Science in an inquiry into the world around us fundamentally based on empirical data and the development of models/theories to describe the patterns we find in this data. For knowledge to be considered scientific it is dependent on fulfilling a number of fundamental requirements, firstly it must correlate with the relevant empirical data repeatedly and consistently over time. Secondly, it must work within some formal framework based on a coherent logic that can abstract way the particular details of the phenomena to describe unknown or future events.

Functioning as a reciprocal process through which on the one hand observation of phenomena within our environment are aggregated, synthesized and abstracted to create hypothesizes. While on the other hand theoretical models derived from formal languages such as mathematics are applied as mechanisms for integrating empirical data into coherent frameworks of understanding.

Thus as opposed to the deductive knowledge of the formal languages of mathematics and logic, that are solely dependent on internal consistency, the scientific method of inquiry is productive in occupying and finding solutions to this uneasy tension between empirical data from the world around us with all its richness of diversity and the theoretical world of the formal languages.

Once we have distinguished this type of inquiry called science the question is how do we approach it what are the fundamental tools, concepts or ways of proceeding, that will be used as a base, in other words, what is the paradigm that will be used. An important question as to simply assume the neutrality of our basic paradigm is a form of naivety.
The world around us appears inherently complex in that there does not appear to be any trivial solution to describing the myriad of observable phenomena. Probably the oldest method for dealing with this complexity is called reductionism otherwise known as divide and conquer, an approach to solving problems by recursively dividing or breaking them down into small parts until they are simple enough to be solved and then recombined them to gain an understanding of the system as a whole.
Reductionism provides a detailed description of the world by creating a hierarchy where an entity can be fully described by its constituent parts one level lower than it, this hierarchy continues down until reaching irreducible elements (atoms in the original sense of the word), with these elements exhibiting a linear deterministic functionality and thus any higher level non-linear or non-deterministic behavior should be theoretically reducible to the combination of these linear atomic elements and the pre-determined causation between them.
Following on from its initial success in classical physics reductionism has proven itself a highly successful method in many ways to become foundational to mainstream science. After centuries of following a more or less coherent trajectory and employing a large community of scientist building upon previous discoveries it now constitutes a large and formidable body of knowledge. With recent discoveries into the origins and development of the universe presenting a somewhat unified picture of the world around us that occupies a central position in the collective conscious of industrial and post-industrial societies whiles also providing a throughput of technical knowledge required in supporting rapid technological growth.

As compelling as this approach to science may seem its failures exist, probably most fundamental to these is the emphasis of reductionism upon the structure or order of a system as being defined solely by its constituent parts which fails to capture how novel patterns of order can emerge, such as on the level of biological systems or within society, that can only be properly understood within their own context. And thus it appears that the more advanced or complex a system becomes the less amenable to the analytical reductionist approach of inquiry it presents its self. As can be seen in the failure to establish a robust social science based on the reductionist logic, leaving a strong disconnect between the ‘soft’ social sciences and the ‘hard’ natural sciences, resulting in a body of knowledge much more advanced in its understanding of the natural environment as opposed to the social world.
The current tensions surrounding synthetic biology may be seen to express this, where our technical capability to manipulate biological systems far exceeds our ethical frameworks for dealing with its cultural implications.
This is where complexity science tries to add another dimension to the mainstream scientific paradigm. Through such concepts as emergence and self-organization complexity science includes the possibility of qualitative levels, where new parameters are defined, essentially tipping points or phase changes within the development of a system beyond which different rules define its makeup, such as the transition from inorganic molecules to living cells where the order that defines biological systems is not simply an extension or derivative of the laws of physics. Many more examples of these irreducible phase changes existing within neuroscience and the social sciences.
Another area that has presented difficulties for traditional science has been non-linear systems, with reductionism resorting to linear approximations to deal with them, although this may work well in many cases when systems become more complex, that is increasingly interconnected, interdependent and non-linear, these models may well fail to capture important information. An example of this might be seen in economics where traditional economics employs much of the vocabulary and toolset of the natural sciences to create reductionist models based on how rational individuals should respond to incentives in a linear fashion, though these models often fail to capture critical information that can result in unpredicted large scale systemic shocks.

Complexity science has arguably proven itself most productive and relevant when dealing with non-linear systems, in particular what are called complex adaptive systems where non-linearity is largely the product of an ‘if/then’ logic possessed by the individual elements within the system, with the structure and order of the system as a whole deriving from the emergent patterns created by the interaction between the parts. This approach, called Agent-Based Modeling is an active area of research within many areas of social science from management and organization studies to economics and sociology, when combined with the capability of current computation it represents a powerful theoretical and practical framework for the social sciences.

The emphases on reductionism within traditional science, as opposed to holism within complexity science, may present itself as diametrically opposed but to think about science as fundamentally being an inquiry into the world around us, both physical and social, requires a broader framework than that presented by reductionist reasoning. Although reductionism may be a powerful mechanism within many domains of science the world is inherently complex and multidimensional requiring, in turn, a multi-dimensional and adaptive framework. The real challenge is not in developing one or the other, reductionism or holism, but the integration of the two to provide both a detailed analytical perspective and a broader vision for integrating this into a deeper understanding of the world we live in.