Until the early 20th century, classical mechanics, as first formulated by Newton and further developed by Laplace and others, was seen as the foundation for science as a whole. It was expected that the observations made by other sciences would sooner or later be reduced to the laws of mechanics. Although that never happened, other disciplines, such as biology, psychology or economics, did adopt a general mechanistic or Newtonian methodology and world view. This influence was so great, that most people with a basic notion of science still implicitly equate "scientific thinking" with "Newtonian thinking".
The reason for this pervasive influence is that the mechanistic paradigm is compelling by its simplicity, coherence and apparent completeness. Moreover, it was not only very successful in its scientific applications, but largely in agreement with intuition and common-sense. Later theories of mechanics, such as relativity theory and quantum mechanics, while at least as successful in the realm of applications, lacked this simplicity and intuitive appeal, and are still plagued by paradoxes, confusions and multiple interpretations.
The logic behind Newtonian science is easy to formulate, although its implications are subtle. Its best known principle, which was formulated by the philosopher-scientist Descartes well before Newton, is that of analysis or reductionism: to understand any complex phenomenon, you need to take it apart, i.e. reduce it to its individual components. If these are still complex, you need to take your analysis one step further, and look at their components.
If you continue this subdivision long enough, you will end up with the smallest possible parts, the atoms (in the original meaning of "indivisibles"), or what we would now call "elementary particles". Particles can be seen as separate pieces of the same hard, permanent substance that is called matter. Newtonian ontology therefore is materialistic: it assumes that all phenomena, whether physical, biological, mental or social, are ultimately constituted of matter.
The only property that fundamentally distinguishes particles is their position in space (which may include dimensions other than the conventional three). Apparently different substances, systems or phenomena are merely different arrangements in space of fundamentally equivalent pieces of matter. Any change, development or evolution is therefore merely a geometrical rearrangement caused by the movement of the components. This movement is governed by deterministic laws of cause and effect. If you know the initial positions and velocities of the particles constituting a system together with the forces acting on those particles (which are themselves determined by the positions of these and other particles), then you can in principle predict the further evolution of the system with complete certainty and accuracy. The trajectory of the system is not only determined towards the future, but towards the past: given its present state, you can in principle reverse the evolution to reconstruct any earlier state it has gone through.
The elements of the Newtonian ontology are matter, the absolute space and time in which that matter moves, and the forces or natural laws that govern movement. No other fundamental categories of being, such as mind, life, organization or purpose, are acknowledged. They are at most to be seen as epiphenomena, as particular arrangements of particles in space and time.
Newtonian epistemology is based on the reflection-correspondence view of knowledge: our knowledge is merely an (imperfect) reflection of the particular arrangements of matter outside of us. The task of science is to make the mapping or correspondence between the external, material objects and the internal, cognitive elements (concepts or symbols) that represent them as accurate as possible. That can be achieved by simple observation, where information about external phenomena is collected and registered, thus further completing the internal picture that is taking shape. In the limit, this should lead to a perfect, objective representation of the world outside us, which would allow us to accurately predict all phenomena.
All these different assumptions can summarized by the principle of distinction conservation (Heylighen, 1990): classical science begins by making as precise as possible distinctions between the different components, properties and states of the system under observation. These distinctions are assumed to be absolute and objective, i.e. the same for all observers. The evolution of the system conserves all these distinctions, as distinct initial states are necessarily mapped onto distinct subsequent states, and vice-versa (this is equivalent to the principle of causality (Heylighen, 1989)). In particular, distinct entities (particles) remain distinct: there is no way for particles to merge, divide, appear or disappear. In other words, in the Newtonian world view there is no place for novelty or creation (Prigogine & Stengers, 1984): everything that exists now has existed from the beginning of time and will continue to exist, albeit in a somewhat different configuration. Knowledge is nothing more than another such distinction-conserving mapping from object to subject: scientific discovery is not a creative process, it is merely an "uncovering" of distinctions that were waiting to be observed.
In essence, the philosophy of Newtonian science is one of simplicity: the complexity of the world is only apparent; to deal with it you need to analyse phenomena into their simplest components. Once you have done that, their evolution will turn out to be perfectly regular, reversible and predictable, while the knowledge you gained will merely be a reflection of that pre-existing order.
Rationality and modernity
Up to this point, Newtonian logic is perfectly consistent--albeit simplistic in retrospect. But if we moreover want to include human agency, we come to a basic contradiction between our intuitive notion of free will and the principle of determinism. The only way Newtonian reasoning can be extended to encompass the idea that people can act purposefully is by postulating the independent category of mind. This reasoning led Descartes to propose the philosophy of dualism, which assumes that while material objects obey mechanical laws, the mind does not. However, while we can easily conceive the mind as a passive receptacle registering observations in order to develop ever more complete knowledge, we cannot explain how the mind can freely act upon those systems without contradicting the determinism of natural law. This explains why classical science ignores all issues of ethics or values: there simply is no place for purposeful action in the Newtonian world view.
At best, economic science has managed to avoid the problem by postulating the principle of rational choice, which assumes that an agent will always choose the option that maximises its utility. Utility is supposed to be an objective measure of the degree of value, "happiness" or "goodness" produced by a state of affairs. Assuming perfect information about the utility of the possible options, the actions of mind then become as determined or predictable as the movements of matter. This allowed social scientists to describe human agency with most of the Newtonian principles intact. Moreover, it led them to a notion of linear progress: the continuous increase in global utility (seen mostly as quantifiable, material welfare) made possible by increases in scientific knowledge. Although such directed change towards the greater good contradicts the Newtonian assumption of reversibility, it maintains the basic assumptions of determinism, materialism and objective knowledge, thus defining what is often called the project of modernity.
The purpose of the Principia Cybernetica Project can be summarized as the development of a world view that is as clear and coherent as the Newtonian one, but that is based on the notion of system or whole, creative evolution, and intelligent action.