Sadly, I am not good enough at maths and logic, so I can't post valid or interesting comments regarding this paradox. What I try to defend is that what keystone wrote is actually a paradox. — javi2541997
I think that if you're not good at maths and logic, I would think that you might not be in a good position to know if this is a valid paradox or just straightforward nonsense — flannel jesus
That's a complicated remark, because the numbers assigned are assigned in a specific context. If the staircase existed in the way that a physical staircase exists, the steps can easily be re-numbered in the new context (wanting to go up, rather than down). In that context, the first step up is numbered, even though it would not be numbered 1 in the context of going down. I think I recognized the problem when I said:-Focus first step up, not last step down- Unfortunately, the stairs are numbered in ascending order from the top down, so the first step up wouldn't be numbered 1. — keystone
My conclusion in the light of what you say is that the staircase up is not the same as the staircase down.But it would be a bad idea for him to ask whether the stairs up were the same stairs as the stairs down, or whether the staircase exists. — Ludwig V
Let's recast Zeno's ideas using contemporary terminology. In his era, the dominant philosophical view was presentism, which posits that only the present moment is real, and it unfolds sequentially, moment by moment. — keystone
No end to the staircase but the end is reached - Yes, this is the very issue I'm trying to highlight. And this has nothing to do with continuous acceleration or motion. — keystone
I suggested that movement was discrete, not that space was discrete. — Michael
The issue we have is that if there is no smallest unit of time then the counter is metaphysically possible, but this entails a paradox as the answer to what the counter shows after 60 seconds is undefined yet the counter will show something after 60 seconds. Assuming that paradoxes are metaphysically impossible then the counter is metaphysically impossible, and that suggests that it's metaphysically impossible for time to be infinitely divisible. — Michael
.What digit does the counter show after 60 seconds?
If there is no answer then perhaps it suggests a metaphysically necessary smallest period of time. — Michael
The paradox is that given the premise(s) what happens at the limit is undefined, and yet something must happen at the limit. — Michael
60 seconds will pass in the universe. The counter is just one thing that exists in the universe and it changes according to the prescribed rules.
So given the prescribed rules, when the universe is 60 seconds older, what digit will the counter show? — Michael
Ah, thank you for that. I sort of remembered the story but not the name/author.Bernadete's Paradox of the Gods: — Michael
But I've been arguing that the above reasoning is fallacious. Yes, each division must be passed, and each division is preceded by other divisions (infinitely many), and yes, from that it can be shown that there is no first division. All that is true even in a physical journey (at least if distance is continuous).It's the same principle as Zeno's dichotomy, albeit Zeno uses distance markers rather than barriers. Given that each division must be passed before any subsequent division, and given that there is no first division, the sequence of events cannot start.
Mathematically it is, and mathematics seems to have no problem with it. Yes, I believe certain axioms must be accepted, but I'm no expert there.The solution, similar to my proposed solution above, is that movement is not infinitely divisible
I don't find that to be a contradiction.If movement is continuous then an object in motion passes through every marker in sequential order, but there is no first marker, so this is a contradiction. — Michael
OK, if you deny the continuous nature of both space and time, then the number of iterations is finite, and the argument falls apart. My arguments presume a more mathematical interpretation: the continuous nature of both. If space is discreet, Achilles passes the tortoise after finite iterations. There would be a last one, after which the tortoise is passed. The conclusion of the inability to overtake doesn't follow because the premise upon which it is based becomes false.The false premise for Zeno is that each distance, and each time period will always be divisible. — Metaphysician Undercover
Presentism is still presentism even if time is continuous. You seem to describe a discreet view there, which runs into problems.In his era, the dominant philosophical view was presentism, which posits that only the present moment is real, and it unfolds sequentially, moment by moment. — keystone
Block view also defines motion as change in position over time, and thus motion is very much meaningful under the view.n this comprehensive perspective, motion is impossible. — keystone
All these are trips from beginning to end. Zeno's initial state (0) to the point where the tortoise is passed (1). In your OP, 0 is time zero, and 1 is time 1-minute.rip from 0 to 1-I don't get it. — keystone
This seems to contradict yourlelf. You say time is discreet, in which case the number of digit changes is finite, and there is an answer. You also seem to deny that the sum of the converging series is not 1, or that time somehow is obligated to stop, which is the same thing.Yes, that is the point. Your expressed conceptualization "60 seconds will pass in the universe" is not consistent with the conceptualization prescribed by the OP. But this conceptualization — Metaphysician Undercover
But I've been arguing that the above reasoning is fallacious. Yes, each division must be passed, and each division is preceded by other divisions (infinitely many), and yes, from that it can be shown that there is no first division. All that is true even in a physical journey (at least if distance is continuous).
But it doesn't follow that the journey thus cannot start, since clearly it can. — noAxioms
For example, Thompson's proposed solution to his Lamp paradox is to accept (i) and (ii) but to reject (iii). — sime
It seems impossible to answer this question. It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off. This is a contradiction.
Yes, in other words rejecting iii), namely the idea that one can finish counting an infinite sequence. — sime
There is a contradiction in the stated scenario: there's an END to the ENDLESS staircase. Better to ask where he is after a minute.Despite the staircase being endless, he reached the bottom of it in just a minute. — keystone
The answer to all those paradoxes is that you haven't defined what happens at the limit.
— fishfry
I think this is a misrepresentation. The paradox is that given the premise(s) what happens at the limit is undefined, and yet something must happen at the limit. This is a contradiction, therefore one or more of the premises must be false. — Michael
Why on earth must there be a behavior defined at the limit? — fishfry
That's the point. There's no paradox. You've simply neglected to tell me what the lamp does at 1, and you're pretending this is a mystery. It's not a mystery. You simply didn't defined the lamp's state at 1. — fishfry
The example is simply: after 30 seconds a single-digit counter increments to 1, after a further 15 seconds it increments to 2, after a further 7.5 seconds it increments to 3, and so on for 60 seconds, resetting to 0 at every tenth increment. — Michael
The problem set-up, which gives the axioms of the system we are working with, does not provide enough information to decide.What digit does the counter show after 60 seconds? — Michael
This seems to be an assertion, not a logical consequence of the premise. In fact it leads to a contradiction of the premise, hence demonstrating that the journey being able to start very much does follow from the premise, unless you can also drive that to contradiction, in which case the premise has been shown to be false.It does follow that the journey cannot start. — Michael
I swear you changed this. You had something that logically followed from your assertion. The conclusion that movement is discreet contradicts Zeno's premise that "That which is in locomotion must arrive at the half-way stage before it arrives at the goal". So by contradiction, the journey not being able to start doesn't follow from the premise.Therefore given that the journey can start then the premise that there is no first division is false.
No, the reals are not countable. The example we've been using is. There is no final count of steps in Zeno's dichotomy, so there is no demonstrated requirement of a 'first step' or any kind of final count of steps. Insistence otherwise seems to be leading to contradictions.Given that each division is some 1/n then such a movement is akin to counting all the real numbers from 0 to 1 in ascending ordering. Such a count cannot start because there is no first number to count after 0. — Michael
Applying this to Zeno's cases, or to the OP: All three seem to be true. I disagree that only two can be.As for the OP, its triad of premises are inconsistent. For only two of the three following premises can be true of a sequence
i) The length of the sequence is infinite.
ii) The sequence is countable
iii) The sequence is exhaustible — sime
By the law of excluded middle and non-contradiction, after 60 seconds the lamp must be either on or off. — Michael
We're being asked what the lamp "does at 1", so you saying that we must be told what the lamp "does at 1" makes no sense. — Michael
Given the defined behaviour of the lamp, will the lamp be on or off after 60 seconds? — Michael
If the answer is undefined, but if the lamp must be either on or off, then the behaviour is metaphysically impossible. — Michael
The paradox is resolved by recognising that the premise is flawed. — Michael
This brings to mind Sagan's quote "extraordinary claims require extraordinary evidence." We start with an extraordinary premise—the existence of infinite stairs and supertasks—and to resolve it, we resort to an equally extraordinary solution: he has infinitely long legs, enabling him to ascend to the top in just one stride. This doesn't strike me as a satisfactory resolution.It's always only a finite number of steps from infinity back to zero — fishfry
What you seem to overlook is that I'm beginning with a premise widely accepted within the mathematical community: the existence of actually infinite objects (like these infinite stairs or the set, N) and the completion of actually infinite operations (such as traversing the stairs or calculating the sum of an infinite series). If you do not accept the concepts of infinite sets or supertasks, then this paradox is not aimed at you. If you claim that an old woman is 2 years old, then you're not basing your argument on any widely accepted concepts of age.You described it as endless, and yet claim he reached the end... The "paradox" is just you choosing to invent a story with contradictory concepts. — flannel jesus
If there is a parallel staircase where the steps start at 1 and increase as you go up, then there must be a point where the step numbers on both staircases align. What would that step number be?But if a staircase down can be created by our, or your, say-so, another one, going up, can be created in the same way. — Ludwig V
Then your argument should be that supertasks are impossible, not that 60 seconds cannot elapse.But the end is not reached. — Metaphysician Undercover
Consider linear motion. If you plot position against time, are you suggesting that the resulting curve, when examined closely, appears stairstepped rather than smooth? If that's the case, what would be the width of these incremental steps? This presents the same issue, as I could always plot a more accurate curve of motion using even smaller incremental steps.I suggested that movement was discrete, not that space was discrete — Michael
This response does not adequately address my reinterpretation of Zeno's ideas.I wouldn't say that. — Metaphysician Undercover
I don't see how Zeno's paradoxes work any differently under presentism than under eternalism. — noAxioms
The issue arises if Achilles toggles Thomson's Lamp with each stride, leading to a contradiction: his feet suggest that the sequence is exhaustible, but his hand indicates it is not.For only two of the three following premises can be true of a sequence: i) The length of the sequence is infinite. ii) The sequence is countable iii) The sequence is exhaustible — sime
First, instead of using decimal, let's switch to binary, where the counter can only be 0 or 1. You suggest that quantum mechanics resolves this by introducing indivisible units, perhaps akin to Planck time. Looking to QM for inspiration is a good idea. However, the idea of Planck time doesn't hold up because in the abstract realm, we can always conceptualize a smaller increment. I propose that the correct solution is that at 60 seconds, the counter is in an unobserved state where its status fundamentally remains unknown. It could be either 0 or 1, so let's say it's in a state of (0 or 1). If we wish to steal technical terms from QM, we might refer to this state as being in superposition.Assuming that paradoxes are metaphysically impossible then the counter is metaphysically impossible, and that suggests that it's metaphysically impossible for time to be infinitely divisible. — Michael
What if the undefined state is fundamentally unobservable? This raises the question similar to "If a tree falls in a forest and no one is around to hear it, does it make a sound?" The limitations I'm suggesting on observation should not be surprising to a generation that has grown up in the era of quantum mechanics.The paradox is that given the premise(s) what happens at the limit is undefined, and yet something must happen at the limit. This is a contradiction, therefore one or more of the premises must be false. — Michael
Yet, it's impossible to determine what this limit might be. Would you argue that there is a limit to the slope of a line?It is metaphysically necessary that there is a limit to how fast something can change — Michael
Suppose that with each flick of the lamp, the lampholder adds another term to a cumulative total: first 1/2, then 1/4, then 1/8, and so forth. What does his calculator show at 60 seconds? Why on earth must we assert that it displays 1? After all, the narrative doesn't specify what his calculator must indicate at 60 seconds. It seems to me that you're contesting the very idea which you support - that infinite series can have definitive sums.Why on earth must there be a behavior defined at the limit? — fishfry
Yeah, that law needs updated. I propose "for every proposition, either this proposition or its negation can be measured to be true." This introduces the possibility of a third, unmeasured state—when we're not observing, the lamp could either be on or off, placing it in a state of being (on or off).By the law of excluded middle and non-contradiction, after 60 seconds the lamp must be either on or off. — Michael
The "ground", thus defined, is a point that cannot be reached from the stairs, being infinitely far below it. Similarly, you cannot reach the stairs from that point, as every stair is infinitely far above it. That's why the man on the "ground" can't see any stairs as described in the OP story. They are all too far away above him. By making such a definition, we are essentially dividing our thought-experiment-world into two parts, neither of which can reach the other. — andrewk
This brings to mind Sagan's quote "extraordinary claims require extraordinary evidence." We start with an extraordinary premise—the existence of infinite stairs and supertasks—and to resolve it, we resort to an equally extraordinary solution: he has infinitely long legs, enabling him to ascend to the top in just one stride. This doesn't strike me as a satisfactory resolution. — keystone
Suppose that with each flick of the lamp, the lampholder adds another term to a cumulative total: first 1/2, then 1/4, then 1/8, and so forth. What does his calculator show at 60 seconds? Why on earth must we assert that it displays 1? — keystone
Presumable it would be at (the number of steps in the first staircase divided by 2). So?If there is a parallel staircase where the steps start at 1 and increase as you go up, then there must be a point where the step numbers on both staircases align. What would that step number be? — keystone
Yes. With a real staircase would exist in both contexts and independently of both of them. Then the first step down is the last step up and the last step down is the first step up. But the last step down is not defined, which means it can't be reached. That's why the game is fascinating and frustrating at the same time, even though it is what I would call, arbitrary.Mathematically it has some meaning, but it never has physical meaning, as several have pointed out. — noAxioms
What you seem to overlook is that I'm beginning with a premise widely accepted within the mathematical community: the existence of actually infinite objects (like these infinite stairs or the set, N) and the completion of actually infinite operations (such as traversing the stairs or calculating the sum of an infinite series). If you do not accept the concepts of infinite sets or supertasks — keystone
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.