In mathematics, a theory is a set of statements closed under logical implication. In mathematical logic, "theory" is the term for a set of well-formed formulae consisting of certain axioms and all theorems provable from said axioms. Gödel's incompleteness theorem; states that no consistent theory, with a finite number of axioms (in a language at least as strong as arithmetic), can include all true statements.
In sciences, a theory is a model or framework for understanding. In physics, the term theory generally is taken to mean mathematical framework derived from a small set of basic principles capable of producing experimental predictions for a given category of physical systems. An example would be "electromagnetic theory", which is usually taken to be synonymous with classical electromagnetism, the specific results of which can be derived from Maxwell's equations.
The term theoretical to describe certain phenomena often indicates that a particular result has been predicted by theory but has not yet been observed. For example, until recently, black holes were considered theoretical. It is not uncommon in the history of physics for theory to produce such predictions that are later confirmed by experiment.
For a given body of theory to be considered part of established knowledge, it is usually necessary for the theory to produce a critical experiment, that is, an experimental result which cannot be predicted by any established theory.
The word ‘theory’ derives from the Greek ‘theorein’, which means ‘to look at’. According to some sources, it was used frequently in terms of ‘looking at’ a theatre stage, which may explain why sometimes the word ‘theory’ is used as something provisional or not quite real. The term ‘teoria’ was already used by the ancient Greeks.
A theory has to be something which is in some way testable; for example, one can theorize that an apple will fall when dropped, and then drop an apple, to see what happens. Many scientists, but not all, argue that religious beliefs are not testable, and thus not theories, because they are matters of faith.
According to Stephen Hawking in A Brief History of Time, "a theory is a good theory if it satisfies two requirements: It must accurately describe a large class of observations on the basis of a model that contains only a few arbitrary elements, and it must make definite predictions about the results of future observations." He goes on to state..."Any physical theory is always provisional, in the sense that it is only a hypothesis; you can never prove it. No matter how many times the results of experiments agree with some theory, you can never be sure that the next time the result will not contradict the theory. On the other hand, you can disprove a theory by finding even a single observation that disagrees with the predictions of the theory."
A theory is different from a theorem. The former is a model of physical events and cannot be proved from basic axioms. The latter is a statement of mathematical fact which logically follows from a set of axioms. A theory is also different from a physical law in that the former is a model of reality whereas the latter is a statement of what has been observed.
Theories can become accepted if they are able to make correct predictions and avoid incorrect ones. Theories which are simpler, and more mathematically elegant, tend to be accepted over theories which are complex. Theories are more likely to be accepted if they connect a wide range of phenonomena. The process of accepting theories, or of extending existing theory, is part of the scientific method.
Theories start out with empirical observations such as “sometimes water turns into ice.” At some point, there is a need or curiosity to find out why this is, which leads to a theoretical/scientific phase. In scientific theories, this then leads to research, in combination with auxiliary and other hypotheses (see scientific method), which may then eventually lead to a theory. Some scientific theories (such as the theory of gravity) are so widely accepted that they are often seen as laws. This, however, rests on a mistaken assumption of what theories and laws are. Theories and laws are not rungs in a ladder of truth, but different sets of data. A law is a general statement based on observations.
Some theories that have been disproved are those such as Lamarckism and the geocentric universe theory. Sufficient evidence has risen to declare these theories false, as they have no evidence supporting them and better explanations have taken their place.
Unfortunately, the usage of the term is muddled by cases such as string theory and "theories of everything," each probably better characterized at present as a bundle of competing hypotheses. A hypothesis, however, is still vastly more reliable than a conjecture, which is at best an untested guess consistent with selected data, and is often a belief based on non-repeatable experiments, anecdotes, popular opinion, "wisdom of the ancients," commercial motivation, or mysticism.