The focus of this paper will be on certain aspects of three concepts, those of existence, explanation, and force, together with some related ideas. Specifically, I want to look at some of the roots of those concepts both in written history and, somewhat speculatively, in more primitive circumstances of the history of the human species; and I want to compare certain aspects of the three concepts with their rational descendants in the theories and expectations of contemporary physics. Although these topics are interesting and important in themselves, and there is much more to say about them, my primary aim in the present essay will be to extract some methodological implications from them regarding the problems and methods of the philosophy of science. Indeed, some of the ways in which the topics are important for philosophers of science will be brought out by the discussion of those methodological implications.

Over the past two centuries the character of science has changed markedly and crucially. When humans had relatively small amounts of information and energy available they perforce had no option but to attempt to describe objectively a world which behaved independently of them. The basic practical problem was to discover what the pattern of the world was so as to adapt themselves as best they may to it, a little in anticipation of events, but mostly in reaction to them. It is these circumstances which give most sense to the empiricist paradigm of objective knowledge.

But the more information and energy is available to us, the more we are able to ascend from merely correct description of what is to a grasp of what is possible. Newton's laws don't merely describe the paths of the actual projectiles that have fallen and will fall, they describe the paths of all possible projectiles.

The Discours de la Méthode and the three essays that were published with it, the Dioptrique, the Météores, and the Géometrie make up a very curious book. The very title page emphasizes the preliminary discourse, and that discourse, the Discours de la Méthode, emphasizes method, the importance that method had for Descartes in making the discoveries he made, the importance that the method Descartes claims to have found will have for the progress of the sciences and for the benefit of humankind as whole. Descartes is not, of course, telling us that we are obligated to follow his method; the Discours is, after all, proposed “as a story, or, if you prefer, as a fable.” (AT VI 4). But Descartes expects that we will all see the light, the light of reason, of course, and follow his example.

Mathematics plays a central role in the description, explanation and manipulation of natural phenomena. To what extent, and how and why mathematics applies to nature is a problem that has long occupied philosophers. Descartes, Leibniz, Kant, Mach and Poincaré, to mention some of the most distinguished names, offer global solutions to this problem that are based on deep-lying metaphysical assumptions. In this essay, I would like to suggest an alternative approach, which is piecemeal rather than global, and historical before it is metaphysical.

I want to propose, first, that the question of applied mathematics be recast as a question about how mathematics and physics, a physics “always already” mathematized, are partially unified at various points in history, in such a way that they can share certain items, problems and methods while nonetheless remaining quite distinct. And, second, I suggest that these unifications may be quite heterogeneous and variable over time.

It is difficult to determine the place of experience in Descartes’ philosophy and science. His a priori, rationalist metaphysics would seem, on the face of it, difficult to reconcile with the explicit appeal he makes to sensory evidence in both his scientific practice and methodological remarks. Although earlier descriptions of Descartes as a pure a priorist in natural science (e.g., Koyré 1978, pp. 89-94) have rightly been rejected, it would be a mistake to embrace the other extreme, as does Clarke (1982), with his bold revisionist thesis that Descartes was, in actuality, an empiricist. And yet the possibility of combining the rationalist and empiricist elements in Descartes’ thought has also seemed problematic, for it has been assumed that Descartes required that experientially based knowledge meet the standard of absolute certainty set by the the method of doubt in the First Meditation (Garber 1978).

In this paper I discuss how, given the complexity of biological systems, reliance on theoretical models in the development and testing of biological theories leads to anti-realism. This is the result of the uniqueness and hence diversity of biological phenomena, in contrast with the uniformity of items in the domain of physics. I have argued elsewhere (Horan, 1986, 1989a, 1989b) that the use of theoretical models creates an unresolvable tension between the explanatory strength and predictive power of hypotheses, and I review this argument here. My discussion is in part motivated by the claims of Nancy Cartwright (1983), who has argued that the use of ceteris paribus laws in physics creates an antagonism between truth and explanation that requires theoretical models to figure centrally in scientific explanation, thereby precluding realism. I argue instead that in biology it is the use of theoretical models that creates this conflict, and conclude that adequate biological explanation cannot rely on the modelling approach alone.

The semantic view of theory structure, as developed by Suppes (1957,1967), Suppe (1974, 1976, 1977, 1988), and van Fraassen (1970, 1972, 1980), represents theories as classes of models or structures. These models are, on the version of the semantic approach used here, defined by specifying their laws, parameters, and variables. The semantic approach to theory structure is simply a method of formalizing the content of scientific theories.

In a series of articles and a book, I have analyzed the structure of modern evolutionary theory using the semantic view as a framework (Lloyd 1984, 1986a, 1986b, 1987a, 1987b, 1988, forthcoming; cf. Thompson 1983, 1985, 1988). I shall briefly recap the analyses I have done, in order to demonstrate the range of problems accessible to the semantic view.

First, I have used the semantic view to analyze the structure of population genetics models, including kin and group selection models.

The semantic conception of theories has a relatively short history the beginnings of which Frederick Suppe has traced to von Neumann (Suppe 1988). Two other early initiators and advocates were Evert Beth in 1948-49 (Beth 1948, 1949; see also 1961) and Patrick Suppes in 1957 in his Introduction to Logic (Suppes, 1957). Beth advanced what has become known as a state space approach while Suppes advanced a set-theoretical predicate approach.

Suppes suggested that scientific theories are more appropriately formalized as set-theoretical predicates. Shortly thereafter in 1961 Robert Stoll in his Set Theory and Logic (Stoll, 1961) made a similar claim about the formalization of informal theories of which scientific theories are instances. Suppes wrote a number of papers during the 1960's in which he indicated the features of theories that were better represented on a set-theoretical predicate conception of theories (see, for example Suppes, 1962).

Two attitudes largely dominate current philosophic thinking about the social sciences. One argues that the social science are not and cannot be real sciences—because the social sciences have not and probably cannot produce anything like the laws of the natural sciences. The other attitude—coming from the hermeneutic tradition—defends the social sciences, but only by arguing that laws are irrelevant to social explanation. Disparate as they are, both traditions share the assumption that laws in the social sciences are beyond reach. In this paper I argue the opposite thesis—viz. that social laws are both a possibility and actuality.

Philosophers have typically approached laws in the social sciences by an old-fashioned route—by means of relatively a priori conceptual analysis. One or several necessary conditions for laws are identified, usually by appeal to intuitions. Then it is argued that no alleged law in the social sciences could have the necessary features.

Explicit or implicit ceteris paribus clauses are pervasive in economics. People do not always buy more of x when the price of x decreases. The generalization holds only “other things being equal” or ceteris paribus. Not everybody wants more wealth, but economists have held that the generalization holds, ceteris paribus. When government imposes price controls, shortages do not always arise, but, ceteris paribus, they do. Ceteris paribus clauses are common in other sciences, but I shall confine my remarks to economics.

Many have found such ceteris paribus clauses problematic. For they are vague, and they seem to insulate theories from empirical criticism and correction. When the observed phenomena are not as the theory predicted, one can (or so it has been alleged) always claim that other things were not “equal”, that there was some “disturbance” or “interference” and thus that the ceteris paribus condition was not met.

We take it that what is intended by “laws” in the title of this session—“Confirming Ceteris Paribus Laws in the Social Sciences”—is reliable rather than fortuitous relations, that is, Millian “tendencies,” with “forcing” properties. These refer to causes which do their work whether or not interfered with, or even counteracted by, other causes. Early econometricians had this in mind when they referred to “autonomous” relations (or behavioral relations). We entirely beg the question of how we know when we are dealing with genuine causes and with the related question how we come by autonomous relations. Instead, we concentrate upon some implications of our supposing that we can identify the experimental conditions necessary to confirm the operation of whatever tendency is in question.

Another expression for these experimental conditions is the steps necessary in order to transform a theoretical model into an estimable one.

Game theory studies the behavior of rational players in interactive situations and its possible outcomes. For such an investigation, the notion of players’ rationality is crucial. While notions of rationality have been extensively discussed in game theory, the epistemic conditions under which a game is played — though implicitly presumed — have seldom been explicitly analyzed and formalized. These conditions involve the players’ reasoning processes and capabilities, as well as their knowledge of the game situation. Game theory treats some aspects of information about chance moves and other players’ moves by means of information partitions in extensive form games. But a player's knowledge of the structure, for example, of information partitions themselves is different from his information about chance moves and other players’ moves. The informational aspects captured by the extensive form games have nothing to do with a player's knowledge of the structure of the game.

Von Neumann and Morgenstern based their theory of games on the representation of individual preferences for outcomes by utilities generated by preferences for gambles over these outcomes. A utility function (an assignment of numbers to outcomes) represents an agent's preferences just in case the agent's preference relation between any two gambles agrees with the numerical relation between their expected utilities (where these expectations are calculated using the objective probabilities specified in the gambles). This representation constrains the utility assignments up to scale transformations (multiplying each value by the same positive number) and adjustments of the zero point (adding the same positive or negative number to each value). This fixes the ratios of differences between utilities of outcomes.

Von Neumann and Morgenstem proposed qualitative constraints on the agent's preferences among these gambles which are equivalent to the assertion that such a utility representation exists. Here is an equivalent formulation of such rationality postulates.

It is by now rather well understood that the notion of common knowledge (first introduced by Lewis (1969) and later formalized by Aumann (1976)) plays a central role in the theory of games. (An event E is common knowledge between two individuals, if each knows E, each knows the other knows E, etc…). Indeed, most justifications of Nash's (1951) equilibrium concept usually include (perhaps only implicitly) the assumption that it is common knowledge among the players that both the Nash equilibrium in question will be played by all and that all players are expected utility maximizers. (We shall henceforth call expected utility maximizers, “rational”.) We hope to illustrate in an informal manner that there is in fact a large class of extensive form games, in which each of which it is not possible for rationality to be common knowledge throughout the game.

One of the most obvious features of empirical science is that scientists attempt to relate their theories to data, that is to the results of observation and experiment. In so doing, they will find it necessary to use the term ‘confirms’ or ‘confirmation’ or at least terms with more or less the same meaning. Thus scientists may think that a theory is well- confirmed by a considerable body of observational evidence. In this case they will regard it as rational to have a high degree of confidence in the theory, and, in particular, to use it in practical applications. Alternatively a theory T may be seen as badly confirmed by the evidence. In this case it would be thought undesirable to use T in practice, and the problem would become that of modifying or replacing T so as to obtain a new theory T’ which is better confirmed than T.

‘A hypothesis can only be received upon the ground of its having been verified by successful prediction’, C.S. Peirce, Collected Papers, Vol. II, p.466.

Hempel, in his classic [1966], raised a problem which already had a long pedigree. Balmer, having examined measurements of the wavelengths of lines in the emission spectrum of hydrogen, constructed a now well-known formula for generating these, and also other lines at that time unknown to him. Hempel asked whether Balmer's formula would have been as strongly confirmed by the new lines had they too formed part of Balmer's database. Hempel's question poses the famous and ancient accommodation-versus-prediction problem: is a hypothesis h, designed among other things to account for data e, confirmed by e as much as h would have been, had it been found to explain e without, however, having been constructed to do so? An unconditionally negative answer has been given by a numerous and influential body of scientists and philosophers; this unconditional view of the matter I shall call predictionism.

For the past century the Central Dogma of confirmation theory has been:

All empirical support (of a theoretical claim) = empirical evidence = empirical data = successful test results = successful predictions or postdictions = true empirical consequences (of the claim plus auxiliary assumptions).

According to the Dogma, all empirical support derives from empirical testing of predicted consequences. I shall attack this pure consequentialism and defend the importance of generative justification in science.

One question which has been widely debated through the years is whether “postdicting” or explaining already known phenomena provides as much epistemic support for an hypothesis as predicting new phenomena or, indeed, whether explanation provides any support at all. I believe that explanation is on a par with prediction, but I want to address a different question here. Both explanation and prediction (as usually construed) involve consequential reasoning.

Imagine a scientist who examines an unending sequence of data about an unknown reality and responds to each datum by announcing a first-order theory in a fixed language L. The scientist hopes to determine the truth or falsity of every sentence of L, but he does not feel obliged to reach a stage in his investigation when all such questions will have been resolved. One way to represent the scientist's project is as follows. Let the unknown reality correspond to the L-structure S. Then the scientist hopes that for every sentence ϕ ∊ L, if S ╞ ϕ, then all but finitely many of the theories he emits, imply ϕ and do not imply - ϕ Following Kelly & Glymour (1987), we say in this case that the scientist “AE-identifies” S. The AE prefix signifies that for all ϕ ∊ L, there is a stage beginning at which the scientist's theories correctly decide ϕ.

Consider the following collection of familiar questions in the philosophy of science.

Underdetermination: What collections of theories can be reliably distinguished from one another on a given sort of evidence presentation? How do differences in evidence presentation, background knowledge, and hypothesis vocabulary affect underdetermination?

Realism: How plausible is it that the methods employed by science are capable of arriving at the truth in a given domain of inquiry? If not, is it plausible that there exist such methods? How does the answer differ from domain to domain?

Scientific progress: Is it possible that scientific knowledge could be continually improved, even if it is never perfectly correct? Is there any sense to be made of the increasing verisimilitude of inquiry?

Methodology: How do standard methodological directives interact? Do different directives interfere with one another or do our favorite ideas about method complement one another? Is experiment really more powerful than passive observation?

The performance of a connectionist learning system on a simple problem has been described by Hinton and is briefly reviewed here: a finite function is learned, and the system generalizes correctly from partial information by finding simple “features” of the environment. For comparison, a very similar problem is formulated in the Gold paradigm of discrete learning functions. Identification in the limit from positive text of a large class of functions including Hinton's is achievable with a trivial, conservative learning strategy. Using Valiant's approach, we place an arbitrary finite bound on function complexity and then we can guarantee text and resource efficiency relative to a probabilistic criterion of success. But the connectionist system generalizes. That is, it uses a non-conservative learning strategy. We define a simple, non-conservative strategy that also generalizes like the connectionist system, finding simple “features” of the environment.