A dynamical law describes how a specified state can evolve. An initial condition selects where the evolution begins. Neither ingredient can generally replace the other, and even their combination needs conditions that make the resulting problem well posed.
One rule, many histories
Consider the scalar equation dx/dt = −kx with k positive. Its solutions are x(t) = x₀ exp(−kt). Fixing k fixes the decay rate; choosing x₀ selects a particular history. At k = 1 per second, preparations with x₀ = 2 and x₀ = 5 retain their ratio throughout the evolution. Their common law does not require them to have the same value.
A single later measurement also need not identify the law. If x(1 second) = 1, every positive k is compatible with a corresponding x₀ = exp(k × 1 second), in the chosen units for x. Additional measurements or independent preparation information are needed to separate the rate from the starting value.
This is an elementary example of identifiability. A flexible family may fit one datum perfectly while leaving many possible mechanisms. Fixing the preparation first turns a later measurement into a stronger test of the rate; estimating both from the same isolated point cannot do that.
What counts as a complete starting state?
For a harmonic oscillator, position alone does not select a trajectory. With angular frequency ω, the solution is x(t) = x₀ cos(ωt) + (v₀/ω) sin(ωt). Both initial position and velocity matter. Two oscillators starting at the same position can subsequently move in opposite directions.
If an observer records only position, the apparent future ambiguity does not establish indeterminism of this full model. It exposes missing state information. The earlier essay on memory for prediction explores a related situation: discarded variables can reappear as dependence on the observed past.
For a spatial field, initial data may be functions across space and may have to satisfy constraints. A finite region also needs boundary information appropriate to the equations. Specifying a number called “the initial state” without identifying all required variables can leave the prediction problem incomplete.
An equation does not guarantee a unique future
In dimensionless variables, consider dx/dt = 2√|x| with x(0) = 0, for t ≥ 0. One solution remains zero forever. For any chosen T ≥ 0, another remains zero until T and then follows x(t) = (t − T)². Before T both sides vanish; after T both equal 2(t − T); at T the derivative is also zero. These are multiple differentiable solutions of the same stated initial-value problem.
The example is mathematical, not evidence that a physical object spontaneously chooses a departure time. The usual local uniqueness theorem needs regularity assumptions, such as local Lipschitz dependence on x, which this right-hand side lacks at zero. The missing hypothesis explains why merely writing a first-order differential equation does not settle determinism.
One should distinguish failure of uniqueness, sensitivity to a slightly uncertain initial condition, and practical inability to solve a unique problem. Each limits prediction for a different reason. The word “unpredictable” alone does not identify which reason applies.
Keep preparation separate from calibration
Suppose a candidate theory has a source parameter, a detector scale and an initial-state distribution. If all are adjusted after each new observation, the fitted curve may conceal how little has been predicted. A stronger procedure fixes a common parameter set using one collection of preparations and checks other trajectories with their independently specified starting conditions.
Initial conditions are not always arbitrary knobs: a larger physical account may explain how the preparation was produced. That adds a preparation model with its own inputs and tests. It does not justify deleting the preparation from the calculation.
The task for a source-based reconstruction
For SMM/DTT, a single source could in principle constrain both the dynamics and admissible preparations. The constraint must be exhibited. It is useful to state separately which rule follows from the source postulate, which parameters are supplied and which particular history has been selected.
The expanded book's auditable model discussion gives a practical direction: use the same construction to calculate several observable responses and reserve some observations for testing. The published completion framework should likewise be read through its explicit assumptions rather than treated as a substitute for a complete initial-value problem.
Sources and further reading
- Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, AMS, 2012, especially the initial-value problem and uniqueness theory.
- Constraints and mechanisms: why a restriction on states is different from an evolution rule.