Navigating the Ancient Discourse: The Debate on Free Will in Philosophy

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Let's now explore how we can apply logic, math, and physics to formalize the relationship between determinism and indeterminism in causality within the monadological framework. First, let's define our basic entities and relations: - Let M be the set of all monads (fundamental psychophysical entities). - Let T be a set of "time points" or "moments." - Let S be a function from M × T to some set of "states," where S(m, t) represents the state of monad m at time t. - Let C be a relation on M × M × T, where (m1, m2, t) ∈ C means monad m1 "causes" or "influences" monad m2 at time t. Now, let's formalize the idea of determinism and indeterminism in causality: - Determinism: ∀m ∈ M, ∀t ∈ T, S(m, t) is uniquely determined by {S(m', t') : (m', m, t') ∈ C}. - Indeterminism: ∃m ∈ M, ∃t ∈ T, such that S(m, t) is not uniquely determined by {S(m', t') : (m', m, t') ∈ C}. In other words, determinism means that the state of each monad at each time is uniquely determined by its causal influences, while indeterminism means that there are some monads whose states are not uniquely determined by their causal influences. We can formalize this further using the mathematical framework of graph theory and probability theory: - Let (M, E) be a directed graph, where E ⊆ M × M represents the "causal edges" between monads. - Let (Ω, F, P) be a probability space, where Ω represents the set of all possible "outcomes" or "histories," F is a σ-algebra on Ω, and P is a probability measure on F. - The determinism and indeterminism of causality can be expressed as: - Determinism: ∀m ∈ M, ∀t ∈ T, ∃f : Ω → S, such that S(m, t) = f(ω) for all ω ∈ Ω. - Indeterminism: ∃m ∈ M, ∃t ∈ T, such that ∀f : Ω → S, P({ω ∈ Ω : S(m, t) ≠ f(ω)}) > 0. Here, determinism is formalized as the existence of a function f that maps each possible outcome ω to a unique state for each monad at each time, while indeterminism is formalized as the non-existence of such a function (i.e., there are some monads whose states have a non-zero probability of differing from any given function). Finally, we can connect this to physics by noting that this formalism is compatible with both deterministic and indeterministic approaches to causality: - Deterministic models like classical mechanics describe the evolution of physical systems as uniquely determined by initial conditions and dynamical laws. - Indeterministic models like quantum mechanics describe the evolution of physical systems as inherently probabilistic, with outcomes determined only probabilistically by initial conditions and dynamical laws. The monadological framework accommodates both perspectives by treating determinism and indeterminism as emergent properties arising from the complex web of causal relations between fundamental monads. In summary, by using tools from logic, math (graph theory and probability theory), and physics (classical and quantum mechanics), we can formalize the both/and nature of determinism and indeterminism in causality within the monadological framework: - Causality is a relational structure arising from the web of causal influences between fundamental monads. - Causality exhibits both deterministic and indeterministic properties, depending on the scale and perspective of observation. - This formalism is compatible with both deterministic and indeterministic approaches to causality in physics. This showcases the potential of the monadological framework to provide a unified language for expressing and reconciling the complex, often seemingly contradictory nature of reality. By embracing a both/and perspective and drawing on the tools of logic, mathematics, and physics, we can develop a more nuanced and comprehensive understanding of the world and our place within it.