Mathematical Realism and the Hard Problem of Consciousness
Mathematical realism is the view that mathematical structures have an existence independent of the mind. At first blush, this view has no relationship to the problems of consciousness — easy or hard. This note is about where a connection might lie. But first, let me establish a context: I’ll argue for mathematical realism and against direct realism for the physical world , and finally tell how the relationship between the phenomenal world (aka “consciousness”) and the brain is analogous to the relationship between a mathematical model, and the theory that describes that model (and I’ll explain what I mean by “models” and “theories” — terms taken from mathematical logic). The existence of consciousness is itself evidence for mathematical realism.
1. An Argument for Mathematical Realism
You can skip this part if you don’t need convincing — or feel you can’t be convinced.
I can contemplate the set of prime factors of 8,678,104,367,575,337 as an object with its properties, for example how many factors there are, or what the biggest one is. The same goes for the set of all possible positions arising from the current state of play in a chess game. I might not know what the properties are, but have no doubt that they are well determined: if I should work them out, I will come to the same answer as anyone else.
Of course, these are not objects in the physical world, but my thinking about them is very similar; they are well-determined independent of my wishes. They are things of some sort in the world external to me, I feel.
But just because I posit something as an object of thought does not mean it exists outside the mind. People are good at dreaming up new sorts of things, naming them, and thinking of them as real. It is this capability that allows for the contemplation of numbers, Chi, astrological forces, electrons, and all the rest as real things. This might be called ontological fertility. Of course, the products of this fertility deserve scrutiny. There are arguments for (explanatory power, compatibility with evidence, logical necessity) and against (internal inconsistency, incompatibility with evidence). The principle of parsimony leads to rejection of ontologies as the default. There is a fourth sort of argument in favor in the case of mathematical structures: not logical necessity, but necessity as a foundation for thought.
Here is that argument. No person decides what the prime factors of 8,678,104,367,575,337 are. If two people look into the matter separately, they come to the same conclusion without any communication between them. The properties come from outside of the mind, it seems. You might say, well these two people with their common conclusion are simply playing the same game of factorization. It is only a matter of a game with numerals with commonly understood rules, not any separately existent number.
However, there are lots of regularities about what happens with numbers, or if you like, numerals — these regularities are to be found in the theorems of number theory. In the anti-realist (nominalist) view these are regularities of the game of mathematics. But to reason about the outcome of this kind of game you need to reason about numbers. So nominalism provides no escape, unless you want to say that all the regularities are inexplicable. You can forswear numerical (and more generally) mathematical thinking. But then it is not just mathematical regularities that become inexplicable (interpreted by the nominalist as game-outcome-regularities), but the major portion of the sciences.
It goes beyond that. All thinking about the future (not just what chess move to make, but whether to take a rain jacket with you tomorrow), involves logical, and sometimes, mathematical considerations (planning your cuts in carpentry, for example, involves the latter). What we know of the physical world is a trace through a landscape of possibilities, and that landscape is not itself a physically existent thing. Finding our own way through the landscape often requires the ability to reason about mathematical structures.
So, if you want to understand the world, even the day-to-day world, you can’t avoid mathematical realism, it seems to me. For some notions of what exists (some ontologies), for example the notion of Chi, or theism, it might be possible to stand outside — to say, “no, the supposed existents mentioned in the ontology can be dismissed”. No such possibility exists for realism for mathematics without, it seems to me, the loss of the ability to think. At least no possibility of standing outside systematically: it is easy to say “numbers don’t exist” at one time, but then let them back into existence when you need them.
The argument that I have just given is similar to the Quine-Putman indispensibility argument, but doesn’t focus as much on the mathematical nature of scientific understanding. The idea that there exists an independent realm of mathematics, and more generally, of objects of logical thought, goes back past the forms of Plato, to the logos of Heraclitus.
We determine the properties of mathematical structures via thought rather than perception in most cases. Not all cases! Mathematical structures are sometimes incarnated in physical objects. This fact no doubt was the origin of our mathematical capability — both in biological and cultural evolution. But now the mathematician, chess player, or skilled carpenter often sees the relevant structures in his mind’s, not his body’s, eye. This is direct realism — no mediation is needed.
For the rest of this note, mathematical realism is assumed. If you choose to use a word other than “existence” for these things with their mind-independent properties, the argument is not changed. Certainly, they are not “out there” in the sense of residing in a “place”.
2. An Argument Against Direct Realism for Physical Objects, and for the Existence of the Phenomenal Realm.
I have argued for direct realism for mathematical objects. What about physical objects?
Is the physical world present directly to me through my senses? When I see a coffee cup, is the physical coffee cup present in my mind, or does perception involve a separate version of the cup in my head?
Neuroscience dictates the latter standpoint (indirect realism). The same would hold for any theory according to which the mind is embedded in the physical world. The presence of objects to my mind requires some sort of explanation, if my mind arises in some fashion from my brain. If I see a coffee cup in front of me, clearly it doesn’t show up as it is in my brain — which remains gooey and free of ceramic content. If one avoids the spooky or theistic, and stays more or less in the naturalist standpoint, one has to say that the pattern of my brain activity holds within it something corresponding to the coffee cup, because if I can talk about it, my vocal chords must be directed to do so via nervous impulses — this cannot happen via some kind of unmediated causation from the outward cup. Direct realism is contradicted. Now there are two coffee cups — one in the world, and one in my mind/brain. The latter is conventionally called the phemonenal coffee cup. For me to say these words, think these thoughts, I need to take on a standpoint from which I look down on my own mind as something in the world, and see the mental version of the coffee cup there. I also need to look down at the physical world as something separate from the world as it is usually present to me. After all my world is the phenomenal world, and there is no way to actually stand outside of it. Nonetheless, in my thinking, I can ask — what is the world as a whole, including me, my experience, and the experience of other people? So, to address this big question, I need to posit a theoretical three part realm, which includes the mathematical, the physical, and the phenomenal. When following this train of thought, I occupy in an abstract way what Thomas Nagel calls a view from nowhere.
3. The Phenomenal as Model
Now we arrive at the connection between mathematical realism and consciousness. This relies on a notion from mathematical logic: the notion of a model of a theory. Let me explain these terms briefly . A theory is a list of sentences in a chosen mathematical language. For example, the theory known as Peano arithemetic has sentences like: “for every natural number n the number n+1 exists”and “For all natural numbers m and n, m = n if and only if m+1= n+1. “. A model of a theory is a mathematical structure in which the sentences in the theory are true. In the case of Peano arithemetic, the theory is a small finite set of sentences, whereas every model of it is an infinite mathematical structure. One of the models, the so called standard model, is the natural numbers themselves. A model of a theory is called its interpretation. The sentences themselves are simply lists of characters. If we are mathematical realists, we believe that the models as well as the theories have a definite sort of existence.
Now, the sentences themselves can and are construed as simple mathematical structures themselves. But, of course, they have physical realizations too: for example as what you see on your screen in quotation marks: “4+5=9” is a physical realization of a simple sentence in the theory of arithmetic. In a computer, the bitstrings in memory and in CPU registers are also physical things to which numbers (the mathematical objects) correspond.
In what I said just now about computers and numbers, I was speaking as a mathematical realist. I regard 4, the number not numeral, as an existent mathematical object. I am able to make connections between physical objects, like marks on a page or computer screen or bits in a computer, and mathematical things.
For the mathematical realist, the physical world is embedded within the realm of mathematical structures. One can visualize a sort of cloud of mathematics around physical things consisting of the mathematical objects that exhibit systematic correlations to them.Consider the cloud around a computer. Everyone who programs or designs computers knows something about that cloud’s nature, and talk about it constantly without noticing where the computer ends and the mathematics begins. The speakers may not regard themselves as mathematical realists, but I would say that they are thinking as mathematical realists, whether they are willing to admit it or not.
Back to theories and models. Consider a computer running theorem proving software. In the computer are the sentences and terms of the theory (perhaps Peano arithmetic), and in the cloud is some portion of the actual mathematical structure to which the theory refers.
Now, at long last, I’ll get to the point: what mathematical cloud surrounds a living brain? We don’t know much about this, at the current primitive state of neuroscience. That is, we are quite ignorant from the outside — our view from nowhere. For a while some people thought that the brain was quite a bit like the theorem proving computer mentioned above, but now this notion is in retreat, if not completely discredited. This does not prevent a theory/model relationship of some sort with brain states as theory, if we take the notion generally. After all, in computers running CAD software, and abacuses, there is a systematic relationship between the physical constituents and mathematical entities. Once neuroscience makes more progress perhaps an extensive realm of mathematical structures that correspond well to brain states might be discovered.
But can we say anything from the inside? Perhaps. The world is present to us as, for one thing, shapes. Shapes have their mathematical aspect, as Euclid was aware. Might it be that our phenomenal awareness consists partly in a set of mathematical structures that are picked out by an interpretation of brain states? If so our phenomenal worlds are built partly from fragments of mathematical reality. Note that the shapes as phenomena are not at all the same as mathematical theorizing about shapes, since the former are direct contents of awareness as we look around us, and not usually the subject matter of our mathematical thoughts, for those of us who have them (a small minority in the history of the human race). There is a separate interpretation for the latter.
Can this approach account for the whole of awareness. No! There are “feels” (qualia) too. That is, there is the structure of our phenomenal world, but also its substance, its particularity — for example the particularity of a color or a texture. I have nothing to propose about this aspect of the mystery, only the structural aspect, upon which the qualia are painted.