Mathematics of All Souls

The non-physical nature of the human soul or self-consciousness as opposed to the physical nature of the human body, has been studied by many thinkers including Descartes, Spinoza, Kant, Hegel, Freud, Lacan, Dennet and Damasio, as described in particular in The Parallax View by Slavoj Zizek.

Lacan's psychoanalytic theory is mathematical in a sense, and on All Souls Day on Nov 2 it is natural to ask if mathematics can offer some understanding of body-soul duality? Let's seek an answer:

It appears that one aspect of the non-physical or virtual nature of soul can be described as sensitivity of future goal satisfaction with respect to perturbation of the present state. In economy this is represented by derivates relating to changes of underlying assets, like options on stocks, with options representing virtual soul and stocks representing real body.

The important aspect is that the derivate has a non-physical virtual character since it relates to changes of an underlying physical asset. For example, the velocity of a physical body as change of position per unit time step, has a virtual non-physical character, while position is physical. You can directly touch position but not velocity. Being is body and becoming is soul, in the spirit of Heidegger.

The soul thus would appear as a sensitivity, which in mathematical optimization is expressed by duality with the sensitivity being the solution to a linearized dual/adjoint equation obtained by linearizing the state equation. The duality body-soul would thus be reflected in the duality of mathematical optimization.

The body would thus be the solution of the state equation, while the soul would be the solution of a corresponding dual/adjoint state equation, expressing sensitivity of a goal function with respect to state perturbations. In particular, this captures the aspect of the soul of planning for the future since the dual/adjoint state equation runs backward in time. The physical body thus moves forward in time, while the non-physical soul moves backward. The body moves from the present towards the future, and the soul moves from the future towards the present.

The dual world of the soul would appear as a partly strange Alice in Wonderland reflection of the real world of the body, since in the dual/adjoint equation in linearization picks up certain aspects of the state equation and distorts them in a backward time process. The dual world is a strange fully virtual world with everything moving backwards in a strange distorted fashion. But it can be captured by a computer and thus is open to inspection and a posteriori rationalization and understanding. The rabbit obsessed with time in Alice in the Wonderland maybe lives in the dual world.

Bodily functions may be controled in unconscious direct feed-back, while the soul by duality

would be able to plan and steer towards future goals as a unique capability of homo sapiens.

The sensitivities could be expressed as feelings giving weight to perturbations of the state equation in the optimization process of human life.

The financial crisis shows the danger of souls with too much belief in option derivate fantasy

reflecting unrealistic future gains without connection to realities of underlying body assets.

Another aspect is soul as simulation of body with the soul housing a brain representation of the body, rather than a brain dual of feelings. When the soul back-tracks from some future goal in planning coming activities, simulation of different possible scenarios in forward time plays an important role. In other words, the soul has a double role of both simulating the state equation in forward time and solving the dual equation in backward time. This may be the reason the human brain is so uniquely big compared to the body...