Here's a rampantly realist description of an experiment that produces a violation of Bell's Theorem. The reason to give a realist description is to make it clear what the commitments of realism are, and to show how they lead to a contradiction with experimental fact. Thus, having swept realism out of the way, we can proceed to see how conceptualism allows us to provide a description of this kinds of experiment that does not lead to contradictions.
By realism I mean [Ray,1999]:
So when a realist talks about atoms or photons or their properties, she is claiming that these things exist intrinsically, as kinds, independently of her interactions with them or awareness of them. A conceptualist talking about the same things, is claiming that the terms "atom", "photon" and so on are names for categories that he has come up with to subsume various particulars based on similarities that are relevant to his purposes. His use of these categories and not others is a result of his purposes as much as it is a result of the reality he's categorizing.
The difference between realism and conceptualism is particularly pointed when it comes to talking about the values that the properties of things have. Properties, such as polarization and direction, are universals, and so to a realist will have the same ontological status as the particulars to which they belong. A universal, to a realist, is just a type of particular, and what we normally think of as particulars are they way they are--are members of the categories they are members of--by virtue of bearing an exemplification relationship to some universals and not others.
Now, "being linearly polarized along the x-axis" is a general term: there are many photons that could have this property. So to a realist, there must be a real, mind-independent category that makes the attribution of this general property true by virtue of its exemplification relationship to it. To a realist, there is part of reality that "just is" a photon, independent of any knowing subject categorizing it that way, and that photon "just has" a linear polarization along the x-axis, independently of any knowing subject categorizing it that way. Furthermore, it must be the case, to a realist, that we do not have any choice as to how to categorize reality: the categories are given by and in reality.
Let's see how these commitments play out when applied to a real case.
Consider a single calcium atom suspended somehow in space, perhaps in a Penning trap. Two beams of light strike the calcium atom and excite it into an atomic state that has zero angular momentum, as does the ground state of calcium. The atom cannot decay from this excited state by emitting a single photon, because photons have one unit of angular momentum and so angular momentum would not be conserved in such a transition because in the excited state the atom has zero angular momentum so the total angular momentum of the system is zero, and in the ground state the atom has zero angular momentum so after the transition the system of photon plus atom would have 1 unit of angular momentum, so some angular momentum would have to appear out of nowhere, which is forbidden by very general conservation laws.
The only way the atom can decay is by emitting two photons, whose angular momenta add up to zero. It turns out that this condition is only met when both photons have the same linear polarization. However, because the original system consisting only of the excited atom is spherically symmetric (this is one of the consequences of have zero angular momentum) there is no preferred direction for the common plane of polarization of the two photons. That is, they must have the same plane of polarization, but there is nothing about the system that causes that plane to be one plane rather than another. The identity of the plane appears to be ambiguous.
Thus far, the commitment to realism in the description above has been mostly implicit, and simply a consequence of the fact that natural language has a heavily realist bias. Now, however, a realist must make an explicit commitment: the common plane of polarization is part of the identity of the individual photons (there are reasons why treating the pair of photons as a single, non-local entity create problems for realism that I'll get to later, but for now we can simply say most realists take "photon" to be a natural kind and "pair of photons" to not be a natural kind, so for the average realist there isn't any choice about treating the photons as separate individuals.)
To reiterate: to a realist, the common plane of polarization is part of the identity of the individual photons. It must be possible to talk about this property of the photons independently of any mind that might want to categorize this part of reality as photons, or, having made that categorization, wanting to know what their polarization might be. The property "having a linear polarization along a particular plane" is something the photons must possess in a realist's view, because on that view things exist in categories independently of anyone who categorizes, and a part of reality that bears an exemplification relation to the category "photon" and bears an exemplification relation to the category "linearly polarized" must also bear an exemplification relation to the category "linearly polarized along some particular plane."
As a final consideration, let us assume that the events described above happen many times: the atom gets re-excited after it emits a pair of photons, and then decays again emitting another pair, and so on, for as long as we want.
So the realist's claims amount to this:
Remember that by "real" in the above the claim is that not just that there are particulars that exist, but that there are particulars that exist as members of the categories that name them, and all that that implies. So to be a real photon is on this view to have a real polarization, independently of anyone categorizing the photon in a way that makes its polarization relevant or meaningful.
Because of this, if any of these claims are false, they are all false. If there is no real plane of polarization, then there is no real photon, because on the realist view identity must be independent of the identifier--there is no room within realism for things that are photons but that don't have any particular plane of linear polarization (readers with a background in quantum theory will realize that this means we don't have to do anything so elaborate as Bell's argument to rule out realism, as Heisenberg in particular was quite aware.)
So do the realist claims above commit us to anything that can be shown to be false? They do, and this is the nature of Bell's argument, which usually gets presented in the form of statistical inequalities. Here, however, I'm just going to point out the significance of the functional form of the results of measurements on the two photons.
If the two photons in each pair have a real common plane of polarization, then we expect that measurements of their polarization will yield a particular functional form. Polarization measurements are typically made by letting a photon pass through a polaroid filter that transmits one linear polarization and absorbs the polarization at 90 degrees to it. For photons with linear polarizations oriented between these two extreme angles, the probability of passing through the polaroid filter goes as the cosine of the angle between the plane of polarization and the direction the polaroid filter transmits.
Furthermore, once a photon has been transmitted through a polaroid, its plane of polarization has been changed to the match the orientation of the transmission direction of the polaroid, so if we put a second polaroid behind the first and orient it at a different angle, the total transmission probability is just the cosine of the angle between the two.
If the two photons in each of the pairs of photons emitted from the calcium atom have a common plane of polarization, and we set up two polaroid filters on opposite sides of the source so that one photon passes through one and one photon passes through the other (we assume for simplicity that the photons always travel in opposite directions, which is roughly correct) then we can predict what the correlation between them will be as a function of the angle between the transmission directions of the two polaroids.
In particular, when the two polaroids are oriented at 90 degrees to each other, we can say that there will still be photons that get through both polaroids about a half of the time. This is because the common plane of polarization, if there is one, is randomly oriented because the initial system is spherically symmetric, so each photon has a better than 50% chance of getting through it's filter. There will be cases, for instance, when the common plane of polarization is at 45 degrees relative to both filters, which are at 90 degrees to each other. In these cases, both photons will be transmitted about 70% of the time (cosine(45) = 0.707).
If realism is true, then we can be sure that when the filters are at 90 degrees to each other, there will still be cases where both the photons are detected. And conversely, when the filters both have the same orientation, we can be sure if realism is true there will still be lots of cases where one photon is detected and the other is not. A plot of the functional form is forthcoming.
So we have an empirical claim that allows us to test a metaphysical doctrine: do things exist with their identity fully specified relative to a set of predefined, mind-independent universal categories?
The answer is unequivocally, "No."
The functional form of the correlation between detections of the two photons is a cosine law: when the polaroids are at 90 degrees to each other, there are no pairs that have both members detected. When the polaroids are lined up with each other, the joint detection rate is 100%. There are of course non-ideal aspects to the experiments, however, the precise numerical agreement with the quantum-mechanical predictions for the experiment--which show a cosine law dependence--clearly rules out realism.