Conjecture (math)

In metamathematics , an assumption is a statement that is not clear or for some time it was not clear whether it was true or not.

Classification of the term

Each math statement falls into one of the following categories:

proven
refuted,
it is not known whether it is provable or not known whether it is refutable, or both,
it has been proven that it is neither provable nor refutable, i.e. logically independent of the accepted system of axioms .

Third grade statements are called open-ended questions . If the experts expect the statement to be correct, they speak of an assumption instead of an open question. The reasons for this can lie in the practical use, in numerical evidence or be purely intuitive in nature.

The designation of specific statements often deviates from this for historical reasons, for example the Bieberbach hypothesis has now been proven.

An important metamathematic assumption is that the mostly accepted formal basis of mathematics, the ZFC set theory, is consistent : If this is the case, then ZFC itself cannot prove it (see Gödel's incompleteness theorem ). If ZFC contains a contradiction, i.e. a statement exists so that both it and its negation can be proven, it immediately follows that every statement that can be formulated in ZFC can be proven ( ex falso quodlibet ).

List of some guesses

Guesswork proved today

Fermat's conjecture was even famous outside of mathematics , which is probably due to the fact that it is both easy to formulate in a way that is understandable for the mathematical layperson and is formally similar to one of the most famous theorems of mathematics, the Pythagorean theorem , having. After more than 300 years and numerous unsuccessful attempts, the conjecture was finally proven and has since been called the great Fermat sentence. Remarkably, the English name has always been "Fermat's last theorem". The proof attempts have promoted numerous developments in number theory . The road to definitive proof in 1993 by Andrew Wiles and Richard Taylor was through the proof of some much more general conjecture.

The four-color conjecture , which clearly deals with the coloring of flat maps and actually has its origin in a practical question about the coloring of maps, was also easy to explain to the non-mathematician . It is noteworthy that historically "evidence" was presented several times, which was only recognized as incorrect after several years. The first valid evidence after more than 100 years, in turn, aroused numerous skeptics because it was based on the use of computers to a large extent. Converted to conventional human-readable evidence, it would have been too extensive to be fully verified by a single person.

The Bieberbach Hypothesis , established by Ludwig Bieberbach in 1916 , was proved by Louis de Branges de Bourcia in 1985 and has therefore been referred to as de Branges ' theorem ever since .

The Burnside conjecture that all finite groups of odd order are solvable was proven in the 1960s by Walter Feit and John Griggs Thompson .

The Poincaré conjecture , which had been open for almost 100 years and was added to the list of Millennium Problems in 2000 , was proven in 2002 by Grigori Jakowlewitsch Perelman .

Kepler's conjecture about the closest packing of spheres , established in 1611, was possibly proven in 1998. The given computer evidence is, however, not fully recognized in the professional world because of the particularly extensive calculations carried out by the computer (approx. 3 gigabytes of data).

Today rebutted suspicions

The Euler's guess is by Leonhard Euler named presumption of number theory and generalizes the Fermat's theorem. It was refuted in 1966 by a counterexample.

Fermat number : Fermat showed that the first five Fermat numbers are prime numbers and suggested in 1637 that this applies to all Fermat numbers. This assumption was refuted by Leonhard Euler in 1732. On the contrary, it has now been assumed that only the first 5 Fermat numbers are prime. Proof of this assumption is still pending. ${\ displaystyle F_ {n} = 2 ^ {(2 ^ {n})} + 1, n \ in \ mathbb {N} _ {0}}$

The Mertens hypothesis states that the inequality holds for the amounts of the partial sums of the series that results from the summation of the Möbius function . The Riemann Hypothesis would follow from the conjecture . The conjecture was formulated by Stieltjes in a letter to Hermite in 1885 and refuted by Odlyzko and Riele in 1985 . ${\ displaystyle \ textstyle \ left | \ sum _ {k = 1} ^ {n} \ mu (k) \ right | <{\ sqrt {n}}}$

For two millennia it was repeatedly assumed that the postulate of parallels , which was introduced by Euclid in the 4th century BC. Chr. Was formulated in its elements , from the other axioms and postulates of its Euclidean geometry can be proven. Euclid left this question open - in the sense of the terminology at the time - by using the statement “required” (postulate) and not as “inevitable and indispensable” (axiom). In the 19th century it could be shown that Euclid's system of axioms (axioms and postulates) is free of contradictions, but not sufficient to prove all the theorems formulated by him. With modern formulations of a geometric axiom system "in the sense of Euclid", for example Hilbert's axiom system of Euclidean geometry , it could be proven that the parallel postulate is independent of the other axioms and postulates of a Euclidean geometry by giving suitable models for non-Euclidean geometries .
The Pólya conjecture 1919 stating that up to any limit there n > 1 through so many numbers with an odd number of at least prime factors are such as those with an even number. The smallest counterexample n = 906,150,257 was found in 1980. This makes it clear that the correctness of a guess for very many (even relatively large) numbers is not a guarantee of general validity.

Evidence unclear

The abc conjecture is a mathematical conjecture established by Joseph Oesterlé and David Masser in 1985 , which can be viewed as a generalization of the now proven large Fermat theorem.
The Collatz problem , also known as the conjecture , was discovered by Lothar Collatz in 1937 . Kurtz and Simon showed in 2006 that a natural generalization of the conjecture is undecidable , i.e. a 4th grade conjecture. Their argument is based on considerations published by John Horton Conway in 1972. ${\ displaystyle (3n + 1)}$

The P-NP problem is an unsolved problem of complexity theory , it was formulated by Stephen Cook and Leonid Levin in the early 1970s . Guess: .

{\ displaystyle P \ neq NP}

The Riemann Hypothesis states that all nontrivial zeros of the zeta function lie on the straight line . This statement has not yet been proven, but numerical calculations support the assumption.

{\ displaystyle \ {s \ in \ mathbb {C} \ mid \ mathrm {Re} \, (s) = 1/2 \}}

Provably logically independent

The axiom of choice is usually taken to the formal foundations of mathematics. One then speaks of the Zermelo-Fraenkel set theory with axiom of choice , or ZFC for short. Without the axiom of choice, set theory is called ZF for short. The axiom of choice is independent of ZF and ZFC is free of contradictions if and when ZF is free of contradictions.

The continuum hypothesis was formulated by Georg Cantor in 1878 , and its logical independence from ZFC was proven by Paul Cohen in the 1960s . Kurt Gödel had previously proven the relative consistency even for the constructability axiom , which is really stronger than the continuum hypothesis - from ZFC, the continuum hypothesis can be proven by adding the constructibility axiom, but not vice versa.

The axiom of parallels used in geometry since Euclid has long been considered “superfluous”; that is, it has been suggested that it should be inferred from the remaining axioms. However, for over 2000 years all attempts to prove this conjecture have failed. It was only around 1826 that Lobachevsky and Bolyai succeeded independently of one another in showing the independence of the parallel axiom from the other axioms of Euclid. Depending on whether one accepts the original axiom or one or the other variant of its negation as an axiom, one obtains different geometries: Euclidean geometry , hyperbolic geometry and spherical geometry .

Using guesswork as a hypothesis

There are numerous works that presuppose conjecture, such as the Riemann Hypothesis. Strictly speaking, sentences of the form " If the (e.g. Riemannian) conjecture is correct, then ..." is proven in such papers .

Different cases can now arise:

If the conjecture is proven at some point, all implications of such work are proven in one fell swoop.
Should the presumption assumed as a hypothesis be refuted at some point, such work can become practically worthless if the conclusions cannot be proven from special cases or weaker presumptions. It is also conceivable that the presumption is refuted in exactly this way, namely if a statement that has been proven under the assumption of the presumption is refuted. It is also conceivable, however, that due to attractive conclusions from the conjecture, the axioms and conclusions used for the refutation are put to the test and modified in such a way that the conjecture can still be consistent.
If the presumption turns out to be a logically independent statement, it is in principle free to accept the presumption or its opposite. If one wants to adopt the results of such work, one would have to formally accept the conjecture as an axiom. Then there can be a split into Euclidean and non-Euclidean geometry or ZF with or without axiom of choice.

In principle, it is of course permissible to only use special cases that have already been proven. Popular solutions to the so-called Lucifer riddle often refer to the - so far unproven - Goldbach conjecture as an auxiliary proposition, but only use the conjecture statement in (relatively) few, long-checked special cases.

literature

Kurt Gödel: About formally undecidable sentences of the Principia Mathematica and related systems , 1930, in: MONTHS FOR MATHEMATICS AND PHYSICS 38 (1931), 173–198
Kurt Gödel: The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory , in: Annals of Mathematical Studies, Volume 3, Princeton University Press, Princeton, NJ, 1940, ed. In: Collected Works II , Oxford 1990, 33-101
David Hilbert : Basics of theoretical logic , Berlin, Heidelberg, 6th edition, 1972
David Hilbert: Mathematical Problems , 1900, in: Archives for Mathematics and Physics, 3rd series, Volume I (1901),
David Hilbert: New foundation of mathematics , 1922, in: Treatises from the Mathematical Seminar of the Hamburg University, Volume I (1922), 157–177

Web links

Wiktionary: Assumption - explanations of meanings, word origins, synonyms, translations

Wikiquote: Guess - Quotes

Individual evidence

↑ Stuart A. Kurtz, Janos Simon: The Undecidability of the Generalized Collatz Problem in Proceedings of the 4th International Conference on Theory and Applications of Models of Computation, TAMC 2007, held in Shanghai, China in May 2007 , pages 542-553 ISBN = 3540725032.
^ JH Conway: Unpredictable Iterations in Proceedings of the 1972 Number Theory Conference , University of Colorado, Boulder, Colorado, August 14-18, 1972 University of Colorado, Boulder, 1972, pages 49-52.
^ Paul Taylor: Foundations for Computable Topology . S. 10 ( paultaylor.eu [PDF]).

[1] Stuart A. Kurtz, Janos Simon: The Undecidability of the Generalized Collatz Problem in Proceedings of the 4th International Conference on Theory and Applications of Models of Computation, TAMC 2007, held in Shanghai, China in May 2007 , pages 542-553 ISBN = 3540725032.

[2] JH Conway: Unpredictable Iterations in Proceedings of the 1972 Number Theory Conference , University of Colorado, Boulder, Colorado, August 14-18, 1972 University of Colorado, Boulder, 1972, pages 49-52.

[3] Paul Taylor: Foundations for Computable Topology . S. 10 ( paultaylor.eu [PDF]).