IMPROPERLY POSED PROBLEMS

Many physical phenomena are described by models that are not properly posed in any reasonable sense; certain initial conditions may fail to produce a solution in the model (although the phenomenon itself certainly has an outcome) and/or solutions may not depend continuously on their initial data. One such phenomenon is, say, the tossing of a coin on a table. Assuming we could measure with great precision the initial position and velocity of the coin, we could integrate the differential equations describing the motion and predict the outcome (head or tails); however, it is obvious that for certain trajectories, arbitrarily small errors in the measurement of initial position and velocity will produce radical changes in the predicted outcome. In studying phenomena like these we are then forced to abandon the deterministic point of view and to obtain information by other means (e.g., by probabilistic arguments). This was known to the creators of probability theory and was explicitly stated by Poincaré almost a century ago.