Irreducibility

In philosophy, a phenomenona is governed by the principle of irreducibility when a complete account of an entity is not possible at lower levels of explanation because the phenomenona exhibits novel properties beyond prediction and explanation in terms of lower levels. Another way to state this is that Occam's razor requires the elimination of only those entities that are unnecessary, not as many entities as could conceivably be eliminated. 
In hard irreducibility, such as materialist explanations of the human soul, it's a matter of legitimate dispute whether the (purportedly) emergent phenomenona (the soul) inheres in any way at the lower level of explanation (materialism). 
In soft irreducibility, such as situating the entire theory of chemistry on top of the physics of the atom, the emergent phenomenon (chemistry) is believed to inhere in the underlying level of explanation (physics), but the complexity of going from one level to the other is so formidable as to force dealing with both levels on their own terms, in the majority of cases, even though one level, in principle, determines the other, in the example, as chemistry arising directly and entirely from physics. 
Vygotsky treatment.
Lev Vygotsky provides the following illustration of the idea, in his "Thought and Language":
"Two essentially different modes of analysis are possible in the study of psychological structures. It seems to us that one of them is responsible for all the failures that have beset former investigators of the old problem, which we are about to tackle in our turn, and that the other is the only correct way to approach it.
The first method analyzes complex psychological wholes into "elements". It may be compared to the chemical analysis of water into hydrogen and oxygen, neither of which possesses the properties of the whole and each of which possesses properties not present in the whole. The student applying this method in looking for an explanation of some property of water — why it extinguishes fire, for example — will find to his surprise that hydrogen burns and oxygen sustains fire ...
In our opinion the right course to follow is to use the other type of analysis, which may be called "analysis into units". By "unit", we mean a product of analysis which, unlike elements, retains all the basic properties of the whole, and which cannot be further divided without losing them. Not the chemical composition of water, but its molecules and their behaviour, are the key to the understanding of the properties of water ..."
In other words: to conserve the properties under investigation, it is necessary to remain within a certain level of complexity. Irreducibility is most often deployed in defence of the reality of human subjectivity and/or free will, against those who treat such things as folk psychology, such as Paul and Patricia Churchland.
Polanyi treatment.
In "Life's Irreducible Structure", Micheal Polanyi argues that higher-order sciences such as biology and other chemical processes can not be explained by the fundamentalist perspective. Indeed, machines and the humans that make them fall into "boundary conditions" that produce phenomena that itself can not be reduced further. He writes The lowest level is the production of a voice; the second, the utterance of words; the third, the joining of words that make sentences; the fourth, the working of sentences into a style; the fifth, and the highest, the composition of the text. The principles of each level operate under the control of the next-higher level. The voice you produce is shaped into words by a vocabulary; a given vocabulary is shaped into sentences in accordance with a grammar; and the sentences are fitted into a style, which in turn is made to convey the ideas of the composition. Thus each level is subject to dual control: (i) control in accordance with the laws that apply to its elements in themselves, and (ii) control in accordance with the laws of the powers that control the comprehensive entity formed by these elementsIt is worth noting that reductionism has not been ruled out as an explanation of these tipping-point, emergent behaviors.