Scientific theories

A theory (from Greek theorein, “contemplate, observe, examine”) is a coherent set of explanations, notions or ideas on a specific subject, which may include laws and hypotheses, induced by the accumulation of facts from of observation, experimentation or, in the case of mathematics, deduced from a given axiomatic basis: theory of matrices, torsors, probabilities. It should not be confused with a philosophical principle contrary to the principles observed and provisionally admitted suggested by experience, nor with a hypothesis. The term theoretician, who is a scientist, is opposed to that of technician who is charged with putting into practice a particular science.

In the philosophy of science, a scientific theory must meet several criteria, such as the correspondence between the theoretical principles and the phenomena observed. A theory must also make it possible to make predictions about what will be observed. Finally, the theory must resist the experience and be compatible with new facts that can be added over time, or remain valid in new areas not yet explored during its first elaboration. If this is not the case, the theory must be corrected or invalidated outside its first domain.

Thus, it is in the long term that the strength of a theory is judged because it must be able to remain compatible with new facts, resist experiments that would like to demonstrate its invalidity, and ensure the accuracy of its predictions.

In common language, the term “theory” is sometimes used to designate a set of speculations without any real foundation, contrary to the meaning accepted by scientists. The word hypothesis is then more appropriate. The principle, on the other hand, is an observation which can be explained or not according to the period considered (e.g., Fermat’s principle), but in all cases of which we do not possess – or, to neglect any track, not yet – of counterexample (e.g., second principle of thermodynamics).

Mathematics

When mathematics is formalized, in mathematical logic, a theory is a set of affirmations, some of which are axioms and the others theorems demonstrable from these axioms and by means of rules of logic. The first incompleteness theorem of Gödel, coined by Kurt Gödel, states that any coherent theory, having a finite number of axioms (or axiom schemas) in a language that can describe arithmetic, and that demonstrates some simple arithmetic statements, will always contain undecidable propositions, that is to say propositions that theory does not allow to demonstrate or refute (using of course only the axioms of this theory). This is the case of Peano’s arithmetic but also of set theory. In this case it is permissible to arbitrarily put these propositions as true or false. The best-known example is the axiom of choice (or its equivalent, the Zorn theorem), which is undecidable in Zermelo’s set theory: it is therefore possible to add either the axiom of choice or its negation to Zermelo’s axioms to obtain two equally coherent mathematical theories.

Science

“A theory without facts is only a fantasy, but facts without theory are only chaos.”
– Charles Otis Whitman, 1894.

In science, a theory is a model or framework for understanding nature and the human. In physics, the term “theory” generally refers to mathematical support, derived from a small set of basic principles and equations, to produce experimental predictions for a given class of physical systems. An example is the “electromagnetic theory”, usually confused with classical electromagnetism, whose specific results are obtained from Maxwell’s equations.

(Precession of the perihelion of Mercury (exaggerated). The deviation in Mercury’s position from the Newtonian prediction is about 43 arc-seconds (about two-thirds of 1/60 of a degree) per century.)

The adjective “theoretical”, which is added to the description of a phenomenon, often indicates that a particular result has been predicted by a theory, but has not yet been found. For example, until the mid-1960s, black holes were considered as theoretical objects. The existence of Uranus and then of Neptune were also supposed (or predicted) by abduction by means of Newtonian theory, and later confirmed by observation. Similarly, the laser/maser effect was postulated in 1917 by Albert Einstein and realized only in 1953.

For a theory to be considered as part of the established knowledge, it is usually necessary for it to produce a critical experience, that is, an experimental result that can not be predicted by another established theory. An example was the apparent deviation of light rays observed during an eclipse, a deviation predicted in 1915 by general relativity, and partially noted for the first time on May 29, 1919.

• If the expected consequences are not contradicted by the observed and measured reality, then the theory and its principles are confirmed (Bayesian inference).
• If there are observed and measured facts that the theory does not provide, then either the theory must be modified or its limits clarified. Thus Newton’s theory of gravitation, for very high velocities or gravitational fields, requires not only a corrective, but a complete reinterpretation of the relations between space and time which constitutes relativity.
• If the theory predicts effects, then we must try to observe and measure them. For example, predictive astrophysical theories confirm that there are laws, rules, models that apply to the behavior of the universe. So :
  • the laws of conservation (see Noether’s theorem)
  • the principles of maxima and minima, like those of Maupertuis and Hamilton