SD,
Do you know .999 repeating can be expressed as
9/10 + 9/100 + 9/1000 + 9 /10, 000 + ....
which has sum
9/10 / (1 - 1/10) = 1
Do you also rejected .999.... = 1 ?
Smiling Dave:You forgot about the parenthetic statement in step 2
Which one? "[in the sense of the sum of an infinite series]"?
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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.
*sigh
Dave, you need to pay attention to what I'm saying. This isn't my first go-round on this particular topic and I actually have a minor in mathematics, so it's not like I don't have any clue what I'm talking about.
The picture is just a summarization of the deductive argument. Look again, it's not "just a picture" - it's a schematic.
1. PRQ is similar to TPS
2. Therefore, the tan of PQR (= PR/PQ) and the tan of TPS (=TP/TS) are the same
4. Therefore, the proportion of PS:QR is equal to the infinite sum constructed along the base of TS through nesting similar triangles ad infinitum – 1/(1-r) = 1 + r + r2 + …
The equal-sign here is unqualified. And that’s the point. Who cares whether you like or dislike the ellipsis or the use of “ad infinitum”, the point is that the ellipsis’d expression is actually and fully equal to the closed-form equation. To venture going to the extreme, the two expressions are just different ways of writing the same thing.
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.
Not at all - the whole picture is there from the beginning, see step 4 above.
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?
You're confusing summation with sub-division. In the second figure, we are not summing anything, we're dividing a known length infinitely many times. The key to the inductive method lies in proving that the "next step in the chain" must necessarily follow based on the existing, specified condition. Once that is proved, then the entire (infinite) chain of logic must hold or we must throw out logic itself.
Clayton -
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.
True. But the left hand side of that equation is not a sum. It is a least upper bound. The equation states that the least upper bound of the series is a/1-r. Not that it "actually sums" to a/1-r.
Certainly it is not saying that if you fill up the square of side 1 with an infinity of boxes that they will ecatly fill up the square, as in your picture. The whole reason Hardy goes on and on and on in Chapter Four is to emphasize that he is not saying the latter at all, and that the equation is not to be understood as saying that.
I am not misreading Hardy. You have misquoted him. The most egregious mistake is your insertion of the word "also". The equation is not also saying something about partial sums. It is saying only something about partial sums.
It doesn't matter so long as the formal structure of the sum equation is provably correct.
But it's not provably correct. There is no proof.
And I am amazed to see you dismissing all of analysis as pointless hand wringing. In fact, I don't see the point in further discussion of someone who thinks that, because it shows a Grand Canyon sized lack of understanding.
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?
I made the effort. You should to. Unless you want to refute him without actualy knowing what he said.
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It's easy to refute an argument if you first misrepresent it. William Keizer
Freidmanite,
1. It's not me that has a hard time believing a series can equal a finite value. It's G. H. Hardy, and every other mathematician who ever lived or lives, starting from the nineteenth century.
Clayton gave an excellent mental picture to clarify what is going on. Say you have a square of area 1. You lay on that square little squares of size 1/2. 1/4. etc, all of them. Does the equation you see in math texts, that 1/2 + 1/4+ ...=1 assert that the big square will be exactly filled? I claim, and have quoted chapter and verse here from Hardy, and nobody has yet found any mathematician who disagrees, that it most certainly does not assert that.
2. And once again, Zeno admits that there may exist a chain of reasoning [such as one you may have provided] that will prove Achilles passes the tortoise.
But he is not looking for what you are providing. He is asking a different question altogether. His question is what is the flaw in the chain of reasoning that he has presented? And you have not shown the flaw in his reasoning. You have merely provided alternate reasoning.
Hardy's book is available free online right here:
http://www.gutenberg.org/ebooks/38769
Autolykos: since points have zero extension (no length, area, volume, etc.), there's no way to determine the "next point".
since points have zero extension (no length, area, volume, etc.), there's no way to determine the "next point".
Now you're getting somewhere. The key, my friend, is that there are no points at all! This will help explain a bit: http://en.wikipedia.org/wiki/Formalism_(mathematics). I think you might find it interesting as you are a semantics relativist.
Basically, people are giving me a hard time because I haven't figured a solution out within the system of mathematics. But to do so leads nowhere, as the rest of the members in this thread have implicitly found out. Without admitting that all of reality is one thing, and change is illusion, as Zeno posited (being an apologist of Parmenides who thought the same thing), there is no mathematical solution to the paradox.
People who are trying to mathematically prove that Achilles will catch up to and surpass the tortoise are clinging to a Platonic belief that numbers exist. http://en.wikipedia.org/wiki/Philosophy_of_mathematics#Platonism That's why they will not give up, for to give up means to admit that their system is unreal, which I maintain it most certainly is, merely, a subjective system.
Perhaps the words of Parmenides, Zeno's supposed teacher or partner, will offer ample context as to Zeno's intention:
One path only is left for us to speak of, namely, that It is. In it are very many tokens that what is, is uncreated and indestructible, alone, complete, immovable and without end. Nor was it ever, nor will it be; for now it is, all at once, a continuous one.
I am amazed to see you dismissing all of analysis
Read what I wrote again - of course I do not dismiss analysis. The point is that analysis is the wrong tool for investigating methodological questions such as whether it is appropriate to treat infinity as a proper mathematical object, whether it is appropriate to use mathematical induction (which, by the way, analysis does do), and so on.
Every time you use the number pi, you are implicitly using mathematical induction since there is no closed-form expression for the number pi. That unassuming little Greek letter masks an infinite series - are you telling me that no math equation that uses pi expresses anything more than some kind of least-upper-bound? epi*i merely approaches -1?? Seriously?
As an aside, the most elegant expansion of pi is, in my opinion, that given by Leibniz, perhaps the most ingenious man who ever lived.
Smiling Dave,
You believe convergent infinite series are bounded above, right? Well clearly, they're also bounded below. So what you're saying is that an infinite series can be less than say 1 and greater than 0, and so lies in the open interval
(0, 1)
Do you know a number between 0 and 1 that is not finite?
Do you also rejected .999.... = 1
Why do you make this personal? It's not me, it's all mathematicians.
In any case, I accept it as saying that the least upper bound of the series on the left hand side is 1.
Yes.
Did you even do a quick google search? I don't know of a single mathematician in the world who doesn't accept
.999..... = 1
Since nobody has conceded to Vive's and my interpretation, one that is validated by empiricism and one which nobody in their right mind would bet against, perhaps a few of you would like to see how Aristotle approached Zeno's Achilles paradox. He did so by challenging the discrete moments of time that Zeno was assuming existed:
"One way to describe Zeno's argument is to time the passage of the object as it passes between the marked points. Suppose we believe that it takes a certain amount of time for the object to pass from A to B. Then (assuming travel at a constant speed), it will take 1/2 of the time to pass from C to B, and 1/4 = (1/2)(1/2) = (1/2)^2 of the time to pass from D to C, and 1/8 = (1/2)(1/4) = (1/2)^3 of the time to pass from E to D, and so on. In particular, the time periods that elapse between successive pairs of points must be shrinking in size. This does not conform to a discrete view of time, for if time were made up of indivisible instants, it would be impossible to consider a time period any shorter than that indivisible instant. As the time periods in the argument will eventually become shorter than any given indivisible instant, no such indivisible could exist."
http://cerebro.xu.edu/math/math147/02f/zeno/zenonotes.html
Basically, Dave wants to rewind the clock to 500 BC when the Pythagoreans were up in arms about the square root of 2. Dave, I have a challenge for you - use a compass and straight-edge to draw a square. Now, place the straight-edge across opposite corners of the square and draw a line. What is ratio between the length of the diagonal and a side of the square? Oh noes!!!
There is no answer to that question that does not involve: an infinite number of steps, an infinite number of digits, the use of an infinite magnitude, etc. This is why the Pythagoreans were so upset. On the one hand, infinity is apeiron - chaos. On the other hand, it's staring you right in the face. That innocent little diagonal is standing there saying "nyah nyah nyah nyah, you can't make me finite."
The picture is just a summarization of the deductive argument.
OK, let's examine the deductive argument.
Begging the question. You are not proving 4, just asserting it. In particular, the phrase "infinite sum" has to be defined before it is used. Which is exactly the problem you are ignoring. The best and only defintion the greatest mathematical minds could come up with was that an infinite sum is a least upper bound. I quoted chapter and verse several times.
The equal-sign here is unqualified.
This is your own personal opinion, but that picture certainly does not prove anything, as you admit. And I claim that your personal opinion is not shared by anyone. I hope you don't expect me to quote every single math book ever written. I showed you Hardy. Show me anyone who disagrees.
In the second figure, we are not summing anything, we're dividing a known length infinitely many times.
The advantage of a picture is that it represents something that can actually be seen. But no one has actually sub divided anything infinitely many times, obviously. And all the verbiage you added on is beside the point. The inductive method cannot be used here, neither scientific nor mathematical induction.
If you think it can, set it up please. What is the statement to be proven by induction? Prove it is true for n=1, then prove that if it true for n, it is also true for n+1. You will find this impossible to do.
You started out claiming you had a proof with no epsilons etc. You then admit that you just have a possibly thought provoking picture, but certainly not one that actually proves anything. You have yet to present a proof. I claim that none exists.