The mysteries of the Paradox

[Jon]

I have always had a fascination of paradoxes. Their mysteries begged to be unlocked, yet each time you try you are left wanting.

To illustrate, take 1/3. That is 0.3 recurring. 0.3 recurring x 3 = 0.9 recurring. But 1/3 x 3 = 1 Therefore, 0.9 recurring = 1. Yet, intuitively, 0.9 recurring does not equal 1. A paradox.

What are your favourite paradoxes?

 

[The_Doc_Man]

Zeno's paradox is fun. That's the one where you have a distance to travel. So you travel half the distance on the first day and half the remaining distance the second day, then half of what is left the third day. Do you ever get where you are going?

I.e. you have to travel X miles. So you travel X/2 + X/4 + X/8 + X/16 + .... but do you ever reach X? No, according to math.

Jon, the mathematical version of your problem is best stated as a "limits" problem. The LIMIT of X = 3 x ( 0.3/1 + 0.3/10 + 0.3/100 + 0.3/1000...) is 1 but if you truncate the infinite series, it will not be at the limit - because you stopped it from more closely approaching infinity.

On the other hand, SCREW that math paradoxes. How about literary paradoxes? I'm thinking of Robert A Heinlein's "By His Bootstraps" - a time travel story that will make your head hurt incredibly much.

 

[Jon]

Actually Doc, I would like to respectfully challenge the x/2 + x/4 + x/8 to infinity being a case of not reaching x.

Consider dropping a coin. That coin travels half distance between the point of release and the table top. That is x/2. Then it passes another half distance, x/4 and so on. There are an infinite number of half distances between the release and successive diminishing distances. The empirical proof is that the coin reaches the table, while mathematically it can be represented as a x/2 + x/4 + ...to infinity. Thus, while it seems like it shouldn't, we have empirical proof that it does. This may also explain why 0.9 recurring equals 1.

From Wikipedia:

This number is equal to 1. In other words, "0.999..." and "1" represent the same number.

https://en.wikipedia.org/wiki/0.999...

because you stopped it from more closely approaching infinity.

You can't approach infinity, because you will always remain infinitely distant from it. (new insight)

 

[The_Doc_Man]

Wikipedia, if they make that statement, is technically incorrect. The number in question approaches 1 and for all PRACTICAL purposes IS 1 - but technically it is not 1. The infinite series will always be infinitesimally less than one.

Yes, I just challenged the accuracy of a statement in Wikipedia. But the challenge is on the difference between theoretic science and applied science.

Your "coin drop" experiment fails because the dropping coin doesn't travel incrementally. The dropped coin reaches the table. But it doesn't do so because of a flaw in the math. It fails because math isn't a good model of the real world. I.e. the flaw is not in the math but in the experiment that says that math model is correct for the physical process.

Let me explain with a different experiment to illustrate the point I am trying to make here.

If you take a shotgun and shoot it at graph paper, then plot out the points represented by the contact of each piece of lead that hit the paper, you can get an (x,y) coordinate list that you can then use to generate a linear regression to determine the straight line that goes with the shot pattern. And you WILL get an answer. But it won't matter UNLESS you were testing something about a possibly skewed gun barrel. If the shotgun barrel is a true cylinder (best for this experiment), there should be no long-term pattern. I.e. repeat the experiment several times. You should get different lines out of each attempt at linear regression.

Why? Because this is an experiment that SHOULD give an approximately random distribution. To then apply a linear regression formula to it, you run into the issue that the formula WILL give you a number regardless of whether the number MEANS anything. I.e. YOU are the one saying that there is a straight line to be found. You say that by using the formula. But you COULD be wrong. You would want to run some other statistical formulas against the data set to see the quality of the data. And you should see that the Standard Error of the Estimate for that straight line is bigger than the slope or intercept you computed. That happens because of this rule: To apply a formula to a problem, a person SAYS that the formula applies. The formula, being an inanimate object, doesn't get to say "Time out, I don't work for this case."

That is what I mean when I say that the math can say anything but if the math really didn't apply, the fault doesn't lie with the math. It lies with the person who incorrectly applied the math to an incompatible problem.

In a different thread we explored the discussion about global climate change and claims that it is man-made in origin. Is it? I don't know, but many of the complaints I have seen suggest that the problem lies in the formulas and models used for global atmospheric interactions. If you apply bad math, you get bad results. And that is the basis of some of my objections.

 

[Jon]

It is an interesting position Doc and I think I might have found the source for your comments on limits. Forgive me if I am wrong.

http://mathforum.org/library/drmath/view/52507.html

What is interesting, is that Doctor Mike and Doctor Anthony both have a different perspective, but what we can agree on is that it is one of Zeno's paradoxes. We can also agree that you have to match theory to reality, or A priori reasoning with A posteriori reasoning. Then again we may have a problem, because in my case I believe reality is proving the mathematical formula, while in your case you believe it is the wrong formula to apply!

 

[The_Doc_Man]

The reason that Zeno's paradox doesn't apply is because the end goal is incorrectly chosen. You don't care about the finish line. You want one step OVER the finish line. Now, here is where the original paradox fails. I can take my original stopping point and add an inch to it and decide to stop there instead. As I add inches and continue to strive for my stopping point, at some moment in this process I will have crossed the "stationary" finish line and thus will have completed the race.

Your falling coin will add some distance to be traveled to its "goal" as each second passes. Eventually that "goal" will be lower than the top of the table and you suddenly cannot reach the goal because something else is in the way.

The problem is therefore your choice of how to describe the motion of the coin. Zeno's Paradox fails here because eventually the time interval between those ever-diminishing increments becomes impractically small - and there is the key word: Impractical.

Applying a theoretical paradox to a real-world process is always suspect. Oh, it might work perfectly will in some contexts. It is worth a chuckle or two in situations like these. But it is still the wrong model of dropping a coin.

 

[Jon]

In my description of the falling coin, I never factor in time. I see it as irrelevant. It is merely about the distance. If there is an finite number of half steps between the falling coin and the table, the coin never reaches the table. But there is an infinite number of half steps. With a finite number, you converge on the limit, but never reach it. But with an infinite number of half steps, you do reach it! One proof of this is that 0.9 recurring = 1. QED!

Perhaps one way to look at it is to understand that there is a binary difference between finite and infinite. They are distinct. So while a finite number will always converge to very close, it won't...just...quite...get there! But hop over to infinity and you have made that leap of faith, a transition from very close to 100% there.