User:ZuluPapa5/Existence (Mathematics)

Existence in mathematics is an ontological postulate. A significant question is do mathematical constant such as pi or e, lead to an persistent and absolute universe assumption which is independent of the human consciousness? Next, there is the question of the existence of solutions to mathematical problems. Platonist mathematics exists in axiomatic faith. Formally free from contradiction and intuitively constructed by man's discovery of truth.

Objectivity in referring requires a definition of truth. According to metaphysical objectivists, an object may truthfully be said to have this or that attribute, as in the statement "This object exists," whereas the statement "This object is true" or "false" is meaningless. Thus, only propositions have truth values. In the philosophy of language and metaphysics, an existential commitment is necessary to make a proposition where the existence of one thing is presupposed or implied by asserting the existence of another. Intuitionistic logic, or constructive logic, is a symbolic logic system that preserves justification, rather than truth, across transformations yielding derived propositions. From a practical view point, the motivation for applying intuitionistic logic, is that it has the existence property, making it suitable for other forms of mathematical constructivism. The existence property is fundamental to understanding in what sense proofs can be considered to have content: the essence of the discussion of existence theorems. A strict formalist does not require meaning or content as sufficient for existence.

In philosophy, an objective fact means a truth that remains true everywhere, independently of human thought or feelings, as existence implies. To be termed scientific, a method of inquiry must be based on gathering observable, empirical and measurable evidence subject to specific principles of reasoning.

Philosophy of science looks at the underpinning logic of the scientific method, at what separates science from non-science, and the ethic that is implicit in science. There are basic assumptions derived from philosophy that form the base of the scientific method - namely, that reality is both objective and consistent, that humans have the capacity to perceive reality accurately, and that rational explanations exist for elements of the real world. These assumptions from methodological naturalism form the basis on which science is grounded. Logical Positivist, empiricist, falsificationist, and other theories have claimed to give a definitive account of the logic of science, but each has in turn been criticized.

In mathematical logic existence is a quantifier, the "existential quantifier", symbolized by ∃, a backwards capital E. To symbolize "Four leaf clovers exist," mathematicians would first define predicates, P(x) = "x is a clover" and Q(x) = "x has four leaves", and then form the well-formed formula (∃x)(P(x) and Q(x)). In naive set theory, the empty set is a primitive notion because to assert that it exists is an implicit axiom in the axiom of empty set. In Zeroth Order Logic, existence is either true or false and can represented by the (1,0) binary group. First-order logic is a formal logic used in mathematics, philosophy, linguistics, and computer science. First-order logic requires at least one additional rule of inference to obtain completeness. It is distinguished by applying quantifiers, such that each interpretation of first-order logic includes a domain of discourse over which the quantifiers range. Logical connectives are applied to restrict the domain of discourse to fulfill a given predicate. In mathematical logic existence is a quantifier, the "existential quantifier", symbolized by ∃, a backwards capital E. To symbolize "Four leaf clovers exist", mathematicians would first define predicates, P(x) = "x is a clover" and Q(x) = "x has four leaves", and then form the well-formed formula (∃x)(P(x) and Q(x)). While ∃ can mean "for some", to contrast in symbolic logic, the "universal quantifier" (typically, ∀) is the symbol used to denote universal quantification, and is often informally read as "given any" or "for all". These statements may be applied in an existence proof or existence theorem.

In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") 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 existence, according to constructivists. A mathematical singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. Second-order logic is more expressive than first-order logic and it is possible to write formal sentences that quantify "the domain is finite" for existence, which is theoretically impossible in first order logic. In ordinary language, such second-order forms use either grammatical plurals or terms such as “set of” or “group of”.