Transcendental - Definition and Overview

Transcendental in philosophical contexts

In philosophy, transcendental experiences are experiences of an exclusively human nature that are other-worldly or beyond the human realm of understanding.

Things sometimes considered transcendental are religion, parts of philosophy (especially metaphysics and ontology), humour, death and more.

See also metaphysics.

Transcendental field elements in mathematics

In mathematics, a transcendental function is a function which is not expressible as a composition of a finite number of elementary operations, or inverses of functions so constructible, where the elementary operations consist of addition, multiplication, taking additive or multiplicative inverses, and integer root extraction. Transcendental functions include all the trigonometric functions and logarithmic functions, along with most other special functions in mathematics.

A transcendental element ξ of a field extension K over the field F is an element that is not the solution of a polynomial equation with coefficients in F, i.e., if there exists no polynomial

P(x) = a_n xⁿ + ... + a₁ x + a₀,

with all a_i ∈ F, such that P(ξ) = 0.

In the case of the field C of complex numbers or the field R of real numbers, a transcendental number is a number which is transcendental over the field Q of rational numbers.

See also: (spiritual) transcendentalism, (technological) transhumanism