Noumenon
In the philosophy of Immanuel Kant, a noumenon or thing in itself (German Ding an sich) is an unknowable, indescribable reality that in some way underlies observed phenomena. The etymology of the word ultimately reflects the Greek nous (mind).
Some writers also refer to noumena (the plural form), though the very notion of individuating items in the noumenal category seems problematic, since the very notions of number and individuality appear among the categories of understanding, so that individuality itself is a noumenon. Phenomenon serves as a (contrasting) technical term in Kant's philosophy, meaning the world as experienced. Kant wondered if practical reason could enter immediate contact with the "noumenal real". Then metaphysics would be capable of establishing its validity, just as the impact of experience grounds the objective validity of the sciences.
Explaining the relationship between the noumenal and phenomenal worlds forms one of the most difficult problems for Kant's philosophy. On Kant's view as expressed in his Critique of Pure Reason, reality is structured by "concepts of the understanding", or innate categories that the mind brings to make sense of raw unstructured experience. Since these categories include causality and number, it becomes problematic to say that many noumena exist that individually cause us to have perceptions of phenomena. But if the noumenal does not cause the phenomenal, then what is the relationship? The suggested answer is that the noumenal and phenomenal coexist simultaneously; we cannot say that either causes the other.
It can be said that on Kant's view the noumenal is radically unknowable. Whatever concept we might want to use to categorize some noumenon or noumena, that is only a way of categorizing phenomena, so that the act of knowing a noumenon must itself be defined by a noumenon, a situation that is unresolvable.
Schopenhauer
Schopenhauer claimed that Kant used the word incorrectly. He explained in his Criticism of the Kantian Philosophy.
"But it was just this difference between abstract knowledge and knowledge of perception, entirely overlooked by Kant, which the ancient philosophers denoted by noumena and phenomena. (See Sextus Empiricus, Outlines of Pyrrhonism, Book I, Chapter 13, ' What is thought (noumena) is opposed to what appears or is perceived (phenomena).' ) This contrast and utter disproportion greatly occupied these philosophers in the philosophemes of the Eleatics, in Plato's doctrine of the Ideas, in the dialectic of the Megarics, and later the scholastics in the dispute between nominalism and realism, whose seed, so late in developing, was already contained in the opposite mental tendencies of Plato and Aristotle. But Kant who, in an unwarrantable manner, entirely neglected the thing for the expression of which those words phenomena and noumena had already been taken, now takes possession of the words, as if they were still unclaimed, in order to denote by them his things-in-themselves and his phenomena."
Noumenon, which stems from the Greek, is linguistically unrelated to "numinous," a term coined by Rudolf Otto and based on the Latin numen (deity).