Creating a mathematical theory of consciousness is a challenging and ambitious task, but it can be approached by drawing on existing theories and models in neuroscience, psychology, and information theory. Here’s a basic outline of how such a theory might be structured, including some foundational assumptions, key variables, and proposed equations. The goal would be to describe and predict aspects of consciousness using mathematical formalisms.

Foundational Assumptions

1. Consciousness as Information Integration: Following theories like Tononi’s Integrated Information Theory (IIT), assume consciousness correlates with the capacity of a system to integrate information.
2. Quantifiable Neural Activity: Assume that neural activity and its interactions can be quantitatively measured and modeled.
3. States of Consciousness: Define consciousness states that can be differentiated mathematically, such as wakefulness, sleep, and various altered states.

Key Variables

• Φ (Phi): A measure of integrated information, representing the capacity of a system to integrate information.
• S: The state matrix of the system, where each element represents the state of a neuron or a group of neurons.
• T: A tensor representing the connectivity and interaction strength between the elements in S.
• C: A complexity measure, possibly derived from the diversity and distribution of network states.

Proposed Equations

Information Integration

[ \Phi(S) = \sum_{i=1}^n \left( H(S) – H(S | S_{-i}) \right) ]
Where:

• ( H(S) ) is the entropy of the system state S, measuring the uncertainty or randomness in the system.
• ( H(S | S_{-i}) ) is the conditional entropy of S given the state of all parts except part i, indicating how much additional information is contributed by part i.

Consciousness Threshold

Define a threshold ( \Phi_0 ) such that if ( \Phi(S) > \Phi_0 ), the system is considered conscious.

Temporal Dynamics of Consciousness

[ \frac{dS}{dt} = F(S, T) ]
Where:

• ( \frac{dS}{dt} ) describes the change in the neural state matrix over time.
• ( F(S, T) ) is a function describing the dynamics of the system based on current states and their connectivity.

Complexity and Consciousness

[ C = -\sum_{i=1}^n p_i \log p_i ]
Where ( p_i ) is the probability of the system being in state i, and the sum is over all possible states, reflecting the distribution and diversity of states contributing to consciousness.

Hypothetical Predictions

• Systems with higher ( \Phi ) values exhibit richer and more diverse conscious experiences.
• Changes in ( T ) (connectivity patterns) can predict changes in consciousness, such as transitions into sleep or altered states.
• Systems that maintain high complexity ( C ) under various conditions are more likely to exhibit adaptive and robust conscious behavior.

Validation and Implications

This theory would need to be validated through simulations and empirical data from neuroimaging and behavioral studies. If successful, it could provide a powerful tool for diagnosing consciousness in different biological and artificial systems and could guide interventions to alter or manage states of consciousness in clinical settings.

By mathematically formalizing aspects of consciousness, such a theory can help bridge the gap between empirical observations and theoretical models, facilitating deeper understanding and practical applications in neuroscience, psychology, and artificial intelligence.

Sources: InnerIGPT