Alexander Weber
Alexander Weber
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String theory is bullshit!

Currently there’s a claim going round that the sum of 1 + 2 + 3 + 4 + … is -1/12. The source of the claim is found in the following video:

https://www.youtube.com/watch?v=w-I6XTVZXww

I saw this a few months ago already and declared it bullshit and now George Takei also posted it too and I still declare it bullshit. But it is apparently used in string theory and I’m just a basic software engineer with simple school education in physics and mathematics, so I must be wrong, mustn’t I? Ok, let me present my reasoning:

In the video they create three sums first:

S1 = 1 - 1 + 1 - 1 + …
S2 = 1 - 2 + 3 - 4 + …
S  = 1 + 2 + 3 + 4 + …

S1 is oscillating between 0 and 1. They claim the „value“ of this sum is therefore 1/2. That’s the first problem. You can’t determine a limit for a sequence which is oscillating between 2 values. The limit of such a sequence is undetermined. Period. So the rest of the video is already based on this false assumption and therefore bullshit. But let’s assume you actually can do this and continue…

Next comes S2. They take two copies of S2 and add it to itself. The trick here is that they shift the second summand:

2 * S2 = 1 - 2 + 3 - 4 + …
         +   1 - 2 + 3 - 4 + …

So they get

2 * S2 = 1 - 1 + 1 - 1 + …

and they claim the sum is therefore again 1/2 as calculated above for S1. This is wrong again. If you shift the second summand, you have to take into account that the resulting summands are 1 and -1 except for the last one which is the last value of S2. Of course that makes no sense for an infinite sequence, but as we’re calculating the partial sums, the sequence is oscillating again, but with different numbers than S1. We get:

1
+   1 =
1 + 1 = 2

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

1 - 2 + 3
+   1 - 2 + 3 =
1 - 1 + 1 + 3 = 4

1 - 2 + 3 - 4
+   1 - 2 + 3 - 4 =
1 - 1 + 1 - 1 - 4 = -4

if we continue this, we get

2, -2, 4, -4, 6, -6, 8, -8, …

The „limit“ here is an oscillation towards infinity and -infinity. You can use the trick above for determining a „fake“ sum by taking the mean value, but that would be 0, not 1/2.

Now they calculate that

S - S2 = 4 * S
(see the video for how they came up with this, which is valid)

We have determined the „fake“ value of S2 as 0, not as 1/4 like they did. So we get instead:

S = 4 * S

This is true for sums with the values of 0, infinity and -infinity. And this includes the intuitive result of infinity, but absolutely not -1/12.

So why can the result of -1/12 still make sense in string theory? I have no idea, because I’m absolutely no expert in string theory. But I think a lot of physics theories are based on mathematics, be it good mathematics or bad mathematics. If string theory is based on bad mathematics, then this just might be a bad theory IMO. They are saying there’s no infinity in nature. They are right. So why in science’s name are they using infinite sums in string theory? I don’t know. Maybe string theory is just bullshit. Yes, I said that. There.

Posted on 2014-04-16 20:40:00
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