In fact, I posted this question:
http://math.stackexchange.com/questions/149373/does-the-fact-that-sum-n-0-infty-frac12n-converges-to-1-mean-t
Now let's just see what them mathematicians say.
That link is to finite sums, Clayton. We are talking about infinite sums.
So? You challenged me to demonstrate that mathematical induction is a valid deductive principle (despite its confusing name) and the link gives a simple, universally accepted example of the use of mathematical induction to prove an infinite number of true facts from only a base case and an induction step. Then you shift the goal-posts. Nice.
As for 0.999..., please explain where this proof goes wrong:
0.9999... × 10 = 9.999999...
0.9999... × (9+1) = 9.999999...
0.9999... × 9 + 0.99999.... × 1 = 9.99999....
0.9999... × 9 = 9.9999....-0.999999 = 9
0.9999... × 9 = 9
0.9999... = 1
*sigh (bracing self)
It is true that once you re-arrange the terms or decompose an infinite sum, it is no longer algebraically identical to its numerical summand. Treating the sum this way leads to all sorts of absurdities. But for naive algebraic manipulation, it is never wrong to treat the infinite sum as algebraically identical to its summand so long as you wrap it in parentheses and do not rearrange its terms in any way. If you want to move to more sophisticated manipulations, then you need to observe the rules of convergence, which analysis has helped us discover.
But to say that a convergent infinite sum that has not been decomposed or rearranged in any way is not identically equal to its numerical summand is ridiculous. No serious mathematician except perhaps a few finitists would ever say such a thing. Contrary to popular belief, analysis does not asterisk the equality between the infinite sum and its summand, it simply analyzes the behavior of the partial sums.
And I did not misquote Hardy - I intentionally inserted the "also" for emphasis. It is you, not me, who is misreading him.
Clayton -
Have you ever studied the construction of the real numbers from the integers?
Yes.
You challenged me to demonstrate that mathematical induction is a valid deductive principle...
No. I challenged you to use it to prove 1/2 + 1/4 ...= 1. I know about math induction, have used it many times.
...please explain where this proof goes wrong:
It goes wrong in the very first line, because two of the symbols are undefined [i.e. 0.9999... and 9.9999....], unless you go with all mathematicians everywhere and define them as least upper bounds. Then the whole thing becomes a proof about least upper bounds, with the final result being that the l.u.b. of .9999... is 1.
But to say that a convergent infinite sum that has not been decomposed or rearranged in any way is not identically equal to its numerical summand is ridiculous.
I agree, with the proviso [that you deny] that a "convergent infinite sum" means, by definition, a least upper bound. But no more than that. Corrolary: The sqaure 1x1 has not been proven anywhere that it will be filled by dropping in squares of 1/2 , 1/4, etc.
No serious mathematician except perhaps a few finitists would ever say such a thing.
So Hardy wasn't a serious mathematician. And every mathematician on the subject since the 19th century is also not serious. Clayton, time to put up or shut up. Show me one, just one, book, website, link, something, that says what you do.
Contrary to popular belief, analysis does not asterisk the equality between the infinite sum and its summand, it simply analyzes the behavior of the partial sums.
Contrary to popular belief [as exhibited in this thread], analysis does not define an infinite sum in any other way but as a least upper bound. There is no other infinite sum. The infinite sum is distinct from its summand, because the infinite sum is the limit of the summand [which is an infinite sequence].
The mistake lies in an ignorance of the meaning of a summation sign that has an infinite index. Whenever you see one, it means only one thing, by definition. It means the limit of a certain sequence.
And I did not misquote Hardy - I intentionally inserted the "also"...
Oh. OK. You mean you did not realize that inserting that innocent looking word negates everything he actually wrote.
It is you, not me, who is misreading him.
OK then. I leave it to whoever is reading this to go straight to the master [=Hardy], sit at his feet [=read Chapter 4 and beginning of Chapter 8] and see what he says.
We are done here, Clayton. If we disagree on the meaning of what Hardy actually said, there is no room for further discussion.
My humble blog
It's easy to refute an argument if you first misrepresent it. William Keizer
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I think I solved the paradox. It appears that the confusion emerges from analyzing the problem at too-high a level and sliding a few mistakes in such as believing that the time intervals are constant.
NOTE: OPEN IMAGES IN A NEW WINDOW TO VIEW THEM IN PARALLEL WITH THE EXPLANATION.
Here is the problem graphically:
http://i.imgur.com/uHNe7.gif
At 1 you have your beginning situation. A is at 0 and B is at 8.
Say the speed of B is 2/sec and A is 8/sec.
2) Now A has moved 8 units in a second to where B was before. B has moved only 2 units ahead.
3) Now, B moves for a quarter of a second, so it goes 0.5 away, while A travels 2, to where B was before.
4) B moves 0.125 while A moves 0.5 (in 1/16 of a second)
You may wonder why B is moving less and less each time. Well, for A to not have caught up to B yet, B can only move an amount of time in which A will not have had the chance to catch up with him.
What you see in the picture is that infinity is indeed involved. The problem is, not an infinity of time, but of divisions of space. To follow the conditions in the problem (ie, Achilles not catching up), we must make the time period of each motion smaller and smaller (this way Achilles can't catch up). This means that eventually, at infinity time intervals, the two might converge. Thing is, add up all of these infinity of time intervals and you will get a constant time. What that means is that at a given, constant-valued time, A and B will converge after going through infinitely many intervals of time (yet still a finite sum).
Does the math support this answer? In fact, yes.
First, let's look at the "naive" method of doing this, with which the paradox has a problem:
http://i.imgur.com/Wm6zE.gif
The position of B is as shown and A is as shown, both dependent on time t.
Setting them equal, we figure out that t=4/3 when they meet, and the distance from the origin is 32/8.
Now, the paradox says "that's not true!"
Taking the insights I provided before with the first diagram (infinite time intervals, not infinite time), let's see what series and calculus tell us:
http://i.imgur.com/N3TX0.gif
We start by figuring out where B will end up. It begins at 8, so that is the first term. Next, we have to add an infinite geometric series. The first term is 2, so 2 is the numerator. The ratio is 1/4, so 1-1/4 is in the denominator. Solving, we get 32/3.
For A, it starts at 0 and its first geometric term is 8 with ratio 1/4, evaluating, we have 32/3.
This provides support for the idea that the problem is in the infinite time intervals, not actually infinite time.
The solution might be made more rigorous if I used dt instead of t, but the idea is there.
Yes, by the time Achilles reaches the turtle's starting point the turtle is ahead, but much closer than initially. This keeps going and becomes smaller and smaller, until they converge. But note, too, that for Achilles to be always behind, the time period needs to shorten every time. The paradox arises by having the reader think that the time period of the movements is the same. It's not. It's a geometric series with a ratio < 1, so it converges to a constant. It has infinite time intervals, but converges to a constant.
I feel proud.
I'm going to resurrect my syllogism because you've slyly dismissed it out-of-hand without dealing with the substance of its argument.
1. PRQ is similar to TPS
2. Therefore, the tan of PQR (= PR/PQ) and the tan of TPS (=TP/TS) are the same
3. The base of TS is first divided into two segments - the segment marked '1' and the remaining segments. Let's name that point (directly below Q and R) U. Let's name the point dividing the segment along TS marked r and the sgement marked r2 V
4. TUR is similar to TPS. Therefore, the ratio SU:UT is the same as the ratio of UV:VT.
5. Therefore, another similar triangle can be constructed that will be in the same ratio as the similar triangles TUR:TPS, ad infinitum - (this is the inductive step). Please show me how this point violates some fact about analysis or mathematical induction or any law of mathematics.
6. Therefore, the proportion of PS:QR is equal to the infinite sum constructed along TS, that is, the ratio of the segment marked 'r' to the remaining segments through nesting similar triangles ad infinitum (from point 5): 1/(1-r) = 1 + r + r2 + …
Please show me how this point violates some fact about analysis or mathematical induction or any law of mathematics
It violates the accepted standard of mathematical proof. Mathematics is conducted within the framework of ordinary logical thinking. Much thought has been invested into what the rules of logical thinking are, which rules of deduction are valid, and which are invalid.
For example, say someone writes a proof that contains the following step:
"I count to three, clap my hands and make a wish. The tooth fairy then tells me that the answer is 5" QED.
This proof would not be accepted, because consulting the tooth fairy, though possibly totally reliable, has not been accepted by the mathematical community as valid.
The same is true of your step 5. Writing "ad infinitum" is exactly like consulting the tooth fairy, as far as validity of a proof is concerned. The mathematical community, after many disappointments, has rejected as inadmissable all proofs with the words "ad infinitum" in them, just as it has rejected proofs from the tooth fairy, and proofs based on pictures.
Now you may ask, but what about mathematical induction? Isn't writing "ad infinitum" the very thing that one does when performing mathematical induction?
The answer is a resounding no. I wrote in an earlier post the very strict criteria required to correctly use mathematical induction. Step 5 fails those criteria decisively.
Friedmanite: In fact, I posted this question: http://math.stackexchange.com/questions/149373/does-the-fact-that-sum-n-0-infty-frac12n-converges-to-1-mean-t Now let's just see what them mathematicians say.
I also used the search on there. This is what I gathered from it: some formulas were offered, but this guy below broke down the point of the entire question. Historically, he's correct, but they've squabbled with themselves over it too, just more akin to mathematically than we have.
I think the mathematical "explanation" of the Zeno's paradox (convergence of infinite series) is quite unsatisfying. Assuming that each term in the series corresponds to one step of Achilles's and considering that he indeed overtakes the turtle in finite time, which of Achilles feet is forward at the moment when he reaches the turtle?
Or a slightly different, but equivalent, presentation of the paradox: assume that the turtle changes direction at each discrete instant of time Achilles reaches her previous position, alternatively moving NE and SE. Achilles just follows her path. What direction is the turtle facing the moment Achilles reaches her?
Achilles's and the turtle is no paradox at all, but a refutation of the hypotheses that the space is continuous. Zeno's arrow paradox is a refutation of the hypothesis that the space is discrete. Together they form a paradox and an explanation is probably not easy. For Zeno the explanation was that what we perceive as motion is an illusion. In any case, I don't think that convergent infinite series have anything to do with the heart of Zeno's paradoxes.
EDIT: The same argument can be made point-like particles, only assuming that physical reality is continuous and infinitely divisible. Imagine a photon travelling between an infinite sequence of mirrors placed in a zig-zag shape with distance between mirrors decreasing at a geometric rate. So the photon bounces from a NE to SE direction and back, with the distance travelled decreasing "fast". Since the length of the total path is finite (sum of a geometric series), the photon will emerge from the sequence of mirrors in finite time. What direction will it travel? The heart of the Zeno's argument is that there is no logical way to decide that. You may argue that it is impossible to build such a sequence of mirrors, however this is just conceding Zeno's point that physical reality is NOT continuous and infinitely divisible.
I think the mathematical model of the Zeno's paradox is a great pedagogical tool in first year calculus, probably could be made even earlier in high school, but it misses an important aspect of Zeno's argument. Granted, this argument lies at the boundary of math, physics and perhaps philosophy.
http://math.stackexchange.com/questions/142932/achilles-and-the-tortoise-paradox
This is brilliant. It is a response to Achilles vs. the Tortoise made by Dragon Lady on www.objectivismonline.com.
"...the entire nature of Zeno's paradox means you're treating mathematics as if they inform physics and not the other way around. Just because it's possible to do something mathematically doesn't mean it's possible to do it physically."
I think that sums it up nicely.
Objectivism is braindead.
PS:
has anyone read Godel Escher Bach?
"As in a kaleidoscope, the constellation of forces operating in the system as a whole is ever changing." - Ludwig Lachmann
"When A Man Dies A World Goes Out of Existence" - GLS Shackle
Clayton: Objectivism is braindead.
Prove it.
vive la insurrection: PS: has anyone read Godel Escher Bach?
Looks very interesting, Vive. I'll check it out.
You just proved it for me.