To clarify what I said in one of the previous posts about consciousness and quantum mechanics. (This is from a very limited and probably very erroneous understanding of this topic. Alternatively, you can just skip the mumbo-jumbo below and just watch this — no discussion of four-dimensional waves can happen without a reference to this scene.)
From what I understand, the idea is that measurement affects what "decision" the electron makes. According to some people, measurement is congruent with human consciousness. In reality, that's not necessarily the case, because just measurement alone (i.e., interaction with a non-conscious measuring device), without the consciousness, is enough to collapse the wave.
The same goes for Schrödinger's Cat. According to the "paradox", until you open the box with the cat, the cat exists as both alive and dead, and then your measurement (and, some people would say, consciousness) influences the reality.
(By the way, Schrödinger himself never postulated this as the state of events — he used his Cat as an argument ad absurdum to show that our understanding and mathematical apparatus of quantum mechanics are incomplete. But, interestingly enough, around the same time, Einstein, Podolsky and Rosen published a paper, in which they showed that, following mathematics of quantum mechanics, you can have a state of events when particles are separated spatially and yet are instantaneously linked to each other. I.e., just because they are far apart from each other doesn't mean they are distinct, independent entities. This was also an ad absurdum paper, but this phenomenon was shown to be true — it is now known as quantum entanglement. So, not all ad absurdums are absurd.)
Anyway, the point is that it doesn't require consciousness to collapse a function. What actually is required is measurement, but why that is the case is unclear. Some people say it's the interaction with the "macro-world" which collapses the function (in which case, Schrödinger's cat is exclusively dead or exclusively alive as soon as the electron interacts with the poisonous gas machine). But, recently they've done another experiment, in which they "labeled" the electron itself as having passed through either right or left slit (this way, one would know, once the electron reaches the screen, which slit it had passed through). Well, if you do that, you, as expected, collapse the interference pattern.
But then, if you put a "quantum eraser" between the slits and the screen, which will erase the "label" from the electron, you actually restore the interference pattern. So, when the electron interacts with the macro-world once, this collapses the wave function. But, if it interacts with the world twice, this restores the wave function! So, in the end, what's important is whether the electron's passage through one or the other slit is measured. If it is measured, the wave collapses. If the measurement information is erased, the wave is restored.
Again, this doesn't necc. say anything about the importance of consciousness. I think that we just have no idea what really happens when the electron wave collapses.
The unique thing about the Quantum Eraser experiment is that once electron wave collapses at the slit, there is no way that interaction of the electron's wave with itself can happen — but then, if you put up the quantum eraser, it restores the interference pattern... meaning, it is as if it told the electron: "Now go back to the past and actually interfere with yourself".
So, some people say that this shows that an electron is actually a four-dimensional wave. Its interaction with quantum labeler changes its energy such that it becomes collapsed. Its interaction with quantum eraser restores the same energy level and restores the electron's wave -- but, it restores it back in four dimensions (including the time), such that the electron is now a wave "again", not just in the space, but also in the time, including the point in the past where it passed through the slit. (I think this would be a good moshol for teshuva that Tzvi Freeman et al. could use. When a person makes a teshuva, he goes back and erases his sin — in fact, turns his sin into a mitzva. This is because, although to our perception, we exist as a point on the time line, always moving forward, in reality, we exists in all possible dimensions at the same time. And, by making teshuva, we interact, with G-d's help, with ourselves in the past.)
If you want an illustration of this concept, just watch this scene.
But, another interpretation of the Quantum Eraser experiment could be that when a wave function collapses, it does not collapse "completely", so to speak. It just passes from the state where it influences reality to the state where it does not, but it (and its interaction with itself) always exists in potential, alongside with the electron's "collapsed" path.
So, imagine that when electron passes through the two slits, all the possible paths it can take exist simultaneously — and interfere with each other in a wave-like fashion. Then, when the electron is "labeled", its wave function collapses, and only one of the paths exists "in reality". The other paths, however, do not disappear. They are still out there, hanging like punctured lines, "in potential".
If the electron is allowed to reach the screen, the potential paths and their potential interactions are never allowed to influence the reality of the electron's trajectory.
If, however, the electron interacts with the quantum eraser, the latter reverses whatever effect the quantum labeler did (by restoring the energy levels to the former, etc.). This restores the electron's state as a wave and all the "potential" possible paths become real (all at once) again — as well as their, so far only potential, interactions.
In this interpretation, there is no need to do something as esoteric as imagining the electron as a four-dimensional wave interacting with itself not only in space, but also in time. Although, perhaps the concept of potential paths existing in a state of semi-reality is even more esoteric.
Update: Regarding probabilities, a conversation in the kitchen:
— Did you know that on average, a human foot sweats a cup a day?
— Not my foot.
— I said “on average”.
— Just sayin’. My feet don’t sweat no cups per day.