I obviously have a great deal of sympathy for the scientific perspective in general. I am on the side of the scientific community in their ongoing dispute with mystics over the rational comprehensibility of Reality. But we part company over the belief among many scientists that Reality is fundamentally a material domain.

The typical hardcore materialist insists that there is nothing beyond the material world. A common response to this position is to trot out phenomena such as hope, love or spirit as counterarguments. My initial response is to ask if the concepts that form the basis of the intelligibility of the material world are material phenomena. If so, what are the masses, locations, momenta or charges of the numbers we use to quantify the mass, location, momentum and change of matter? For that matter, what are the sizes, masses, locations, momenta and charges of the ideas we call size, mass, location, momentum and charge? Unlike hope, love, spirit and other such arguably mental manifestations, there are quantitative concepts that exist independently of the mind.

Before I prove my conjecture let me state that for the sake of this argument I am willing to concede that as a phenomenon that emerges from interactions among material phenomena (i.e., neurons, molecules, atoms, fermions, etc.), the mind can be characterized as residing in the material domain. In light of this concession, the standard materialist counterargument to my position is that even if ideas are not material phenomena, they cannot exist without minds. Being contingent upon occupants of the material domain means ideas are also encompassed by that domain. These materialists would argue, for example, that the idea of mass did not exist, although massive phenomena did, prior to conscious beings manifesting and creating the idea.

My subsequent counterargument is based on the theory of real numbers. Mathematicians theorize that the vast majority of real numbers can only be referenced by an infinite string of symbols or operations used to calculate them. This means that only an infinitesimal fraction of real numbers can be referenced in any finite sense (which is the only sense available to us as finite beings). To put this in perspective, consider that if the continuum of real numbers is viewed as the infinite set of all possible integers, the referencible values in it correspond to the number 1.

To appreciate the unreferencible nature of most real numbers, consider the value we get by executing the following algorithm. As we sequentially scan each of the theoretically limitless decimal places of the value π, wherever we encounter a digit that is greater than or equal to 5, we leave it alone. Where we come upon a digit that is less than 5, we flip a “fair coin”. If the coin lands with the heads side up, we double the digit and if it lands with tails up, we add 1 to it. The specific value that would emerge from the theoretical completion of this algorithm is essentially unreferencible since it can only be uniquely referenced by a limitless series of numbers or operations.

Every real number whose only reference is comprised of an infinite sequence of digits is essentially unreferencible. In other words, each such value represents a target without a reference. But the reference to a target is the means by which it manifests in a conscious mind. For example, when you think about an elephant, it is not an actual elephant that manifests in your mind; it is merely a reference to one that is created by your mind.

A matter-based mind can only create references to targets that manifest in some comprehensible manner in the material domain. For instance, the reference to a mythical creature such as a unicorn can manifest in our minds because a finite combination of physical phenomena that captures its uniqueness is available to us (e.g., shaped like a horse with a single horn protruding from its forehead…). By contrast, no finite combination of material manifestations captures the uniqueness of a particular unreferencible number. This is evident in that the algorithm mentioned earlier does not refer to a particular unreferencible number since it will never produce the same number twice (the probability of this happening is one in infinity, which makes it infinitely improbable). Where the mind cannot create a reference to a target entity, that entity cannot manifest in the mind. This only happens where there is no finite combination of phenomena in the material domain that encompasses the target’s uniqueness.

The unreferencible nature of virtually all real numbers means they cannot be used to refer to anything else. In other words nothing can have a size or manifest in a quantity that corresponds to an unreferencible number. This means that unreferencible numbers cannot refer to entities in the material domain.

But, one might ask, if unreferencible numbers cannot manifest in our minds, do they exist at all? In a mathematical sense, unreferencible numbers must exist since without them the continuum of real numbers is indistinguishable from the discrete system of rational numbers. In other words, if unreferencible numbers do not exist, then there are no real numbers.

To summarize: unreferencible numbers are not material manifestations (i.e., they have no material properties such as mass, charge, momentum, etc.), they are not individually referred to by occupants of the material domain (e.g. conscious minds) and they do not refer to any occupants of the material domain (i.e., they do not correspond to the size or quantity of any material manifestation). This means that these numbers are completely uncoupled from the material domain. Since they are not in any way connected to the material domain, unreferencible numbers represent completely non-material phenomena.

Where unreferencible numbers actually reside is beyond the scope of this essay (though it is explained in my framework). My conclusion that there are phenomena that reside completely beyond the material domain is why I reject pure materialism and the label of materialist, regardless of my frequent disagreements with mystics.