Today's key features of logical positivism (or logical empiricism; see also constructive empiricism), as originally created by A. Comte (19th century) and later adapted and corrected by Karl Popper, are:
1. A focus on science as a product, a linguistic or numerical set of statements;
2. A concern with axiomatization, that is, with demonstrating the logical structure and coherence of these statements (Göedel's 1921 and 1951 demonstrations of the essential insufficiency of many axiomatic systems, have largely reshaped and structured this vision);
3. An insistence on at least some of these statements being testable, that is amenable to being verified, confirmed, or falsified by the empirical observation of reality; statements that would, by their nature, be regarded as untestable included the teleological; (Thus positivism rejects much of classical metaphysics.)
4. The belief that science is markedly cumulative;
5. The belief that science is predominantly transcultural;
6. The belief that science rests on specific results that are dissociated from the personality and social position of the investigator;
7. The belief that science contains theories or research traditions that are largely commensurable;
8. The belief that science sometimes incorporates new ideas that are discontinuous from old ones;
9. The belief that science involves the idea of the unity of science, that there is, underlying the various scientific disciplines, basically one science about one real world;
10. The belief that "all true knowledge is scientific"[14];
11. The belief that all things are ultimately measurable;
12. The belief that "entities of one kind... are reducible to entities of another,"[14] such as societies to numbers, or mental events to chemical events (reductionism).
What's new
Major progress over this picture came, at the end of 20th century, from the science (i.e. mathematics) of complexity. It is now clear that the scaling, up or down, of a phenomenum usually produces new laws, that essentially account for new, qualitatively different, phenomena. This essencially challenges the 12th point, above.
In this sense, although macro-processes can, indeed, be "reducible to physiological, physical or chemical events,"[14] and "social processes are reducible to relationships between and actions of individuals,"[14] or "biological organisms are reducible to physical systems"[14] . It is no longer believed that ALL laws of the former phenomena can be tracked back or inferred up, from the later. In a parallel to Goedel's finding, about the incompleteness of most axiomatic mathematical systems, there is now a perception of an essencial insufficiency of micro laws, to explain macro phenomena.
Simple programs, for instance, are capable of a remarkable range of complex behavior. Some have been proven to be universal computers, others exhibit properties familiar from traditional science, such as thermodynamic behavior, continuum behavior, conserved quantities, percolation, sensitive dependence on initial conditions, and others. They have been used as models of traffic, material fracture, crystal growth, biological growth, and various sociological, geological, and ecological phenomena.
Stephen Wolfram, in A New Kind of Science argues that, in order to capture the essence of almost any complex system it is necessary to systematically explore these systems and document what they do. He believes this study should become a new branch of science, like physics or chemistry. The basic goal of this field is to understand and characterize the computational universe using experimental methods.
The proposed new branch of scientific exploration admits many different forms of scientific production. For instance, qualitative classifications like those found in biology are often the results of initial forays into the computational jungle. On the other hand, explicit proofs that certain systems compute this or that function are also admissible. There are also some forms of production that are in some ways unique to this field of study. For instance, the discovery of computational mechanisms that emerge in different systems but in bizarrely different forms.
What's wrong
As of the first decade of the 21st century, the main challenge posed to Positivism (by its own ranks; "metaphisical" and teleological claims being, naturally, disqualified a priori) is the emergence of unsolved paradoxes from within seemingly "well-constructed" theories. Namely, Quantum Phisics and Bayesian Statistics result in disturbing, logic-defying results, that have generated a lot of havoc and schism within the positivist community.
