Mathematics has no generally accepted definition. Different schools of thought, particularly in philosophy, have put forth radically different definitions. All are controversial. [1] [2]
Aristotle defined mathematics as: [3]
The science of quantity.
In Aristotle's classification of the sciences, discrete quantities were studied by arithmetic, continuous quantities by geometry. [4] Aristotle also thought that quantity alone does not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart. [5]
Auguste Comte's definition tried to explain the role of mathematics in coordinating phenomena in all other fields: [6]
The science of indirect measurement. [3] Auguste Comte 1851
The "indirectness" in Comte's definition refers to determining quantities that cannot be measured directly, such as the distance to planets or the size of atoms, by means of their relations to quantities that can be measured directly. [7]
The preceding kinds of definitions, which had prevailed since Aristotle's time, [4] were abandoned in the 19th century as new branches of mathematics were developed, which bore no obvious relation to measurement or the physical world, such as group theory, projective geometry, [3] and non-Euclidean geometry. [8]
Three leading types of definition of mathematics today are called logicist, intuitionist, and formalist, each reflecting a different philosophy of mathematics. Each has its own flaws, none has achieved mainstream consensus, and all three appear irreconcilable. [9]
With mathematicians pursuing greater rigor and more abstract foundations, some proposed defining mathematics purely in terms of deduction and logic:
Mathematics is the science that draws necessary conclusions. [10] Benjamin Peirce 1870
All Mathematics is Symbolic Logic. [8] Bertrand Russell 1903
Peirce did not think that mathematics is the same as logic, since he thought mathematics makes only hypothetical assertions, not categorical ones. [11] Russell's definition, on the other hand, expresses the logicist view without reservation. [9]
Rather than characterize mathematics by deductive logic, intuitionism views mathematics as primarily about the construction of ideas in the mind: [9]
The only possible foundation of mathematics must be sought in this construction under the obligation carefully to watch which constructions intuition allows and which not. [12] L. E. J. Brouwer 1907
... intuitionist mathematics is nothing more nor less than an investigation of the utmost limits which the intellect can attain in its self-unfolding. [12] Arend Heyting 1968
Intuitionism sprang from the philosophy of mathematician L. E. J. Brouwer and also led to the development of a modified intuitionistic logic. As a result, intuitionism has generated some genuinely different results that, while coherent and valid, differ from some theorems grounded in classical logic. [9]
Formalism denies logical or intuitive meanings altogether, making the symbols and rules themselves the objects of study. [9] A formalist definition:
Mathematics is the science of formal systems. [13] Haskell Curry 1951
Still other definitions emphasize pattern, order, or structure. For example:
Mathematics is the classification and study of all possible patterns. [14] Walter Warwick Sawyer, 1955
Yet another approach makes abstraction the defining criterion:
Mathematics is a broad-ranging field of study in which the properties and interactions of idealized objects are examined. [15]
Most contemporary reference works define mathematics by summarizing its main topics and methods:
The abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra. [16] Oxford English Dictionary, 1933
The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols. [17] American Heritage Dictionary, 2000
The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. [18] Encyclopædia Britannica, 2006
Bertrand Russell wrote this famous tongue-in-cheek definition, describing the way all terms in mathematics are ultimately defined by reference to undefined terms:
The subject in which we never know what we are talking about, nor whether what we are saying is true. [19] Bertrand Russell 1901
Many other attempts to characterize mathematics have led to humor or poetic prose:
A mathematician is a blind man in a dark room looking for a black cat which isn't there. [20] Charles Darwin [21]
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. [22] G. H. Hardy, 1940
Mathematics is the art of giving the same name to different things. [10] Henri Poincaré
Mathematics is the science of skillful operations with concepts and rules invented just for this purpose. [this purpose being the skillful operation ....] [23] Eugene Wigner
Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer's gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness of life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud cell, and is forever ready to burst forth into new forms of vegetable and animal existence. [24] James Joseph Sylvester
What is mathematics? What is it for? What are mathematicians doing nowadays? Wasn't it all finished long ago? How many new numbers can you invent anyway? Is today's mathematics just a matter of huge calculations, with the mathematician as a kind of zookeeper, making sure the precious computers are fed and watered? If it's not, what is it other than the incomprehensible outpourings of superpowered brainboxes with their heads in the clouds and their feet dangling from the lofty balconies of their ivory towers? Mathematics is all of these, and none. Mostly, it's just different. It's not what you expect it to be, you turn your back for a moment and it's changed. It's certainly not just a fixed body of knowledge, its growth is not confined to inventing new numbers, and its hidden tendrils pervade every aspect of modern life. [24] Ian Stewart
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.
In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction, and the law of identity; however, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law / principleof the excluded third, in Latin principium tertii exclusi. Another Latin designation for this law is tertium non datur or "no third [possibility] is given". In classical logic, the law is a tautology.
Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory, algebra, geometry, analysis, and set theory.
In the philosophy of mathematics, intuitionism, or neointuitionism, is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.
In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory, relations between primitive notions are restricted by axioms. Some authors refer to the latter as "defining" primitive notions by one or more axioms, but this can be misleading. Formal theories cannot dispense with primitive notions, under pain of infinite regress.
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship with other human activities.
Friedrich Ludwig Gottlob Frege was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, logic, and mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics ever.
The history of logic deals with the study of the development of the science of valid inference (logic). Formal logics developed in ancient times in India, China, and Greece. Greek methods, particularly Aristotelian logic as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia. The Stoics, especially Chrysippus, began the development of predicate logic.
Analytic philosophy is an analysis focused, broad, contemporary movement or tradition within Western philosophy, especially anglophone philosophy. Analytic philosophy is characterized by a clarity of prose; rigor in arguments; and making use of formal logic and mathematics, and, to a lesser degree, the natural sciences. It is further characterized by an interest in language and meaning known as the linguistic turn. It has developed several new branches of philosophy and logic, notably philosophy of language, philosophy of mathematics, philosophy of science, modern predicate logic and mathematical logic.
Foundations of mathematics is the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and, in particular, to have reliable concepts of theorems, proofs, algorithms, etc. This may also include the philosophical study of the relation of this framework with reality.
Luitzen Egbertus Jan "Bertus" Brouwer was a Dutch mathematician and philosopher who worked in topology, set theory, measure theory and complex analysis. Regarded as one of the greatest mathematicians of the 20th century, he is known as one of the founders of modern topology, particularly for establishing his fixed-point theorem and the topological invariance of dimension.
In the philosophy of mathematics, the pre-intuitionists is the name given by L. E. J. Brouwer to several influential mathematicians who shared similar opinions on the nature of mathematics. The term was introduced by Brouwer in his 1951 lectures at Cambridge where he described the differences between his philosophy of intuitionism and its predecessors:
Of a totally different orientation [from the "Old Formalist School" of Dedekind, Cantor, Peano, Zermelo, and Couturat, etc.] was the Pre-Intuitionist School, mainly led by Poincaré, Borel and Lebesgue. These thinkers seem to have maintained a modified observational standpoint for the introduction of natural numbers, for the principle of complete induction [...] For these, even for such theorems as were deduced by means of classical logic, they postulated an existence and exactness independent of language and logic and regarded its non-contradictority as certain, even without logical proof. For the continuum, however, they seem not to have sought an origin strictly extraneous to language and logic.
In logic, the logical form of a statement is a precisely-specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal formal language, the meaning of a logical form can be determined unambiguously from syntax alone. Logical forms are semantic, not syntactic constructs; therefore, there may be more than one string that represents the same logical form in a given language.
In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess." According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all. Rather, mathematical statements are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation. In contrast to mathematical realism, logicism, or intuitionism, formalism's contours are less defined due to broad approaches that can be categorized as formalist.
The Brouwer–Hilbert controversy was a debate in twentieth-century mathematics over fundamental questions about the consistency of axioms and the role of semantics and syntax in mathematics. L. E. J. Brouwer, a proponent of the constructivist school of intuitionism, opposed David Hilbert, a proponent of formalism. Much of the controversy took place while both were involved with Mathematische Annalen, the leading mathematical journal of the time, with Hilbert as editor-in-chief and Brouwer as a member of its editorial board. In 1920 Hilbert had Brouwer removed from the editorial board of Mathematische Annalen.
A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an object is anything that has been formally defined, and with which one may do deductive reasoning and mathematical proofs. Typically, a mathematical object can be a value that can be assigned to a variable, and therefore can be involved in formulas. Commonly encountered mathematical objects include numbers, sets, functions, expressions, geometric objects, transformations of other mathematical objects, and spaces. Mathematical objects can be very complex; for example, theorems, proofs, and even theories are considered as mathematical objects in proof theory.
Ludwig Wittgenstein considered his chief contribution to be in the philosophy of mathematics, a topic to which he devoted much of his work between 1929 and 1944. As with his philosophy of language, Wittgenstein's views on mathematics evolved from the period of the Tractatus Logico-Philosophicus: with him changing from logicism towards a general anti-foundationalism and constructivism that was not readily accepted by the mathematical community. The success of Wittgenstein's general philosophy has tended to displace the real debates on more technical issues.
The mathematical concept of a function dates from the 17th century in connection with the development of calculus; for example, the slope of a graph at a point was regarded as a function of the x-coordinate of the point. Functions were not explicitly considered in antiquity, but some precursors of the concept can perhaps be seen in the work of medieval philosophers and mathematicians such as Oresme.
Logical Intuition, or mathematical intuition or rational intuition, is a series of instinctive foresight, know-how, and savviness often associated with the ability to perceive logical or mathematical truth—and the ability to solve mathematical challenges efficiently. Humans apply logical intuition in proving mathematical theorems, validating logical arguments, developing algorithms and heuristics, and in related contexts where mathematical challenges are involved. The ability to recognize logical or mathematical truth and identify viable methods may vary from person to person, and may even be a result of knowledge and experience, which are subject to cultivation. The ability may not be realizable in a computer program by means other than genetic programming or evolutionary programming.
It is first necessary to ask what is meant by mathematics in general. Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form.
