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Functionalism, in philosophy of consciousness, is the idea that consciousness arises from the functional role that parts of a system play within that system.

More specifically, as I interpret it here:

A mental state is a condition of holding certain concepts in relation to each other. The relation is one of potential causation, i.e. the potential of each concept to act upon and with the others to produce a result. A concept corresponds to a part of a physical system. Any physical systems whose parts stand in the same relation have the same mental state.

We look for an isomorphism of potential causation between the parts of two systems; if such an isomorphism does exist, the two systems are in the same mental state.

Mathematically all this language is very unclear. To mathematize it, at least the following terms would need rigorous definitions:

  • Physical system
  • part of a system
  • potential causation between parts of a system
  • isomorphism of potential causation

Is there a source, within functionalism, that has taken a similar approach to this, and succeeded in mathematically expressing at least some of these terms?

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This is an attempt to a formal, not mathematical description. Like mathematics and logic, systems are part of metaphysics, but there is no agreement on the formal description of systems. Mind maps, predicate logic or UMLv2 are valid formal systems of description of systems (sorry for the redundancy, but it is valid). But essentially, you just need lines, arrows and circles, the basic UML traces. Here, I will use circles and lines.

System: A system is a set of interrelated parts. That is, a system is a set. A set of what? Of parts, which are also systems. And which have the same structure. So, from an abstract view, a system is a fractal, and like fractals, its definition is recursive. To visualize the system fractal, draw a circle with or more circles inside, and on each circle, do the same. Repeat that an infinite number of times. That is a system, that drawing can easily represent a bike, a glass of water or a rainbow.

Relationships: the system family is the set of persons that are related by an ideal set of bounds: family links. Notice that what makes the family is not the collection of members, but instead, the relationships. What makes a molecule of water is not so much the atoms, but the relationships. To visualize a relationship: draw two circles and join them with a line.

Physical systems: strictly, it can't be said that there are physical systems. Systems are subjective ideals. If a house is a set of parts, one person might divide it in terrain, construction materials and legal assets. Another can divide it in a kitchen, bedrooms, etc. Assuming that the system is a physical set of parts implies assuming that the noumenon has a internal structure, which is by definition wrong.

Physical relationships: what we subjectively perceive as physical relationships are normally inputs and outputs. A coffee machine is not a closed system: it is open because it relates with humans in a specific form: we provide it of coins as inputs and get coffee cups and garbage as outputs. To visualize inputs and outputs: draw two circles A and B, and draw an arrow from A to B. A has an output, B has an input.

Causality: it is also an ideal(phenomenon), not a noumenal fact. Physically, all things interact with all other things (e.g. A butterfly that creates a storm in the opposite side of the world; any atom interacts with all other atoms), but we learn to create mental abstractions: cause: coin, consequence: coffee cup. To visualize: draw a circle with two arrows: one towards the circle, the opposite in the other direction. The one which goes towards the circle is the coin. The other is the coffee cup. Voilà: causality.

According to functionalism, fear would be not the set of elements (parts) that constitute it (e.g. the location, the impression, the sensation, the intensity, etc.) but moreover the functional relationships that hold the parts as a whole, in general, the causes (inputs) and the consequences (outputs). So, in order to survive, an entity needs to reject what is destructive and get attracted to what is constructive. So, all living entities need to have equivalent functions. What is called pain in humans should have an equivalent on animals, and perhaps even in plants, although the movement of plants is much, much slower, but the stimulus, stress and reaction should be equivalent.

Then, according to your interpretation, which seems consistent, consciousness in general would be a function of a holistic goal. And what constitutes the system fear itself are a set of functional relationships that raise it.

Opinion: I agree with such position; in my research I propose that such final goal is survival. So, from a macroscopic perspective, the system fear follows a supra systemic goal, survival.

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