Continued from the last post.

Discounting the Bohmian song and dance, we are led to conclude that each electron does in some sense pass through both slits. But in what sense? Saying that an electron went through both slits cannot be equivalent to saying that the electron went through L and that it went through R, for to ascertain the truth of a conjunction we must individually ascertain the truths of its components, and we never find that an electron launched at G and detected at D has taken the left slit and that it has taken the right slit.

Nor can saying that an electron went through both slits mean that a part of the electron went through L while another part went through R. Analogous experiments have been performed with C₆₀ molecules using a grating with slits 50 nanometers wide and spaced 100 nanometers apart. The sixty carbon nuclei of C₆₀ are arranged like the corners of an old-fashioned soccer ball having a diameter of just 0.7 nanometers. We do not picture parts of such a molecule as getting separated by many times 100 nanometers and then reassemble into a ball less than a nanometer across.

Saying that an electron went through both slits can only mean that it went through L&R — the two cutouts in the slit plate considered as an undivided whole. Whenever Rule B applies, the distinction we make between L and R is a distinction that has no reality as far as the electron is concerned. In other words, the distinction between "the electron went through L" and "the electron went through R" is a distinction that Nature does not make — it corresponds to nothing in the actual world.

If we consider the fuzzy position of the electron relative to the nucleus in the 2p0 state of atomic hydrogen, this should not come as a surprise:

The electron is neither to the left nor to the right of the nucleus, which is situated at the center. Its fuzzy position (relative to the nucleus) is the entire "cloud", which can (in principle) be observed as a distribution over the outcomes of a sufficiently precise position measurement performed on a large number of  atoms in the 2p0 state.

Now ask yourself this: Does the expanse over which this relative position is "smeared out" have parts? If it had, the position itself would have parts; it would be divided by the parts of space. But this makes no sense. One can divide an object, and thereby create as many positions as there are parts (one for each part), but one cannot divide a position. The expanse over which a fuzzy position extends lacks parts. Hence if we think of space as this expanse, we must think of it as intrinsically undivided.

To thinkers from Aristotle to Kant and Gauss this was self-evident. They considered space as an expanse that made it possible to divide material objects, rather than as something that was intrinsically divided. They only made the mistake of thinking that synchronic multiplicity — the multiplicity of things in space at any one time — was realized through boundaries (delimiting geometrical surfaces). "Space is essentially one", Kant wrote, "the manifold in it ... arises entirely from the introduction of limits". In reality, as described by quantum mechanics, synchronic multiplicity is realized through the fuzzy spatial relations (relative positions and relative orientations) that exist between material objects.

There are neurobiological reasons why it is hard not to believe that matter is carved up by geometrical boundaries just as rolled-out pastry is carved up by cookie cutters. For the way in which the brain processes visual information guarantees that the result — the visual world — is a world of objects whose shapes are bounding surfaces. In reality, the shapes of things are sets of more or less fuzzy internal spatial relations, while positions are sets of more or less fuzzy external spatial relations. (Internal = within the same object, external = between different objects.) Following the evidence where it leads, we conclude that fundamental particles, lacking internal relations, are formless.

Which makes room for another way of thinking about space. Instead of attributing the property of being spatially extended (of "occupying space") to an expanse to which spatial relations owe their spatial character, we may attribute it directly to the spatial relations themselves. Space is then simply the set of all spatial relations existing between material objects. It contains — in the proper, set-theoretic sense of containment — the forms of all things that have forms, inasmuch as forms are sets of spatial relations, while it does not contain the fundamental particles, inasmuch as these are formless. Talk about weirdness!

Whereas the aforementioned path-breaking thinkers also insisted that infinity should be conceived as merely potential, in the second half of the nineteenth century mathematics shifted to the conception of an actually completed infinity and to dealing with the continuum as a set of points. As far as pure mathematics is concerned, this is a legit way of widening the playing field, but when a prominent philosopher of science defines "a field theory in physics" as "a theory which associates certain properties with every point of space and time", problems arise. (For example: The geometrical concept of a point presuppose a surrounding space. How  can this space consist of points?)

Here, too, it seems to be our neurobiology that is leading us up the garden path. Although we readily agree that red, round, or a smile cannot exist without a red or round object or a smiling face, we seem to just as readily believe that positions can exist without being properties of material objects. What makes positions that special? A possible explanation is that the role that position plays in perception is analogous to the role that substance plays in conception.

As to conception, among the ideas that philosophers have associated with the word "substance", the following is relevant here: while a property is that in the world which corresponds to the predicate in a sentence composed of a subject and a predicate, a substance is that in the world which corresponds to the subject. It objectifies the manner in which a conjunction of predicative sentences with the same subject term bundles predicates.

As to perception, our visual cortex is teeming with feature maps. (A feature map is a layer of the cerebral cortex in which cells map a particular phenomenal variable, such as hue, brightness, shape, motion, or texture, in such a way that adjacent cells generally correspond to adjacent locations in the visual field.) Every phenomenal variable has at least one separate map except location, which is present in all maps. If there is a green box here and a red ball there, "green here" and "red there" are signaled by neurons from one feature map, and "boxy here" and "round there" are signaled by neurons from another feature map. "Here" and "there" are present in both maps, and this is how we know that green goes with boxy and red goes with round. Position is the integrating factor. Whereas substance serves as the "conceptual glue" that binds an object's properties, position serves as the "perceptual glue" that binds an object's phenomenal features. In the brain, and consequently in the visual world, positions pre-exist — in the brain at the scale of neurons, in the visual world at visually accessible scales. They exist in advance of visual objects, and this makes us think that they also exist in advance of physical objects, not only at the scale of neurons or at visually accessible scales, but also at the scales of atoms and subatomic particles.

Moreover, since features present in the same place get neurally integrated into a single object, while features present in different places get neurally integrated into different objects (or different parts of the same object), we are convinced by default that one and the same object cannot be in different places, and that different objects cannot be in the same place, although both possibilities exist in the quantum world. (The two electrons in the ground state of helium have exactly the same fuzzy position, while a single electron can be simultaneously in two places — the latter because the distinction we make between the two places is real for the electron only under the conditions stipulated by Rule A.)