International Encyclopedia of Systems and Cybernetics

A property not affected by some specified transformation.

The following classes of invariance can be distinguished:

- Shift invariance, by displacement of the item in conceptual space, as for example the group of permutations in arithmetics.

- Scale invariance, similarity at different scales, i.e. homomorphisms; fractals

- Time invariance, as for example regular periodicities.

- Symmetry invariance, through the rotation of some perfectly symmetrical figure as in geometry.

Conceptual invariance could be added, as for example isomorphisms.

B. WALLISER extends the concept to more global qualitative invariance, which maintains the basic characteristics of a system. He gives the genetic code in an individual as an example (1977, p.56).

An invariance can be external to the system: something which exists independently of it.

It can conversely exist or be constructed within the system.

This concept, translated into psychology, is a cornerstone of constructivism.

E.von GLASERSFELD writes: "A rule, no matter what it says or does, posits an actual or projecting regularity in our experience, something that is experienced more than once, something that is repeated and thus in some sense an invariance" (1976, p.115).

J. PIAGET showed how children progressively construct their perception of permanent objects (1970b) and later on, permanent concepts (1970a), i.e. how we acquire invariances, which become the basis of our mental and conceptual organizational closure.

In von GLASERSFELD words: "… it is the active organism that constructs an invariant item out of two or more experiences by holding on to certain parts of the experience and discarding others" (Ibid).

As explained by S. KATZ, the constructivist view is related to the neural computer analogy: "… (complex) invariances can be established in neural computers by means of weighted summations of neural currents arising from the activities of many endings. Hence, the invariances emerge as a result of neural computation and are not simply identical to, or transformations of, structures presented ready made to the nervous system. They are, in Gestalt terms, "wholes" that are more than the sum of their "parts", more than the innumerable local effects of sensory endings" (1976, p.44).