You drop a ball. You drop a key. You drop a piece of lead. They each slip from your hand and tumble to the ground. But you have an orderly mind (you're a physicist) and, to you, falling things don't seem haphazard — they seem neat. The unruly world has rules, laws. So instead of saying "Bummer!" you say, "Of course! How beautiful! How logical!"

Galileo thought this way. He got a bunch of weights, dropped them and discovered just such a rule. (I* *would never have expected* *it. When I drop an object, it feels like a drama, a short story that will never repeat the same way. But I'm not a physicist — I am whatever the opposite of a physicist is.)

Here's what Galileo did. (You can do it, too. In his new book, *The Accidental Universe*, physicist Alan Lightman tells us how.)

First, "drop a weight to the floor from a height of 4 feet, and time the duration of its fall," Alan says:

*Robert Krulwich/NPR*

You should get about 0.5 seconds.

Then try it again, this time from a height of 8 feet. That should take, Alan says, about 0.7 seconds:

*Robert Krulwich/NPR*

Again — but now from a height of 16 feet. Duration: about 1 second.

*Robert Krulwich/NPR*

Says Alan: "Repeat [this experiment] from several more heights and you will discover the rule that the time exactly doubles with every quadrupling of height."

This always happens. Four times higher, the falling time doubles. Always.** **"With this rule," says Alan, you can now predict the time to fall from any height. You have witnessed, firsthand," he says triumphantly, "the lawfulness of nature." As you watch things drop, now you can say, "Of course. How beautiful! How logical!"** **

Mathematicians believe (actually, they know) that there are rules out there in the real world — patterns that don't vary. If you follow the math, the equations can lead you to otherwise invisible objects in the universe, things that logic tells you must be there. The math says, "Look here." You look, and bingo! There it is. (That's how we found the planet Uranus.)

But sometimes the math does something even stranger. Spookier. Instead of "Of course!" it makes you say, "Huh?"** **(Or, in less polite circles, "WTF?")

What you're about to see is a deeply beautiful "Huh?" story.

**This Is A Queer Universe, Says The Math**

The math isn't hard. It comes in simple steps. But it leads to the strangest, most impossible-sounding conclusion. Yet every step you take makes perfect sense until you get to the end — and go, "No way!"

I'm a total math dummy, and yet I followed this puzzle almost to the end. It's one of the few times I've ever been able to feel what it's like to be a mathematician exploring the universe.

Go ahead. Even if you think you'll hate it, take the chance. It's only a few minutes long. Give yourself a "Huh?"

## Comments [13]

The key to understand why this "proof" is junk is at the 1:54 min mark, when the person says "we need to attach a number". That's the key of the whole "proof". There is nothing to prove. The video should have ended there, when the physicist in essence defined a specific number as the sum of a divergent series (it is the same as saying that a power boat that runs forever is called a motorcycle). Conventions do not need mathematical proof.

How do you represent infinity? We defined a symbol for it. What is the sum of (1,2,3,4,5...)? Some physicists defined it as -1/12. Nothing to prove, no amazing insight. Just a convention.

The rest of the video is a there to distract the viewer from the convention made at the 1:54min mark.

"S1 can be 1 or 0 so we will just take S1=1/2". Is this guy for real? S1 does not converge to 1/2. From then on the rest of the "proof" is junk.

I too am very disappointed that Radiolab has given credence to this "proof" that is as much credible as the "proof" that 1=2 that I learned in high school algebra (as a lesson in fallacious proofs). As soon as the huckster shifted the numbers one digit, he committed a mathematical and logical crime. You see, we only saw it shifted at the LEFT end...the right end DOESN'T EXIST. It stretches into infinity, and negates the entire idea. I saw this about a month ago and can't believe the likes of Krulwich and Abumrad would share such a fraud. (and if I've misspelled their names, all the better. It's not as bad as this abomination of mathematical logic.) Bah, I say! Please publish a retraction!

You can move the line to the right because when you are adding two sums, you can add the terms of the sum in any order you want. It looks incorrect because we are used to addition having to line up in columns of ones, tens, etc but with sums it doesn't matter. For instance, let's say we are adding 94 plus 23. If you add them normally, you'd have to do:

94

+23

---

117

But if you write it as adding two sums 9(10)+4 plus 2(10)+3.. you can add them in whatever order you want.

9(10) + 4

+ 2(10) + 3

------------------

90 + 24 + 3 = 117

Ok, I'm with several other people that got lost at the point that we shifted the equation to get S2. Can someone help me grasp where that came from?

I'm skeptic. Look at the longer version with the alternate proof. He makes a very blatent error in multiplying exponents to get to his result. How should we trust this result when he's messing up high school math in order to get into quantum physics and string theory?

I look forward to more good articles and I think we all love to thank so many good articles, blog to share with us.Really great post nice work i love your work.Thanks

While evaluating the three sums, what rule of mathematics allows him to "shift" the second writing of S2 over to come up with 2S2?

John

you really are "whatever the opposite of a physicist is" (a moron? liberal arts major? same thing really). i will never get back the time it took to read through this drivel. american education is really that bad that a discussion of the experimental basis of galileo's discovery and subsequent formulation of a theoretical model is worthy of a post?

take this garbage to wired

Isn't the definition of infinity that you can always add one more number and and the sum gets bigger -- regardless of what the number may be?

Since the finite sum would always be (n + (n+1) )/2 I can always find a bigger value of n where n is the term number. This sum does not converge. Nor does it alternate among several fixed values.

Your result seems to depend upon a notion of average values which makes sense for a sum which alternates among a few values, but the sum 1+2+3+4+5+6+...

does not alternate between repeating values -- it always gets bigger as the number of terms increases. As n approaches infinity then sum too approaches infinity.

Why didn't Tony explain why he moved the number line to the right in example 2 or why he needed to involve S2 to solve the original question? I didn't get the connection tbh.

It isn't all true, rather. The result is correct, but the method is a hack. There is actually a link in that video to another which shows a proper proof.

Sounds like this video isn't true at all... Always maintain skepticism!

http://scientopia.org/blogs/goodmath/2014/01/17/bad-math-from-the-bad-astronomer/

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