Proof

What is Proof

In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true and can be rigorously justified through logical reasoning. A proof is fundamentally a logical argument, not an empirical one, which differentiates it from observations or experiments conducted in the natural sciences. That is, one must consistently demonstrate that a proposition holds true in all conceivable cases, following a systematic methodology, before it is considered a theorem of mathematics. This rigorous requirement ensures that mathematical knowledge is built upon a solid foundation of verified truths. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture, representing an educated guess that invites further investigation and exploration. Conjectures often motivate mathematical research, as they can lead to the discovery of new patterns, techniques, and ultimately, theorems that enrich the field of mathematics.

Proofs employ logic but usually include some amount of natural language, which typically admits some ambiguity, as the nuances of linguistic expression can introduce various interpretations. In fact, the vast majority of proofs in written mathematics can be considered as applications of informal logic, which allows mathematicians to convey their reasoning in a more intuitive manner. Purely formal proofs, on the other hand, are carefully constructed and rigorously defined within the realm of proof theory, often requiring a deep understanding of specific symbols and notations. The distinction between formal and informal proofs has led to significant examination of current and historical mathematical practice, prompting discussions on how these approaches affect learning and understanding. This dichotomy has also contributed to the exploration of quasi-empiricism in mathematics, wherein empirical methods are sometimes applied in a more informal context. Furthermore, the concept of folk mathematics, encompassing both the intuitive reasoning used by laypersons and the established practices of professional mathematicians, continues to intrigue researchers as it reveals how mathematical ideas are communicated and understood across different contexts and audiences, and reflects the diverse approaches individuals take towards solving mathematical problems.

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:

Consider two even integers x and y. Since they are even, they can be written as x = 2a and y = 2b respectively for integers a and b. Then the sum x + y = 2a + 2b = 2(a + b). From this it is clear that 2 is a factor of x + y, so the sum of two even integers is always even.
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 cannot 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 { 1, 2, 3, 4, … } be the set of natural numbers and P(n) be a mathematical statement involving the natural number n belonging to N such that (i) P(1) is true, ie, P(n) is true for n=1 (ii) P(m+1) is true whenever P(m) is true, ie, P(m) is true implies that P(m+1) is true. Then P(n) is true for the set of natural numbers N.

Proof by transposition

Proof by Transposition establishes the conclusion “if p then q” by proving the equivalent contrapositive statement “if not q then not p”.

Proof by contradiction

In proof by contradiction (also known as reductio ad absurdum, Latin for “reduction into the absurd”), it is shown that if some statement were false, a logical contradiction occurs, hence the statement must be true. This method is perhaps the most prevalent of mathematical proofs. A famous example of a proof by contradiction shows that \sqrt{2} is irrational:

Suppose that \sqrt{2} is rational, so \sqrt{2} = {a\over b} where a and b are non-zero integers with no common factor (definition of rational number). Thus, b\sqrt{2} = a. Squaring both sides yields 2b2 = a2. Since 2 divides the left hand side, 2 must also divide the right hand side (as they are equal and both integers). So a2 is even, which implies that a must also be even. So we can write a = 2c, where c is also an integer. Substitution into the original equation yields 2b2 = (2c)2 = 4c2. Dividing both sides by 2 yields b2 = 2c2. But then, by the same argument as before, 2 divides b2, so b must be even. However, if a and b are both even, they share a factor, namely 2. This contradicts our assumption, so we are forced to conclude that \sqrt{2} is irrational.

Proof by construction

Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists. For example, in certain geometrical problems and proofs, usually the kind found in engineering and mechanical problems with given lengths and angles, instead of actually attempting a direct and formal proof, we can just substitute the given values to obtain a required proof. For example, let us take a triangle ABC, of which AD is a particular median. Let G be the centroid of the triangle. BG extended meets AC at X. By enlarging or diminishing the ratios of certain triangles and taking all possible cases (isosceles, equilateral, right, acute scalene, and obtuse scalene), we can obtain a direct comparision of AX to CX as 1:5. Solving the same problem by application of geometry would take a considerably longer time.

Proof by exhaustion

In Proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four colour theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked by a computer program, not by hand. The shortest known proof of the four colour theorem today still has over 600 cases.

Probabilistic proof

A probabilistic proof is one in which an example is shown to exist by methods of probability theory – not an argument that a theorem is ‘probably’ true. The latter type of reasoning can be called a ‘plausibility argument’; in the case of the Collatz conjecture it is clear how far that is from a genuine proof. Probabilistic proof, like proof by construction, is one of many ways to show existence theorems.

Combinatorial proof

A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Usually a bijection is used to show that the two interpretations give the same result.

Nonconstructive proof

An nonconstructive proof establishes that a certain mathematical object must exist (e.g. “Some X satisfies f(X)”), without explaining how such an object can be found. Often, this takes the form of a proof by contradiction in which the nonexistence of the object is proven to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it.

Proof nor disproof

There is a class of mathematical formulae for which neither a proof nor disproof exists, using only the standard ZFC axioms. This result is known as Gödel’s (first) incompleteness theorem and examples include the continuum hypothesis. Whether a particular unproven proposition can be proved using a standard set of axioms is not always obvious, and can be extremely technical to determine.

Elementary proof

An elementary proof is (usually) a proof which does not 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.

Mathematical proof