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.
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.”
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.
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.
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.
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.
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.
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.
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).
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.
Those people are irrelevant in the context of our interaction.
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.
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”.
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?
Fair enough - even though it uses the intuitive example of a race, it talks only about reaching points.
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?
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.
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.
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.
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.