out the instructions of its program deductively. Accordingly, Gödel’s incompleteness theorems for formal arithmetic apply to computer E. It follows from these theorems that no program can enable computer E to decide of an arbitrary statement of L whether or not that statement is true. More strongly, there cannot even be a program that will enable E to enumerate the truths of language L one after another. Therefore Leibniz’s characteristica universalis cannot exist. Electronic computers are the first active or ‘live’ mathematical systems. They are the latest addition to a long historical series of mathematical tools for inquiry: geometry, algebra, calculus and differential equations, probability and statistics, and modern mathematics.
The most effective use of computer programs is to instruct computers in tasks for which they are superior to humans. Computers are being designed and programmed to cooperate with humans so that the calculation, storage, and judgment capabilities of the two are synthesized. The powers of such human–computer combines will increase at an exponential rate as computers continue to become faster, more powerful, and easier to use, while at the same time becoming smaller and cheaper. The social implications of this are very important.
The modern electronic computer is a new tool for the logic of discovery (Peirce’s abduction). An inquirer (or inquirers) operating a computer interactively can use it as a universal simulator, dynamically modeling systems that are too complex to study by traditional mathematical methods, including non-linear systems. Simulation is used to explain known empirical results, and also to develop new hypotheses to be tested by observation. Computer models and simulations are unique in several ways: complexity, dynamism, controllability, and visual presentability. These properties make them important new tools for modeling and thereby relevant to some important philosophical problems.
A human–computer combine is especially suited for the study of complex holistic and hierarchical systems with feedback (cf. cybernetics), including adaptive goal-directed systems. A hierarchical-feedback system is a dynamic structure organized into several levels, with the compounds of one level being the atoms or building blocks of the next higher level, and with cyclic paths of influence operating both on and between levels. For example, a complex human institution has several levels, and the people in it are themselves hierarchical organizations of selfcopying chemicals, cells, organs, and such systems as the pulmonary and the central nervous system.
The behaviors of these systems are in general much more complex than, e.g., the behaviors of traditional systems of mechanics. Contrast an organism, society, or ecology with our planetary system as characterized by Kepler and Newton. Simple formulas (ellipses) describe the orbits of the planets. More basically, the planetary system is stable in the sense that a small perturbation of it produces a relatively small variation in its subsequent history. In contrast, a small change in the state of a holistic hierarchical feedback system often amplifies into a very large difference in behavior, a concern of chaos theory. For this reason it is helpful to model such systems on a computer and run sample histories. The operator searches for representative cases, interesting phenomena, and general principles of operation. The human–computer method of inquiry should be a useful tool for the study of biological evolution, the actual historical development of complex adaptive goal-directed systems. Evolution is a logical and communication process as well as a physical and chemical process. But evolution is statistical rather than deterministic, because a single temporal state of the system results in a probabilistic distribution of histories, rather than in a single history. The genetic operators of mutation and crossover, e.g., are probabilistic operators. But though it is stochastic, evolution cannot be understood in terms of limiting relative frequencies, for the important developments are the repeated emergence of new phenomena, and there may be no evolutionary convergence toward a final state or limit. Rather, to understand evolution the investigator must simulate the statistical spectra of histories covering critical stages of the process. Many important evolutionary phenomena should be studied by using simulation along with observation and experiment. Evolution has produced a succession of levels of organization: selfcopying chemicals, self-reproducing cells, communities of cells, simple organisms, haploid sexual reproduction, diploid sexuality with genetic dominance and recessiveness, organisms composed of organs, societies of organisms, humans, and societies of humans. Most of these systems are complex hierarchical feedback systems, and it is of interest to understand how they emerged from earlier systems. Also, the interaction of competition and cooperation at all stages of evolution is an important subject, of relevance to social philosophy and ethics. Some basic epistemological and metaphysical concepts enter into computer modeling. A model is a well-developed concept of its object, representing characteristics like structure and function. A model is similar to its object in important respects, but simpler; in mathematical terminology, a model is homomorphic to its object but not isomorphic to it. However, it is often useful to think of a model as isomorphic to an embedded subsystem of the system it models. For example, a gas is a complicated system of microstates of particles, but these microstates can be grouped into macrostates, each with a pressure, volume, and temperature satisfying the gas law PV % kT. The derivation of this law from the detailed mechanics of the gas is a reduction of the embedded subsystem to the underlying system. In many cases it is adequate to work with the simpler embedded subsystem, but in other cases one must work with the more complex but complete underlying system.
The law of an embedded subsystem may be different in kind from the law of the underlying system. Consider, e.g., a machine tossing a coin randomly. The sequence of tosses obeys a simple probability law, while the complex underlying mechanical system is deterministic. The random sequence of tosses is a probabilistic system embedded in a deterministic system, and a mathematical account of this embedding relation constitutes a reduction of the probabilistic system to a deterministic system. Compare the compatibilist’s claim that free choice can be embedded in a deterministic system. Compare also a pseudorandom sequence, which is a deterministic sequence with adequate randomness for a given (finite) simulation. Note finally that the probabilistic system of quantum mechanics underlies the deterministic system of mechanics.
The ways in which models are used by goaldirected systems to solve problems and adapt to their environments are currently being modeled by human–computer combines. Since computer software can be converted into hardware, successful simulations of adaptive uses of models could be incorporated into the design of a robot. Human intentionality involves the use of a model of oneself in relation to others and the environment. A problem-solving robot using such a model would constitute an important step toward a robot with full human powers.
These considerations lead to the central thesis of the philosophy of logical mechanism: a finite deterministic automaton can perform all human functions. This seems plausible in principle (and is treated in detail in Merrilee Salmon, ed., The Philosophy of Logical Mechanism: Essays in Honor of Arthur W. Burks, 1990). A digital computer has reasoning and memory powers. Robots have sensory inputs for collecting information from the environment, and they have moving and acting devices. To obtain a robot with human powers, one would need to put these abilities under the direction of a system of desires, purposes, and goals. Logical mechanism is a form of mechanism or materialism, but differs from traditional forms of these doctrines in its reliance on the logical powers of computers and the logical nature of evolution and its products. The modern computer is a kind of complex hierarchical physical system, a system with memory, processor, and control that employs a hierarchy of programming languages. Humans are complex hierarchical systems designed by evolution – with structural levels of chemicals, cells, organs, and systems (e.g., circulatory, neural, immune) and linguistic levels of genes, enzymes, neural signals, and immune recognition. Traditional materialists did not have this model of a computer nor the contemporary understanding of evolution, and never gave an adequate account of logic and reasoning and such phenomena as goaldirectedness and self-modeling. See also ARTIFICIAL INTELLIGENCE , CYBER- NETICS , DETERMINISM , GÖDEL’ S INCOMPLETE – NESS THEOREMS , SELF – REPRODUCING AUTOM – ATON , TURING MACHIN. A.W.B.