Reductionism is an approach to building descriptions of systems out of the descriptions of the subsystems that a system is composed of, and ignoring the relationships between them.
For example, considering a biological system to be composed of molecules with certain structures, or considering a molecule to be composed of atoms.
Note that once positions of atoms in a molecule are specified, relationships between the atoms are also specified. Similarly, once the relative locations of molecules are specified in a biological organism, relationships between these molecules are also specified. Taking into account such relationships is beyond a purely reductionist approach.
However, it is also considered "reductionist" to consider specifying the positions of atoms of a biological organism a specification of the organism. Some consider this reductionist because it is difficult to obtain macroscopically relevant information from this approach, or because this approach is not practical, others because of the notion of strong emergence, that there is more to a system than the specification of parts and their relationships.
The mathematics of reductionism is, at its most basic, the use of fractions which holds that dividing and multiplying are opposites. The statement 2(1/2)=1 is the idea that dividing a system into two parts and then putting them back together restores the original system. This is valid for systems where only the total mass of a system matters, such as for weights, or for collections of small weakly interacting particles, like a pound of flour. The existence of many systems that can be effectively described in this way is important for the study of complex systems as it provides one aspect of a contrasting notion of a simple system. However, it is not generally true.
Related concepts: complexity at different scales, complexity profile, information, algorithmic complexity, emergence and complexity, emergence
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