What is "irreducibility"

Irreducibility \Ir`re*du`ci*bil"i*ty\, n. The state or quality of being irreducible.

n. 1 (context uncountable English) The quality or degree of being irreducible. 2 (context countable English) Something that is irreducible.

In mathematics, the concept of irreducibility is used in several ways.

• In abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial.
• In representation theory, an irreducible representation is a nontrivial representation with no nontrivial proper subrepresentations. Similarly, an irreducible module is another name for a simple module.
• Absolutely irreducible is a term applied to mean irreducible, even after any finite extension of the field of coefficients. It applies in various situations, for example to irreducibility of a linear representation, or of an algebraic variety; where it means just the same as irreducible over an algebraic closure.
• In commutative algebra, a commutative ring R is irreducible if its prime spectrum, that is, the topological space Spec R, is an irreducible topological space.
• A matrix is irreducible if it is not similar via a permutation to a block upper triangular matrix (that has more than one block of positive size). (Replacing non-zero entries in the matrix by one, and viewing the matrix as the adjacency matrix of a directed graph, the matrix is irreducible if and only if such directed graph is strongly connected.)
• Also, a Markov chain is irreducible if there is a non-zero probability of transitioning (even if in more than one step) from any state to any other state.
• In the theory of manifolds, an n-manifold is irreducible if any embedded (n − 1)-sphere bounds an embedded n-ball. Implicit in this definition is the use of a suitable category, such as the category of differentiable manifolds or the category of piecewise-linear manifolds. The notions of irreducibility in algebra and manifold theory are related. An n-manifold is called prime, if it cannot be written as a connected sum of two n-manifolds (neither of which is an n-sphere). An irreducible manifold is thus prime, although the converse does not hold. From an algebraist's perspective, prime manifolds should be called "irreducible"; however, the topologist (in particular the 3-manifold topologist) finds the definition above more useful. The only compact, connected 3-manifolds that are prime but not irreducible are the trivial 2-sphere bundle over S and the twisted 2-sphere bundle over S. See, for example, Prime decomposition (3-manifold).
• A topological space is irreducible if it is not the union of two proper closed subsets. This notion is used in algebraic geometry, where spaces are equipped with the Zariski topology; it is not of much significance for Hausdorff spaces. See also irreducible component, algebraic variety.
• In universal algebra, irreducible can refer to the inability to represent an algebraic structure as a composition of simpler structures using a product construction; for example subdirectly irreducible.
• A 3-manifold is P²-irreducible if it is irreducible and contains no 2-sided $\mathbb RP^2$ ( real projective plane).
• An irreducible fraction (or fraction in lowest terms) is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent fraction.

Irreducibility is the philosophical principle that a complete account of an entity is not be possible at lower levels of explanation.

Irreducibility may also refer to:

• Biological irreducibility, a creationist objection to evolution
• Irreducibility (mathematics), a concept in mathematics

The principle of Irreducibility, in philosophy, has the sense that a complete account of an entity will not be possible at lower levels of explanation and which has novel properties beyond prediction and explanation. 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. 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.