On The Possibility of Conscious Numbers

“ Type-F monism is the view that consciousness is constituted by the intrinsic properties of fundamental physical entities… On this view, phenomenal or protophenomenal properties are located at the fundamental level of physical reality, and in a certain sense, underlie physical reality itself.”
-David J. Chalmers, Consciousness and its Place in Nature

Chalmers gives a coherent argument for the problems that come from a complete reduction of phenomenal properties to material interactions. The view that he finds most convincing, in opposition to a complete materialism, is one that locates consciousness at the fundamental level of material reality, which would, by extension, give consciousness to everything it composes, from atomic particles, up to us (and, even further, to the entire universe!)

This seems reasonable, when approached from a top-down direction. We do not, even intuitively, think of consciousness as operating at the same level in all things. Consciousness does not emerge fully formed, as we experience it, in an instant. It is not intuitive to think that dogs have the same “level” of consciousness that we do, even though it is intuitive to many to say that they do have some level of consciousness. Similarly, there are more complex consciousnesses, and less complex ones, according to Chalmers. Chalmers says that the levels of complexity, or perhaps it’s better to say simplicity, go all the way down to the level of a single photon, which has an inconceivably simple consciousness, but consciousness nonetheless.

I think it’s safe to say that most philosophers of mind who have followed recent developments in cognitive science and AI research agree that whatever the mind is, it is fundamental to a certain pattern or ongoing process carried out by our brains, that is to say, it is not dependent on our brains themselves. This seems clear when we entertain the idea of a world in which every carbon atom in our brain was replaced with a separate molecule, Q, that behaves exactly like carbon. It seems absurd to think that such a world would have no consciousness, simply by virtue of not having carbon atoms. The takeaway is that organic matter does not have a privileged place in philosophy of mind.

AI research tells us that the only important factor in the creation of intelligence is the pattern that is hosted on our wetware (our brains). Whatever pattern this is, as far as we know, it is Turing complete. By this, we mean that the pattern that makes you, you, would work the same if it was hosted on a computer made of meat, as it does, or of silicon, as transhumanists hope it one day will be. What’s important about this is that the pattern of your mind can be described, as can any computer program, as a single, gigantic mathematical function (since all these functions can be combined through function composition).

Sensationalist as it may be, there are many who believe that our universe is a computer simulation. Regardless of whether it is true, it is still possible. This is because, like the mind, the universe is also Turing complete (to the best of our knowledge).

Many philosophers and mathematicians, notably Frege and Gödel, have argued for mathematical platonism. This is the view that mathematical entities exist independently of us and our conception of them. If this is true, it seems to imply that all possible mathematical function, including those that describe minds and universes, do exist. This is the view that leads some, like Max Tegmark, to believe that our universe is a simulation, only it is not being run on any computer, rather, it exists simply by virtue of being a mathematical object.

All of the points thus far seem reasonable, or at least possible. However, when we bring these views together, we find that they lead us to a new belief that I believe has not been considered before: conscious numbers.

All of Chalmers arguments that locate phenomenal properties at the fundamental level of physical reality should also apply to the mathematical functions that describe this reality; they are functionally identical, indistinguishable from the inside, in the same way that would lead us to believe we cannot tell if we are living in a simulation. Here the only difference is that no physical computer need actually run the mathematical function for us, on the inside, to exist. There is, in a way that I would like to develop further elsewhere, a way that a perfect map doesn’t just perfectly reflect a territory; it is the territory.

If we look for the fundamental level of mathematical reality, most (perhaps all) mathematicians would direct us to set theory, of our very own beloved Bertrand Russell. Without delving to deeply, we begin with the null set, {}, which we would call 0, and then we get the set containing the null set, { {} }, which we would call 1, and then the set containing the null set as well as the set containing the null set { {}, { {} } }, which is 2, etc. This is the fundamental level of reality where Chalmers says consciousness is located.

If we wrote out the truly humongous functions that perfectly describe our universe, we would end up with an absolutely enormous composite of sets. Significantly, our everyday mathematical objects, 2’s 100’s, Φ and π, are all made up of the same material, sets. This implies, as Chalmers says of conscious photons, that even these relatively simple objects are conscious on a low level.

In summary, if we accept the arguments of Chalmers and Tegmark, together we end up with the unintuitive conclusion that numbers, like 2 and π, are also conscious.