Equality (mathematics)

[4] Equality is often considered a kind of primitive notion, meaning, it is not formally defined, but rather informally said to be "a relation each thing bears to itself and nothing else".

Basic properties about equality like reflexivity, symmetry, and transitivity have been understood intuitively since at least the ancient Greeks, but weren't symbolically stated as general properties of relations until the late 19th century by Giuseppe Peano.

Other properties like substitution and function application weren't formally stated until the development of symbolic logic.

There are generally two ways that equality is formalized in mathematics: through logic or through set theory.

[6] The word entered Middle English around the 14th century, borrowed from Old French equalité (modern égalité).

[9] Diophantus's use of ⟨ἴσ⟩, short for ἴσος (ísos 'equals'), in Arithmetica (c. 250 AD) is considered one of the first uses of an equals sign.

[16] The operation-application property was also stated in Peano's Arithmetices principia,[14] however, it had been common practice in algebra since at least Diophantus (c. 250 AD).

Equation solving is the problem of finding values of some variable, called unknown, for which the specified equality is true.

[19] More specifically, an equation represents a binary relation (i.e., a two-argument predicate) which may produce a truth value (true or false) from its arguments.

There is no standard notation that distinguishes an equation from an identity, or other use of the equality relation: one has to guess an appropriate interpretation from the semantics of expressions and the context.

The first recorded symbolic use of "Equal by definition" appeared in Logica Matematica (1894) by Cesare Burali-Forti, an Italian mathematician.

[30] This tradition can be traced back to at least 350 BC by Aristotle: in his Categories, he defines the notion of quantity in terms of a more primitive equality (distinct from identity or similarity), stating:[31]"The most distinctive mark of quantity is that equality and inequality are predicated of it.

For instance, one solid is said to be equal or unequal to another; number, too, and time can have these terms applied to them, indeed can all those kinds of quantity that have been mentioned.

- (Translated by E. M. Edghill)Aristotle had separate categories for quantities (number, length, volume) and qualities (temperature, density, pressure), now called intensive and extensive properties.

The Scholastics, particularly Richard Swineshead and other Oxford Calculators in the 14th century, began seriously thinking about kinematics and quantitative treatment of qualities, a major development in the quantification of physics, and which broadened the scope of mathematical objects beyond numbers and magnitudes.

This trend lead to the axiomatization of equality through the law of identity and the substitution property especially in mathematical logic[11][34] and analytic philosophy.

[36] Its introduction to logic, and first symbolic formulation is due to Bertrand Russell and Alfred Whitehead in their Principia Mathematica (1910), who credit Leibniz for the idea.

Outside of pure math, the identity of indiscernibles has attracted much controversy and criticism, especially from corpuscular philosophy and quantum mechanics.

José Ferreirós credits Richard Dedekind for being the first to explicitly state the principle, (although he does not assert it as a definition):"It very frequently happens that different things a, b, c ... considered for any reason under a common point of view, are collected together in the mind, and one then says that they form a system S; one calls the things a, b, c ... the elements of the system S, they are contained in S; conversely, S consists of these elements.

- Richard Dedekind, 1888 (Translated by José Ferreirós)Around the turn of the 20th century, mathematics faced several paradoxes and counter-intuitive results.

For example, Russell's paradox showed a contradiction of naive set theory, it was shown that the parallel postulate cannot be proved, the existence of mathematical objects that cannot be computed or explicitly described, and the existence of theorems of arithmetic that cannot be proved with Peano arithmetic.

[49] The logical term was introduced to set theory in 1893, Gottlob Frege attempted to use this idea of an extension formally in his Foundations of Arithmetic, where, if

[51] The specific term for "Extensionality" used by Zermelo was "Bestimmtheit".The specific English term "extensionality" only became common in mathematical and logical texts in the 1920s and 1930s,[52] particularly with the formalization of logic and set theory by figures like Alfred Tarski and John von Neumann.

Incorporating half of the work into the first-order logic may be regarded as a mere matter of convenience, as noted by Azriel Lévy.

[55] Or, equivalently, one may choose to define equality in a way that mimics, the substitution property explicitly, as the conjunction of all atomic formuals:[56] In either case, the Axiom of Extensionality based on first-order logic without equality states: Numerical approximation is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis.

Other equivalence relations, while less restrictive, often generalize equality by identifying elements based on shared properties or transformations, such as congruence in modular arithmetic or similarity in geometry.

[60] In mathematics, especially in abstract algebra and category theory, it is common to deal with objects that already have some internal structure.

An isomorphism describes a kind of structure-preserving correspondence between two objects, establishing them as essentially identical in their structure or properties.

When two objects or systems are isomorphic, they are considered indistinguishable in terms of their internal structure, even though their elements or representations may differ.

Isomorphisms facilitate the classification of mathematical entities and enable the transfer of results and techniques between similar systems.