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Longman Dictionary of Contemporary English
irreducible
adjective
EXAMPLES FROM CORPUS
▪ Amid all these fantasies and equivocations, however, there were two irreducible facts: death and Judith.
▪ From the point of view of demand management, therefore, frictional and structural unemployment is an irreducible minimum unemployment rate.
▪ In James, it is civilized, social man negotiating and experiencing a world of irreducible ambivalence and complexity.
▪ It is imagination, and the irreducible sovereignty of the individual which engender disequilibrium and tension.
▪ It is the irreducible seat of strength.
▪ To follow this, an irreducible minimum of biochemical knowledge is necessary.
The Collaborative International Dictionary
Irreducible

Irreducible \Ir`re*du"ci*ble\, a.

  1. Incapable of being reduced, or brought into a different state; incapable of restoration to its proper or normal condition; as, an irreducible hernia.

  2. (Math.) Incapable of being reduced to a simpler form of expression; as, an irreducible formula.

    Irreducible case (Alg.), a particular case in the solution of a cubic equation, in which the formula commonly employed contains an imaginary quantity, and therefore fails in its application. -- Ir`re*du"ci*ble*ness, n. -- -- Ir`re*du"ci*bly, adv.

Douglas Harper's Etymology Dictionary
irreducible

1530s, from assimilated form of in- (1) "not, opposite of" + reducible. Related: Irreducibly; irreducibility.

Wiktionary
irreducible

a. 1 Not able to be reduced or lessened. 2 Not able to be brought to a simpler or reduced form. 3 (context mathematics of a polynomial English) Unable to be factorized into polynomials of lower degree, as ''(x^2 + 1)''. 4 (context mathematics of an integer English) Unable to be factored into smaller integers; prime. 5 (context topology of a manifold English) Not containing a sphere of codimension 1 that is not the boundary of a ball. 6 (context group theory of a representation English) impossible to divide further into representations of lower dimension by means of any similarity transformation n. (context mathematics English) Such a polynomial

WordNet
irreducible

adj. incapable of being made smaller or simpler; "an irreducible minimum"; "an irreducible formula"; "an irreducible hernia" [ant: reducible]

Wikipedia
Irreducible (representation theory)

Usage examples of "irreducible".

In order that the living being can live, there must exist several functional structures, all irreducible one to another, and also an uninterrupted movement between each one of those structures and the air it breathes, the water it drinks, the food it absorbs.

The latter, as we have seen, took as its foundation a triple theory of irreducible needs, the objectivity of labour, and the end of history.

Proponents of this view, for example, Paul Church-land and Stephen Stich, argue that subjectively experienced mental states should be regarded as nonexistent, on the grounds that the descriptions of such states are irreducible to the language of neuroscience.

The paradox confronting the participants was that from the first-person perspective, consciousness is a prime irreducible datum, but from the third-person scientific perspective there is no way of investigating it directly.

If this principle of denying the validity of one area of experience on the grounds that it is irreducible to another were to be applied within other domains of science, a number of our present scientific theories would have to be abandoned.

Mental states are ontologically irreducible, they exist only as subjective, first-person phenomena, and they almost always have a content.

The bacterial cilium that Behe presents as one of his cases of irreducible complexity is a whiplike rotary paddle used for propulsion, driven by an intricate molecular machine that includes an acid-powered engine, stator housing, O-rings, bushings, and a drive shaft, and is built from over 40 interacting proteins, every one of them essential.

Eric Stokes has convincingly shown, utilitarianism combined with the legacies of liberalism and evangelicalism as philosophies of British rule in the East stressed the rational importance of a strong executive armed with various legal and penal codes, a system of doctrines on such matters as frontiers and land rents, and everywhere an irreducible supervisory imperial authority.

But honest human thought will seek above all to determine what are the veritable irreducible mysteries.

On the contrary, it was the unconsciousness of the mass, compelled to act in self-defence against thoughts too intrinsically, individually human to satisfy the irreducible exigencies of life on this earth.

Foissac gives various strange examples of the persistent, inexplicable, fundamental, pre-ordained, irreducible iniquity in which many existences are steeped.

But, this reproach once made--and it is the cardinal reproach against irreducible injustice--they have no further cause of complaint.

This temporarily irreducible condition can often be cured and always prevented from growing worse if the proper truss is fitted and worn.

They met each day in the yard to, it was said, decide the important questions relating to our lives and-if you bought into the view that Diamond Bar was the purest expression of a carceral universe, the irreducible distillate of the essential human condition-the lives of everyone on the planet.

We discover these two--Matter and Idea--by sheer force of our reasoning which distinguishes continually in pursuit of the simplex, the irreducible, working on, until it can go no further, towards the ultimate in the subject of enquiry.