The individual substances, of which the universe is composed, fall into three great irreducible kinds: nature, God, man.
On British railways the duty of the companies to provide all practicable safeguards and to educate and caution the servants may be said to have been faithfully performed, and the accident totals must be taken as being somewhat near the " irreducible minimum" - unless some of the infirmities of the human mind can be cured.
There remains too the consequence that there are different substances, separate from but related to one another; and these substances of three irreducible kinds, natural, supernatural, human.
The peace of Bromsebro was the first of the long series of treaties, extending down to our own days, which mark the progressive shrinkage of Danish territory into an irreducible minimum.
Or again, within the latter in his admission of a duality of thought and " the given " in knowledge, which within knowledge was apparently irreducible, concurrently with hints as to the possibility, upon a wider view, of the sublation of their disparateness at least hypothetically and speculatively.
Thus in the true idea of things there is no irreducible residuum of matter: mind is the Alpha and Omega, at once the initial postulate and the final truth of reality.
In the place of Maura he found a more pliant minister for the colonies, Seor Abarzuza, who framed a Cuban Reform Bill so much short of what his predecessor had thought an irreducible minimum of concessions, that it was censured in Havana by all the colonial Liberals and home rulers, and by their representatives in Madrid.
In respect to them, the final result was found in a series of irreducible notions or categories, the prima possibilia, the analysis and elucidation of which was specifically the business of philosophy or metaphysics.
An irreducible element of indetermination, e.g.
Indeed, if thought admits irreducible units, what can unite?
It is true matter can never be completely resolved into motion, but the irreducible remainder may be treated like the 7rpc.)T fl van of Aristotle as an abstraction which we asymptotically approach but never reach.
It follows from them that the relation of a real ground to that which is thereby posited or denied cannot be expressed by a judgment but only by means of a notion, which by analysis may certainly be reduced to yet simpler notions of real grounds, but yet in such a way that the final resort of all our cognition in this regard must be found in simple and irreducible notions of real grounds, the relation of which to their consequents cannot be made clear."
In his previous essays, Kant, while likewise maintaining that such pure, irreducible notions existed, had asserted in general terms that they applied to experience, and that their applicability or justification rested on experience itself, but had not raised the question as to the ground of such justification.
Clearer ideas of imaginary quantities and the " irreducible case " were subsequently published by Bombelli, in a work of which the dedication is dated 1572, though the book was not published until 1579.