Modeling Terminology


Terms


Definition


Alternate Hypothesis

The alternative hypothesis is the hypothesis used in hypothesis testing that is contrary to the null hypothesis. It is usually taken to be that the observations are the result of a real effect (with some amount of chance variation superposed).

Ansatz

An ansatz is an assumed form for a mathematical statement that is not based on any underlying theory or principle. An example from physics is the Bethe Ansatz (Müller).

Axiom

An axiom is a proposition regarded as self-evidently true without proof. The word "axiom" is a slightly archaic synonym for postulate. Compare conjecture or hypothesis, both of which connote apparently true but not self-evident statements.

Conjecture

A proposition which is consistent with known data, but has neither been verified nor shown to be false. It is synonymous with hypothesis.

Corollary

An immediate consequence of a result already proved. Corollaries usually state more complicated theorems in a language simpler to use and apply.

Hypothesis

A hypothesis is a proposition that is consistent with known data, but has been neither verified nor shown to be false. In statistics, a hypothesis (sometimes called a statistical hypothesis) refers to a statement on which hypothesis testing will be based. Particularly important statistical hypotheses include the null hypothesis and alternative hypothesis. In symbolic logic, a hypothesis is the first part of an implication (with the second part being known as the predicate). In general mathematical usage, "hypothesis" is roughly synonymous with "conjecture."

Lemma

A short theorem used in proving a larger theorem. Related concepts are the axiom, porism, postulate, principle, and theorem. The late mathematician P. Erdos has often been associated with the observation that "a mathematician is a machine for converting coffee into theorems" (e.g., Hoffman 1998, p. 7). However, this characterization appears to be due to his friend, Alfred Rényi (MacTutor, Malkevitch). This thought was developed further by Erdos' friend and Hungarian mathematician Paul Turán, who suggested that weak coffee was suitable "only for lemmas" (MacTutor, Malkevitch).

Logic

The formal mathematical study of the methods, structure, and validity of mathematical deduction and proof. In Hilbert's day, formal logic sought to devise a complete, consistent formulation of mathematics such that propositions could be formally stated and proved using a small number of symbols with well defined meanings. The difficulty of formal logic was demonstrated in the monumental Principia Mathematica (1925) of Whitehead and Russell's, in which hundreds of pages of symbols were required before the statement could be deduced. The foundations of this program were obliterated in the mid 1930s when Gödel unexpectedly proved a result now known as Gödel's incompleteness theorem. This theorem not only showed Hilbert's goal to be impossible, but also proved to be only the first in a series of deep and counterintuitive statements about rigor and provability in mathematics. A very simple form of logic is the study of "truth tables" and digital logic circuits in which one or more outputs depend on a combination of circuit elements (AND, OR, NAND, NOR, NOT, XOR, etc.; "gates") and the input values. In such a circuit, values at each point can take on values of only true (1) or false (0). de Morgan's duality law is a useful principle for the analysis and simplification of such circuits. A generalization of this simple type of logic in which possible values are true, false, and "undecided" is called three-valued logic. A further generalization called fuzzy logic treats "truth" as a continuous quantity ranging from 0 to 1.

Metamathematics

Metamathematics is another word for proof theory. The branch of logic dealing with the study of the combination and application of mathematical symbols is also sometimes called metamathematics or metalogic.

Metatheorem

A statement about theorems. It usually gives a criterion for getting a new theorem from an old one, either by changing its objects according to a rule (duality principle), or by transferring it to another area (from the theory of categories to the theory of groups) or to another context within the same area (from linear transformations to matrices).

Null Hypothesis

A null hypothesis is a statistical hypothesis that is tested for possible rejection under the assumption that it is true (usually that observations are the result of chance). The concept was introduced by R. A. Fisher. The hypothesis contrary to the null hypothesis, usually that the observations are the result of a real effect, is known as the alternative hypothesis.

Porism

The term "porism" is an archaic type of mathematical proposition whose historical purpose is not entirely known. It is used instead of "theorem" by some authors for a small number of results for historical reasons. However, two meanings predominate in nonhistorical usage. The first is 'corollary,' a usage now mostly superseded by that term itself. The second (which may now be considered the 'modern' usage) is, "A proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate, or capable of innumerable solutions" (Playfair 1792). Unfortunately, this definition is slightly inaccurate, because the proposition actually states the conditions, rather than affirming the possibility of finding them.

Postulate

A statement, also known as an axiom, which is taken to be true without proof. Postulates are the basic structure from which lemmas and theorems are derived. The whole of Euclidean geometry, for example, is based on five postulates known as Euclid's postulates.

Principle

A loose term for a true statement which may be a postulate, theorem, etc.

Problem

A problem is an exercise whose solution is desired. Mathematical "problems" may therefore range from simple puzzles to examination and contest problems to propositions whose proofs require insightful analysis. Although not absolutely standard, The Greeks distinguished between "problems" (roughly, the construction of various figures) and "theorems" (establishing the properties of said figures; Heath 1956, pp. 252, 262, and 264). There are many unsolved problems in mathematics. Two famous problems which have recently been solved include Fermat's last theorem (by Andrew Wiles) and the Kepler conjecture (by T. C.Hales). Among the most prominent of remaining unsolved problems are the Goldbach conjecture, Riemann hypothesis, Poincaré conjecture, the conjecture that there are an infinite number of twin primes, as well as many more. K.S. Brown, D. Eppstein, S. Finch, and C. Kimberling maintain extensive pages of unsolved problems in mathematics.

Proof

A rigorous mathematical argument which unequivocally demonstrates the truth of a given proposition. A mathematical statement that has been proven is called a theorem. According to Hardy (1999, pp. 15-16), "all physicists, and a good many quite respectable mathematicians, are contemptuous about proof. I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite uninteresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for proof.... [This opinion], with which I am sure that almost all physicists agree at the bottom of their hearts, is one to which a mathematician ought to have some reply." To prove Hardy's assertion, Feynman is reported to have commented, "A great deal more is known than has been proved" (Derbyshire 2004, p. 291). There is some debate among mathematicians as to just what constitutes a proof. The four-color theorem is an example of this debate, since its "proof" relies on an exhaustive computer testing of many individual cases which cannot be verified "by hand." While many mathematicians regard computer-assisted proofs as valid, some purists do not. There are several computer systems currently under development for automated theorem proving, among them, THEOREMA. A page of proof-related humor is maintained by Chalmers.

Proposition

A proposition is a mathematical statement such as "3 is greater than 4," "an infinite set exists," or "7 is prime." An axiom is a proposition that is assumed to be true. With sufficient information, mathematical logic can often categorize a proposition as true or false, although there are various exceptions (e.g., "This statement is false").

Theorem

A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is an embodiment of some general principle that makes it part of a larger theory. The process of showing a theorem to be correct is called a proof. Although not absolutely standard, the Greeks distinguished between "problems" (roughly, the construction of various figures) and "theorems" (establishing the properties of said figures; Heath 1956, pp. 252, 262, and 264). According to the Nobel Prize-winning physicist Richard Feynman (1985), any theorem, no matter how difficult to prove in the first place, is viewed as "trivial" by mathematicians once it has been proven. Therefore, there are exactly two types of mathematical objects: trivial ones, and those which have not yet been proven. The late mathematician P. Erdos has often been associated with the observation that "a mathematician is a machine for converting coffee into theorems" (e.g., Hoffman 1998, p. 7). However, this characterization appears to be due to his friend, Alfred Rényi (MacTutor, Malkevitch). This thought was developed further by Erdos' friend and Hungarian mathematician Paul Turán), who suggested that weak coffee was suitable "only for lemmas" (MacTutor, Malkevitch). R. Graham has estimated that upwards of 250000 mathematical theorems are published each year (Hoffman 1998, p. 204).


Some definitions were obtained from Foundations of Mathematics at Wolfram MathWord.

 

 
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