Mathematical realism (also called mathematical platonism) is the view that mathematical structures exist independent of the mind. Other positions view mathematics and its objects as, in one sense or another, human creations or interpretations. Nominalists deny the existence of mathematical objects, and provide reformulations of mathematical statements in other terms. Intuitionists regard mathematical objects as mental constructions. Mathematical realism is not monolithic. For example, finitists take a realistic view of finite mathematical objects such as trillion digit numbers or the monster group, but not of completed infinities. Note that even the realism of finitists is wild in a way, in that it posits the existence of unbounded numbers of mathematical structures that are bigger and more complex than the physical universe.

Although I am a mathematical realist myself, I will not argue the case here, but discuss a consequence of realism : it provides a way of thinking about the substance of which consciousness consists.

A mathematical realist can visualize a sort of cloud of mathematics around physical things. This cloud consists of the mathematical objects that exhibit systematic correlations to the things in question. This supports the following way of talking about the mathematical structure of physical reality. One interprets “physical thing X exhibits mathematical structure Y”, as meaning “there exists the mathematical structure Y that exhibits such and such correlation to the physical object X”. This way of putting matters takes the realistic view towards the mathematical object: it is an existent in its own right which correlates to the physical. For example, for each elementary particle there is the corresponding Lie group. Actually, physicists almost always use the formulations of mathematical realism. This may seem like a distinction without a difference, but please read on.

Part of the cloud arises from mathematical formulations of basic physical laws, as exemplified just now. But there are many mathematical regularities exhibited by particular physical things at a more specific level. For example, a computer exhibits many mathematical regularities other than the general physical laws that apply to its particles. There are, for example, regularities to do with the inputs and outputs of the arithmetic unit. There are also the regularities to do with the software that the computer is executing. Anyone who programs computers knows a lot about these regularities, and talks about them constantly without noticing where the computer ends and the mathematics begins.

For illustration, let us consider a more specific case: a computer running graphics software, and, even more specifically, a situation in which the graphics software is modeling and displaying the shape of a coffee cup. Here is a picture:

There is also a cloud of mathematical structures correlated to the activity of a human brain, very likely with many layers, as is the case for computers (not that the brain is a computer!). Not much is known about this cloud, but, to the mathematical realist, it nonetheless exists.

I say that not much is known about the cloud, but I mean this from the point of view of neuroscience — that is, from the outside. Here is the hypothesis: suppose that our consciousness is itself a part of the cloud of mathematical structures, which, according to the mathematical realist, is associated with the brain. Then, some of the mathematical objects, such as the mathematical shape of the coffee cup, would themselves reside in consciousness. But this is the case! An intrinsically mathematical entity, the 3D shape of the cup, is present to my awareness. The 3D shape, that is the mathematical aspect of my awareness, does not by any means exhaust that awareness (more about that later). Also, it is available to awareness in a particular way. It is not that I am aware of the equations of the shape. Instead, the shape has the “feel” of something with width, depth, and height, and it can be bumped into by other shapes, and mentally rotated. My awareness of the cup contains the operations that apply to mathematical shapes (only selected operations, of course). So it is not too outlandish to imagine that the cup-awareness is such a mathematical shape.

Before going further, let me emphasize a point. Where mathematical realism comes into the matter is that, according to realism, the cloud of mathematics around the physical, of which consciousness might be a part, needs no interpreter to endow it with existence — it simply is. Without the realism, some sort of interpreter would be needed, and this runs in the direction of theism.

There are two huge issues.

First, only a tiny bit of the mathematical cloud that correlates with the brain can constitute consciousness. How might this fraction be selected?

I don’t have my own answer to this, but there are relevant proposals in current theorizing about consciousness : first, Tononi’s integrated information theory (IIT), and more recently Hoel’s effective information (EI). IIT posits that consciousness exists where there is an integrated causal system of activity — integrated in the sense that it cannot be partitioned into non-interacting subsystems without changing its behavior. Actually, “integrated” is not a true or false distinction, but a number, called Φ (phi), which measures the degree of integration. According to IIT, consciousness is everywhere, but the systems with high Φ have a high degree of consciousness. The theory is mathematical — it posits a way to compute the Φ of a system if its details are known. The computation is difficult, but in each case there is a particular number that applies to any system.

In order to apply IIT, a system needs to be characterized in a particular way, in which the basic elements and causal relationships are defined. This in turn requires the choice of a level of description. Considering the example computational system pictured above, with its four levels, Φ would take on different values at different levels. Viewed from the level of elementary particles, for example, there would be a profusion of causally independent subsystems (since not every particle’s exact state affects the rest of the system). Thus Φ would be zero. Only at the higher levels would integration appear. The same is true of the brain, presumably. IIT takes this into account by identifying the level of consciousness of a system with the maximum value that Φ takes on over differing characterizations (e.g. levels of description) of the system.

So the IIT answer to our question about where in the mathematical cloud consciousness is to be found is this: in the parts of the cloud with high Φ.

I am not advocating this particular answer, but it does show a way forward. Φ cannot be the final answer, because while high Φ might be a necessary condition for consciousness, it is not sufficient. Scott Aaronson has shown this by exhibiting a simple family of systems that have arbitrarily high Φ, but that cannot possibly be conscious. (Not everyone is convinced by Aaronson’s argument, but I am). That is, IIT can generate false positives.

Hoel’s effective information, similarly, is a number which measures the integration of a causal structure, formulated in terms mutual information between past and future. Again the causal structure can be defined at different levels of description, and higher levels of description can yield more effective information than lower ones. The measure is formulated in information-theoretic terms, and is aimed at the general question of causation at different levels of description, rather than at consciousness specifically (and it is a far simpler measure than Φ). Given IIT’s false positives there is in any case a need for additional filters in the search for conscious systems. EI is a candidate for the initial pre-filter search. We are in early days on this topic!

Φ or EI locate consciousness within a causal system at a particular level of description. But there is a further distinction to be made: the mechanisms of consciousness versus the contents of consciousness. The latter consists of the things of which I am aware, to which I can attend, can remember, anticipate, and be aware of my awareness of. They are the phenomena, the things that appear. According to one theory, a global workspace commonly available to the mechanisms is what defines the contents.The causal mechanisms underlying awareness are for the most part missing from awareness. For example, I don’t see the mechanisms by which my attention moves from this to that, nor how I choose the next word in the sentence I am writing.

Still, the contents of consciousness constitute a subset of the part of the mathematical cloud isolated by, for example, EI or Φ. It is another challenge to draw the line enclosing just the contents, and excluding the invisible mechanisms, but however that line is drawn, mathematical realism is what grants uninterpreted existence to both contents and mechanisms.

There is a technical point here to do with how the contents fit into the causal structure : namely, contents appear as part of the formulation of the state space underlying the causal structure. It is far more parsimonious to formulate that state space and its transitions in terms of contents (e.g. as general rules for transitions based the position and shape of the cup), than say, as a vast table . So, the contents of consciousness will be captured by a criterion which selects, for any given causal structure under consideration, a formulation of the state space and its transitions having minimal complexity. This is a criterion in addition to the numerical EI or Φ measures themselves. (Actually, one would need this sort of criterion anyway in a full formulation of “mathematical structure correlated to a physical structure” — any finite mathematical structure correlates to any physical structure, if one is given no bounds on the complexity of the correlation. )

Now to the second issue: the “hard problem”. Why should it feel like anything to be a node of integration (say) in the brain’s mathematical cloud? And why do the contents of consciousness, for example sensations of red or pain (“qualia”) have the particular essence that they do? This is the explanatory gap that yawns between any purported non-mental substrate, and the “being there” of our own awareness. In this case the proposed substrate is mathematical rather than strictly physical, but the gap remains as wide. Clearly, neither the physical nor the mathematical can fully constitute consciousness. This is a problem that our minds seem ill-equipped to ponder. I have nothing to propose. But, as argued above with the shape of a cup as example, it does seem that the contents of consciousness are partly constituted by mathematical structure — structure upon which the qualia are somehow painted, perhaps.