Mathematical Conundrum or Not? Number Two
This one is probably more well known than the other one, so if it turns out to be too easy I'll move on to the next one.
This is one of Zeno’s paradoxes, I'll paraphrase and simplify it for those that like the short and sweet:
Say I want to cross the room, then I must first cover half the distance. Then I must cover half the remaining distance. Then I must cover half the remaining distance and so on to the end of time. So I can never get to the other side of the room. This would actually make all motion impossible, as to get anywhere I must first cover an infinite number of small intermediate distances.
For those that want the longer version here is the original:
This is one of Zeno’s paradoxes, I'll paraphrase and simplify it for those that like the short and sweet:
Say I want to cross the room, then I must first cover half the distance. Then I must cover half the remaining distance. Then I must cover half the remaining distance and so on to the end of time. So I can never get to the other side of the room. This would actually make all motion impossible, as to get anywhere I must first cover an infinite number of small intermediate distances.
For those that want the longer version here is the original:
The Tortoise challenged Achilles to a race, claiming that he would win as long as Achilles gave him a small head start. Achilles laughed at this, for of course he was a mighty warrior and swift of foot, whereas the Tortoise was heavy and slow.
“How big a head start do you need?” he asked the Tortoise with a smile.
“Ten meters,” the latter replied.
Achilles laughed louder than ever. “You will surely lose, my friend, in that case,” he told the Tortoise, “but let us race, if you wish it.”
“On the contrary,” said the Tortoise, “I will win, and I can prove it to you by a simple argument.”
“Go on then,” Achilles replied, with less confidence than he felt before. He knew he was the superior athlete, but he also knew the Tortoise had the sharper wits, and he had lost many a bewildering argument with him before this.
“Suppose,” began the Tortoise, “that you give me a 10-meter head start. Would you say that you could cover that 10 meters between us very quickly?”
“Very quickly,” Achilles affirmed.
“And in that time, how far should I have gone, do you think?”
“Perhaps a meter—no more,” said Achilles after a moment’s thought.
“Very well,” replied the Tortoise, “so now there is a meter between us. And you would catch up that distance very quickly?”
“Very quickly indeed!”
“And yet, in that time I shall have gone a little way farther, so that now you must catch that distance up, yes?”
“Ye-es,” said Achilles slowly.
“And while you are doing so, I shall have gone a little way farther, so that you must then catch up the new distance,” the Tortoise continued smoothly.
Achilles said nothing.
“And so you see, in each moment you must be catching up the distance between us, and yet I—at the same time—will be adding a new distance, however small, for you to catch up again.”
“Indeed, it must be so,” said Achilles wearily.
“And so you can never catch up,” the Tortoise concluded sympathetically.
“You are right, as always,” said Achilles sadly—and conceded the race.
Comments (163)
It is a very important paradox concerning series and calculus. Ask yourself: How is that you can have an infinite number of distances to cover, but yet still be able to cross the room?
He just said movement could be discrete. There would therefore not be an infinite number of distances to cover. Think Planck length.
In theory length is not discrete, only our ability to measure it is.
https://arxiv.org/abs/1608.08506
http://iopscience.iop.org/article/10.1088/1742-6596/701/1/012014/pdf
And so on.
I want to cross 10 meters but to do so first I have to get to the half way point.
10/2 = 5
Then I have to get to the half way point of 5 meters.
5/2 = 2.5
And so on.
2.5/2 = 1.25
1.25/2 = 0.625
0.625/2 = 0.3125
0.3125/2 = 0.15625
0.15625/2 = 0.078125
etc....
That will go on forever, it is called a series.
We have a discipline called physics which provides a potential solution to the problem as outlined. Have a read of the papers.
This is actually sequences and series that leads to an important mathematical concept that I studied in the college classroom and was tested on. Did you take those classes? Did you read the book? Here the answer is in this book. . .
https://www.amazon.com/Calculus-Early-Transcendentals-MultiVariable-Rogawski/dp/1464171750/ref=pd_lpo_sbs_14_img_0?_encoding=UTF8&psc=1&refRID=V13FMHA65YWJKE428B0F
That is essentially what you are doing, it is something people do when they don't want to have an actual discussion but they want to claim someone is wrong. If it helps focus on the fractions and forget we are talking about length.
It might be better if you have this conversation with yourself. That way you can get the answer you want.
Yes, I know you are eager to put me in place because you don't like me.
The explanation of Zeno's paradox in that article is useful here:
Given that motion is possible, either motion isn't a supertask (so it's discrete, not continuous) or supertasks are possible. Thomson's lamp is an attempt to show that supertasks aren't possible, and (apparently) supertasks being possible would contradict the Church-Turing thesis (although admittedly I don't know how sure the Church-Turing thesis is).
I don't understand, if you know the answer, why ask the question. To summarize - the sum of an infinite series can have a finite value. The proof that Zeno's particular series does add to a finite value is that we can walk through the door. See this video prepared by the University of Helsinki Institute for Finding Out Things, which I've used before....
...and likely will use again. Those Finns are great practical philosophers.
When I was a kid, I was taught, like Jeremiah here, that limits and convergent series and calculus "solve" Zeno's paradox. Greeks just didn't have as much as math as we do. Of course they didn't teach me about computability when I was 17.
Zeno's paradox in mathematics lends to the idea of infinite partial sums, which is the sum of infinite parts. While it can be divided an infinite number of times, it still sums up to a whole, hence we can cross the room. I am not sure how that fits in with supertask, but even if we decide Zeno's paradox is wrong, as clearly we can cross the room, it still highlights important mathematical concepts.
And I think that's wrong. That the sum of an infinite series can be finite doesn't explain how a supertask can actually be completed (or in this case even started; which point is the first point to move to)?
And I agree. (Should have made that clear. The computability approach actually makes more sense.)
What's curious is that even in a high school science class there's likely a kid who'll argue that you can't subdivide matter infinitely -- everyone's heard something about particles, even if it's hard to understand. But the idea that space is granular just seems crazy!
From the web - The Planck length is the scale at which classical ideas about gravity and space-time cease to be valid, and quantum effects dominate. This is the quantum of length, the smallest measurement of length with any meaning. And roughly equal to 1.6 x 10-35 m or about 10-20 times the size of a proton.
Obviously, mathematicians can keep on dividing smaller than that.
Using Excel I see that we get to that length after only about 120 steps in our series. As I said, Excel allows us to continue on smaller and smaller than that.
This doesn't follow. A good analogy to Zeno's paradox is that of counting the real numbers from 0 to 1. Before we can count to 1 we have to count to 0.5. Before we count to 0.5 we have to count to 0.25. And so on. But that we can sum this series isn't that we can actually count it.
The issue has nothing to do with the suggestion that it would take an infinite amount of time but with the fact that there isn't a first number to count to after 0. We're stuck at 0 with nowhere to proceed. It's the same logic with Zeno's paradox; there's no first position after the start point. We're stuck standing still. Calculus can't solve this problem.
The answer is sequence. If you ask me to get between 0.1 and 1 and give me a path that goes between 0.1 and 0.999999999999....n, of course I won't ever reach 1.
Under natural circumstances, movement is not measurement. The action of moving from one position in the sequence to the other does not alter the sequence itself, and does not provide a step for subdivision of the furthest point. That's why when I want to reach 1 from 0.1, I get to reach 1 after I've gone through a certain sequence in the serie.
And since infinite series are also sequential, they are no more an obstacle to reaching a specific endpoint.
?
I was referring to the Planck length in my previous response.
Ah. Cool.
"As you're indicating, you need to accept some arbitrary discrete amount in order to move"
"In other words, if we hypothosized that space were infinitely subdividable (i.e. not discrete), Zeno still makes it to the finish line, right?"
The track of land is basically an arc length and we are looking for the net change from point a to point b. That's calculus.
But that doesn't address Zeno's paradox at all.
Sure it does, Zeno's paradox is about getting from point a (the start of the race) to point b (the turtle) over infinite small intermediate distances. This is exactly what calculus is.
Again consider the analogy of counting the real numbers between 0 and 1. Calculus can't show that it's possible to count each of the infinite reals between them. It doesn't matter than we can sum a geometric series.
Calculus is about net change, so we'd be looking for the net change from 0 to 1. You can put that on a number line call it F(b) - F(a) and you have the Fundamental Theorem of Calculus.
We I get done with work, I can do up a visual I think which may help.
Treating a supertask just as a geometric series is misplaced.
I am not concerned with the parts as the parts are infinite, that is not the point, I am interested in the sum of the parts which is finite.
Which doesn't have anything to do with solving Zeno's paradox.
Yes it does, this distance from Achilles to the Turtle is finite, even after accounting for the movement of the Turtle. The only thing Zeno's paradox does is create an infinite number of points on a finite number line.
The number line between 0 and 1 is finite, and yet it is still impossible to count the real numbers between them.
But now let's apply this reasoning to my example of counting the reals:
It will take me some fixed time to count half the numbers between 0 and 1, say 2 seconds. How long will it take to count half the remaining numbers? Half as long—only 1 second. Counting half of the remaining numbers (an eighth of the total) will take only half a second. And so on. And once I have counted all the infinitely many reals and added up all the time it took to count them? Only 4 seconds, and there I am, having counted them after all.
There are two issues with this. The first is that it begs the question by asserting as a premise that counting the reals is possible (assume it takes 2 seconds to reach 0.5) and the second is that this premise must be false given that there isn't a first number to count to after 0.
The paradox is less to do with reaching the end and more to do with starting at all.
Just for clarity's sake: the problem you're pointing up is that the reals are uncountable. You could look at the rationals and say, there is no first one after zero, but this doesn't actually matter, because the rationals can be re-arranged into a list. You can just pick what to count as the first, the second, etc. You cannot do this with the reals.
Yes, rationals. That's the right term.
Quoting Srap Tasmaner
Except to keep this analogous to movement the counting has to be in ascending order. We don't jump to the half-way point and then back to some earlier point.
I'd have to brush up on this to answer properly, but my instinct is that that's an interpretation problem, essentially a matter of labeling. There's the standard interpretation, associated with the number line, of what order numbers are in, but they don't have to be. That may not look like much of an answer.
And infinite tasks of any kind are still beyond the capabilities of finite beings. The difference between the rationals and the reals is that even if you had infinite time or could count infinitely fast, you still couldn't count the reals.
Of course if space is granular, then our task is finite, yes?
Then to make it simpler, imagine a machine moving from A to B, where B is 1 metre from A. At set intervals the machine records the distance in metres it has travelled such that it can be mapped to the geometric series:
[math]0m[/math] [math]... \frac{1}{8}m, \frac{1}{4}m, \frac{1}{2}m, 1m[/math]
What is the first position on its recording after the start point, i.e. the first count after 0? There can't be one. It's impossible to even begin such a count, and so it is impossible to even begin such a movement.
We're concerned with solving Zeno's paradox. Summing a geometric series doesn't solve Zeno's paradox.
Not necessarily space but movement. I don't see a problem with continuous space but with movement occurring by "jumping" from one position to the next without passing through some half-way point. It seems the only way for movement to make sense at all. It might be unintuitive, but that's better than the incoherence that continuous movement seems to be.
This is getting confusing, so big thanks to @Jeremiah!
What you're pointing out now, I think, is that the rationals (or, I guess [math]\{ \frac{1}{2^n}\}[/math]) are not well-ordered under [math]<[/math], and that's true.
The reason we care is because we're talking about movement, and movement looks like a matter of going from one place to the next, where "next" is already defined in a particular way.
Which gives rise to another paradox? Given an infinite amount of time I could hop to all the rationals between my starting point and any destination (inclusive), but I cannot do them in order from closest to where I start to farthest (i.e., at my destination). So whatever that is, it doesn't look much like movement in the usual sense.
Getting back to our finite world, any finite subset of [math]\{ \frac{1}{2^n}\}[/math] is well-ordered under [math]<[/math], so that's what we're looking for I guess.
I am unclear on whether progress is being made, which is pretty freaking ironic.
Could you, though? Is there a way to prove it without begging the question, as this "solution" does, and taking as a premise that it takes [math]n[/math] seconds to count some percentage of them?
As with any supertask, I'd say that completing it is impossible. Unlike the ordered variation, where it's impossible to start, in this case it's impossible to finish. It would be a variation of Thomson's lamp where for each number counted you toggle the switch. Is the lamp on or off when you finish?
But isn't it simply the case that you are confusing the continuum with reality?
Don't tortoises and demi-gods obey the laws of physics rather than the rules of certain branches of mathematics?
Map/territory confusion.
The paradoxical claim is:
Quoting Jeremiah
The realm is ultimately physics. If the mathematical models (maps) cause paradoxes so much the worse for their application in this instance.
Quoting Michael
This doesn't seem consistent with your continuing focus on the maths.
I don't think so. It seems much more like an unfamiliarity with the laws that govern reality, and the mistaken assumption that these laws admit the (decoherent) continuum.
Quoting Baden
The statement of what exists in reality, how it behaves, and why, can be expressed in any language. Mathematics lends itself to a particularly efficient expression of the laws of reality, in a form amenable to testing. It's not a map, it's an assertion.
If it took me 1 second for each hop, it would take me countably many seconds to do all the rationals, the same number of seconds it would take to hop to all the natural numbers in order. The only point here is that you cannot hop to all the reals in countably many seconds.
A supertask has countably many steps completed in a finite amount of time. Seems like we could get go "infinitely fast" and get a finite amount of time, instead of taking an infinite amount of time at a finite speed. I'm not clear whether the definition of "supertask" precludes going infinitely fast in this sense. Maybe "infinitely fast" doesn't even make sense the way "infinite amount of time" does. Maybe that's part of the point?
Countably, sure, but still infinite. You never actually finish.
Quoting Srap Tasmaner
Thomson's lamp paradox shows that this leads to a contradiction.
At the end of what? When it reaches the largest integer?
That is the different between a convergent series and a divergent series. We know the distance between Achilles and the turtle converges to a finite number, so you are kind of arguing a moot point.
Also it should be pointed out that infinity is just an amount too big to measure. It could be discrete; it does not necessarily have to be continuous; such as, there are an infinite number of stars in the heavens.
Physics is ultimately in the realm of mathematics. The entire backbone of science is mathematics, as well as the rest of the skeleton. Basic human comprehension is rooted in mathematics, everything is in the realm of mathematics. That's the very reason I study it.
Maybe the concept of a supertask is what is being misplaced here. 0 is your starting point, now all I need is the end point, let's pick 1. Net change on a straight line is 1.
I was wondering whether it makes sense, yes.
I'm not sure your code tests that exactly. I mean, you specify that it runs infinitely fast, but the question you ask is just whether the last integer is even. There's no last integer.
Does this show that no matter how fast you go, counting integers takes an infinite amount of time? If so, that's interesting. It does quash the assumption that doing anything faster makes it take less time -- not true if the task is infinite.
Is this our conclusion?
So is it possible to count the rational numbers between them? No. And for the same reason it's impossible for any object to pass through all the [math]\frac{1}{2^n}[/math] points between 0m and 1m. That one can sum a geometric series doesn't say otherwise in either case.
It shows that no matter how fast you go you can never finish.
Who cares if I can count an infinite number of points or not, I don't need to I have calculus.
This is exactly what I said here. And given that completing a supertask is demonstrably impossible, it must be that motion isn't a supertask. But for motion to not be a supertask it must be that Zeno's premise that an object must pass through an infinite number of half-way points is false. Motion is possible only if it is discrete rather than continuous.
An object moving through an infinite number of half-way points in succession is like counting the rational numbers between 0 and 1. If the latter is impossible then so is the former. Being able to sum a geometric series doesn't mean that we can count the rational numbers, and nor does it mean that an object can move through an infinite number of half-way points. Calculus can't solve Zeno's paradox.
Ya, I read your post the linked pages.
Quoting Michael
It must mean motion is not a supertask OR supertasks are not impossible. At anyrate it is moot.
Quoting Jeremiah
Look at the first part of that sentence: given that completing a supertask is demonstrably impossible....
Maybe you should go back and read your own links.
Form one of your own links on Zeno's paradox.
Hey man, I am just citing the reference you gave me, maybe next time don't use Wikipedia.
Actually the conclusion in Wikipedia makes more sense than yours.
P1. Zeno's paradox shows that either motion is not a supertask or supertasks are possible.
P2. Thomson's lamp shows that supertasks are not possible.
C. Therefore, motion is not a supertask.
--
Now back on topic...
We already know that motion is possible and we know there are infinite parts between Achilles and the turtle. Whether you want to say those parts are continuous or discrete, that does not really matter, as you can't count them anyways, so they are effectively continuous and effectively infinite. We know Achilles can pass the turtle. This sounds like a job for calculus.
Thing is, we don't know that space is infinitely divisible. Atomicity versus Infinite Divisibility is still up for debate in science, and things like Zeno's paradox suggest that there is a smallest possible division of space.
Currently, I believe, Planck lengths are theoretically the smallest possible distances.
It truly does not matter. Infinite is just another word for more than I can count. Which is why we use calculus.
How can it not matter? If space is only finitely divisible, it solves the paradox.
No. Infinitesimal numbers are infinite not because it's more than what you can count in principle.
Infinitesimal infinite parts. There are more parts than I can count without approximating somehow or using calculus.
What laws of physics do Achilles and the turtle obey? What do these LAWS say will happen?
Does P2 show that all supertasks are impossible, or just a certain class of tasks?
Are we not labouring under the conflation that certain abstract attributes are the same thing as physical attributes that share the same name? Why should the abstract idea of infinity and its properties determine what can and what cannot happen in reality?
If the laws of physics tell us that Achilles will cross an uncountably infinite number of points in a finite time to catch the tortoise, then that is what he will do. Nothing physically infinite has occurred.
So,
1. What Achilles can and cannot do is not deducible from mathematics.
2. Stop confusing abstract infinity with physical infinity.
This is ambiguous. By "laws of physics" are you referring to our models or the way things actually behave? If the latter then I would argue that, if Zeno's paradox is correct in calling into question the notion of infinite divisibility of movement, then things don't actually move in a continuous manner but in a discrete manner. If the former then I would argue that, if Zeno's paradox is correct in calling into question the notion of infinite divisibility of movement, then our models are faulty.
Where it goes wrong, is to translate this mathematical argument into one about physical reality. Zeno wasn't arguing for a physical theory of motion or distance but was making points about mathematics by using an allegory, which shouldn't be taken literally. There isn't a paradox because mathematically he's correct.
For physical motion it's s = d/t and Zeno's formulas are simply the wrong ones to use.
There is precisely zero evidence for a discrete space-time.
General Relativity is a theory of a continuous space-time, and Quantum Field Theory os a theory of continuous fields in a continuous space-time.
And, despite your claim, String Theory also takes place in a continuous space-time.
Loop quantum gravity is a theory of discrete space-time, and causal sets is a theory of discrete space-time.
These are viable alternatives to quantum field theory and general relativity (themselves known to be incompatible and incomplete); they're not pseudo-science.
We don't a have a theory of everything. Nothing explains everything; every theory has shortcomings.
Quoting tom
That's not sufficient to dismiss any theory that suggests that it is discrete, hence why such theories are studied. And I would argue that the logic of Zeno's paradox and supertasks would suggest that such theories must be correct.
Quoting tom
My mistake.
The real numbers are represented in terms of length or distance with a number line. The number line is a model for real numbers. But the reverse is not true. [I]Physical[/i] length/distance can't be mapped onto the real numbers.
So, Zeno's paradox arises from confusing the model (length/number line) with the actual stuff, the real numbers.
Numbers not on the number line are imagery numbers and infinity, as infinity is not a number. If you mean something else I would suggest you don't use the phrase real numbers.
But THAT is not the paradox.
The series obviously converges, but how can it ever reach the limit, if the entity performing the infinite sum is a finite creature in finite time? That is the (apparent) paradox.
You will never encounter a distance expressed in anything more than the set of computable numbers.
That's the flawed assumption of Zeno's Paradox, that is what sequence and series resolved, a finite being can converge an infinite partial sums.
Watch, I a finite creature will converge an infinite partial sums to a finite number.
.3 +.03 + .003 + .0003 + .00003 . . .
On forever. That is an infinite series, do you recgonize it?
I could rewire it this way. .33333....
Which is 1/3 a finite sum of the infinite parts.
Why stop there?
Because in math ". . ." means the pattern repeats forever and that was easier to type. You know shorthand.
I am sorry, but this is where I lose respect for philosophy. Zeno's paradox certainly doesn't prove such theories "must be correct". That is a very bold claim off the back of a few conceptual paradoxes whose relation is a subjective classification called supertask. As far as I can tell, there is not even consensus that all supertask are impossible, or even relevant.
In mathematics Zeno's paradox does not prove partial sums, it lends to the notion yes, but those are backed by formal proofs that have been through the works. Deciding something must be correct just off the back of this paradox is sloppy and lazy. These paradoxes are guides, not proof.
Though experiments in physics are very real things.
I know they are a real thing, but then they are backed by real evidence.
You mean literally FOREVER? Literally infinite time?
Yes they are, and the whole point of Zeno's paradox is that it purports to be one, but it is not.
Partial sums only explains how theoretically in math you can have infinite points in finite space. It doesn't solve the paradox, rather it lends itself to justifying the paradox.
Yes. Unless a number is given at which to stop.
2, 4, 6, 8, ...
Implies all even numbers to infinity.
2, 4, 6, 8, ... , 200
Implies all even numbers until 200.
Sheesh, I must have missed that when I did my maths degree.
Not sure what you're getting at? Maybe your math degree needed to be supplemented by some English classes so you could learn how to express yourself clearly.
How does Achilles achieve a net change with infinte small intermediate distances to cross. Zeno's paradox only existed because no one knew how to sum infinite parts to a finite amount. Now we do, so paradox resloved.
Even if the net sum is finite, if it were infinitely divisible, you'd always have one more halfway point to reach. In fact, the paradox would damn us all to complete inertia, because there's halfway points between us and the halfway points, and halfway points to those, etc.
Quite. The paradox has nothing whatsoever to do with the finiteness of the sum, it has to do with a finite entity being unable to perform the infinite sum in finite time.
If my memory serves me correctly, there is a similar paradox referring to the impossibility of firing an arrow, or taking a single step.
Yes, there is.
Anyway, that's why I suggest that Planck units solve the paradox--space is not infinitely divisible.
Kant explains in the Critique of Pure Reason why it's hard for us to accept finite divisibility--it's outside of anything humans ever experience, so we can't wrap out heads around it.
Correct me if I'm wrong, but to converge, in mathematics, means a<1. To be able to complete the movement it would have to be a=1.
As a mathematical argument it's quite good and easily imagined but the allegory is just an aid for understanding the mathematical argument not intended as to say anything sensible about the real world. So once you realise it isn't about physical reality, the paradox disappears.
Not a smart move. All physical theories that work require the continuum.
I am not sure what you are referring to , perhaps you are thinking about one of the test for convergence. I would have to review them to be sure.
I don't know how to input math notation into here so you'll have to bear with my lack of proper notation but by definition the sum of an infinite series is the limit S= lim_n?? S_n, if the limit exist then the infinite series converges to the sum S. If the limit does not exist the infinite series diverges.
:grin: :up:
Suppose Zeus is reciting all of the natural numbers infinitely quickly.
What does it mean to say that he never finishes?
Does he ever recite the largest integer? No. There isn't one. That's a point for not finishing.
But is there any integer he never gets to? Nope. That's a point for finishing.
Suppose we do it differently: Give Zeus one second to recite "1", half a second to recite "2", a quarter to recite "3", and so on. Is there any natural number he hasn't recited after 2 seconds have passed?
He will get to any integer but at no point has he ever gotten to every integer.
Quoting Srap Tasmaner
This is Thomson's lamp paradox, except the question in this case will be "was the last number odd or even?" I suppose the paradox shows that in principle there must be some fundamental division of time that cannot be halved.
It really doesn't make sense for this program to ever terminate, and so it must be that one of the premises that leads to the conclusion that it does is necessarily false. That’s a proof by contradiction (which is exactly what Zeno’s paradox is).
Planck UNITS don't solve the "problem".
So to reiterate, what can and what cannot happen in Reality cannot be deduced from mathematics. In particular, the assumption that our understanding of the abstract infinity tells us anything about what is finite or infinite or possible in Reality is a mistake.
But of course, it is interesting to look at what our best theories tell us about motion, and how it happens.
I think these calculus solutions are just a bewitchment. They work as pure maths, but they can't apply to an actual supertask.
I suggested there are two criteria for "having finished a task":
(1) Having performed the last step;
(2) Having performed all of the steps, in some specified order.
For finite tasks, these are the same: "last" can just be defined as "all but one have already been performed."
But what about infinite tasks?
I see your argument as something like this:
1. If you have recited all the members of a set, there is some member of the set that is the last one you recited.
2. Zeus has recited the natural numbers in order. At step n, he recited the natural number "n".
3. By (1) and (2), there is a natural number z that was the last one Zeus recited.
4. By (2) and (3), z is the largest natural number.
5. Since there is no largest natural number, (2) is false.
I'm questioning step (5). We have the option of discarding premise (1) instead of (2).
Look at how criterion (1) works with finite tasks. Each time you perform a step, the number of steps remaining to be performed is one smaller. You're done when that number is 0. But this is just not true for infinite tasks. The number of natural numbers remaining to be recited is the same after reciting any finite number.
In fact, it looks to me like (1) is derivative of (2). We need a closer look at what it means to specify a task.
Suppose I give you a jar of marbles and tell you to count them. I come back half an hour later to find you haven't even started. Your explanation is that I didn't tell you what order to count them in. Fine. I know order doesn't matter, but evidently you don't, so I instruct you to pick one, take it out of the jar, add 1 to your running total, then pick any remaining marble as the next one. Go on until there are no marbles left.
Is it reasonable now to say you cannot count the marbles because I didn't tell you which one is the last one? No, of course not, because my recursive specification is enough. Here's how to start; here's how to continue; here's how to know when you're done.
Quoting Michael
I think we could play around with "first" as I have been with "last", but for many cases recursive specifications are exactly what we want, so I can just as well say that what you describe here is not a task at all.
Quoting Michael
I think if there's an intuition pump in the room, it's not calculus but Thompson's lamp.
I think it could be that some tasks we specify by specifying the last step -- maybe that's all we care about and are indifferent about what steps are or aren't taken. Really that seems more like a direction just to bring about a certain state of affairs.
But some tasks we naturally specify using recursion, and the infinite tasks we're talking about are clearly that kind. (Counting all the marbles is not the same as making the jar empty; the jar being empty is just how you know you're done.)
So is there an argument for (1), or an argument that it is not just a special case of (2)?
Counting up from 1 is a task but counting down to 1 isn't? Why is that?
Quoting Srap Tasmaner
I don't know. It seems a truism that if one has completed a series of consecutive tasks then some task was the final task. I know it's not much of an argument, but I honestly can't make sense of it being any other way.
Surely "I finished all the tasks" is the same as "I completed the final task"? But you're saying that one can have finished all the tasks without having completed a final task?
Which is why it can't ever be finished. So the simple answer to Thomson's lamp and Zeus counting is that it is incoherent for a supertask to occur in a way described by a convergent series.
Just in the sense that I don't know how to specify that task recursively. Is there a way? If not, is there some other way?
Quoting Michael
And my argument is that you're generalizing something that happens to be true of finite tasks. In effect your claim is that "infinite task" is a contradiction. My claim is that this is false, because (1) is a side-effect, and we needn't consider it part of the very definition of "task".
No, my claim is that completing an infinite task is a contradiction. If a task has been completed then, by definition, it wasn't infinite.
Can you specify the task of movement recursively? “Move to the first half-way point”? It’s a lot like counting the rational numbers between 0 and 1 in order.
Those are not the same.
Moving by half the remaining distance can be specified recursively; doing the rationals between 0 and 1 in order cannot be if by "in order" you mean "smallest to largest". There's no smallest rational > 0 to be the first.
There's also no smallest half-way point (by which I mean [math]\frac{1}{2^n}m[/math]). You can't move to 0.5m before moving to 0.25m, just as you can't count to 0.5 before counting to 0.25.
Oh I see -- my spec is "move each time by half the distance remaining to be covered" and that works recursively. Your way makes it impossible to start. I'm content for the moment with defining task as something
1. I know how to start,
2. I know what the next step is, and
3. I know when I'm done.
Just off-the-shelf recursion.
Edit: nevermind. I see your argument. Hang on.
You've switched back to talking about movement, where there is a strong intuition that each step in the task of moving from A to B can be subdivided into just as many steps as the original task. (I.e., a lot.)
I was talking about Zeus reciting the natural numbers (with geometrically increasing speed).
If you now want to say that each step can be divided into, let's say, "starting" and "finishing", then we'd be back in your regress of being unable to start. Do we like that argument? Now the problem is that every task is an infinite task, and we needn't worry about whether any of them can be finished because none of them can even be started.
I'm still leaving aside movement. My point for the moment is that if anyone can do anything, then Zeus can recite all the natural numbers in 2 seconds.
So you're saying that because we can sum a geometric series then we can coherently talk about this program terminating (although again; what is the value of [math]$state[/math] after the 2 seconds)?
This is where I think we have to treat this as a proof by contradiction to show that it doesn't make sense to suggest that speed can increase geometrically without end. Convergent series are necessarily inapplicable to supertasks.
Oh I think that's probably true, even though I'm feeling a bit uncertain about how supertasks should be analyzed.
(One reason I've been going through this is to get clearer about what your thinking is. Sometimes your objection is that a given task can't be finished, sometimes that it can't be started, sometimes, as here, that if it could be completed then something else you don't like could also be possible.)
Zeus, it should have been clear, is here as a stand-in for the power of mathematics itself, which isn't bound by many of the usual considerations. He is by stipulation magic. Thus if you encountered this on a test in a math class
17. If Zeus takes 1 second to say "1", one half a second to say "2", one quarter to say "3", and so on, how many seconds does it take Zeus to say all the natural numbers?
you'd answer "2 seconds", and you'd be right. I'm not arguing for informal pedagogy as serious philosophy, but I am interested in how the Zeus story does make perfect and uncontroversial sense in the right context.
*
Getting back to Zeno ... What turns out to be wrong with this family of arguments? It's not just about movement, for instance, but about there being any sort of change at all, about anyone, as I said before, ever doing anything. So what's going wrong here?
As to the lamp, its speed of switching would be infinite which could be interpreted to mean multiple things. Thomson had the premise that at any given moment the lamp is either on or off that he never questioned. Is the lamp on or off the moment it's switched? That's the answer to whether it's on or off at two minutes.
Yes, if you're counting up the natural numbers then it can't be finished; if you're counting down the natural numbers (or up or down the rational numbers) then it can't be started. The math of a convergent series can tentatively be used in either case to show how it could take 2 seconds despite the fact that there's still the fundamental problem of not being able to start or finish, and so the takeaway is that a convergent series can't actually apply to these supertasks.
Quoting Srap Tasmaner
Infinite divisibility is the problem, which was Zeno's target all along (although in his case he wanted to argue that all is one, whereas I'm suggesting that there must be some fundamental unit of space/time (or at least movement) that cannot be halved).
It's not about time. It's about there not being a first position to move to. Even if you assume you could count at infinite speed (whatever that would mean) you can't count the rational numbers between 0 and 1. There's no first number to count to after 0. And even if you could move at infinite speed, there's no first [math]\frac{1}{2^n}m[/math] point to move to after the start.
So it's not that it would take too long to finish but that starting at all is impossible.
It needs to be there regardless. Even if we considered the idea of instantaneous counting then it would still be impossible to count the rational numbers in order because there's no first number. This has nothing to do with speed or time but the very logic of starting a task that has no start.
Also, this is false, as movement isn't instantaneous. It takes time to travel from one point to the next.
Quoting BlueBanana
Or, in the case of locations, the locations between the two points, i.e. teleportation.
So, to clarify, if I count the natural numbers from 1 to 2 and say "one, two," that takes time, but if I move from point A to point B, without moving through the locations between, it takes no time, assuming I don't stay at the locations.
I say solid state in a metaphorical sense.
Pretty much this.
So you're agreeing with me that movement is possible if it is discrete rather than continuous. The issue is where one argues for continuous movement, and so that in moving from point A to point B one must move through all the locations between.
As for taking no time, I disagree. If one can move from A to B instantly and from B to C instantly and from C to D instantly and so on then one can move from one point to any arbitrarily distant point instantly, which empirically isn't the case. So even with discrete movement it takes time to move from one point to the next.
You don't have any empirical proof that movement is either discrete or continuous; however, mathematically, calculus can show net change and rate of speed across differences on a continuous line. Your supertask, on the other hand, is nothing but a subjective classification of ill fitting paradoxes.
What law of motion are you using? If you are using Newton's laws, then you might want to reconsider.
Anyway, when we consider even the classical equation for the time evolution of the state variables (p,q) or any function of them F(p,q,t), one is not immediately struck by the impossibility of motion, or the need to render space as discrete. Everything takes place in the continuum.
[math]
\begin{align}\frac{\mathrm{d}F}{\mathrm{d}t} &= \{F,H\} + \frac{\partial F}{\partial t}\end{align}
[/math]
If we leave Hamilton's equation behind and take a look at the quantum equivalent, the continuum is still there, but there are also some other features that might be worth noticing.
[math]
\begin{align}\frac{\mathrm{d}\hat F}{\mathrm{d}t} &= -\frac{i}{\hbar}[\hat F,\hat H] + \frac{\partial \hat F}{\partial t}\end{align}
[/math]
This is an equation for the time evolution of operators, so, instead of the law of motion being about a particular value of interest, it is about a matrix of values. This indicates, to me at least, that what is going on in reality is quite different from what our classical intuition tells us.
Perhaps the most famous implications of the Heisenberg equation are the uncertainty relations, which have to have some bearing on Zeno's paradox.
As I have mentioned before, you can make deductions about Reality from physical laws, you are mistaken if you think you can make similar deductions from abstract mathematical ideas.
Our best theories tell us space(-time) is continuous.
I see. The proposal makes even less sense than I had previously thought. Thanks.
Are you saying that Zeno's argument is sound, and that it shows that if space-time is continuous, then motion is impossible?
What about other variants, like the "starting and finishing" one I proposed?
Zeno's argument is mathematically consistent, there is a path from 0.1 to 0.9...n, and if I do walk this path, I will never reach the end of it. But it requires that I walk a very specific path. Most paths are consistently sequential in nature, meaning if you didn't subdivide n+1 by 1/2 by the second step, you have no reason to believe you are going to do so at the last step.
I'm totally not following this. Could you take another run at it?
Can someone remind me why anyone should assume that the abstract properties of an abstract idea (in this case infinity) should have any bearing on physical Reality?
Only the laws of physics can tell us what is physically infinite, and they tell us that when motion occurs, nothing physically infinite happens.
Think of it as a barrier to understanding, if you like. Until I know what's wrong with Zeno's argument, I don't really understand the physics.
There is nothing "wrong" with Zeno's argument beyond the PRESUMPTION that the properties of abstract entities are identical the the properties of real entities that bear the same name.
If Zeno is complaining about the mathematical notion of infinity, then we can refer him to Cantor. If he is complaining that motion does not make sense, then we can refer him to the laws of physics. That only leaves the complaint that what is mathematically infinite and what is physically infinite are not the same thing. Why should they be the same?
Only the laws of physics, whether you understand them or not, can tell us what is finite or infinite in Reality.
It's a mild disappointment though, that no one seems interested in examining what the laws of physics tell us about motion. I suppose they already know.