In 1980, J.S. Bell wrote a famous paper with the title "Bertlmann's socks and the nature of reality". In this document, I want to highlight some possible non-typo error's that I found and make a proposal for a general solution.

1. The Bertlmann’s Socks conceptual mistakes
Chris De Corte
chrisdecorte@yahoo.com
July 8, 2015
1 Key-Words
EPR Paradox, Einstein-Podolsky-Rosen, Bohr, Heisenberg, Jordan, Spooky action at a dis-
tance, local causality, Bell inequality, determinism. [1] .
2 Introduction
In 1980, J.S. Bell wrote a famous paper with the title “Bertlmann’s socks and the nature of
reality” [1]. In this document, I want to highlight some possible non-typo error’s that I found
and make a proposal for a general solution. In the rest of the document, when I write “the
author writes”, I mean J.S. Bell. When I refer to pages or figures, I mean in Bell’s document.
3 The error’s
First of all, it is important for the formula’s, (3) on page 6, about the probabilties in the ad
hoc model used in the document to add the useful validity:
P(up, up) = P(down, down) =
|a − b|
π
for 0 ≤ |a − b| ≤ π (1)
P(up, down) = P(down, up) =
1
2
−
|a − b|
π
for 0 ≤ |a − b| ≤ π (2)
On the second part of page 7, the author starts a story about washing socks at different degrees
to derive the Bell inequalities. Later in the document (page 9), he will seamlessly replace
1

2. 3 THE ERROR’S
the degrees in temperature with degrees in angle, completely neglecting the fact that for the
washing experiment, the expectations are linear with the temperature while for the EPRB
gedanken experiment the probabilities follow a square sinus rule with the angle (formula 4
page 7)!
After a complicated logic of probabilities (where the down particles are blocked), extending
from the second part of page 7 to the first part of page 9, the author suddenly drops all what
was said and postulates formula 9 where the wording “is not less than” is in my opinion wrong
and this for the following reason: suppose P(x) is the probability to pass x degrees, then, it
is obvious that:
45
0
P(x)dx +
90
45
P(x)dx =
90
0
P(x)dx (3)
So, the “is not less than” should be “is equal to”.
Then, on the first part of page 10, the author claims that the formula for one particle to pass
a magnet with orientation a and the other to pass a magnet with orientation b is the formula
for P(up,up). However, an angle a will cause the left particle to move more up and an extra
angle b will cause the right particle to move even more down resulting in a clear P(up,down)
situation. What we would get then is:
[
1
2
−
1
2
(sin(22.5))2
] + [
1
2
−
1
2
(sin(22.5))2
] ≥ [
1
2
−
1
2
(sin(45))2
] (4)
or:
0.8536 ≥ 0.25 (5)
This result is not a violation of the Bell inequality.
The confusion can not become bigger as on page 9 formula 9 of the document, Bell writes:
”the probability to pass at 0° and NOT able at 45°” (so pas/not pas) while on page 10 as an
explanation of the sin2 term he writes: ”probability to pass a magnet with orientation a and
the other to pass a magnet with orientation b” (so pas/pas). So, in stead of using sin2 as he
does, I would have used (1−sin2) which would not have caused the violation as demonstrated
above.
In fact, if we would pick out single angles , the formula for P(up,up) would always be violated
and the formula for P(up,down) would always pass as can be seen from our own table 1.
On page 11, the author starts talking about joint probabilities P(A, B | a, b) with regard to
some experiment and assumes on page 12 that a product formula for individual probabilities
can be used to compare with the joint probability formula. He assumes that if A and B are
not correlated, one can write: P(A, B | a, b) = P(A, a)P(B, b). However, a product formula
2

3. 3 THE ERROR’S
can only be used if an experiment is conducted in series and not in parallel. But the detectors
in figure 7 of the author’s document are positioned in parallel. Therefore, the joint probability
should be written as a sum and not as a product. If we assume a more general but still linear
case, including an extra general factor like e.a. the temperature T (We refer to figure 7 in
the author’s document):
A = α1a + β1b + γ1T (6)
similarly:
B = α2a + β2b + γ2T (7)
Then, we would get:
P(A, B | a, b) =
(α1 + α2)a + (β1 + β2)b + (γ1 + γ2)T
a + b
(8)
In general, we can see immediately that the above equation would not split into:
P(A, B | a, b) = P(A | a)P(B | b) (9)
Even in the extreme case where A and B are far away from each other and we assume local
causality and we assume that the impact from the temperature is the same for A and B, we
get:
P(A, B | a, b) =
αa + βb + 2γT
a + b
(10)
Still, this equation would not split according to formula 9.
On page 13, the author then supposes that there might be local variables λ, which, if we
knew them, they would allow a decoupling like in formula 9.
This is a crazy thought process. This is really like saying: hey, we have here an electronic
circuit with 2 resistors in series but if we assume some local variable, we can see them as
being in parallel (formula 11)! Or, if we turn it into a joke: here is a cat, but if we could find
the proper hidden variables, it might turn out to be a dog!
R = R1 + R2 ⇐⇒ R =
R1R2
R1 + R2
(11)
3

4. 3 THE ERROR’S
However, we continue reading the author’s document.
The author then introduces a probability distribution ρ(λ) which he chooses such that:
dλρ(λ) = 1 (12)
On the second part of page 14, the author again chooses to pick out one situation on the left
side of the equation and one situation on the right side of the equation for which the equality
doesn’t hold. However, one is not allowed to do that.
Furthermore, he splits a situation like this:
dλρ(λ){| ¯B(b, λ) ± ¯B(b , λ)| + | ¯B(b, λ) ¯B(b , λ)|} (13)
as follows:
= {| ¯B(b, λ) ± ¯B(b , λ)| + | ¯B(b, λ) ¯B(b , λ)|} · dλρ(λ) ≤ 2 · 1 ≤ 2 (14)
This last step of coarse might not be correct, especially since ρ might temporary reach high
values at moments when the second part of formula 13 reaches high values as well while ρ
might become very low (or perhaps negative) at moments when the second part of formula
13 becomes very small, even though formula 12 still holds.
On page 18, in the prove of the Clauser-Holt-Horne-Shimony inequality, the author writes
“From (18)”:
≤ however, this should be = (15)
Actually, in (19), the author writes | ¯A(a, λ)| ≤ 1 where ¯A is written as probabilities whereas
the author actually means “expectations”. I wander if temporary negative expectations are
not possible while still having a total probability of 1. Especially since angles are going to
90° or higher, it would actually mean that the particles are send back to the source which
could mean, a negative expectation. This would then be a case for the formula (19) not to
be correct.
Later on page 18, the author assumes the following:
dλρ(λ) ¯A(a, λ)( ¯B(b, λ) ± ¯B(b , λ)) ≤ dλρ(λ)( ¯B(b, λ) ± ¯B(b , λ)) (16)
based on the his formula 9:
| ¯A(a, λ)| ≤ 1 (17)
4

5. 4 ORIGIN OF THE CONCEPTUAL MISTAKE
But I would not have a lot of confidence in this step.
In the proof for the Clauser-Holt-Horne-Shimony inequality, the author writes on page 18:
because:
0 ≤ P1 ≤ 1 ; 0 ≤ P2 ≤ 1 , it follows that : | ¯A(a, λ)| ≤ 1 ; | ¯B(b, λ)| ≤ 1 (18)
According to me, this is not the case. I could agree that:
| ¯A(a, λ)|dλ ≤ 1 and | ¯B(b, λ)|dλ ≤ 1 (19)
But the individual values could probably achieve temporary values that are higher.
This whole complicated mathematical reasoning the author starts from page 11 towards the
end of his document is actually to have a scientific foundation to conduct statistical tests on
physical outcomes in order to test if some situation having parallel detectors could be put
into a formula used for detectors in series. However, we had seen from the first glance that
this would not have been possible (our formula 11).
4 Origin of the conceptual mistake
To summarize the origin of the conceptual mistake, we will demonstrate what happened with
an analogous example (see figure 1) using a Venn diagram. We consider 3 sets A, B and C
which all are intersected with each other. A has the sets a,b,c and d. B has the sets e, b,
d, f. And C has the sets g, c, d, f. Based on the Venn diagram, we can quickly come to an
equation which is similar to the Bell inequality:
(a + c) + (b + e) ≥ (a + b) (20)
We test this with an example:
((1 + 9) + 3) + ((2 + 8) + 5) ≥ ((1 + 9) + (2 + 8)) (21)
When testing this inequality, it always seems to hold. However, what some people (like the
author) are doing is using a special subset of a and b for the left side of the equation and
another special subset for the same a and b for the right side of the equation (or for another
part of the formula like Clauser-Holt-Horne-Shimony). They achieve this special subset by
looking at the situation from a different angle on the left side than on the right side. By doing
this, they can arrive to a situation whereby the inequality no longer holds. Example:
(1 + 3) + (2 + 5) ≥ (9 + 8) which is false (22)
5

6. 6 REFERENCES
5 In or out of the box?
In the previous section, I have been describing how the bell inequality could be violated by
the use of special subsets on the left and on the right side of the equations. Based on this,
the author is looking for hidden variables to explain the situation. I think the reason why
he can not find the hidden variables (lambda) is because he is looking for them ”in the box”
while I think they are ”outside the box”. To explain this, we suppose we have written the
following equation: 1+2=3. Suppose we paint the ”2” with some holographic paint so that it
doesn’t display under an angle alpha. We move our paper with this angle alpha and we get:
1=3 which is a violation! But what happened? Some phenomenon ”outside our box” was the
cause for this. Not any reasoning within mathematics could have explained this except for
this new concept outside our box.
This situation is comparable with the missing mass problem in astronomy. There a solution
has been to postulate a ”dark matter” to bring back equilibrium. Maybe, a similar thing
needs to be done in this case by postulating something extra outside the box to bring back
equilibrium with regard to the Bell inequalities.
6 References
1. J.S. Bell. Bertlmann’s Socks and the nature of reality. Ref.TH.2926-CERN. CERN -
Geneva. Page 1-25.
6

7. 6 REFERENCES
Table 1: In this table we show that the Bell inequality always seems to hold for the subset of
P(up,down) while it does not hold for the subset P(up,up). This illustrates the fact that
one is not allowed to pick out a specified subset and draw conclusions from the results.
P(up,up)
α β γ 1
2sin2(β−α
2 ) + 1
2sin2(γ−β
2 ) <=> 1
2sin2(γ−α
2 )
0 45 90 0,0732+0,0732=0,1464 < 0,2500
44 60 90 0,0097+0,0335=0,0432 < 0,0763
70 80 90 0,0038+0,0038=0,0076 < 0,0151
P(up,down)
α β γ 1
2cos2(β−α
2 ) + 1
2cos2(γ−β
2 ) <=> 1
2cos2(γ−α
2 )
0 45 90 0,4268+0,4268=0,8536 > 0,2500
44 60 90 0,4903+0,4665=0,9568 > 0,4237
70 80 90 0,4962+0,4962=0,9924 > 0,4849
Figure 1: In this figure we quickly derive the Bell inequality and demonstrate by way of example
how by chosing different subsets, we can simulate “inexplicable” violations of the Bell in-
equality and can suppose that this is related to a phenomenon called “quantum mechanical
entanglement“.
7