Theorem

As the axioms are often abstractions of properties of the physical world, theorems may be considered as expressing some truth, but in contrast to the notion of a scientific law, which is experimental, the justification of the truth of a theorem is purely deductive.

Until the end of the 19th century and the foundational crisis of mathematics, all mathematical theories were built from a few basic properties that were considered as self-evident; for example, the facts that every natural number has a successor, and that there is exactly one line that passes through two given distinct points.

For example, the sum of the interior angles of a triangle equals 180°, and this was considered as an undoubtable fact.

So, the property "the sum of the angles of a triangle equals 180°" is either true or false, depending whether Euclid's fifth postulate is assumed or denied.

Similarly, the use of "evident" basic properties of sets leads to the contradiction of Russell's paradox.

In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses.

The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed.

In addition to the better readability, informal arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true.

[7] On the other hand, a deep theorem may be stated simply, but its proof may involve surprising and subtle connections between disparate areas of mathematics.

In order for a theorem to be proved, it must be in principle expressible as a precise, formal statement.

However, theorems are usually expressed in natural language rather than in a completely symbolic form—with the presumption that a formal statement can be derived from the informal one.

These hypotheses form the foundational basis of the theory and are called axioms or postulates.

Some, on the other hand, may be called "deep", because their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics.

[12][page needed] 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 experiments.

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.

[6] 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.

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 Riemann hypothesis has been verified to hold for the first 10 trillion non-trivial zeroes of the zeta function.

Although most mathematicians can tolerate supposing that the conjecture and the hypothesis are true, neither of these propositions is considered proved.

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.

[17] The classification of finite simple groups is regarded by some to be the longest proof of a theorem.

Some accounts define a theory to be closed under the semantic consequence relation (

Different deductive systems can yield other interpretations, depending on the presumptions of the derivation rules (i.e. belief, justification or other modalities).

The Pythagorean theorem has at least 370 known proofs. [ 1 ]
A planar map with five colors such that no two regions with the same color meet. It can actually be colored in this way with only four colors. The four color theorem states that such colorings are possible for any planar map, but every known proof involves a computational search that is too long to check by hand.
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 ) resembles the Mandelbrot set .
This diagram shows the syntactic entities that can be constructed from formal languages . The symbols and strings of symbols may be broadly divided into nonsense and well-formed formulas . A formal language can be thought of as identical to the set of its well-formed formulas. The set of well-formed formulas may be broadly divided into theorems and non-theorems.