Computation and the Philosophy of Science
Paul Thagard
Philosophy Department
University of Waterloo
Waterloo, Ontario, N2L 3G1

What do philosophers do? Twenty years ago, one might have heard such answers to this question as "analyze concepts" or "evaluate arguments". The answer "write computer programs" would have inspired a blank stare, and even a decade ago I wrote that computational philosophy of science might sound like the most self-contradictory enterprise in philosophy since business ethics (Thagard 1988). But computer use has since become much more common in philosophy, and computational modeling can be seen as a useful addition to philosophical method, not as the abandonment of it. I will try in this paper to summarize how computational models are making substantial contributions to the philosophy of science.
If philosophy consisted primarily of conceptual analysis, or mental self-examination, or generation of a priori truths, then computer modeling would indeed be alien to the enterprise. But I prefer a different picture of philosophy, as primarily concerned with producing and evaluating theories, for example theories of knowledge (epistemology), reality (metaphysics), and right and wrong (ethics). The primary function of a theory of knowledge is to explain how knowledge grows, which requires both describing the structure of knowledge and the inferential procedures by which knowledge can be increased. Although epistemologists often focus on mundane knowledge, the most impressive knowledge gained by human beings comes through the operation of science: experimentation, systematic observation, and theorizing concerning the experimental and observational results. Hence at the core of epistemology is the need to understand the structure and growth of scientific knowledge, a project for which computational models can be very useful.
In attempting to understand the structure and development of scientific knowledge, philosophers of science have traditionally employed a number of methods such as logical analysis and historical case studies. Computational modeling provides an additional method that has already advanced understanding of such traditional problems in the philosophy of science as theory evaluation and scientific discovery. This paper will review the progress made on such issues by three distinct computational approaches: cognitive modeling, engineering artificial intelligence, and theory of computation.
The aim of cognitive modeling is to simulate aspects of human thinking; for philosophy of science, this becomes the aim to simulate the thinking that scientists use in the construction and evaluation of hypotheses. Much artificial intelligence research, however, is not concerned with modeling human thinking, but with constructing algorithms that perform well on difficult tasks independently of whether the algorithms correspond to human thinking. Similarly, the engineering AI approach to philosophy of science seeks to develop computational models of discovery and evaluation independently of questions of human psychology. Computational philosophy of science has thus developed two streams that reflect the two streams in artificial intelligence research, one concerned with modeling human performance and the other with machine intelligence. A third stream of research uses abstract mathematical analysis and applies the theory of computation to problems in the philosophy of science.

1. Cognitive Modeling
Cognitive science is the interdisciplinary study of mind, embracing philosophy, psychology, artificial intelligence, neuroscience, linguistics, and anthropology. From its modern origins in the 1950s, cognitive science has primarily worked with the computational-representational understanding of mind: we can understand human thinking by postulating mental representations akin to computational data structures and mental procedures akin to algorithms (Thagard 1996). The cognitive-modeling stream of computational philosophy of science views topics such as discovery and evaluation as open to investigation using the same techniques employed in cognitive science. To understand how scientists discover and evaluate hypotheses, we can develop computer models that employ data structures and algorithms intended to be analogous to human mental representations and procedures. The cognitive modeling stream of computational philosophy of science can be viewed as part of naturalistic epistemology, which sees the study of knowledge as closely tied to human psychology, not as an abstract logical exercise.
In the 1960s and 1970s, philosophers of science discussed whether there is a "logic of discovery" and whether discovery (as opposed to evaluation) is a legitimate topic of philosophical (as opposed to psychological) investigation. In the 1980s, these debates were superseded by computational research on discovery that showed how actual cases of scientific discovery can be modeled algorithmically. Although the models that have been produced to date clearly fall well short of simulating all the thought processes of creative scientists, they provide substantial insights into how scientific thinking can be viewed computationally.
Because of the enormous number of possible solutions involved in any scientific problem, the algorithms involved in scientific discovery cannot guarantee that optimal discoveries will be made from input provided. Instead, computer models of discovery employ heuristics, approximate methods for attempting to cut through data complexity and find patterns. The pioneering step in this direction was the BACON project of Pat Langley, Herbert Simon and their colleagues (Langley et al. 1987). BACON is a program that uses heuristics to discover mathematical laws from quantitative data, for example discovering Kepler's third law of planetary motion. Although BACON has been criticized for assuming an over-simple account of human thinking, Qin and Simon (1990) found that human subjects could generate laws from numerical data in ways quite similar to BACON.
Scientific discovery produces qualitative as well as quantitative laws. Kulkarni and Simon (1988) produced a computational model of Krebs' discovery of the urea cycle. Their program, KEKADA, reacts to anomalies, formulates explanations, and carries out simulated experiments in much the way described in Hans Krebs laboratory notebooks.
Not all scientific discoveries are as data-driven as the ones so far discussed. They often involve the generation of new concepts and hypotheses that are intended to refer to non-observable entities. Thagard (1988) developed computational models of conceptual combination, in which new theoretical concepts such as sound wave are generated, and of abduction, in which new hypotheses are generated to explain puzzling phenomena. Darden (1990, this volume) has investigated computationally how theories that have empirical problems can be repaired.
One of the most important cognitive mechanisms for discovery is analogy, since scientists often make discoveries by adapting previous knowledge to a new problem. Analogy played a role in some of the most important discoveries ever made, such as Darwin's theory of evolution and Maxwell's theory of electromagnetism. During the 1980s, the study of analogy went well beyond previous philosophical accounts through the development of powerful computational models of how analogs are retrieved from memory and mapped to current problems to provide solutions. Falkenhainer, Forbus, and Gentner (1989) produced SME, the Structure Mapping Engine, and this program was used to model analogical explanations of evaporation and osmosis (Falkenhainer 1990). Holyoak and Thagard (1989) used different computational methods to produce ACME, the Analogical Constraint Mapping Engine, which was generalized into a theory of analogical thinking that applies to scientific as well as everyday thinking (Holyoak and Thagard 1995).
Space does not permit further discussion of computational models of human discovery, but the above research projects illustrate how thought processes such as those involved in numerical law generation, theoretical concept formation, and analogy can be understood computationally. Examples of non-psychological investigations of scientific discovery are described in section 2 and 3.
How scientific hypotheses are evaluated has been a central problem in philosophy of science since the nineteenth century debates between John Stuart Mill and William Whewell. Work in the logical positivist tradition has centered on the concept of confirmation, asking what it is for hypotheses to be confirmed by observations. More recently, various philosophers of science have taken a Bayesian approach to hypothesis evaluation, using probability theory to analyze scientific reasoning. In contrast, I have developed an approach to hypothesis evaluation that combines philosophical ideas about explanatory coherence with a connectionist (neural network) computational model.
Coherence theories of knowledge, ethics, and even truth have been popular among philosophers, but the notion of coherence is usually left rather vague. Hence coherence theories look unrigorous compared to theories couched more formally using deductive logic or probability theory. But connectionist models show how coherence ideas can be precisely and efficiently implemented. Since the mid-1980s, connectionist (neural network, PDP) models have been very influential in cognitive science. Loosely analogous to the operation of the brain, such models have numerous units that are roughly like neurons, connected to each other by excitatory and inhibitory links of varying strengths. Each unit has an activation value that is affected by the activations of the units to which it is linked, and learning algorithms are available for adjusting the strengths on links in response to experience.
My connectionist computational model of explanatory coherence, ECHO, uses units to represent propositions that can be hypotheses or descriptions of evidence, and links between units to represent coherence relations. For example, if a hypothesis explains a piece of evidence, then ECHO places an excitatory link between the unit representing the hypothesis and the unit representing the evidence. If two hypotheses are contradictory or competing, then ECHO places an inhibitory link between the units representing the two hypotheses. Repeatedly adjusting the activations of the units based on their links with other units results in a resting state in which some units are on (hypotheses accepted) and other units are off (hypotheses rejected). ECHO has been used to model many important cases in the history of science (Nowak and Thagard 1992a, 1992b; Thagard 1991, 1992, in press). Eliasmith and Thagard (in press) argue that ECHO provides a better account of hypothesis evaluation than available Bayesian accounts.
A different connectionist account of inference to best explanation is given by Churchland (1989). He conjectures that abductive discovery and inference to the best explanation can both be understood in terms of prototype activation in distributed connectionist models, i. e. ones in which concepts and hypotheses are not represented by individual units but by patterns of activation across multiple units. There is considerable psychological evidence that distributed representations and prototypes are important in human cognition, but no one has yet produced a running computational model of hypothesis evaluation using these ideas. Non-connectionist models of hypothesis evaluation, including probabilistic ones, are discussed in the next section.

2. Engineering AI
As the references to my own work in the last section indicate, I pursue the cognitive modeling approach to computational philosophy of science, allying philosophy of science with cognitive science and naturalistic epistemology. But much valuable work in AI and philosophy has been done that makes no claims to psychological plausibility. One can set out to build a scientist without trying to reverse engineer a human scientist. The engineering AI approach to computational philosophy of science is allied, not with naturalistic, psychologistic epistemology, but with what has been called "android epistemology", the epistemology of machines that may or may not be built like humans (Ford, Glymour, and Hayes 1995). This approach is particularly useful when it exploits such differences between digital computers and humans as computers' capacity for very fast searches to perform tasks that human scientists cannot do very well.

One goal of engineering AI is to produce programs that can make discoveries that have eluded humans. Bruce Buchanan, who was originally trained as a philosopher before moving into AI research, reviewed over a dozen AI programs that formulate hypotheses to explain empirical data (Buchanan 1983). One of the earliest and most impressive programs was DENDRAL which performed chemical analysis. Given spectroscopic data from an unknown organic chemical sample, it determined the molecular structure of the sample (Lindsay et. al. 1980). The program META-DENDRAL pushed the discovery task one step farther back: given a collection of analytic data from a mass spectrometer, it discovered rules explaining the fragmentation behavior of chemical samples. A more recent program for chemical discovery is MECHEM, which automates the task of finding mechanism for chemical reactions: given experimental evidence about a reaction, the program searches for the simplest mechanism consistent with theory and experiment (Valdes-Peres, 1994).
Discovery programs have also been written for problems in biology, physics, and other scientific domains. In order to model biologists' discoveries concerning gene regulation in bacteria, Karp (1990) wrote a pair of programs, GENSIM and HYPGENE. GENSIM was used to represent a theory of bacterial gene regulation, and HYPGENE formulates hypotheses that improve the predictive power of GENSIM theories given experimental data. More recently, he has shifted from modeling historical discoveries to the attempt to write programs that make original discoveries from large scientific databases such as ones containing information about enzymes, proteins, and metabolic pathways (Karp and Mavrovouniotis 1994). Cheeseman (1990) used a program that applied Bayesian probability theory to discover previously unsuspected fine structure in the infrared spectra of stars. Machine learning techniques are also relevant to social science research, particularly the problem of inferring causal models from social data. The TETRAD program looks at statistical data in fields such as industrial development and voting behavior and builds causal models in the form of a directed graph of hypothetical causal relationships (Glymour et al., 1987).
One of the fastest growing areas of artificial intelligence is "data mining", in which machine learning techniques are used to discover regularities in large computer data bases such as the terabytes of image data collected by astronomical surveys (Fayyad, Piatetsky-Shapiro, and Smyth 1996). Data mining is being applied with commercial success by companies that wish to learn more about their operations, and similar machine learning techniques may have applications to large scientific data bases such as those being produced by the human genome project.
The topic of how scientific theories can be evaluated can also be discussed from a computational perspective. Many philosophers of science (e.g. Howson and Urbach 1989) adopt a Bayesian approach to questions of hypothesis evaluation, attempting to use probability theory to describe and prescribe how scientific theories are assessed. But computational investigations of probabilistic reasoning must deal with important problems involving tractability that are usually ignored by philosophers. A full-blown probabilistic approach to a problem of scientific inference would need to establish a full joint distribution of probabilities for all propositions representing hypotheses and evidence, which would require 2n probabilities for n hypotheses, quickly exhausting the storage and processing capacities of any computer. Ingenious methods have been developed by computer scientists to avoid this problem by using causal networks to restrict the number of probabilities required and to simplify the processing involved (Pearl 1988, Neapolitain 1990). Surprisingly such methods have not been explored by probabilistic philosophers of science who have tended to ignore the substantial problem of the intractability of Bayesian algorithms.
Theory evaluation in the context of medical reasoning has been investigated by a group of artificial intelligence researchers at Ohio State University (Josephson and Josephson 1994). They developed a knowledge-based system called RED that uses data concerning a patient's blood sample to infer what red-cell antibodies are present in the patient. RED performs an automated version of inference to the best explanation, using heuristics to form a composite hypothesis concerning what antibodies are present in a sample. Interestingly, Johnson and Chen (1996) compared the performance of RED with the performance of my explanatory coherence program ECHO on a set of 48 cases interpreted by clinical experts. Whereas RED produced the experts' judgments in 58% of the cases, ECHO was successful in 73% of the cases. Hence although the engineering AI approach to scientific discovery has some evident advantages over the cognitive modeling approach in dealing with some problems such as making mining hypotheses from large data bases, the cognitive modeling approach exemplified by ECHO has not yet been surpassed by a probabilistic or other program that surpasses human performance.

3. Theory of Computation
Both the cognitive modeling and engineering AI approaches to philosophy of science involve writing and experimenting with running computer programs. But it is also possible to take a more theoretical approach to computational issues in the philosophy of science, exploiting results in the theory of computation to reach conclusions about processes of discovery and evaluation.
Scientific discovery can be viewed as a problem in formal learning theory, in which the goal is to identify a language given a string of inputs (Gold 1968). Analogously, a scientist can be thought of as a function that takes as input a sequence of formulas representing observations of the environment and produces as output a set of formulas that represent the structure of the world (Kelly 1995, Kelly and Glymour 1989, Osherson and Weinstein 1989). Although formal learning theory has produced some interesting theorems, they are limited in their relevance to the philosophy of science in several respects. Formal learning theory assumes a fixed language and therefore ignores the conceptual and terminological creativity that is important to scientific development. In addition, formal learning theory tends to view hypotheses produced as a function of input data, rather than as a much more complex function of the data and the background concepts and theories possessed by a scientist. Formal learning theory also overemphasizes the goal of science to produce true descriptions, neglecting the important role of explanatory theories and hypothetical entities in scientific progress.
The theory of computational complexity has provided some interesting results concerning hypothesis evaluation. If you have n hypotheses and want to evaluate all the ways in which combinations of them can be accepted and rejected, you have to consider 2n possibilities, an impossibly large number for large n. Bylander et al. (1991) gave a formal definition of an abduction problem consisting of a set of data to be explained and a set of hypotheses to explain them. They then showed that the problem of picking the best explanation, is NP-hard, i.e. it belongs to a class of problems that are generally agreed by computational theorists to be intractable in that the amount of time to compute them increases exponentially as the problems grow in size. Similarly, Thagard and Verbeurgt (1996) generalized explanatory coherence into a mathematical coherence problem that is NP-hard. What these results show is that theory evaluation, whether it is conceived in terms of Bayesian probabilities, heuristic assembly of hypotheses, or explanatory coherence, must handled by computational approximation, not an exhaustive algorithm. So far, the theoretical results concerning scientific evaluation have been largely negative, but they serve to provide outline the limits within which computational modeling must work.

4. What Computation Adds to Philosophy of Science
Almost twenty years ago, Aaron Sloman (1978) published an audacious book, The Computer Revolution in Philosophy, which predicted that within a few years any philosopher not familiar with the main developments of artificial intelligence could fairly be accused of professional incompetence. Since then, computational ideas have had a substantial impact on the philosophy of mind, but a much smaller impact on epistemology and philosophy of science. Why? One reason, I conjecture, is the kind of training that most philosophers have, which includes little preparation for actually doing computational work. Philosophers of mind have often been able to learn enough about artificial intelligence to discuss it, but for epistemology and philosophy of science it is much more useful to perform computations rather than just to talk about them. To conclude this review, I shall attempt to summarize what is gained by adding computational modelling to the philosophical tool kit.
Bringing artificial intelligence into philosophy of science introduces new conceptual resources for dealing with the structure and growth of scientific knowledge. Instead of being restricted to the usual representational schemes based on formal logic and ordinary language, computational approaches to the structure of scientific knowledge can include many useful representations such as prototypical concepts, concept hierarchies, production rules, causal networks, mental images, and so on. Philosophers concerned with the growth of scientific knowledge from a computational perspective can go beyond the narrow resources of inductive logic to consider algorithms for generating numerical laws, discovering causal networks, forming concepts and hypotheses, and evaluating competing explanatory theories.
In addition to the new conceptual resources that AI brings to philosophy of science, it also brings a new methodology involving the construction and testing of computational models. This methodology typically has numerous advantages over pencil-and-paper constructions. First, it requires considerable precision, in that to produce a running program the structures and algorithms postulated as part of scientific cognition need to be specified. Second, getting a program to run provides a test of the feasibility of its assumptions about the structure and processes of scientific development. Contrary to the popular view that clever programmers can get a program to do whatever they want, producing a program that mimics aspects of scientific cognition is often very challenging, and production of a program provides a minimal test of computational feasibility. Moreover, the program can then be used for testing the underlying theoretical ideas by examining how well the program works on numerous examples of different kinds. Comparative evaluation becomes possible when different programs accomplish a task in different ways: running the programs on the same data allows evaluation of their computational models and background theoretical ideas. Third, if the program is intended as part of a cognitive model, it can be assessed concerning how well it models human thinking.
The assessment of cognitive models can address questions such as the following:
1. Genuineness. Is the model a genuine instantiation of the theoretical ideas about the structure and growth of scientific knowledge, and is the program a genuine implementation of the model?
2. Breadth of application. Does the model apply to lots of different examples, not just a few that have been cooked up to make the program work?
3. Scaling. Does the model scale up to examples that are considerably larger and more complex than the ones to which it has been applied?
4. Qualitative fit. Does the computational model perform the same kinds of tasks that people do in approximately the same way?
5. Quantitative fit. Can the computational model simulate quantitative aspects of psychological experiments, e.g. ease of recall and mapping in analogy problems?
6. Compatibility. Does the computational model simulate representations and processes that are compatible with those found in theoretical accounts and computational models of other kinds of cognition?
Computational models of the thought processes of sciences that satisfy these criteria have the potential to greatly increase our understanding of the scientific mind. Engineering AI need not address questions of qualitative and quantitative fit with the results of psychological experiments, but should employ the other four standards of assessment.
There are numerous issues connecting computation and the philosophy of science that I have not touched on in this review. Computer science can itself be a subject of philosophical investigation, and some work has been done discussing epistemological issues that arise on computer research (see e.g. Fetzer, this volume; Thagard, 1993). In particular, the philosophy of artificial intelligence and cognitive science are fertile areas of philosophy of science. My concern has been more narrow, with how computational models can contribute to philosophy of science. I conclude with a list of open problems that seem amenable to computational/philosophical investigation:
1. In scientific discovery, how are new questions generated? Formulating a useful question such as "How might species evolve?" or "Why do the planets revolve around the sun?" is often a prerequisite to more data-driven and focused processes of scientific discovery, but no computational account of scientific question generation has yet been given.
2. What role does visual imagery play in the structure and growth of scientific knowledge? Although various philosophers, historians, and psychologists have documented the importance of visual representations in scientific thought, existing computational techniques have not been well suited for providing detailed models of the cognitive role of pictorial mental images (see e.g. Shelley 1996).
3. How is consensus formed in science? All the computational models discusses in this paper have concerned the thinking of individual scientists, but it might also be possible to develop models of social processes such as consensus formation along the lines of the field known as distributed artificial intelligence which considers the potential interactions of multiple intelligent agents (Thagard 1993).
Perhaps problems such as these will, like other issues concerning discovery and evaluation, yield to computational approaches that involve cognitive modeling, engineering AI, and the theory of computation.
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