1 The Impossibility of Free Will

Proof. Suppose the universe is deterministic. Let \(S(t_0)\) be the complete state of the universe at some time \(t_0\) before the agent’s birth. By the definition of determinism, \(S(t_0)\) together with the laws of nature \(L\) uniquely determines the state \(S(t)\) at all later times \(t > t_0\), including the time \(t_a\) at which the agent performs action \(A\).

The agent’s action \(A\) at time \(t_a\) is part of the state \(S(t_a)\). Since \(S(t_a)\) is uniquely determined by \(S(t_0)\) and \(L\), and since \(t_0\) precedes the agent’s existence, the action \(A\) is determined by factors that existed before the agent existed and over which the agent had no control.

By Definition 1.3, free will requires that the action not be “determined by prior causes external to the agent’s will.” But \(S(t_0)\) is prior to and external to the agent’s will (since the agent did not exist at \(t_0\)). Therefore, the agent does not have free will with respect to action \(A\).

Since \(A\) was arbitrary, no action of the agent is free. Since the agent was arbitrary, no agent in the deterministic universe has free will. ◻

Proof. Let \(E\) be a random event—that is, an event not determined by any prior causes. By Definition 1.2, this means the complete state of the universe prior to \(E\) does not determine whether \(E\) occurs.

Suppose \(E\) is an action of agent \(A\). For \(E\) to be an exercise of free will, by Definition 1.3(c), it must be “determined by the agent’s own free choice.”

But if \(E\) is determined by the agent’s choice \(C\), then \(C\) is a prior cause of \(E\), contradicting the assumption that \(E\) is random (not determined by any prior causes).

If \(E\) is not determined by any choice of the agent, then \(E\) is not an exercise of the agent’s will at all—it is something that happens to the agent, not something the agent does.

Therefore, random events cannot constitute exercises of free will. ◻

Proof. Let \(E\) be any event in such a universe. Either \(E\) is determined by prior causes, or it is not.

Case 1: \(E\) is determined by prior causes. Then by Theorem 1.1, \(E\) is not an exercise of free will, since it was determined by factors external to any agent’s will.

Case 2: \(E\) is not determined by prior causes. Then \(E\) is random (by Definition 1.2), and by Theorem 1.2, \(E\) is not an exercise of free will.

Since these cases are exhaustive and mutually exclusive, no event \(E\) is an exercise of free will. ◻

Proof. Every logically possible universe is either wholly deterministic, wholly random, or a combination of the two (this exhausts the logical space). By Theorems 1.1, 1.2, and 1.3, free will is impossible in each case. Therefore, free will is impossible in any logically possible universe. ◻

Proof. Suppose agent \(A\) makes a free choice \(C\) at time \(t\). By Definition 1.3, \(C\) is not determined by prior causes. Let \(S(t^-)\) be the state of the universe immediately before \(t\), and let \(S'(t^+)\) be the state immediately after \(C\).

If \(C\) is genuinely free, then \(S'(t^+)\) is not uniquely determined by \(S(t^-)\) together with the laws of nature. The laws, applied to \(S(t^-)\), would produce some state \(S(t^+)\); but the free choice \(C\) produces a different state \(S'(t^+) \neq S(t^+)\).

This means the agent has altered the trajectory of the universe away from what natural law would dictate. This is precisely what we mean by a miracle: a deviation from the natural order that cannot be explained by natural causes alone.

Moreover, this miracle is not a one-time occurrence but must happen at every moment of free choice. Free will requires a continuous stream of miracles, each one redirecting the universe from its lawful course. ◻

Proof. Let \(|\Psi(t)\rangle\) be the universal wave function at time \(t\). Under standard quantum mechanics, \(|\Psi(t)\rangle\) evolves unitarily: \[|\Psi(t + \Delta t)\rangle = \hat{U}(\Delta t)|\Psi(t)\rangle = e^{-i\hat{H}\Delta t/\hbar}|\Psi(t)\rangle \tag{1.1}\] where \(\hat{H}\) is the Hamiltonian of the universe.

Suppose an agent with free will makes a choice that influences the physical world—say, by affecting which outcome occurs in a quantum measurement or by influencing neural processes that lead to bodily action.

Let \(|\Psi_{\text{natural}}(t + \Delta t)\rangle = \hat{U}(\Delta t)|\Psi(t)\rangle\) be the state that would result from unitary evolution alone. Let \(|\Psi_{\text{actual}}(t + \Delta t)\rangle\) be the state that actually results after the free choice.

If the choice is genuinely free (not determined by \(|\Psi(t)\rangle\) and \(\hat{H}\)), then \(|\Psi_{\text{actual}}(t + \Delta t)\rangle \neq |\Psi_{\text{natural}}(t + \Delta t)\rangle\).

But then the evolution from \(|\Psi(t)\rangle\) to \(|\Psi_{\text{actual}}(t + \Delta t)\rangle\) is not given by \(\hat{U}(\Delta t)\). The actual evolution violates unitarity.

Since unitarity is required for conservation of probability (and hence conservation of energy, momentum, etc.), free will would violate fundamental conservation laws. ◻

Proof. Suppose agents \(A\) and \(B\) both have free will. At time \(t\), let \(A\) freely choose that the universe enter state \(S_A\) at time \(t + \Delta t\), and let \(B\) freely choose that the universe enter state \(S_B\) at time \(t + \Delta t\), where \(S_A \neq S_B\).

The universe can only be in one state at \(t + \Delta t\). Therefore, at least one of the following must occur:

1. The universe enters \(S_A\), and \(B\)’s choice is not realized.

2. The universe enters \(S_B\), and \(A\)’s choice is not realized.

3. The universe enters some third state \(S_C \neq S_A, S_B\), and neither choice is realized.

In each case, at least one agent’s free choice fails to determine the outcome. But if an agent’s “free choice” can be overridden by external factors (another agent’s choice, or some mechanism for adjudicating conflicts), then the agent does not have ultimate control over the outcome—contradicting the requirement of free will. ◻

Proof. Let \(H\) be a human and \(G\) be God. Suppose \(H\) freely wills that event \(E\) occur, and \(G\) freely wills that event \(E\) not occur. By Theorem 1.5, at least one will must fail to be realized.

If God’s will always prevails, then humans do not have free will in any robust sense—their choices can always be overridden by divine fiat.

If human wills can sometimes prevail against God’s will, then God is not omnipotent in the traditional sense.

If there is some higher mechanism that adjudicates between divine and human wills, then that mechanism—not the agents—determines outcomes, and neither truly has free will.

This trilemma shows that the simultaneous existence of divine and human free will is incoherent. ◻

Proof. Consider an entangled state of two particles: \[|\Psi\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle_A|0\rangle_B + |1\rangle_A|1\rangle_B\right) \tag{1.2}\] where \(A\) is Alice’s particle and \(B\) is Bob’s particle, spatially separated.

Suppose Alice, by free will, influences which outcome (\(|0\rangle_A\) or \(|1\rangle_A\)) occurs when she measures her particle. Before Alice’s measurement, Bob’s particle is in a mixed state (reduced density matrix \(\rho_B = \frac{1}{2}(|0\rangle\langle 0| + |1\rangle\langle 1|)\)).

After Alice’s measurement and free choice, Bob’s particle is in a definite state (\(|0\rangle_B\) or \(|1\rangle_B\), correlated with Alice’s outcome). Alice’s free choice has thus changed the quantum state of Bob’s particle—instantaneously, regardless of the spatial separation.

This is a nonlocal influence: Alice’s local choice affects Bob’s distant particle without any signal propagating between them. ◻