"Complete" does not apply to infinite series, by definition. — Ludwig V
Yes. I discovered that after the tea-time hour, it turns into a grumpy tortoise.Everyone knows that tea is taken at at the tea time hour and that one is not to dawdle still drinking it, not even hypothetically, not even gedankenishly, past the tea time hour. — TonesInDeepFreeze
Since then, however, it has been discovered that citizens will still get themselves into a hopeless muddle even if they practice all day. So the betting industry is safe.too much public exercise of arithmetic would allow citizens to become too number savvy — TonesInDeepFreeze
We need only take it for granted that it does change at the rate stated in the puzzle. — TonesInDeepFreeze
Yes. I was careless.The lamp puzzle doesn't require anything to occur in an infinitely small amount of time. — TonesInDeepFreeze
But here's my problem. If I take one step, do I execute one task, or many? The argument of the paradox is that in order to take my step, I either must execute infinitely many tasks in a finite duration or fail to complete (or even begin) my step. I maintain that the issue is about how you choose to represent my step, and representing my step as composed of infinitely many segments is only one of many representations.But I do understand Thomson's point that there cannot be infinitely many task steps executed in a finite duration. — TonesInDeepFreeze
I think everyone agrees that there's misuse of something going on here. There's disagreement about what is being misused and how.So it seems your analogy is between misuse of imaginary numbers and misuse of infinite numbers. — TonesInDeepFreeze
I never meant to deny that.However, whatever you mean by 'complete', there are infinite series that have a sum. — TonesInDeepFreeze
Did Thomson make that argument? Was that part of his answer to the paradox? — TonesInDeepFreeze
If something is infinitely divisible, and you are to say into how many parts it shall be divided, you have alternatives from which to choose. This is not to say that is one of them. And if something is infinitely divisible, then the operation of halving it or halving some part of it can be performed infinitely often. This is not to say that the operation can have been performed infinitely often.
Mathematics doesn't say there is no limit to the ways objects may be divided. — TonesInDeepFreeze
I hate to be difficult, and I'm not really disagreeing, just amplifying. But I would like to add that if the pipe is cut in half lengthways, neither half is a (newly individuated) pipe. You have two gutters (or that is what I call them). And that if I paint half the pipe blue and half red, the halves do not become objects in their own right, but remain halves of the same pipe, even though they are of different colours."One half" in practise does not have the same meaning as "1/2" in theory. — Metaphysician Undercover
And that if I paint half the pipe blue and half red, the halves do not become objects in their own right, but remain halves of the same pipe, even though they are of different colours. — Ludwig V
Perhaps I wasn't clear. I never intended to say that. I thought this was crystal clear:-Why do you say that the two gutters are not distinct objects. — Metaphysician Undercover
You have two gutters (or that is what I call them). — Ludwig V
It depends what you mean by "divide" and by "object".However, dividing an object in two always produces two new objects (as well as the waste material). — Metaphysician Undercover
This is not exactly wrong, but requires that you recognize that "division" and/or "object" may change their meaning in some contexts. That's why I said:-However, dividing an object in two always produces two new objects (as well as the waste material). — Metaphysician Undercover
I said that the two painted halves do not become objects in their own right, meaning separate, distinct objects. You may argue that this is not dividing the pipe, or that each half becomes a distinct object. I don't mind what you choose. This shouldn't be too difficult for you, since you said earlier:-... if I paint half the pipe blue and half red, the halves do not become objects in their own right, but remain halves of the same pipe, even though they are of different colours. — Ludwig V
And, divisibility is dependent on the type of thing to be divided. Therefore, when it comes to division one standard does not fit all things, and the principles of division must be specifically designed for the different type of things to be divided. — Metaphysician Undercover
Oh, there's no doubt that no-one could actually cut the pipe into halves, and then divide one of the halves into halves ad infinitum. But painting the pipe shows that it depends what you mean by "divide" and/or "object". You could say that painting the pipe is a theoretical, not a practical division; that would be a bit at odds with ordinary language, but we are not speaking ordinary language here.But the fact of waste in any act of division nullifies the validity of the supertask. — Metaphysician Undercover
I said that the two painted halves do not become objects in their own right, meaning separate, distinct objects. You may argue that this is not dividing the pipe, or that each half becomes a distinct object. I don't mind what you choose. This shouldn't be too difficult for you, since you said earlier:- — Ludwig V
But painting the pipe shows that it depends what you mean by "divide" and/or "object". — Ludwig V
OK. So I guess measuring an object would count as "distinguishing different parts" of it even if the line that I draw does not correspond to any pre-existing difference or discontinuity in the object.I would say that painting a pipe two different colours is not a case of dividing the pipe. To use your terminology, you are distinguishing two halves without separating them. This does not qualify as "dividing". When I look at an object I can distinguish different parts of the object, and even draw lines on its surface, and all this is done without dividing the object. — Metaphysician Undercover
I maintain that the issue is about how you choose to represent my step, and representing my step as composed of infinitely many segments is only one of many representations. — Ludwig V
he only argued that "talk of super-tasks is senseless." — Michael
Mathematics doesn't say there is no limit to the ways objects may be divided.
— TonesInDeepFreeze
That really depends on what you mean by "object". If you mean "physical object" then mathematics doesn't say anything about them at all, and whether or not some physical object is infinitely divisible is a matter for empirical investigation. — Michael
As it stands one cannot have half a photon (or any point particle) and if spacetime is quantized then there is a limit to how far one can divide some section of it into two. — Michael
OK. So I guess measuring an object would count as "distinguishing different parts" of it even if the line that I draw does not correspond to any pre-existing difference or discontinuity in the object. — Ludwig V
The poster claimed that I equivocate about this. On the contrary, I am clear of quite clear of the distinction and none of my comments employ any equivocation regarding it. — TonesInDeepFreeze
The paradoxes discussed don't require splitting material objects. — TonesInDeepFreeze
Where in the paper does Thompson say that? — TonesInDeepFreeze
And one may take it as a premise or as an established fact that there is a shortest distance and a shortest duration. But perhaps one may also logically take a hypothetical premise that that is not the case. — TonesInDeepFreeze
Sticking to the supposition of this puzzle creates confusion. The only possible solution is to look at it differently, not being hypnotized by 1/2, 1/4, ..... But I accept that it is your choice.Of course, if you deny the supposition of the puzzle, then it may be easy to dispense the puzzle. But one may wish not to take the easy way out but instead grapple with the puzzle under the suppositions it makes. — TonesInDeepFreeze
That's right. The difficulty is, I think, the assumption that "divide" means exactly the same thing in all contexts, taking the case of cutting something into pieces as the model. It obviously doesn't apply to numbers, or to space or time.There are not two x/2, each one a separate object made by dividing x. — TonesInDeepFreeze
No, it's confusing theory with practice, abstract with concrete and not understanding that infinity means endless (but not necessarily limited)It is taking this hypothetical premise – that there is no smallest unit of space and time – that gives rise to such things as Zeno's Paradox, Bernadete's Paradox of the Gods, and Thomson's lamp. — Michael
Sticking to the supposition of this puzzle creates confusion. The only possible solution is to look at it differently, not being hypnotized by 1/2, 1/4, ..... But I accept that it is your choice. — Ludwig V
The difficulty is, I think, the assumption that "divide" means exactly the same thing in all contexts — Ludwig V
not understanding that infinity means endless (but not necessarily limited) — Ludwig V
And one may take it as a premise or as an established fact that there is a shortest distance and a shortest duration. But perhaps one may also logically take a hypothetical premise that that is not the case.
— TonesInDeepFreeze
It is taking this hypothetical premise – that there is no smallest unit of space and time – that gives rise to such things as Zeno's Paradox, Bernadete's Paradox of the Gods, and Thomson's lamp.
So if these arguments prove that contradictions follow if we assume that there is no smallest unit of space and time then as a refutation by contradiction it is proven that there is some smallest unit of space or time. — Michael
But that has no bearing on the principle of division. — TonesInDeepFreeze
It is incorrect to infer that infinitely many tasks may be completed in finite time from the premise that there is no finite upper bound to how many task may be completed in finite time. I would put it this way: For for any finite number of tasks, there may be a completion of all the tasks. But that does not imply that there may be a completion of all of infinitely many tasks. — TonesInDeepFreeze
(1) We may question P2. — TonesInDeepFreeze
(2) C1 doesn't follow from P1 and P2. — TonesInDeepFreeze
But that does not imply that there may be a completion of all of infinitely many tasks. — TonesInDeepFreeze
S has a lower bound in S if and only if there is a member of S that is less than or equal to every member of S.
S has a lower bound if and only if there is an x such that x is less than or equal to every member of S. — TonesInDeepFreeze
Thank you for the clarification. I must admit, I was a bit puzzled by "bound". I'm used to "limit". This clarifies something that was puzzling me - how one could describe the relationship between the 0 and 1 to the steps of the series. This seems to work very well.The range of the sequence 1, 1/2, 1/4 ... has no lower bound in the range, but it has a lower bound (the greatest lower bound is 0). — TonesInDeepFreeze
Are you suggesting that it might be the case that all of infinitely many tasks can be completed? What would the last task be?Again, even if there is no completion of all of infinitely many subtasks, it is not entailed that there is a finite upper bound to how many may be completed, so, a fortiori, it is not entailed that each of the subtasks is not completed. — TonesInDeepFreeze
There is the possibility that he doesn't recognize metaphysical possibility. Not everyone does.If I'm not mistaken, Thompson recognizes physical possibility and logical possibility, which are at least fairly well understood, but he doesn't mention metaphysical possibility. That's not to say that the notion of metaphysical possibility should be ruled out, but only that it requires explication. — TonesInDeepFreeze
P1. If (A) the first task is performed at 11:00, the second at 11:30, the third at 11:45, and so on, then (B) infinitely many tasks have been completed by 12:00
P2. B is impossible
C1. Therefore, A is impossible — Michael
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