E. Jackson

Paper #: 95-01-001

During the period of approximately 1570-1790 the first metamorphosis of science transformed the operational foundations of science, that were largely the heritage from the time of Aristotle, into its modern form. These new foundations consisted of the use of (1) Physical Experiments and the use of (2) Mathematical Models, involving differential equations. This metamorphosis was largely due to Brahe, Kepler, Galileo, Newton, Leibniz, Euler, and Lagrange. These operational methods were accompanied by the development of several philosophical attitudes and beliefs. One attitude (an implicit but operative frame of mind) arose from the loss of concern about the limitations of the encoding and decoding of information between experiments and mathematics; that is, an increased identification of the physical world with the mathematical models used to make very limited predictions of that world. The more overt philosophical beliefs related to the fundamental character of reductionistic knowledge, thereby obtaining a theory of the universe, capable of predicting all phenomena in nature. The idea that there is indeed any such thing as one set of laws which govern the behavior of the universe has its origins in antiquity, where the laws often referred to the desires of a god. This belief was immeasurably reinforced within Science by the success of Newton's “universal law” of gravitation. The blending of these persuasions was perhaps captured best, intended or not, in Einstein's famous remark, “I shall never believe that God plays dice with the world.” Over the past century the character and structure of science has been going through a second process of fundamental change which has been brought about by two classes of discoveries. The first group of discoveries concern mathematical results which are directly related to the limitations in what we can learn about the dynamic behaviors in nature from these mathematical models. The discoveries of these limitations should have produced a “loss-of-innocence” era, but they have largely gone unnoticed by scientists, even though they have profound implications concerning the future character of science. Among these discoveries, which apply to essentially all mathematical models of physical systems, are our inherent limitations to make: (1) analytic mathematical deductions, (2) deterministic physical predictions, and (3) structurally stable models of closed systems. In addition, other mathematical discoveries were made which struck at the basic idea that mathematical systems are consistent, and can establish any result which is true. Each of these discoveries have, or will have, a direct impact on the philosophical preconceptions of many scientists, since they shatter widely held beliefs about our ability to extract (deduce) information about nature only through the use of mathematical modeling and analysis, together with physical experiments. Physical experiments also stimulated other mathematical studies that further confirmed this lack of physical predictability. These results establish that our knowledge about the behavior of complex systems is limited to “comprehensible” sets of observables (windows of comprehension), that involved “bounded reducibility,” and also establish that we are only capable of bounded predictability of physical events (proportional to our bounded information about the state of the system). In the second half of this century a totally different class of opportunities and discoveries have been made possible by a new operational basis for scientific investigations, due to the digital computer. This expanded the operational basis of (1) Physical Experiments, and (2) Mathematical Models, established during the first metamorphosis, to include the third operational basis for obtaining knowledge about nature, that will be referred to as (3) Computer Experiments. Operations carried out in each of these three areas can yield independent knowledge relevant to our understanding of nature, and can interface directly with either of the other two areas. In particular, the lost deductive dreams concerning mathematical models, the information encoding-decoding limitations that exist between physical experiments and mathematical solutions, and the bifurcation processes of dynamic models, have already been clarified and extended by Computer Experiments. Some of the numerous potentials for this interplay between the three operational methods is discussed below. For example, information about the dynamics of physical systems can be achieved using the interface between Computer Experiments and Physical Experiments, which does not rely on any mathematical modeling (or algorithms) related to the observables of the physical system. Moreover, using only this experimental information, limited predictions can be made without any dynamic models. Also certain types of local and global mathematical models can be sought, which is a simple form of computer/human induction. Computer Experiments also allow for the search of dynamics (algorithms) which scientists have not been capable of imagining in the inductive dreams; a more profound form of the computer inductive process is related to ongoing developments of computer ‘genetic’ dynamics. Computer Experiments can graphically represent incomprehensible data sets, and can search for ‘emergent’ properties of complex systems. Also there is a branch of research that is exploring fundamental issues related to the possible finite-informational character of all dynamic phenomena, and its natural association with both reconstruction methods and Computer Experiments. With the recognition of deductive limitations and the expanded operational foundation, the general character of science is going though a process of metamorphosis which is both exciting in its richness, and impossible to presently define with any precision. Some aspects of this metamorphosis, and their bearing on widely held philosophical dreams of scientists, will be discussed.