Ordinary mathematics
Encyclopedia
In the philosophy of mathematics
, ordinary mathematics is an inexact term, used to distinguish the body of most mathematical work from that of, for example, constructivist
, intuitionist
, or finitist
mathematics.
Ordinary mathematics is usually studied within the universe
SN, or sometimes Vω+ω (see Von Neumann universe
). Contrast with finitist mathematics, which limits to the study of Vω (see hereditarily finite set
s), or with metamathematics
and the study of large cardinals, which study objects contained in a larger universe.
Ordinary mathematicians generally assume the axiom of choice (at least, because it makes their work easier), whereas constructivists reject it on the grounds that it is "non-constructive", and also reject the law of excluded middle
, which can be derived from it.
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
, ordinary mathematics is an inexact term, used to distinguish the body of most mathematical work from that of, for example, constructivist
Constructivism (mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its...
, intuitionist
Intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism , is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied...
, or finitist
Finitism
In the philosophy of mathematics, one of the varieties of finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps...
mathematics.
Ordinary mathematics is usually studied within the universe
Universe (mathematics)
In mathematics, and particularly in set theory and the foundations of mathematics, a universe is a class that contains all the entities one wishes to consider in a given situation...
SN, or sometimes Vω+ω (see Von Neumann universe
Von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets...
). Contrast with finitist mathematics, which limits to the study of Vω (see hereditarily finite set
Hereditarily finite set
In mathematics and set theory, hereditarily finite sets are defined recursively as finite sets consisting of 0 or more hereditarily finite sets.-Formal definition:...
s), or with metamathematics
Metamathematics
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories...
and the study of large cardinals, which study objects contained in a larger universe.
Ordinary mathematicians generally assume the axiom of choice (at least, because it makes their work easier), whereas constructivists reject it on the grounds that it is "non-constructive", and also reject the law of excluded middle
Law of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
, which can be derived from it.