Conjecture

Encyclopedia

A

but is thought to be true and has not been disproven. Karl Popper

pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis

(hence

, a conjecture is an unproven proposition

or theorem

that appears correct.

The Langlands program

is a far-reaching web of these ideas of 'unifying conjectures' that link different subfields of mathematics, e.g. number theory

and representation theory

of Lie group

s; some of these conjectures have since been proved.

would immediately bring down the conjecture. Conjectures disproven through counterexample are sometimes referred to as

).

Mathematical journals sometimes publish the minor results of research teams having extended a given search farther than previously done. For instance, the Collatz conjecture

, which concerns whether or not certain sequence

s of integer

s terminate, has been tested for all integers up to 1.2 × 10

, rather than meaningful contributions to formal mathematics.

is a conjecture from number theory

that (amongst other things) makes predictions about the distribution of prime number

s. Few number theorists doubt that the Riemann hypothesis is true (it is said that Atle Selberg

was once a sceptic, and J. E. Littlewood always was). In anticipation of its eventual proof, some have proceeded to develop further proofs which are contingent on the truth of this conjecture. These are called

These "proofs", however, would fall apart if it turned out that the hypothesis was false, so there is considerable interest in verifying the truth or falsity of conjectures of this type.

, which tries to ascertain the relative cardinality

of certain infinite sets, was eventually shown to be undecidable

(or independent) from the generally accepted set of axioms of set theory. It is therefore possible to adopt this statement, or its negation, as a new axiom

in a consistent manner (much as we can take Euclid

's parallel postulate

as either true or false).

In this case, if a proof uses this statement, researchers will often look for a new proof that

be proved using only the axioms of neutral geometry, i.e. no parallel postulate.) The one major exception to this in practice is the axiom of choice—unless studying this axiom in particular, the majority of researchers do not usually worry whether a result requires the axiom of choice.

**conjecture**is a proposition that is unprovenFormal proof

A formal proof or derivation is a finite sequence of sentences each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system...

but is thought to be true and has not been disproven. Karl Popper

Karl Popper

Sir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...

pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis

Hypothesis

A hypothesis is a proposed explanation for a phenomenon. The term derives from the Greek, ὑποτιθέναι – hypotithenai meaning "to put under" or "to suppose". For a hypothesis to be put forward as a scientific hypothesis, the scientific method requires that one can test it...

(hence

*theory*

, axiom

, principle

), which is a testable statement based on accepted grounds. In mathematicsTheory

The English word theory was derived from a technical term in Ancient Greek philosophy. The word theoria, , meant "a looking at, viewing, beholding", and referring to contemplation or speculation, as opposed to action...

, axiom

Axiom

In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

, principle

Principle

A principle is a law or rule that has to be, or usually is to be followed, or can be desirably followed, or is an inevitable consequence of something, such as the laws observed in nature or the way that a system is constructed...

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a conjecture is an unproven proposition

Proposition

In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...

or theorem

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...

that appears correct.

## Famous conjectures

- Beal's conjectureBeal's conjectureBeal's conjecture is a conjecture in number theory proposed by Andrew Beal in about 1993; a similar conjecture was suggested independently at about the same time by Andrew Granville....
- The Poincaré theorem (proven by Grigori PerelmanGrigori PerelmanGrigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...

) - Goldbach's conjectureGoldbach's conjectureGoldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:A Goldbach number is a number that can be expressed as the sum of two odd primes...

The Langlands program

Langlands program

The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....

is a far-reaching web of these ideas of 'unifying conjectures' that link different subfields of mathematics, e.g. number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

and representation theory

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...

of Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

s; some of these conjectures have since been proved.

## Counterexamples

Formal mathematics is based on*provable*truth. In mathematics, any number of cases supporting a conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexampleCounterexample

In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....

would immediately bring down the conjecture. Conjectures disproven through counterexample are sometimes referred to as

*false conjectures*(cf. Pólya conjecturePólya conjecture

In number theory, the Pólya conjecture stated that 'most' of the natural numbers less than any given number have an odd number of prime factors. The conjecture was posited by the Hungarian mathematician George Pólya in 1919, and proved false in 1958...

).

Mathematical journals sometimes publish the minor results of research teams having extended a given search farther than previously done. For instance, the Collatz conjecture

Collatz conjecture

The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture , Kakutani's problem , the Thwaites conjecture , Hasse's algorithm The Collatz conjecture is a...

, which concerns whether or not certain sequence

Sequence

In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

s of integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s terminate, has been tested for all integers up to 1.2 × 10

^{12}(over a million millions). In practice, however, it is extremely rare for this type of work to yield a counterexample and such efforts are generally regarded as mere displays of computing powerComputational science

Computational science is the field of study concerned with constructing mathematical models and quantitative analysis techniques and using computers to analyze and solve scientific problems...

, rather than meaningful contributions to formal mathematics.

## Use of conjectures in conditional proofs

Sometimes a conjecture is called a*hypothesis*when it is used frequently and repeatedly as an assumption in proofs of other results. For example, the Riemann hypothesisRiemann hypothesis

In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...

is a conjecture from number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

that (amongst other things) makes predictions about the distribution of prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s. Few number theorists doubt that the Riemann hypothesis is true (it is said that Atle Selberg

Atle Selberg

Atle Selberg was a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory...

was once a sceptic, and J. E. Littlewood always was). In anticipation of its eventual proof, some have proceeded to develop further proofs which are contingent on the truth of this conjecture. These are called

*conditional proof*

s: the conjectures assumed appear in the hypotheses of the theorem, for the time being.Conditional proof

A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent....

s

These "proofs", however, would fall apart if it turned out that the hypothesis was false, so there is considerable interest in verifying the truth or falsity of conjectures of this type.

## Undecidable conjectures

Not every conjecture ends up being proven true or false. The continuum hypothesisContinuum hypothesis

In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...

, which tries to ascertain the relative cardinality

Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...

of certain infinite sets, was eventually shown to be undecidable

Independence (mathematical logic)

In mathematical logic, independence refers to the unprovability of a sentence from other sentences.A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that...

(or independent) from the generally accepted set of axioms of set theory. It is therefore possible to adopt this statement, or its negation, as a new axiom

Axiom

In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

in a consistent manner (much as we can take Euclid

Euclid

Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

's parallel postulate

Parallel postulate

In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...

as either true or false).

In this case, if a proof uses this statement, researchers will often look for a new proof that

*doesn't*require the hypothesis (in the same way that it is desirable that statements in Euclidean geometryEuclidean geometry

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

be proved using only the axioms of neutral geometry, i.e. no parallel postulate.) The one major exception to this in practice is the axiom of choice—unless studying this axiom in particular, the majority of researchers do not usually worry whether a result requires the axiom of choice.