Smiling Dave:Reread the OP carefully.
Fair enough - even though it uses the intuitive example of a race, it talks only about reaching points.
Smiling Dave: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.
The immovable arrow paradox concerns time, not space. But in any case, all of these paradoxes are bound to the notion of extensions being composed of points, while at the same time to the notion of points having zero length, area, volume, etc. Another way of saying this is that they're bound to the notion that an infinitude can become a finitude.
I'd like to touch on something you said in an earlier post. You said that length can be seen as a function that maps between (necessarily infinite) sets of points and real numbers. But how are these infinite sets of points to be distinguished from one another in the first place?
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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?
Of course. The challenge facing us in a paradox is that they all look rock solid and we do not know what to abandon.
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But how are these infinite sets of points to be distinguished from one another in the first place?
If we are talking about the world of math, you make initial assumptions, say the axioms of settheory. You prove from them the existence of real numbers. You call each real number a point. Two points a and b are the same if they satisfy the equation a=b. If not they are different.
@Dave: Mathematical reasoning admits the use of arguments which have an infinite number of logical steps in order to reach a conclusion, so long as the terminus of that infinite chain of reasoning can be proven using some other method than blindly applying each step. I don't think Zeno believed that and I think that's the root cause of his paradoxes.
For example, consider the sum of any geometric series, such as 1/2+1/4+1/8... (a=1, r=1/2). Its sum does not merely "approach" a/(1-r), it is actually equal to a/(1-r) unless there's something wrong with the axioms of mathematics or the deductive reasoning used in the proof. My favorite infinite sum - because it is so shocking - was independently proved by Euler and Ramanujan using completely different methods: 1+2+3+4+... = -1/12.
One way to see why infinite chains of reasoning are admissable is to consider the very definition of natural numbers in terms of the successor function (Peano arithmetic). How many natural numbers are there? Well, there's at least one, 1, because we assume it. But we can make another: 2 = s(1). And another, 3 = s(2). In order that there be a finite number of natural numbers, we would have to show that this s() function "breaks down" at some point. Otherwise, there's no reason it shouldn't always produce another greater number each time it is applied, y = s(x), y>x. Hence, we have an infinite chain of s()'s extending as far as the mind's eye can see and beyond and there's nothing wrong with it because one property of a natural number is that - no matter how large it is - it always has a successor. There are two possible answers to the question "how many natural numbers are there?" - if you don't admit the existence of infinity the answer is "there is no number that describes how many natural numbers there are" but if you do admit the existence of infinity, then the answer is "infinity, in particular, aleph-0."
Once you admit the existence of infinity (which the ancient Greeks, to my knowledge, did not), all of Zeno's paradoxes disappear because he assumes you cannot have a completed/real/actual infinity.
Clayton -
First paragraph: Doesn't show exactly where his mistake is. Set up his syllogysm, point out the line that is in error, and why it is in error. Vague handwaving about him not grasping infinity doesn't cut it. What exactly did he not understand? At which exact line of his reasoning did he err? Why is he wrong at that particular line?
Second paragraph: I've quoted you chapter and verse from a universally respected math book that says quite clearly that you are mistaken.
The axioms of math do not say nor lead to a conclusion that the sum is a/1-r, unless we redefine sum to be least upper bound. Hardy spelled it all out for you. I challenge you to show me the chain of reasoning that proves from the axioms of math that it "actually is" a/1-r. If it's true, you should be able to dig it up rather easily.
Third paragraph: irrelevant [see comment on fourth paragraph].
Fourth paragraph: Existence of infinity does not imply Zeno's paradoxes disappear. He does not assume the existtence or nonexistence of completed/real/actual infinity. Whether it exists or not is beside the point. Completed real actual infinity will not come to your rescue and show that a geometric series actually sums to a/1-r.
a/1-r is still the least upper bound, nothing more nothing less. The epsilons and inequalities cannot be wiped away.
Like I say, show me anyone who says otherwise. Here is the set of all math books and mathemeticians who agree with you: {}.
@Dave: You're confusing analysis with proof. Sum_n( a*rn) = a/(1-r) actually, fully and without qualification. It's not "equal in the limit", it's not "equal to an arbitrarily small difference", it's completely, totally and exactly equal. The fact that n=0->oo is neither here nor there so long as you accept the exsitence of infinity.
You are misreading Hardy - far from saying that an infinite sum does not converge to its limit value. He says (in other words): "When we say that an infinite sum converges to its limit value, we also mean such-and-such about the partial sums of the series..." That's your own quote so I don't see why you're choosing to refute yourself and then imagine that that somehow is refuting us.
In my opinion where you and the analysis people go wrong is in trying to impart numerical "meaning" to every step of an infinite sum. Who cares about the partial sums? Maybe they don't even make sense. It doesn't matter so long as the formal structure of the sum equation is provably correct. If that's the case, then the rest is just pointless hand-wringing about the weirdness of numbers which could just be an indictment of our intuitive visualizations of numerical space. For example, the Koch snowflake encloses a finite area with a border of infinite length. Is that intuitive? Does that "make sense"? Not really, but it's true which is why I prefer to let the symbols do what the symbols want to do and avoid worrying about how to interpet the spatial relationships - if any - between them.
Can you link me to a summary of Zeno's argument which you would accept as close enough to syllogistic that I can respond to that instead of trying to read all of Zeno and rephrase in my own words?
Here is an example of two geometric proofs that infinite sums actually converge to the final value... no epsilons, no deltas, no limits, no fudge factors, absolute and total equality. These figures are the schematic answer to Zeno.
Smiling Dave:First paragraph: Doesn't show exactly where his mistake is. Set up his syllogysm, point out the line that is in error, and why it is in error. Vague handwaving about him not grasping infinity doesn't cut it. What exactly did he not understand? At which exact line of his reasoning did he err? Why is he wrong at that particular line? [Emphasis added.]
Dave, why do you make this demand of others in this thread when you haven't done this yourself?
I did, in the very first reply to this thread. I may have omitted that the flaw is in step 2 of Zeno's syllogysm, because i considered it obvious.
Are you talking about step 2 of the syllogism you present in this post? If so, then I'd say that steps 1 and 2 of the syllogism contradict one another. Step 1 says that the sum of an infinite series can never be reached by a finite partial sum, yet step 2 says that it can.
Smiling Dave,
Notice that Achilles never reaches the tortoise if and only if that series does not converge. It's not my problem if you don't understand what the series' physical interpretation is. Also, I sort of realized I made my argument more complicated than necessary. The same result can be found using the common linear kinematics formula
X = Xo + Vot + 1/2*at^2
So given constant velocity, we set the position of achilles and the tortoise equal.
V1*t = d + V2*t
and solve for t which is
d/(V1 - V2)
I just presented a more simpler proof above. But back to my original (long one), I think, for whatever reason, you have a hard time believing a series can equal a finite value. OKAY, but I know you at least accept that its bounded above by a finite value. That's pretty much all we need. The sum of the times for Achilles to move from position to position is bounded. Hence for any time greater than or equal to this bound, Achilles is at least as far as Tortoise in the race, but given that his speed is greater, he overtakes him.
Clayton,
1. Go to a university professor of mathematics that you trust and respect, and ask him if those pictures constitue a valid mathematical proof. He will explain that pictures have been rejected by the mathematical community for several hundred years as a valid method of proving anything. He will explain why they were rejected. He will explain that the universally accepted criteria for an acceptable proof for the last few hundred years is but one thing: A valid chain of deductive logic. No pictures allowed.
2. Even if we accept the notion that pictures can prove anything, the ones you linked to do not prove anything, because they are not accurate pictures. The flaw lies in those chains of dots that appear in all pictures of the infinite. You see, three dots in a picture mean but one thing: I, the artist, am not drawing something here. So you are presenting a fragment of a picture and claiming you have presented a complete picture. You rely on the reader to fill in the missing piece.
3. All those pictures are guilty of the fallacy known as begging the question.
For example, the picture that claims to prove that 1/2+1/4...=1 is guilty of begging the question. It assumes that the length of the side of the square composed of those infinite squares piled atop one another to be one. How does he know this?
But like I say, the main point here is 1. Points 2 and 3 are icing on the cake. Take care of 1. before bothering about the other two.
Here's a quasi-syllogism of the paradox (page 6):
• Achilles is in a footrace with the tortoise. • Achilles allows the tortoise a head start of 100 metres. • each racer starts runs at some constant speed • afer some fnite tme, Achilles will have run 100 metres, • During this tme, the tortoise has run 10 metres. • afer some fnite tme, Achilles will have run 10 more metres, • During this tme, the tortoise has run 1 more metre • ETC • Whenever Achilles reaches somewhere the tortoise has been, he stll has farther to go. • There are an infnite number of points Achilles must reach wherethe tortoise has already been. • He can never overtake the tortoise.
• Achilles allows the tortoise a head start of 100 metres.
• each racer starts runs at some constant speed
• afer some fnite tme, Achilles will have run 100 metres,
• During this tme, the tortoise has run 10 metres.
• afer some fnite tme, Achilles will have run 10 more metres,
• During this tme, the tortoise has run 1 more metre
• ETC
• Whenever Achilles reaches somewhere the tortoise has been, he stll has farther to go.
• There are an infnite number of points Achilles must reach wherethe tortoise has already been.
• He can never overtake the tortoise.
As far as I can tell, this can be reduced to the following:
1. There are an infinite number of points that a person must reach to cover any distance (i.e. before reaching any other point).
2. To reach an infinite number of points requires an infinite amount of time.
3. Therefore a person can never cover any distance.
I submit that Zeno conveyed it the way he did, with an ever-decreasing amount of distance to yet reach, in order to make more intuitive the notion of a finite distance being composed of an infinite number of points. Strictly speaking, then, the geometric series isn't necessary. I maintain that the whole thing hinges on the notion that points exist as entities in the real world.
...yet step 2 says that it can.
You forgot about the parenthetic statement in step 2