Editing Theorem (section)

==Relation with scientific theories==

Theorems in mathematics and theories in science are fundamentally different in their [[epistemology]]. A scientific theory cannot be proved; its key attribute is that it is [[falsifiable]], that is, it makes predictions about the natural world that are testable by [[experiment]]s. Any disagreement between prediction and experiment demonstrates the incorrectness of the scientific theory, or at least limits its accuracy or domain of validity. Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in the same way such evidence is used to support scientific theories.<ref name=":0"/>

[[File:CollatzFractal.png|thumb|250px|right|The [[Collatz conjecture]]: one way to illustrate its complexity is to extend the iteration from the natural numbers to the complex numbers. The result is a [[fractal]], which (in accordance with [[universality (dynamical systems)|universality]]) resembles the [[Mandelbrot set]].]]
Nonetheless, there is some degree of empiricism and data collection involved in the discovery of mathematical theorems. By establishing a pattern, sometimes with the use of a powerful computer, mathematicians may have an idea of what to prove, and in some cases even a plan for how to set about doing the proof. It is also possible to find a single counter-example and so establish the impossibility of a proof for the proposition as-stated, and possibly suggest restricted forms of the original proposition that might have feasible proofs.

For example, both the [[Collatz conjecture]] and the [[Riemann hypothesis]] are well-known unsolved problems; they have been extensively studied through empirical checks, but remain unproven. The [[Collatz conjecture]] has been verified for start values up to about 2.88&nbsp;×&nbsp;10<sup>18</sup>. The [[Riemann hypothesis]] has been verified to hold for the first 10&nbsp;trillion non-trivial zeroes of the [[Riemann zeta function|zeta function]]. Although most mathematicians can tolerate supposing that the conjecture and the hypothesis are true, neither of these propositions is considered proved.

Such evidence does not constitute proof. For example, the [[Mertens conjecture]] is a statement about natural numbers that is now known to be false, but no explicit counterexample (i.e., a natural number ''n'' for which the Mertens function ''M''(''n'') equals or exceeds the square root of ''n'') is known: all numbers less than 10<sup>14</sup> have the Mertens property, and the smallest number that does not have this property is only known to be less than the [[exponential function|exponential]] of 1.59&nbsp;×&nbsp;10<sup>40</sup>, which is approximately 10 to the power 4.3&nbsp;×&nbsp;10<sup>39</sup>. Since the number of particles in the universe is generally considered less than 10 to the power 100 (a [[googol]]), there is no hope to find an explicit counterexample by [[exhaustive search]].

The word "theory" also exists in mathematics, to denote a body of mathematical axioms, definitions and theorems, as in, for example, [[group theory]] (see [[mathematical theory]]). There are also "theorems" in science, particularly physics, and in engineering, but they often have statements and proofs in which physical assumptions and intuition play an important role; the physical axioms on which such "theorems" are based are themselves falsifiable.