true.
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It's easy to refute an argument if you first misrepresent it. William Keizer
I do admit my earlier argument was a bit hand-wavy. I believe I've solved it now (original Tortoise/Achilles paradox). Here it is.
Let V1 and V2 be the speeds of Achilles and the tortoise respectively. Of course we assume that V1 > V2 and the tortoise has head start of d from the starting line. The original paradox goes like this. By the time, Achilles reaches d, the tortoise will have traveled to some distance d + a1 for some a1 > 0, by the time Achilles reaches d+a1, the tortoise will have reached d + a1 + a2. Since this goes on forever, Achilles never actually reaches the tortoise.
It takes a finite amount of time for Achilles to go from one point to another (to where the tortoise once was). So for instance, let t_1 denote the time it takes for Achilles to reach d, t_2 the time it takes for Achilles to reach d + a1, and so on. If we could show that
Sum (i = 1 to infty) t_i is finite, that shows that in finite time, Achilles reaches the tortoise.
By simple linear kinematics, t_1 = d/V1. (Note I am just using velocity * time = distance). During that time, the tortoise travels a distance of
t_1 * V2 = d/V1*V2. The time it takes for Achilles to travel to d/V1*V2 (from d) is precisely d/V1^2 *V2. Proceeding inductively, we obtain that
t_i = d*V2^(i-1)/V1^(i). Therefore,
Sum( i = 1 to infty ) t_i = Sum(i =1 to infty) d/V1*(V2/V1)^(i-1).
Notice that V2/V1 < 1 by assumption, hence we have a geometric series with the radius less than 1. The above sum converges to
d/V1 / (1 - V2/V1) - (the formula for the value of a converging geometric series).
In essence, we are done. What we have shown is that the sum of the times for Achilles to move from his position to the Tortoise's is a finite value (beginning with Achilles moving from the starting line to d). Hence, in finite time (the value of the geometric series), Achilles will be in the same position as the tortoise. With his velocity greater, he overtakes the tortoise.
Converging to a finite sum doesn't help us at all. Convergence is a statement about the existence of a least upper bound. A series with infinitely many non zero terms does not actually add up to it's least upper bound. Zeno is asking how do they actually add up enough for Achilles to win. Put it this way. A mathemetician can define a new type of operation, finding the least upper bound of an infinite series, and calling that operation by the misleading name of addition. But Zeno is arguing that Achilles does not perform the act of finding a least upper bound. He does simple plodding ordinary arithmetical addition, as he takes a step at a time. And doing that will never sum to the least upper bound. That is his question.
Put it this way. A mathemetician can define a new type of operation, finding the least upper bound of an infinite series, and calling that operation by the misleading name of addition. But Zeno is arguing that Achilles does not perform the act of finding a least upper bound. He does simple plodding ordinary arithmetical addition, as he takes a step at a time. And doing that will never sum to the least upper bound. That is his question.
Hmm, I think you're introducing hidden assumptions, such as that an addition requires some non-zero amount of time or effort. There is nothing inconceivable about an actual infinitude of sums. In fact, we can construct a perfectly consistent arithmetic which incorporates infinity as a number (cardinal or ordinal). It turns out that not only is there the ordinary infinity we are all familiar with but there are actually higher orders of infinity. There are infinitely many infinite cardinals, each strictly greater than all others before. This is the subject of study of large cardinal theory.
To summarize: Tell me why 1/2+1/4+1/8+... does not actually converge to 1.0 rather than just a "least upper bound" that "approaches" 1.0 "arbitrarily closely."
Clayton -
In other words, the series doesn't "add up" to s in the normal arithmatical sense.
Your "in other words" simply contradicts Hardy - he doesn't say that it doesn't add up to s, merely that when we say "this adds up to s" we are also say something about the partial sums of the series.
Infinite series are very non-mysterious. To wring your hands over infinity or infinite sums is like doubting the mathematical status of zero or negative numbers. No numbers exist. We say they "exist" only because they are useful abstractions that follow certain rules that we find interesting (because we have human brains that evolved in the physical world and which "package" the physical world in certain ways). Did you know there was a substantial philosophical argument regarding the use of zero? After all, zero denotes nothing yet whatever denotes the nothing is itself something, which is a contradiction. We can say the same for infinity - whatever denotes infinity is itself finite, which is a contradiction. But the fact is that it isn't a contradiction at all because we can differentiate between an object (abstract or real) and its representation.
In essence, we are done. What we have shown is that the sum of the times for Achilles to move from his position to the Tortoise's is a finite value....
I think there is still a flaw here, the same one as before. Let us say that your series sums to T. What you have shown is that every partial sum is less than T and that for any given epsilon, one can find an N such that the Nth partial sum [and all partial sums from the Nth one on] is greater than T+ epsilon.
But you haven't shown that Achilles actually is in the race for time T. Now we could argue that all we need do is keep an eye on a stopwatch and see that he is out there for T seconds or minutes or whatever, but that is not the point. Zeno is arguing that his analysis shows that Achilles doesn't live until time T, an dhe is asking where the flaw lies in his analysis. He is not asking for an alternate analysis, but for the flaw in his.
Hmm, I think you're introducing hidden assumptions, such as that an addition requires some non-zero amount of time or effort.
Where is this assumption hidden exactly? You are not even sure if it is time or effort. Pin it down. No handwaving allowed.
There is nothing inconceivable about an actual infinitude of sums.
And yet no mathematician has ever dared declare that, but limited himself to defining a sum of a positive infinite series as a least upper bound. Inany case, coinceivable or not, that is not the question. The question is qhat is the flaw in Zeno's analysis, which you have not shown. All you are saying is maybe there is another way of looking at it. Maybe there is [of which more later]. But the question is, what is the mistake, the flaw, the error, in his way?
. In fact, we can construct a perfectly consistent arithmetic which incorporates infinity as a number (cardinal or ordinal). It turns out that not only is there the ordinary infinity we are all familiar with but there are actually higher orders of infinity. There are infinitely many infinite cardinals, each strictly greater than all others before. This is the subject of study of large cardinal theory.
Irrelevant. Large cardinal theory has nothing to do with the issue at hand. See above.
The burden of proof is not on me, but on you. And you have increased the burden, because now you have to explain two things. First the original q, what is the flaw in Zeno's reasoning, and second why did Hardy [and every mathematician from Weirstrass to the present day] insist on emphasizing that all we get from summing a series is a least upper bound that appraoches the sum arbitrarily closely? Hint: there are logical difficulties doing it any other way. Mainly, they would have a burden of proof which they could not fulfill to provge that it actually sums to the sum.
Funny that Hardy and every other other mathemetician from Weirstrass on neglect to make such an important statement as your assertion. I mean, why were they so shy? Why did they never come right out and say that it actually sums to s? I mean, it's kinda important [if it is true, which it isn't]. And more, why did they bother with all this epsilon and N>n stuff in the first place? I mean, who cares? What really counts is that it actually sums to s. The answer, of course, is that least upper bound and all that is the best they could do.
Infinite series are very non-mysterious.
Agreed. Their "sum" is, by definition, the least upper bound [or accumalation point if there are negative terms as well]. Nothing more than that. But they do not ever mysteriously "actually converge", as you put it, to that least upper bound.
All the rest of the post is irrelevant, aka handwaving.
A bit more from Hardy:
the limit itself need not (and in general will not) be the value of the function for any value of n. This is sufficiently obvious in the case of φ(n) = 1/n. The limit is zero; but the function is never equal to zero for any value of n. The reader cannot impress these facts too strongly on his mind. A limit is not a value of the function: it is something quite distinct from these values,
More:
Definition I. The function φ(n) is said to tend to the limit l as n tends to ∞, if, however small be the positive number , φ(n) differs from l by less than for sufficiently large values of n; that is to say if, however small be the positive number , we can determine a number n0 ( ) corresponding to , such that φ(n) differs from l by less than for all values of n greater than or equal to n0 ( ).
[And of course, the sum of a series is merely the limit of its partial sums, by definition.]
He goes on and on, for pages and pages and pages [see Chapter 4], worrying himself about these epsilons. Why not come right out and tell us the important stuff, that the limit is actually acheived by some magic? Short answer: Because it cannot be shown that it does. Of course, you could run a foot race and see that Achilles does catch up to the tortoise, but that is not a mathematical proof.
A historical note:
Newton believed in "actually approaches". But he had no explanation for it. Bishop Berkeley pointed out the ridiculous flaws in Newton's "actually approaches" thesis, doing such a good job that the "actually approaches" approach was dead and buried from that moment on, never to be resurrected.
The least upper bound, epsilon, and n>N defintions of a sum of a series was not just an idle curiosity, but the only way possible to give a logical foundation for the concept of an infinite series. Only by stopping short of claiming that the series "actually sums" to its sum could they save themselves from the good Bishop's intense, justified, ridicule.
Friedmanite: The Achilles and the Tortoise paradox is a variant of : To walk x meters, you must walk x/2 meters, to walk x/2 meters you must walk x/4 meters, and so on. Since this goes on forever, I'm supposed to think I can't walk x meters? Wrong! An infinite number of half steps can be finite! As shown by the following geometric series, This thread should close now, but again people will continue talking out of their behind.
The Achilles and the Tortoise paradox is a variant of :
To walk x meters, you must walk x/2 meters, to walk x/2 meters you must walk x/4 meters, and so on. Since this goes on forever, I'm supposed to think I can't walk x meters? Wrong!
An infinite number of half steps can be finite! As shown by the following geometric series,
This thread should close now, but again people will continue talking out of their behind.
The highest math I took in college was statistics after college algebra. I don't understand the principle of which you are speaking. How can an infinite number of half steps be finite?
Take it a different way from Zeno's paradox. Let's say I have to get from point A to point B by only going in half increments the entire way. The distance between A and B is 10ft. The 1st 1/2 increment is 5ft. I move again and reach 2.5ft. I go again and reach 1.25ft. Yet again is 0.625ft. So on and so on to infinity without ever reaching point B.
I know my logic is sound, so I'm trying to understand the principle of your formula.
Smiling Dave: trik, The question never was about who will win the race. The question was about human reasoning. Zeno is giving an analysis of the situation in which every step of his argument, as well as the way he ties it all together, is rock solid common sense. And yet the conclusion is obviously false. Which leads to the real question. Obviously we must conclude that something is wrong with his argument. And yet we don't know what it is. What is the flaw? We aren't asking whether Achilles will win, but rather, why will he win. Saying that it's all subjective and other hand waving arguments get us nowhere, because we rely on our reasoning [if we have moved beyond the stage of couch potato] all the time. We would like to know what the flaw is in Zeno's rfeasoning, so that we do not make the same mistake in another situation where the outcome is not so obvious, and might lead to disastrous consequences. Saying it's all subjective anyway is another way of saying don't bother with math or logical thinking anymore, they give you unreliable results. Which is exactly the conclusion we want to avoid. We want to be able to rely on math and logic in the future, not sit around pointing with our hands like cavemen.
trik,
The question never was about who will win the race. The question was about human reasoning. Zeno is giving an analysis of the situation in which every step of his argument, as well as the way he ties it all together, is rock solid common sense. And yet the conclusion is obviously false.
Which leads to the real question. Obviously we must conclude that something is wrong with his argument. And yet we don't know what it is. What is the flaw? We aren't asking whether Achilles will win, but rather, why will he win.
Saying that it's all subjective and other hand waving arguments get us nowhere, because we rely on our reasoning [if we have moved beyond the stage of couch potato] all the time. We would like to know what the flaw is in Zeno's rfeasoning, so that we do not make the same mistake in another situation where the outcome is not so obvious, and might lead to disastrous consequences.
Saying it's all subjective anyway is another way of saying don't bother with math or logical thinking anymore, they give you unreliable results. Which is exactly the conclusion we want to avoid. We want to be able to rely on math and logic in the future, not sit around pointing with our hands like cavemen.
I was researching different solutions to Zeno's paradoxes, and one thing led to another, I found an ulterior page that discussed a twofold division in philosophies of mathematics: Platonism vs. formalism. Apparently, I'm a formalist. http://en.wikipedia.org/wiki/Philosophy_of_mathematics#Formalism
Perhaps this will better explain it. I had no idea that such a thing existed, but it makes sense. I disagree with a lot of Plato (although I do believe he and Socrates, from what we can tell, were extraordinarily intelligent. Thus I don't believe integers, numbers or real numbers exist at all. A man named David Hilbert beat me to it.
I might open a separate thread to see who is a mathematical Platonist or formalist here (granted there is simply that duality and no more options).
Smiling Dave:Sorry, you lack the background to grasp my response.
Sorry, I don't believe I do. So why don't you try again? Or else I might conclude that you can't respond to what I wrote.
Smiling Dave:Hey, plenty of other people here know math enough to answer your q's.
Those people are irrelevant in the context of our interaction.
Smiling Dave:But I'll address one point, just cause /i'm a nice guy. You were saying that 1/[2X2X2x2...] , where 2 is multiplied by itself N times in the denominator of that fraction, eventually becomes zero. But it never does. It's an immediate result of the axiom of Archimedes, which itself is the result of the least upper bound axiom, which itself is the result of the construction of the real numbers from the integers. All of which is taught in standard math courses. Oops. It's an immediate result of arithmetic. say 1/A=zero for some value of A. Multiply both sides by A and get 1=0, a contradiction. Therefore there is no such A.
Oops. It's an immediate result of arithmetic. say 1/A=zero for some value of A. Multiply both sides by A and get 1=0, a contradiction. Therefore there is no such A.
Yes, that certainly works out. I googled "one over infinity" and found this. So it only makes sense to say that the limit of 1/n as n approaches infinity is zero. That's not the same as saying 1/infinity = 0, which (as you point out) is an arithmetic statement. From the standpoint of arithmetic, 1/infinity is undefined, as infinity is not a number in the arithmetic sense.
At this point, then, I understand and agree with what you wrote here and here.
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Smiling Dave:Only if you define addition of a [positive termed] infinite series as finding its least upper bound, which Zeno argues that a person walking around on planet Earth does not do. His act of walking does not find least upper bounds, but does ordinary arithmatical addition, which is undefined for an infinite number of terms. Read the second paragraph of my post that explains.
Hang on a minute. If we assume that Achilles is running at a constant velocity, doesn't that mean every step he takes is the same length? If so, then there's no sense in saying that Achilles must first run half a step, then a quarter of a step, then an eighth of a step, and so on. Essentially, then, Zeno is equivocating over the term "run" - either it simply means "move", or it refers to human bipedal locomotion where both feet are off the ground before either foot touches the ground.
I think this goes along with my earlier point about extension. At the start of the race, how many points are between Achilles and the tortoise? Infinity. When Achilles has run 100 meters, how many points are between him and the tortoise? Infinity. When Achilles has run an additional 10 meters, how many points are between him and the tortoise? Infinity. And since points have zero extension (no length, area, volume, etc.), there's no way to determine the "next point".
If we assume that Achilles is running at a constant velocity, doesn't that mean every step he takes is the same length?
Reread the OP carefully.
As for your last paragraph, it is indeed another one of Zeno's paradoxes, the immovable arrow paradox. Of course, one paradox doesn't resolve another.
Dave, if a set of premises, taken together, lead to mutually contradictory conclusions, doesn't logic dictate that one or more of the premises must be abandoned?