The key difference lies in their purpose, level of proof, and application: a law describes what happens, a theory explains why it happens and needs to be proven, and a theorem is a statement that has been proven.
Here's a breakdown in table format:
| Feature | Law | Theory | Theorem |
|---|---|---|---|
| Definition | Universal principle describing the fundamental nature of something. | An idea that needs to be proven or contradicted. | A statement which is already proven. |
| Purpose | Describes observations; tells what happens. | Explains phenomena; tells why something happens. | Establishes a proven relationship. |
| Proof Status | Generally accepted as universally true based on repeated observations and experiments. | Requires evidence and testing to support or refute it. Can evolve or be replaced by new theories. | Rigorously proven to be true based on established axioms, other theorems, and logical deduction. |
| Mutability | Rarely changes, but can be refined with new discoveries that provide a deeper understanding. | Subject to change based on new evidence and interpretations. | Remains true once proven, though the proof may be simplified or generalized. |
| Examples | Law of Gravity, Laws of Thermodynamics. | Theory of Evolution, String Theory. | Pythagorean Theorem, Fundamental Theorem of Calculus. |
Elaborating on the Differences
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Law: A scientific law is a statement based on repeated experimental observations that describes some aspect of the universe. It is always true if the conditions are right. Laws are descriptive, but they don't offer explanations. From the reference, a mathematical law is a universal principle describing the fundamental nature of something.
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Theory: A scientific theory is a well-substantiated explanation of some aspect of the natural world that can incorporate facts, laws, inferences, and tested hypotheses. Theories are explanatory and predictive. The reference states a theory is an idea which needs to be proved/contradicted.
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Theorem: In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. A theorem is a logical consequence of the axioms. According to the reference, a theorem is a statement which is already proven.
