Toward Formalizing Non-monotonic Reasoning in Physics: The use of Kolmogorov Complexity to Formalize the Notions of "Typically" and "Normally"

When a physicist writes down equations, or formulates a theory in any other terms, he usually means not only that these equations are true for the real world, but also that the model corresponding to the real world is “typical” among all the solutions of these equations. This type of argument is used when physicists conclude that some property is true by showing that it is true for “almost all” cases. There are formalisms that partially capture this type of reasoning, e.g., techniques based on the Kolmogorov-Martin-Löf definition of a random sequence. The existing formalisms, however, have difficulty formalizing, e.g., the standard physicists’ argument that a kettle on a cold stove cannot start boiling by itself, because the probability of this event is too small. We present a new formalism that can formalize this type of reasoning. This formalism also explains “physical induction” (if some property is true in sufficiently many cases, then it is always true), and many other types of physical reasoning. In the current mathematical formalizations of physics, physically impossible events are sometimes mathematically possible. From the physical and engineering viewpoints, a cold kettle placed on a cold stove will never start boiling by itself. However, from the traditional probabilistic viewpoint, there is a positive probability that it will start boiling, so a mathematician might say that this boiling event is rare but still possible. In the current formalizations, physically possible indirect measurements are often mathematically impossible. In engineering and in physics, we often cannot directly measure the desired quantity; instead, we measure related properties and then use the measurement results to reconstruct the measured values. In mathematical terms, the corresponding reconstruction problem is called the inverse problem. In practice, this problem is efficiently used to reconstruct the signal from noise, to find the faults within a metal plate, etc. However, from the purely mathematical viewpoint, most inverse problems are ill-defined meaning that we cannot really reconstruct the desired values without making some additional assumptions.