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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.

Humans construct theories in order to explain, predict and master phenomena (e.g. inanimate things, events, or the behaviour of animals). In many instances, it is seen to be a model of reality. A theory makes generalizations about observations and consists of an interrelated, coherent set of ideas.

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."

There are two types of theories; a supposition which is not backed by observation is known as a conjecture, and if backed by observation it is a hypothesis. 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.

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. The word theoretical is derived from theory, and is used to describe that which has not yet been observed. For example, until recently, black holes were theoretical.

Theories can become accepted if they are able to make correct predictions and avoid incorrect ones. Theories which are simplier, 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 is part of the scientific method.

Further explanation of a scientific theory

In common usage a theory is often viewed as little more than a guess or a hypothesis. But in science and generally in academic usage, a theory is much more than that. A theory is an established paradigm that explains all or many of the data we have and offers valid predictions that can be tested. In science, a theory can never be proven true, because we can never assume we know all there is to know. Instead, theories remain standing until they are disproven, at which point they are thrown out altogether or modified slightly.

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.

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.

Often the statement "Well, it's just a theory," is used to dismiss controversial theories such as evolution, but in science a theory usually

  1. is supported by many strands of evidence rather than a single foundation, ensuring that it probably is a good approximation if not totally correct,
  2. has survived many critical tests that could have proven it false,
  3. makes predictions that might someday be used to disprove the theory, and
  4. is the best known explanation, in the sense of Occam's Razor, of the infinite variety of alternative explanations for the same data.

This is true of such established theories as evolution, special and general relativity, quantum mechanics (with minimal interpretation), plate tectonics, etc.

Theories exist not only in the so-call 'hard sciences' but in all fields of academic study, from philosophy to music to literature.

Unfortunately the usage of the term is muddled by cases like 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, "wisom of the ancients," commercial motivation, or mysticism.

A good example of a non-scientific "theory" is Intelligent Design. Likewise, other claims such as homeopathy are also not theories.

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