>
> So if it takes infinite steps to reach that finite distance towards the apple, then it necessarily follows that those infinite steps are achieved in finite time as well, and there’s no paradox.
You began by saying “_if it takes infinite steps_”. Who says it takes infinite steps? In a universe where there exists an indivisible unit of length, the number of steps is finite.
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### Author:  [@Rolken](https://boards.straightdope.com/u/Rolken)
#### Post date: [May 19, 2009, 2:00pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/27 "2009-05-19T14:00:29Z")
I was aiming to demonstrate that your argument against the divisibility of space was erroneous, not that space is necessarily continuous.
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### Author:  [@straggler](https://boards.straightdope.com/u/straggler)
#### Post date: [May 19, 2009, 2:12pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/28 "2009-05-19T14:12:47Z")
> [@Rolken](#):
>
> It can be mathematically demonstrated that the sum of the infinite series of halves from your OP is double the original distance. In what sense is that not real?
How can something be “mathematically proven” when the key starting premise, ie, there exists between two points an infinite amount of half distances, is itself unproven?
> [@](#):
>
> Or to take another tack on it: I assume your point is that your hand is getting closer and closer to the apple but never reaching it, yes? Thus, your hand inches over a finite distance and gets closer and closer but never passes the apple.
No, my point is that my hand _does_ reach the apple. Hence, in a finite distance, there must exist a finite number of half-distances. There must exist an indivisible unit of distance/space that is either full or empty, reached or not reached.
> [@](#):
>
> But the flaw in your model there is that the time spent reaching for the apple _also_ is diminishing in each step. So if it takes infinite steps to reach that finite distance towards the apple, then it necessarily follows that those infinite steps are achieved in finite time as well, and there’s no paradox.
Importantly, you said, _if it takes infinite steps to reach that finite distance towards the apple_. I hold that this sentence is meaningless. You can not reach something after an infinite amount of steps that must be taken sequentially, if each step takes _some_ time.
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### Author:  [@Rolken](https://boards.straightdope.com/u/Rolken)
#### Post date: [May 19, 2009, 2:31pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/29 "2009-05-19T14:31:40Z")
> [@straggler](#):
>
> How can something be “mathematically proven” when the key starting premise, ie, there exists between two points an infinite amount of half distances, is itself unproven?
What’s proven is that it is possible for an infinite list of numbers to add up to a finite amount.
1 + 1/2 + 1/4 + 1/8 + … = 2.
The jury is still very much out on whether reality boils down to continuity or discontinuity, but as a pure mathematical exercise, it is entirely possible. Since the laws of physical reality as we understand them have always been mathematical, it’s entirely plausible (but yes, unproven) for reality to work that way as well. But ultimately we just don’t know, and both options have been explored.
> [@straggler](#):
>
> Importantly, you said, _if it takes infinite steps to reach that finite distance towards the apple_. I hold that this sentence is meaningless. You can not reach something after an infinite amount of steps that must be taken sequentially, if each step takes _some_ time.
You’re not treating time and space congruently in your example. I think you’re thinking of time in terms of the time you’re spending thinking about it, not in terms of the actual time that would be spent reaching.
You asserted in your example that those infinite steps translates to a finite distance short of reaching the apple in an infinite amount of time. You can’t have it both ways like that. Either it’s an infinite time and it can reach infinitely far, or it’s a finite time for a finite distance, unless you have a solid reason to treat time and space differently. “Things don’t work that way” doesn’t count. 🙂
edit: I mean that paragraph in the context of your OP that asserts that continuous space is impossible. There is of course nothing wrong with conjecturing about time and space being different, but that’s not the same as disproving continuous space, and it doesn’t rise to the level of a “scientific breakthrough” until you have a powerful set of arguments arranged behind it.
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### Author:  [@straggler](https://boards.straightdope.com/u/straggler)
#### Post date: [May 19, 2009, 2:44pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/30 "2009-05-19T14:44:43Z")
> [@Rolken](#):
>
> What’s proven is that it is possible for an infinite list of numbers to add up to a finite amount.
>
> 1 + 1/2 + 1/4 + 1/8 + … = 2.
But this is theory. You can’t do this in reality. You can’t add an infinite amount of anything in reality. You can’t even write the equation, which is (one reason) why you have to trail it off with dots.
> [@](#):
>
> The jury is still very much out on whether reality boils down to continuity or discontinuity, but as a pure mathematical exercise, it is entirely possible. Since the laws of physical reality as we understand them have always been mathematical, it’s entirely plausible (but yes, unproven) for reality to work that way as well. But ultimately we just don’t know, and both options have been explored.
But in my example, I ask the question: If something requires an infinite amount of steps, and each step must be taken sequentially, and each step takes _some_ time, then the “result” is unreachable. This is self evidently obvious.
> [@](#):
>
> You’re not treating time and space congruently in your example. I think you’re thinking of time in terms of the time you’re spending thinking about it, not in terms of the actual time that would be spent reaching.
>
> You asserted in your example that those infinite steps translates to a finite distance short of reaching the apple in an infinite amount of time. You can’t have it both ways like that. Either it’s an infinite time and it can reach infinitely far, or it’s a finite time for a finite distance, unless you have a solid reason to treat time and space differently. “Things don’t work that way” doesn’t count. 🙂
In my examples where I say things like “an infinite number of steps” or “an infinite number of half-distances”.. I am using those terms to prove that they aren’t possible. I believe that between any two points in space, there exists a finite number of indivisible units of distance (at least, this is the theory I am testing).
This is why Zeno’s Paradox is not a paradox at all. In my arm-to-apple example, eventually my arm is moving at such small increments, that no smaller increment of movement is possible. If the apple is one meter away from me, perhaps I achieve the smallest possible incremental step at the 13,387,980,857,098,936,856,490,988,478,309th half distance.
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### Author:  [@Rolken](https://boards.straightdope.com/u/Rolken)
#### Post date: [May 19, 2009, 3:01pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/31 "2009-05-19T15:01:10Z")
> [@straggler](#):
>
> But this is theory. You can’t do this in reality. You can’t add an infinite amount of anything in reality. You can’t even write the equation, which is (one reason) why you have to trail it off with dots.
Why does it matter whether I can write out the equation? The concern isn’t whether I can do it, it’s whether the physical laws can express it. And theory has a pretty good track record too.
> [@straggler](#):
>
> But in my example, I ask the question: If something requires an infinite amount of steps, and each step must be taken sequentially, and each step takes _some_ time, then the “result” is unreachable. This is self evidently obvious.
A lot of self-evidently obvious things end up being proven false. It’s self-evident that we’re sturdy solid guys, and yet it turns out that we’re almost completely empty space with pinpricks of matter sprinkled in the void. It was self-evident for a long time that the idea of the Earth hurtling through empty space at enormous speeds was ludicrous, yet it is so. When physical theory and physical intuition collide, historically intuition hasn’t had a very good track record.
> [@straggler](#):
>
> In my examples where I say things like “an infinite number of steps” or “an infinite number of half-distances”.. I am using those terms to prove that they aren’t possible. I believe that between any two points in space, there exists a finite number of indivisible units of distance (at least, this is the theory I am testing).
Right, but that’s not enough. You have to be able to explain something that other options cannot. Arguing that you’re right because the alternative is self-evidently impossible doesn’t cut it when the mathematics work out. Acceptance of the seemingly absurd is a necessary precondition to discussing modern physics at all.
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### Author:  [@Kimmy\_Gibbler](https://boards.straightdope.com/u/Kimmy_Gibbler)
#### Post date: [May 19, 2009, 3:06pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/32 "2009-05-19T15:06:47Z")
> [@straggler](#):
>
> But this is theory. You can’t do this in reality. You can’t add an infinite amount of anything in reality. You can’t even write the equation, which is (one reason) why you have to trail it off with dots.
> But in my example, I ask the question: If something requires an infinite amount of steps, and each step must be taken sequentially, and each step takes _some_ time, then the “result” is unreachable. This is self evidently obvious.
None of these are proofs so much as they are unsubstantiated assertions. The infinite series (which could have been written using sigma notation, if you find the ellipsis so untoward) should scotch your pleonastic assertion of “self evidently obvious.”
> [@](#):
>
> In my examples where I say things like “an infinite number of steps” or “an infinite number of half-distances”.. I am using those terms to prove that they aren’t possible. I believe that between any two points in space, there exists a finite number of indivisible units of distance (at least, this is the theory I am testing).
>
> This is why Zeno’s Paradox is not a paradox at all. In my arm-to-apple example, eventually my arm is moving at such small increments, that no smaller increment of movement is possible. If the apple is one meter away from me, perhaps I achieve the smallest possible incremental step at the 13,387,980,857,098,936,856,490,988,478,309th half distance.
Your original proof that this must be the case was predicated on the impossibility of an infinite series of non-zero terms summing to a finite number. That having been disproven, your argument for discontinuous space has been vitiated. It may well be the case; your argument, however, does not demonstrate it.
Also, how does motion occur in your view, seeing as the tip of the arrow, say, can only ever be at an endpoint of a fundamental space unit? Does it dematerialize and rematerialize an FSU away? It certainly can’t simply sail between them without violating your indivisibility premise.
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### Author:  [@straggler](https://boards.straightdope.com/u/straggler)
#### Post date: [May 19, 2009, 3:34pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/33 "2009-05-19T15:34:00Z")
> [@Kimmy\_Gibbler](#):
>
> Your original proof that this must be the case was predicated on the impossibility of an infinite series of non-zero terms summing to a finite number. That having been disproven..
Where was that disproven?
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### Author:  [@straggler](https://boards.straightdope.com/u/straggler)
#### Post date: [May 19, 2009, 3:36pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/34 "2009-05-19T15:36:36Z")
> [@Rolken](#):
>
> A lot of self-evidently obvious things end up being proven false. It’s self-evident that we’re sturdy solid guys, and yet it turns out that we’re almost completely empty space with pinpricks of matter sprinkled in the void. It was self-evident for a long time that the idea of the Earth hurtling through empty space at enormous speeds was ludicrous, yet it is so.
I don’t accept these examples as self-evidently obvious. Even without having the real explanations as to how either of these things are possible, one could speculate a plausible scenario that makes them possible.
I happily open up my example to wild speculation for you explain a way that it might not be true.
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### Author:  [@Lemur866](https://boards.straightdope.com/u/Lemur866)
#### Post date: [May 19, 2009, 3:42pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/35 "2009-05-19T15:42:35Z")
Why can’t it be time that is quantized, rather than space? Then you could have continuous space, but the smallest possible unit of time is 1/jillionth of a second.
Or it could be both, or neither.
Let me ask you a question. What does 1 + 1/2 + 1/4 + 1/8 + 1/16… equal? Does it equal 2, or some other number? If you say, “In theory it is 2, but in reality it’s something else”, then I ask you what evidence you have for that assertion? Because that’s all you’re doing–asserting it. It might be true, but you haven’t proved it must be true, you’ve asserted it must be true.
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### Author:  [@The\_Hamster\_King](https://boards.straightdope.com/u/The_Hamster_King)
#### Post date: [May 19, 2009, 3:53pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/36 "2009-05-19T15:53:51Z")
> [@straggler](#):
>
> But in my example, I ask the question: If something requires an infinite amount of steps, and each step must be taken sequentially, and each step takes _some_ time, then the “result” is unreachable.
Sure, if it takes an infinite amount of steps to get somewhere and each step takes a finite amount of time, then, yes, you will never arrive at your destination.
But if the steps are infinitesimally small then you can take an infinite number of them and still arrive.
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### Author:  [@straggler](https://boards.straightdope.com/u/straggler)
#### Post date: [May 19, 2009, 3:54pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/37 "2009-05-19T15:54:40Z")
> [@Lemur866](#):
>
> Why can’t it be time that is quantized, rather than space? Then you could have continuous space, but the smallest possible unit of time is 1/jillionth of a second.
I don’t see how that’s possible. Time and space are inextricably linked. You couldn’t have two incremental movements through space with no time elapsing.
> [@](#):
>
> Let me ask you a question. What does 1 + 1/2 + 1/4 + 1/8 + 1/16… equal?
It’s a meaningless equation, you could never add an “infinite” number of halves, halves of halves, halves of haves of halves, etc, to the number 1.
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### Author:  [@Kyrie\_Eleison](https://boards.straightdope.com/u/Kyrie_Eleison)
#### Post date: [May 19, 2009, 3:57pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/38 "2009-05-19T15:57:47Z")
> [@straggler](#):
>
> But this is theory. You can’t do this in reality. You can’t add an infinite amount of anything in reality. You can’t even write the equation, which is (one reason) why you have to trail it off with dots.
You can’t add a finite series of numbers in reality either. I’ve never seen a real two in my life, but even if I had a couple of them in my hands, I wouldn’t know how to “plus” them to turn them into a four. If you mean to say that you cannot manipulate the symbols that represent an infinite series algorithmically to arrive at their sum, then you’re incorrect. Sure, it’s a bit more involved than performing rote grammar school addition, but it can be done.
As far as not being able to write the equation goes, **Rolken** provided an example in which he had done exactly that. That’s an acceptable, even conventional, representation of the equation in question. If you mean that he can’t sit down and write out an infinite number of terms, you’re right, but of what consequence is that?
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### Author:  [@straggler](https://boards.straightdope.com/u/straggler)
#### Post date: [May 19, 2009, 4:01pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/39 "2009-05-19T16:01:27Z")
> [@The\_Hamster\_King](#):
>
> Sure, if it takes an infinite amount of steps to get somewhere and each step takes a finite amount of time, then, yes, you will never arrive at your destination.
>
> But if the steps are infinitesimally small then you can take an infinite number of them and still arrive.
How so?
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### Author:  [@straggler](https://boards.straightdope.com/u/straggler)
#### Post date: [May 19, 2009, 4:04pm UTC](https://boards.straightdope.com/t/what-ramifications-will-my-scientific-breakthrough-have/496889/40 "2009-05-19T16:04:17Z")
> [@Kyrie\_Eleison](#):
>
> You can’t add a finite series of numbers in reality either. I’ve never seen a real two in my life, but even if I had a couple of them in my hands, I wouldn’t know how to “plus” them to turn them into a four. If you mean to say that you cannot manipulate the symbols that represent an infinite series algorithmically to arrive at their sum, then you’re incorrect. Sure, it’s a bit more involved than performing rote grammar school addition, but it can be done.
Sorry, I’m not quite able to follow this. You can’t add a finite series of numbers in reality? Give me a finite series of numbers and I’ll add them for you. Or.. I’m not understanding you.
> [@](#):
>
> As far as not being able to write the equation goes, **Rolken** provided an example in which he had done exactly that. That’s an acceptable, even conventional, representation of the equation in question. If you mean that he can’t sit down and write out an infinite number of terms, you’re right, but of what consequence is that?
I was not suggesting that there is no mathematical symbol that “represents” infinite, I was highlighting the point that in reality, you can not sum infinity.
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