For a more detailed account of theories as expressed in formal language as they are studied in mathematical logic see Theory (mathematical logic)

A theory, in the general sense of the word, is an analytic structure designed to explain a set of observations. A theory does two things: it (1) identifies this set of distinct observations as a class of phenomena, and (2) makes assertions about the underlying reality that brings about or affects this class. The term is often used colloquially to refer to any explanatory thought, even fanciful or speculative ones, but in scholarly use it is reserved for ideas which meet baseline requirements about the kinds of observations made, the methods of classification used, and the consistency of the theory in its application among members of that class. These requirements vary across different fields of knowledge, but in general theories are expected to be functional and parsimonious: i.e. a theory should be the simplest possible tool that can be used to effectively address the given class of phenomena.

Theories are distinct from theorems: theorems are derived deductively from theories according to a formal system of rules, generally as a first step in testing or applying the theory in a concrete situation. Theories are abstract and conceptual, and to this end they are never considered right or wrong. Instead, they are supported or challenged by observations in the world. They are 'rigorously tentative', meaning that they are proposed as true but expected to satisfy careful examination to account for the possibility of faulty inference or incorrect observation. Sometimes theories are falsified, meaning that an explicit set of observations contradicts some fundamental assumption of the theory, but more often theories are revised to conform to new observations, by restricting the class of phenomena the theory applies to or changing the assertions made. Sometimes a theory is set aside by scholars because there is no way to examine its assertions analytically; these may continue on in the popular imagination until some means of of examination is found which either refutes or lends credence to the theory.

The word 'theory' is generally considered to derive from Greek θεωρία theoria (Jerome), Greek "contemplation, speculation", from θεωρός "spectator", θέα thea "a view" + ὁρᾶν horan "to see", literally "looking at a show".[1] A second possible etymology traces the word back to το θείον to theion "divine things" instead of thea, reflecting the concept of contemplating the divine organisation (Cosmos) of the nature. The word has been in use in English since at least the late 16th century.[2]

## Theories formally and generally

Theories are analytical tools for understanding and explaining a given subject matter. There are theories in many different fields of study, and of many different types: from purely syntactic or 'formal' extrapolations of mathematics or logic, to evidence-driven constructs typical of the physical sciences, to rational/moral analyses found in the social sciences and certain branches of philosophy, and to the interpretive principles found in many arenas of the arts and humanities. Theories are abstract, and may be constructed independently of any semantic components, as seen in mathematics, or the syntactic elements may be entirely subordinate the semantic thrust, as is found in post-modernist, post-structuralist, and similar philosophical forms. Theories may be expressed mathematically, symbolically, or in common language, but are generally expected to follow principles of rational thought or logic.

A theory is constructed of a set of sentences which consists entirely of true statements about the subject matter under consideration. However, the truth of any of these statements is always relative to the theory. Therefore the same statement may be true with respect to one theory, and not true with respect to another. This is, in ordinary language, where statements such as "He is a terrible person." cannot be judged to be true or false without reference to some interpretation of who "He" is and for that matter what a "terrible person" is under this theory. [3]

Sometimes two theories have exactly the same explanatory power because they make the same predictions. A pair of such theories is called indistinguishable, and the choice between them reduces to convenience or philosophical preference.

The form of theories is studied formally in mathematical logic, especially in model theory. When theories are studied in mathematics, they are usually expressed in some formal language and their statements are closed under application of certain procedures called rules of inference. A special case of this, an axiomatic theory, consists of axioms (or axiom schemata) and rules of inference. A theorem is a statement that can be derived from those axioms by application of these rules of inference. Theories used in applications are abstractions of observed phenomena and the resulting theorems provide solutions to real-world problems. Obvious examples include arithmetic (abstracting concepts of number), geometry (concepts of space), and probability (concepts of randomness and likelihood).

Gödel's incompleteness theorem shows that no consistent, recursively enumerable theory (that is, one whose theorems form a recursively enumerable set) in which the concept of natural numbers can be expressed, can include all true statements about them. As a result, some domains of knowledge cannot be formalized, accurately and completely, as mathematical theories. (Here, formalizing accurately and completely means that all true propositions—and only true propositions—are derivable within the mathematical system.) This limitation, however, in no way precludes the construction of mathematical theories that formalize large bodies of scientific knowledge.

## Philosophical theories

Theories whose subject matter consists not in empirical data, but rather in ideas are in the realm of philosophical theories as contrasted with scientific theories. At least some of the elementary theorems of a philosophical theory are statements whose truth cannot necessarily be scientifically tested through empirical observation.

### Metatheory

One form of philosophical theory is a metatheory or meta-theory. A metatheory is a theory whose subject matter is some other theory. In other words it is a theory about a theory. Statements made in the metatheory about the theory are called 'metatheorems.

### Political theories

A political theory is an ethical theory about the law and government. Often the term "political theory" refers to a general view, or specific ethic, political belief or attitude, about politics.

## Scientific theories

In the sciences generally, theories are constructed from elementary theorems that consist in empirical data about observable phenomena. A scientific theory is used as a plausible general principle or body of principles offered to explain a phenomenon.[4]

A scientific theory is a deductive theory, in that, its content is based on some formal system of logic and that some of its elementary theorems are taken as axioms. In a deductive theory, any sentence which is a logical consequence of one or more of the axioms is also a sentence of that theory.[3]

A major concern in construction of scientific theories is the problem of demarcation, i.e., distinguishing those ideas that are properly studied by the sciences and those that are not.

Theories are intended to be an accurate, predictive description of the natural world. However, it is sometimes not clear whether the conclusions derived from the theory inform us about the nature of the world, or the nature of the theory.

### Theories as models

Theories are constructed to explain, predict, and master phenomena (e.g., inanimate things, events, or behavior of animals). A scientific theory can be thought of as a model of reality, and its statements as axioms of some axiomatic system. The aim of this construction is to create a formal system for which reality is the only model. The world is an interpretation (or model) of such scientific theories, only insofar as the sciences are true.

### Theories in physics

In physics the term theory is generally used for a mathematical framework—derived from a small set of basic postulates (usually symmetries—like equality of locations in space or in time, or identity of electrons, etc.)—which is capable of producing experimental predictions for a given category of physical systems. A good example is classical electromagnetism, which encompasses results derived from gauge symmetry (sometimes called gauge invariance) in a form of a few equations called Maxwell's equations. Note that the specific theoretical aspects of classical electromagnetic theory, which have been consistently and successfully replicated for well over a century, are termed "laws of electromagnetism", reflecting that they are today taken for granted. Within electromagnetic theory generally, there are numerous hypotheses about how electromagnetism applies to specific situations. Many of these hypotheses are already considered to be adequately tested, with new ones always in the making and perhaps untested.

### Pedagogical definition

In pedagogical contexts or in official pronouncements by official organizations of scientists a definition such as the following may be promulgated.

According to the United States National Academy of Sciences,

Some scientific explanations are so well established that no new evidence is likely to alter them. The explanation becomes a scientific theory. In everyday language a theory means a hunch or speculation. Not so in science. In science, the word theory refers to a comprehensive explanation of an important feature of nature supported by facts gathered over time. Theories also allow scientists to make predictions about as yet unobserved phenomena, [5]

A scientific theory is a well-substantiated explanation of some aspect of the natural world, based on a body of facts that have been repeatedly confirmed through observation and experiment. Such fact-supported theories are not "guesses" but reliable accounts of the real world. The theory of biological evolution is more than "just a theory." It is as factual an explanation of the universe as the atomic theory of matter or the germ theory of disease. Our understanding of gravity is still a work in progress. But the phenomenon of gravity, like evolution, is an accepted fact.[6]

The primary advantage enjoyed by this definition is that it firmly marks things termed theories as being well supported by evidence. This would be a disadvantage in interpreting real discourse between scientists who often use the word theory to describe untested but intricate hypotheses in addition to repeatedly confirmed models. However, in an educational or mass media setting it is almost certain that everything of the form X theory is an extremely well supported and well tested theory. This causes the theory/non-theory distinction to much more closely follow the distinctions useful for consumers of science (e.g. should I believe something or not?)

### The term theoretical

The term theoretical is sometimes informally used in lieu of hypothetical to describe a result that is predicted by theory but has not yet been adequately tested by observation or experiment. It is not uncommon for a theory to produce predictions that are later confirmed or proven incorrect by experiment. By inference, a prediction proved incorrect by experiment demonstrates the hypothesis is invalid. This either means the theory is incorrect, or the experimental conjecture was wrong and the theory did not predict the hypothesis.

## Fields of study called "theories"

Fields of study are sometimes named "theory" because their basis is some initial set of assumptions describing the field's approach to a subject matter. These assumptions are the elementary theorems of the particular theory, and can be thought of as the axioms of that field. Some commonly known examples include set theory, game theory, and number theory; however literary theory, critical theory, and music theory are also of the same form.

## Intertheoretic reduction and elimination

If there is a new theory which is better at explaining and predicting phenomena than an older theory (i.e. it has more explanatory power), we are justified in believing that the newer theory describes reality more correctly. This is called an intertheoretic reduction because the terms of the old theory can be reduced to the terms of the new one. For instance, our historical understanding about "sound," "light" and "heat," have today been reduced to "wave compressions and rarefactions," "electromagnetic waves," and "molecular kinetic energy" respectively. These terms which are identified with each other are called intertheoretic identities. When an old theory and a new one are parallel in this way, we can conclude that we are describing the same reality, only more completely.

In cases where a new theory uses new terms which do not reduce to terms of an older one, but rather replace them entirely because they are actually a misrepresentation it is called an intertheoretic elimination. For instance, the obsolete scientific theory that put forward an understanding of heat transfer in terms of the movement of caloric fluid was eliminated when a theory of heat as energy replaced it. Also, the theory that phlogiston is a substance released from burning and rusting material was eliminated with the new understanding of the reactivity of oxygen.

## Underdetermination

A theory is underdetermined (also called indeterminacy of data to theory) if, given the available evidence cited to support the theory, there is a rival theory which is inconsistent with it that is at least as consistent with the evidence. Underdetermination is an epistemological issue about the relation of evidence to conclusions.

## Notes

1. ^ Frisk; derivation from θεός was suggested by Koller Glotta 36, 273ff.
2. ^ Harper, Douglas. "theory". Online Etymology Dictionary. Retrieved 2008-07-18.
3. ^ a b Curry, Haskell, Foundations of Mathematical Logic
4. ^ Merriam-Webster.com Merriam-Webster Dictionary: Theory in Science
5. ^ National Academy of Sciences (2005), Science, Evolution, and Creationism, a brochure on the book of the same title.
6. ^ AAAS Evolution Resources

## References

• Popper, Karl (1963), Conjectures and Refutations, Routledge and Kegan Paul, London, UK, pp. 33–39. Reprinted in Theodore Schick (ed., 2000), Readings in the Philosophy of Science, Mayfield Publishing Company, Mountain View, Calif., pp. 9–13.
• Chairman of Biology and Kennesaw State Ronald Matson's webpage comparing scientific laws and theories
• Hawking, Stephen (1996). "The Illustrated A Brief History of Time" (Updated and expanded ed.). New York: Bantam Books, p. 15.
• Mohr, Johnathon (2008). "Revelations and Implications of the Failure of Pragmatism: The Hijacking of Knowledge Creation by the Ivory Tower". New York: Ballantine Books. pp. 87–192.