The math industry of confusing people

There is a curious argument that 1 + 2 + 3 + 4 + … = -1 / 12  (New York Times February 3 2014).

Some pronounce this as “minus one over twelve” but this weblog proposes “min per ten-two” or “negative per ten-two”. On occasion we employ H = -1, to be pronounced as “eta”. Thus “eta per ten-two” is okay as well. We can also use 1 / 12 = 12H, pronounced as “per ten-two”. (The Germans would pronounce H as “Ha” and we would not want them to be laughing all the time.)

The NY Times article and Numberphile video was debunked by other mathematicians and physicists on the internet, see some links below. However, this weblog looks at issues from the angles of both econometrics and the education of mathematics. From these angles we find:

  1. The article and video do not satisfy the conditions of didactics.
  2. There appears to be a large mathematical industry to confuse people.

Mathematics professor Edward Frenkel is part of the mêlée. He is quoted in above article (and can be heard in some video’s saying similar things):

“This calculation is one of the best-kept secrets in math.”
“No one on the outside knows about it.”

The article states:

In modern terms, Dr. Frenkel explained, the gist of the calculations can be interpreted as saying that the infinite sum has three separate parts: one of which blows up when you go to infinity, one of which goes to zero, and minus 1/12. The infinite term, he said, just gets thrown away.

The latter is rather curious. Why are you allowed to throw infinity away ? If you take something from infinity before you throw infinity away, why would you select -12H and not something else ?

Let us consider the situation, and start with Grandi’s Series. Personally, I was reminded about an approximation to -12H found last year, but since it is only an approximation this comment has been put into Appendix A.

An unwarranted deduction

In Numberphile, Thomsons’s Lamp, there is this video discussion about “Grandi’s SeriesG. That discussion (and on wikipedia retrieved today, see Appendix B) is unwarranted. The proper deduction is:

G = 1 – 1 + 1 – 1 + …. = (1 + 1 + 1 + ….) – (1 + 1 + 1 + …) = ∞ – ∞ = undefined

It is an altogether different question that we can look at the average of the series of partial sums. The Lamp mentions this (to their credit) but uses the same plus-sign which is unwarranted. We should use a different plus sign. Then we find:

G’ = 1 ⊕ H ⊕ 1 ⊕ H ⊕ ….  = 1 – G’    so that   G’ = 2H

Partial sums of G:  1, 0, 1, 0, 1, 0, ….

Summing (again !) those into a series: 1 + 0 + 1 + 0 + 1 + ….

Averaged series G’:   1 / 1,   (1 + 0) / 2 = 2H,   (1 + 0 + 1) / 3 = 2/3,   2 / 4 = 2H, …

The mystery completely disappears.

Divergent series can be operated upon, with differences, sums, averages, until you find something that converges. You might use this to catalog them.

That Lamp video discusses turning on and off an actual lamp, in ever smaller fractions 2^(-n) of a minute, starting at zero, such that the process should stop after two minutes (we can calculate that period mathematically): and then the question is whether the lamp is on or off. This is a badly defined problem. It is the same as the Zeno paradox of Achilles and the hare. A mathematical story using terms from physics doesn’t make it proper physics.

A string theory mystery

I am no physicist and know nothing about string theory, but am a bit perplexed when this other Numberphile video shows that page 22 for 1 + 2 + 3 + 4 + … ⇒ -1 / 12. Note the arrow rather than the equality sign. It remains a question: are they really taking the limit ? Hopefully the deduction in string theory is more to the point than the deduction given in the video. The deduction in that video clearly is not sound. It uses G = 2H but we have shown that only G’ = 2H. Indeed, see below for some links to physics websites that show that the video is crooked.

Page 22 of Joseph Polchinski, “String Theory" (Source: Numberphile video)

Page 22 of Joseph Polchinski, “String Theory” (Source: Numberphile video)

The Numberphile video uses three series. Confusingly it uses the normal plus sign but let us consider the idea that these would concern averages of a series of partial sums (with ⊕ instead of +). Series S1 = G and S2 is another form of ∞ – ∞ = undefined.

Read (+) and (+ H) instead of plus and minus (Source: Numberphile video)

Read (+) and (+ H) instead of plus and minus (Source: Numberphile video)

Let us repeat above procedure for S. Since there are no negative values involved, the series merely explodes, and obviously the outcome cannot be negative.

S’ = 1 ⊕ 2 ⊕ 3 ⊕ 4 + ….  

Partial sums of S:  1, 3, 6, 10, 15, ….

Summing (again !) those into a series: 1 + 3 + 6 + 10 + 15 +  ….

Averaged series S’:   1 / 1,   (1 + 3) 2H = 2 ,   10 3H = 3 + 3H,   16 4H = 4  ….

The Numberphile team has a longer video on the sum of the natural numbers that uses the Euler-Riemann Zeta function to argue their point, supposedly in “proper fashion”. However, they do not discuss the paradoxes here, and thus leave the reader confused. For example, they also refer to the basic geometric series, differentiate this, and then substitute r = -1 to create S2 (calling this “analytic continuation”), but, if the original geometric series is undefined for r = -1 (and then actually generates the Grandi Series again): why do you think that you can do this ?

Geometric series converges for -1 < r < 1 (Source: wikipedia)

Geometric series converges for -1 < r < 1 (Source: wikipedia)

See some physics links

The 1 + 2 + … = -1/12 video got 1.5 million hits and a fair amount of reactions from physicists. Their point is that Riemann and they are doing their job. See Steven Corneliussen in Physics Today and Phil Plait at Slate, for example. Plait has this quote from Jordan Ellenberg:

“It’s not quite right to describe what the video does as “proving” that 1 + 2 + 3 + 4 + …. = -1/12. When we ask “what is the value of the infinite sum,” we’ve made a mistake before we even answer! Infinite sums don’t have values until we assign them a value, and there are different protocols for doing that. We should be asking not what IS the value, but what should we define the value to be? There are different protocols, each with their own strengths and weaknesses. The protocol you learn in calculus class, involving limits, would decline to assign any value at all to the sum in the video.  A different protocol assigns it the value -1/12. Neither answer is more correct than the other.”

This is not entirely correct. Once you have defined “addition” and “equals” then you are stuck with it. Yes, you are free to find another protocol, but, beware of using “addition” and “equals” in general publications and education in another sense than people understand, because then your create confusion.

It seems to me that Physics Buzz is the most enlightening on what the real intention is.

Some nice quotes however

However, to soften our conclusion, the NY Times article by Dennis Overbye provides some nice quotes:

The problem with infinity is that you can’t stop. You never get there. It’s more of a journey than a destination.

Niels Henrik Abel, whose notion of an Abel sum plays a role here, once wrote, “The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever.”

Of course there is also Wigner again:

To him and others, this is just another example of what the eminent physicist Eugene Wigner called the “unreasonable effectiveness of mathematics.” Why should such woolly and abstract concepts as zeta functions or imaginary numbers, the products of a chess game in our minds, have such relevance in describing the world?

However, as mathematics = abstraction, and abstraction = leaving out aspects, it should not be surprising that if you start with the world and leave things out then you still have something. See here for complex numbers, and check the steps in turning around a circle:

1,  i,  H = i²,  H i, 1  (start at 1 = {1, 0}, quarter turn, half turn, three-quarters turn, back to 1)

Appendix A:   φ² / Θ ≈ 5 / 12

Remember that we found that φ2 / Θ ≈ 5 / 12 with an error of 6 per million, where ‘phi’ φ = 1.618033989… is the golden ratio, and where ‘archi’ Θ = 2 π = 6.283185307….

φ2 / Θ   =   0.416673050492137…

5  / 12   =    0.4166666…

φ2 / Θ  –  5 / 12  =  0.00000638382547060161…

φ2 / Θ  –  2H   ≈    -12H                                                    (with the same error)

Thus, the suggestion is that when some physics formula generates the number –12H, look whether this kind of thing might be involved. We came upon this from an application. The relation holds by approximation only, however, and might be abused again to confuse people.

Appendix B:  Wikipedia 2014-10-15 on Grandi’s series contributes to confusion

Wikipedia's discussion today on Grandi's series (Source: wikipedia)

Wikipedia’s discussion today on Grandi’s series (Source: wikipedia)

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