Scientific theoryhttps://en.wikipedia.org/wiki/Scientific_theoryA scientific theory is a well-substantiated explanation of some aspect of the natural world that is acquired through the scientific method and repeatedly tested and confirmed, preferably using a written, pre-defined, protocol of observations and experiments.[1][2] Scientific theories are the most reliable, rigorous, and comprehensive form of scientific knowledge.[3]
It is important to note that the definition of a "scientific theory" (often ambiguously contracted to "theory" for the sake of brevity, including in this page) as used in the disciplines of science is significantly different from, and in contrast to, the common vernacular usage of the word "theory". As used in everyday non-scientific speech, "theory" implies that something is an unsubstantiated and speculative guess, conjecture, idea, or, hypothesis;[4] such a usage is the opposite of the word 'theory' in science. These different usages are comparable to the differing, and often opposing, usages of the term "prediction" in science (less ambiguously called a "scientific prediction") versus "prediction" in vernacular speech, denoting a mere hope.
The strength of a scientific theory is related to the diversity of phenomena it can explain, and to its elegance and simplicity (see Occam's razor). As additional scientific evidence is gathered, a scientific theory may be rejected or modified if it does not fit the new empirical findings; in such circumstances, a more accurate theory is then desired. In certain cases, the less-accurate unmodified scientific theory can still be treated as a theory if it is useful (due to its sheer simplicity) as an approximation under specific conditions (e.g., Newton's laws of motion as an approximation to special relativity at velocities that are small relative to the speed of light).
Scientific theories are usually testable and make falsifiable predictions.[5] They describe the causal elements responsible for a particular natural phenomenon, and are used to explain and predict aspects of the physical universe or specific areas of inquiry (e.g., electricity, chemistry, astronomy). Scientists use theories as a foundation to gain further scientific knowledge, as well as to accomplish goals such as inventing technology or curing disease.
As with most, if not all, forms of scientific knowledge, scientific theories are both deductive and inductive[6][7] in nature and aim for predictive power and explanatory capability.
Axiomhttps://en.wikipedia.org/wiki/AxiomAn axiom or postulate as defined in classic philosophy, is a statement (in mathematics often shown in symbolic form) that is so evident or well-established, that it is accepted without controversy or question. Thus, the axiom can be used as the premise or starting point for further reasoning or arguments, usually in logic or in mathematics.[1] The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.'[2][3]
As used in modern logic, an axiom is simply a premise or starting point for reasoning.[4] Whether it is meaningful (and, if so, what it means) for an axiom, or any mathematical statement, to be "true" is a central question[citation needed] in the philosophy of mathematics, with modern mathematicians[who?] holding a multitude of different opinions.[5]
As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, "axiom", "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. As modern mathematics admits multiple, equally "true" systems of logic, precisely the same thing must be said for logical axioms - they both define and are specific to the particular system of logic that is being invoked. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.
In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Within the system they define, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow otherwise they would be classified as theorems. However, an axiom in one system may be a theorem in another, and vice versa.
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