Introduction: The Structure of Physical Theories and a Definition of a Theoretical Error Home Contact Me

A physical theory resembles a mathematical theory. Both rely on a set of axioms and employ a deductive procedure for yielding theorems, corollaries, etc. The set of axioms and their results are regarded as elements of the structure of the theory. However, unlike a mathematical theory, a physical theory is required to explain existing experimental data and to successfully predict results of new experiments. This distinction between a mathematical theory and a physical theory has several aspects. First, experiments generally do not yield precise values but contain estimates of the associated errors. (Some quantum mechanical data, like spin, are the exception.) It follows that in many cases, a certain numerical difference between theoretical predictions and experimental data is quite acceptable. Next, one does not expect that a physical theory should explain every phenomenon. For example, it is well known that physical theories yield very good predictions for the motion of planets around the sun. On the other hand, nobody expects that a physical theory be able to predict the specific motion of an eagle flying in the sky. This simple example proves that the validity of a physical theory should be evaluated only with respect to a limited set of experiments. The set of experiments which are relevant to a physical theory is called its domain of validity. (A good discussion of this issue can be found in [1], pp. 1-6.) Relations between two physical theories can be deduced from an examination of their domain of validity. In particular, let DA and DB denote the domains of validity of theories A and B, respectively. Now, if DA is just a part of DB then one finds that theory B takes a higher hierarchical rank than theory A (see [1], pp. 3-6). Here theory B is regarded as a theory having a more profound status. However, theory A is not "wrong," plainly because it yields good predictions for experiments belonging to its own (smaller) domain of validity. Generally, theory A takes a simpler mathematical form. Hence, wherever possible, it is used in actual calculations. Moreover, since theory A is good in its validity domain DA and DA is a part of DB then one finds that theory A imposes constraints on theory B, in spite of the fact that B's rank is higher than A's rank. This self-evident relation between lower rank and higher rank theories is called here constraints imposed by a lower rank theory. It is used in this site more than once. Thus, for example, although Newtonian mechanics is good only for cases where the velocity v satisfies v << c, relativistic mechanics should yield formulas which agree with corresponding formulas of Newtonian mechanics, provided v is small enough. And indeed, it is well known that special relativity is consistent with these constraints. The notion called here constraints imposed by a lower rank theory agrees with the broader sense of Bohr's correspondence principle. However, in some textbooks on physics the Bohr correspondence principle is restricted to the relationships between quantum mechanics and classical mechanics. Therefore the usage of the expression restrictions imposed by a lower rank theory looks better. In the literature on the philosophy of science, closely related ideas are called The Generalized Correspondence Principle (see e.g. [2]). Having these ideas in mind, a theoretical error is regarded here as a mathematical part of a theory that yields predictions which are clearly inconsistent with experimental results, where the latter are carried out within the theory's validity domain. The direct meaning of this definition is obvious. It has, however, an indirect aspect too. Assume that a given theory has a certain part, P, which is regarded as well established. Thus, let Q denote another set of axioms and formulas which yield predictions that are inconsistent with P and they apply to the validity domain of P. In such a case, Q is regarded as a theoretical error. (Note that, as explained above, P may belong to a lower rank theory.) An error in the latter sense is analogous to an error in mathematics, where two elements of a theory are inconsistent with each other. This criterion also covers the case of a technical error in the pure mathematical sense. Indeed, such an errorneous formula must be inconsistent with at least one kind of experimental data. A discussion of the role of mathematics as a basis for physical theories can be seen in Wigner's Article The Unreasonable Effectiveness of Mathematics in the Natural Sciences here. There are other aspects of a physical theory which have a certain value but are not well defined. These may be described as neatness, simplicity and physical acceptability of the theory. A general rule considers theory C as simpler (or neater) than theory D if theory C relies on a smaller number of axioms. These properties of a physical theory are relevant to a theory whose status is still undetermined because there is a lack of experimental data required for its acceptance or rejection. The notion of simplicity is closely related to the principle called Occam's razor (see here). The notions of neatness, simplicity and physical acceptability have a subjective nature and so it is unclear how disagreements based on them can be settled. In particular, one should note that ideas concerning physical acceptability changed dramatically during the 20th century. Thus, a 19th century physicist would have regarded many well established elements of contemporary physics as unphysical. An incomplete list of such elements contains the relativity of length and time intervals, the non-Euclidean structure of space-time, the corpuscular-wave properties of pointlike particles, parity violation and the nonlocal nature of quantum mechanics (which is manifested by the EPR effect). For these reasons neatness, simplicity and physical acceptability of a theory have a secondary value. Thus, if there is no further evidence, then these aspects should not be used for taking a final decision concerning the acceptability of a physical theory. In this site properties of a physical theory pertaining to a lack of neatness, simplicity and physical acceptability are mentioned. However, these properties of problems may be helpful for the reader but they should not be regarded as decisive arguments. In this site there is no distinction between neatness and simplicity. Thus, the term neatness is not used. Before concluding these introductory remarks, it should be stated that the erroneous nature of a physical theory E cannot be established merely by showing the existence of a different (or even a contradictory) theory F. This point is obvious. Indeed, if such a situation exists then one may conclude that (at least) either theory E or theory F is wrong. However, assuming that neither E nor F rely on a mathematical error, then one cannot decide on the issue without having an adequate amount of experimental data. Another issue is the usage of models and phenomenological formulas. This approach is very common in cases where there is no established theory or where theoretical formulas are too complicated. This approach is evaluated by its usefulness and not by its theoretical correctness. Hence, it is not discussed in this site. References: [1] F. Rohrlich, Classical Charged Particles, (Addison-wesley, Reading Mass, 1965). [2] H. Radder, Heuristics and The Generalized Correspondence Principle., British Journal for the Philosophy of Science, 42, 195 (1991).