Everything about Mathematical Proof totally explained
In
mathematics, a
proof is a convincing demonstration that some
mathematical statement is necessarily true, within the accepted standards of the field. A proof is a
logically deduced argument, not an
empirical one. That is, the proof must demonstrate that a proposition is true in all cases to which it applies, without a single exception. An unproven proposition believed or strongly suspected to be true is known as a
conjecture.
Proofs employ
logic but usually include some amount of
natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of
informal logic. Purely
formal proofs are considered in
proof theory. The distinction between
formal and informal proofs has led to much examination of current and historical
mathematical practice,
quasi-empiricism in mathematics, and so-called
folk mathematics (in both senses of that term). The
philosophy of mathematics is concerned with the role of language and logic in proofs, and
mathematics as a language.
Regardless of one's attitude to formalism, the result that's proved to be true is a
theorem; in a completely formal proof it would be the final line, and the complete proof shows how it follows from the
axioms alone by the application of the rules of inference. Once a theorem is proved, it can be used as the basis to prove further statements. A theorem may also be referred to as a
lemma if it's used as a stepping stone in the proof of a theorem. The axioms are those statements one cannot, or need not, prove. These were once the primary study of philosophers of mathematics. Today focus is more on
practice, for example acceptable techniques.
Methods of proof
Direct proof
In
direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. For example, direct proof can be used to establish that the sum of two
even integers is always even:
» For any two even integers
and
we can write
and
for some integers
and
, since both
and
are multiples of 2. But the sum
is also a multiple of 2, so it's therefore even by definition.
This proof uses definition of even integers, as well as
distribution law.
Proof by induction
In
proof by induction, first a "base case" is proved, and then an "induction rule" is used to prove a (often
infinite) series of other cases. Since the base case is true, the infinity of other cases must also be true, even if all of them can't be proved directly because of their infinite number. A subset of induction is
Infinite descent. Infinite descent can be used to prove the
irrationality of the square root of two.
The principle of mathematical induction states that:
Let
N = =2, which is thus a rational of the form
Proof nor disproof
There is a class of mathematical statements for which neither a proof nor disproof exists, using only
ZFC, the standard form of
axiomatic set theory. Examples include the
continuum hypothesis; see further
List of statements undecidable in ZFC. Under the assumption that ZFC is
consistent, the existence of such statements follows from
Gödel's (first) incompleteness theorem. Whether a particular unproven proposition can be proved or disproved using a standard set of axioms isn't always obvious, and can be extremely technical to determine.
Elementary proof
An elementary proof is (usually) a proof which doesn't use complex analysis. For some time it was thought that certain theorems, like the
prime number theorem, could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques.
End of a proof
Sometimes, the abbreviation
"Q.E.D." is written to indicate the end of a proof. This abbreviation stands for
"Quod Erat Demonstrandum", which is
Latin for
"that which was to be demonstrated". An alternative is to use a square or a rectangle, such as □ or ∎, known as a "
tombstone" or "halmos".
Further Information
Get more info on 'Mathematical Proof'.
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