transcendental

1. (philosophy) Concerned with the a priori or intuitive basis of knowledge, independent of experience.Examples: "The best way to demonstrate the possibility of something is to show its actuality, for actuality implies possibility. At least since Kant, transcendental philosophies have been on the scene. However, such simple demonstration of the possibility of transcendental philosophy has not been effective and is not likely to be so — so strong is the presumption that transcendental philosophy just could not be possible, or, if it was possible earlier, it is not possible now."; "1999, Robert Stern, 4: On Kant's Response to Hume: The Second Analogy as Transcendental Argument, Robert Stern (editor), Transcendental Arguments: Problems and Prospects, 2003, Oxford University Press (Clarendon Press), Paperback, page 47, Whilst it was once held that transcendental arguments could provide a direct and straightforward refutation of scepticism, this view now seems over-optimistic."; "2007, Steven Crowell, Jeff Malpas, Chapter 1: Introduction Steven Crowell, Jeff Malpas, (editors), Transcendental Heidegger, Stanford University Press, page 1, Not only does Heidegger's early work stand within the framework of transcendental phenomenology as established by Husserl—even though it also contests and revises that framework—but that thinking also stands in a close relationship to the critical philosophy of Immanuel Kant, and specifically to the transcendental project, and modes of argument, of Kant's Critique of Pure Reason."

2. (algebra, number theory, field theory, of a number or an element of an extension field) Not algebraic (i.e., not the root of any polynomial that has positive degree and rational coefficients).Examples: "The theory of transcendental numbers was originated by Liouville in his famous memoir^† of 1844 in which he obtained, for the first time, a class, très-étendue, as it was described in the title of the paper, of numbers that satisfy no algebraic equation with integer coefficients."; "If the distribution of decimal digits of #92;pi (or any other transcendental number) is truly random (suspected but not yet mathematically proven!), given any arbitrary finite sequence of whole numbers, that sequence would be included an infinite number of times in the decimal expansion of #92;pi."

3. (algebra, field theory, of an extension field) That contains elements that are not algebraic.Examples: "2006, Steven Roman, Field Theory, Springer, 2nd Edition, Graduate Texts in Mathematics 158, page 108, Suppose that F<E is purely transcendental. Show that any simple extension of F contained in E (but not equal to F) is transcendental over F."