# Experimental Non-Violation of the Bell Inequality
## 7 Discussion
The notion that distances in physics should be necessarily described by the Euclidean metric is a deeply held intuition, since almost the first thing we learn as babies is a sense of spatial awareness (for the baby to get its hand close to a colourful toy, it has to learn to equate closeness with smallness of Euclidean distance). However, here we are considering distances in state space, not in space time, and where our intuitions may not apply. This raises the question about whether there is a state-space metric where realistic and counterfactual states are actually distant from one another. The toy model example here is too simple to allow such an alternate interpretation. However, using fractal geometry we have shown that there is a model where such states are indeed distant from one another, thus negating the fine-tuning argument. As discussed in the Introduction, a rather beautiful example of how relying on intuition about distance can lead to inconsistency is provided by the Penrose Impossible Triangle. We claim that quantum theory (cf., \(T^{\prime}\)) is similarly inconsistent, even though it is wonderfully accurate and a convenient tool for analytic manipulation and computation. This inconsistency arises from the use of the Euclidean metric forced on us by the assumption that state space is the algebraically closed Hilbert Space. By weakening this assumption, allowing only Hilbert states with rational descriptors as elements of physical reality, the inconsistencies associated with the Bell and other no-go theorems [6], disappear. The key conclusion we can draw from this discussion is that, in Invariant Set Theory, there is no contradiction with the fact that the violations of (2) are insensitive to polariser orientation. Let us now discuss a related issue. Let us fix the orientation of [PERSON]'s measuring device and ask whether, according to Invariant Set Theory, this in any way constrains [PERSON] in orienting her measuring device. If [PERSON] was somehow constrained, she would not only not be a free agent, she would somehow be remotely under the influence of [PERSON]--clearly unacceptable in any theoretical framework purporting to reinstate the notion of Einsteinian local realism. The answer is that, for all practical purposes, [PERSON] is under no such constraint. What 'for all practical purposes' means is that within _any_ neighbourhood of [PERSON]'s celestial sphere, no matter how small, there exist orientations which [PERSON] is free to choose from, providing \(p\) is sufficiently big. That is to say, the set of orientations from which [PERSON] can choose is as dense as one likes, providing \(p\) is big enough. Colloquially, we can indeed say that [PERSON] can set her measuring apparatus as she pleases--she is a free agent in any practical sense of the phrase. Nevertheless, whatever the size of \(p\), Invariant Set Theory requires that the cosine of the relative orientation between [PERSON] and [PERSON]'s measuring apparatuses must be rational. For large \(p\), this is an utter irrelevance in the _design_ of a Bell experiment. However, it is crucially important in the _interpretation_ of a Bell experiment, since, as discussed, the counterfactualstates needed to establish (1) will inevitably lie off \(I_{U}\) and are therefore \(g_{p}\) distant from the states measured by experiment. In conclusion, in making the statement that the orientation of [PERSON]'s polariser does not influence [PERSON]'s choice of polariser orientation, the contrast between the Euclidean metric of space-time and the metric \(g_{p}\) of state space becomes crucial. The author believes that the ongoing failure to synthesise quantum and gravitational physics satisfactorily arises from the fact that quantum theory is inimical to the local realism of general relativity, and a synthesis between these two areas of physics will require a nonlinear theory of quantum physics, less like \(T^{\prime}\) and more like \(T\).