Pure Gibberish

Earlier this afternoon, my brain was destroyed.  I keep hoping that someone will tell me this is an April fools day joke.  I was perplexed, I was astonished, I could not believe what I was seeing.  It is the most astonishing result I have ever seen in any field.  I still don’t know what to make of it.  The result was presented in this YouTube video by Numberphile.

If you watch the video, the basic idea is that the sum of the infinite series of positive integers is -1/12.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + . . . = -1/12

One of the simple proofs that is offered for this idea begins with finding the sum of this series:

1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + 1 + . . . =  1/2

So far so good.  After an even number of values, you have zero, after an odd number of values, you have 1.  It makes sense that the sum of the infinite series would be the average of the two alternating values.  The next series is a bit weirder:

1 – 2 + 3 – 4 + 5 – 6 + 7 – 8 . . . = S2

To find the sum of this series, they add two copies of the series together:

1 – 2 + 3 – 4 + 5 – 6 + 7 – 8 . . . = S2

+

1 – 2 + 3 – 4 + 5 – 6 + 7 – 8 . . . = S2

1 + (1 – 2) + (- 2 +3) + (3 – 4) + (-4 +5) + (5 – 6) + (- 6 +7) + (7 – 8) +   . . . = 2S2

which if you work out the simple additions is:

2S2 = 1 -1 +1 -1 + 1 -1 + 1

but we know from above that this series has the infinite sum 1/2.  If 2S2 = 1/2, then S2 = 1/4.   This is a weird result, but for a series with alternating positive and negative numbers, it is not too bad.  Now comes the weird part:

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + . . . = S = ?

1 – 2 + 3 – 4 + 5 – 6 + 7 – 8 . . .  = S2 = 1/4

(1-1) + (2 -(-2) + 3 -3 + 4 -(-4) +5-5 + 6-(-6) . . .

leaving the series:

4 + 8 + 12 + 16 + 20 + 24 + …

this can be written as:

4 ( 1 + 2 + 3 + 4 + 5 + 6 + . . . )

S – S2 = 4S

S2 = -3S = 1/4

S = -1/12

So  the sum of all the positive integers is a negative fraction with the value of -1/12.

Now at first I thought this result was the artifact of some kind of weird forbidden operation like divide by zero.  There are two reasons to believe that this is a real result, however:

  1. While divide by zero never yields a unique result, -1/12 can be reached by a number of different methods.
  2. In physics, if you see the infinite sum of integers appear in an equation, it behaves as if it had a value of -1/12.

So there are good reasons to believe this result is real and I have been struggling with how to understand it all day.

So the first thing that I grasp onto is that this is an infinite divergent series and that it doesn’t really have a sum in the way that we understand sums.  It is, however, related to other infinite series that do have what we can understand as averaged “pseudo sums”.  If we examine the relationship between those series and their averaged “pseudo sums” and the infinite series of positive integers, we can derive a “pseudo sum” for the set of positive integers.  The value of this “pseudo sum” happens to be -1/12.  Is this really so bad?

Yes it is.  No matter how hard I try, I have to admit that this result is pure gibberish to me and it is the strangest result in any field of study of which I am aware.  Why?  For me it is a matter of familiarity.  Sure quantum particles behave strangely, but what would you expect?  They are unfathomably small and they obey a set of strange mathematical principles.  The reason this result is so bizarre is because it deals with things that look so well behaved.  What could be more friendly then adding a bunch of positive integers together?  I have been doing that since I was in elementary school!  I feel like I have been betrayed by an old friend.

A while back, I was in an exchange with a correspondent who calls himself MyAtheistLife and I argued that the infinite can boggle a poor finite mind and this is why we need to trust God when dealing with issues that we cannot understand.  The only reason I did not mention this extremely counter-intuitive result was because I did not know about it.  If I had heard of this result before, then I pushed it out of my mind as unfathomable.

Now I keep making this point, but I think it is important enough to emphasize it.  There are a number of propositions that are intrinsic to Christianity that make absolutely no sense to a finite human being.  Let us consider some of them:

  1. The best way to enjoy food is not to be a glutton.
  2. The best way to enjoy sex is not to have sex with multiple partners but to focus on a single person for life.
  3. The best way to create an eternal paradise is a world like this one.
  4. Three different persons can be one divine being.

These propositions and others like them do not make any sense to us as finite beings.  And yet we have to be very careful about this.  If we think about it, we would expect that God would have a great number of things to say which would be extremely counter-intuitive to a finite human being with only a few decades worth of experience and a limited understanding of the world.

I have paraphrased the words of Arthur C. Clarke’s saying about magic and technology many times and I will continue to do so. “The results of a thinking process of a sufficiently advanced intelligence with a vastly greater store of knowledge than we have access to is likely to be pure gibberish to us”.  If the result that the sum of all positive integers behaves like a negative fraction doesn’t demonstrate the truth of that proposition, then I don’t know what does.

About Robert V

Former atheist currently living in Toronto.
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