This symbolic structure consisting of the axiomatized mathematical system, which constitutes the theory, is a distinctive language in science. It is different from all other language of science including the realistic semantics of common discourse, the nonmathematical generalizations of descriptive sciences such as physiology, and the phenomenalist semantics of mathematically expressed empirical laws of science such as Kepler’s laws. The language of theory is distinctive from nontheory language, because the nontheory language has a semantics that describes either the phenomenal or real world, while the language of theory does not have these semantics. Instead the semantics of theory language is called “symbolic”, which means that its meaning is a sign of the meanings of the nontheory language. Thus the semantics of science in Duhem’s philosophy is stratified into two levels, in which one represents the other.
The basis for Duhem’s distinguishing the semantics of theory language from that of all other language is the existence of a numerical indeterminacy caused by the fact that measurements, which may occur in the equations of theory, are always approximate. There are two reasons for the indeterminacy between the equations of theory and the nontheoretical language. The first reason is simply the approximate character of all measurements. When measurements are made, a “translation” must also be made from what Duhem called a “practical” fact to a “theoretical” fact. The practical fact describes the observed phenomena and circumstances of the experiment; the theoretical fact is the set of mathematical data that replaces the practical fact in the equations of the theory. Duhem calls the method of measurement the dictionary that enables the physicist to make this translation.
For any practical fact there is always an infinity of potential theoretical facts, even when the degree of indeterminacy is reduced with improved instruments and measurement procedures. So long as the one or several equations of a theory are correct, the numbers that are the solution set for the equations will fall within the range of measurement indeterminacy. Duhem illustrates the semantical duality caused by this numeric indeterminacy in his discussion of the different meanings of the phrase “free fall.” One meaning is contained in a phenomenal description given by any person who knows nothing about physical theory. And a second meaning occurs in the physical theory that includes the idea of uniform acceleration. These are two distinct meanings; the former may be either a realist or phenomenalist meaning, while the latter is called the symbolic meaning. The latter is a sign of the former, so long as the theory is accurate enough to be accepted as true.
However, the numerical indeterminacy that occasions the semantical distinction between practical facts and theoretical facts is not unique to the variables occurring in the equations of theories, the equations that are the conclusions drawn from the hypotheses which are the postulates of the theory. It also occurs in the variables occurring in the equations of empirical laws, the equations that are developed by experimental or other observational judgments. This creates another occasion for numerical indeterminacy, one which exists between the values of the variables in the equations of theory and the values of the corresponding variables in the equations of the empirical laws that a theory orders. Duhem discusses this numerical indeterminacy and the semantical duality to which it gives rise, when he criticizes Newton’s claim that his theory of gravitation is not based on hypotheses.
The basic question is whether or not Newton’s theory was or could be developed empirically by generalizing from Kepler’s laws. Duhem argues that Newton had actually created hypotheses, because the mathematical deduction from these hypotheses produces conclusions that formally contradict Kepler’s observational laws. In other words the solution sets for the empirical law and for the theory are not the same. Kepler’s laws are approximate, and therefore admit to an infinity of small deviations. The measurements by Tycho Brahe permit the theorist to choose a variation of Kepler’s laws, which is also produced by deduction from Newton’s theory. Just as there must be a translation from practical facts to theoretical facts resolving the indeterminacy in measurements, so too there must be a translation from empirical laws such as Kepler’s laws to “symbolic” laws such as Newton’s dynamics. Here again the numeric indeterminacy causes a semantic dualism, and a translation is made in which the new symbolic formulas derived from Newton’s hypotheses, are substituted for the old phenomenalistic formulas, which are Kepler’s observational laws.
Having shown that there are different semantics for theory and nontheory language in science, Duhem then gives two ways in which the meanings of the symbols in theory language differ from the meanings in all the other language of science. The first way, which is most important to him, is that the semantics of theory language is neither realistic nor phenomenalist; it does not describe the world of phenomena as does the semantics of empirical laws like Kepler’s laws, nor does it describe the real world as does the semantics of common-sense discourse. When Duhem states, therefore, that theories represent laws, he means to be taken literally; he means that theories do not represent the world but instead represent the empirical statements, which in turn represent the phenomenal world. Thus he cannot be called an instrumentalist in the sense that he denies that theory language has any semantics. He has stratified the semantics of science such that theory has its own higher level semantics.
He also states that when a theory agrees with experimental laws to the degree of approximation enabled by the measuring procedures employed, and furthermore when the theory correctly predicts the outcome of an experiment before the outcome has occurred, then there is reason to believe that the theory is not merely an economical representation of the experimental laws. Such a theory is also a “natural classification” of these laws in which the logical order in which the theory organizes the experimental laws is a reflection of the metaphysician’s ontological order that underlies the physicist’s phenomenal order. However, professionally the physicist cannot pass judgment on this analogical apprehension of the underlying ontological order, because this order is the proper subject only of metaphysics or natural philosophy.
The second way in which the meanings of the symbols in theory language differ from those in the other language of science is that the meanings of theory are determined by their context, by the statements that constitute the theory itself. Therefore, according to whether the physicist adopts one or another theory, the variables in the symbolic law change their meaning, so that the law may be accepted by one physicist who admits one theory while it may be rejected by another physicist who admits an alternative theory. Duhem illustrates this contextual determination of meaning in theory language in his discussion of Kepler’s observational laws and the symbolic laws of Newton’s theory. The formulas that constitute Kepler’s laws refer to orbits, but when they are replaced by the symbolic formulas that are deduced from Newton’s dynamics, the symbolic law contains variables referring to forces and masses also. The translation from Kepler’s laws into symbolic laws presupposes the physicist’s prior adherence to the hypotheses of the theory. The contextual determination of the meanings of theories is Duhem’s wholistic concept of theory, a concept that is strategic to his views about scientific criticism of theories. With his wholistic view he says theoretical physics is not like a machine but is more like an organism.
Finally it should be noted that although the higher level semantics of theory language is relatively remote from the phenomena described by the semantics of the nontheory language, nevertheless theory is not remote from the experimental situation. He states that an experiment in physics is not simply the observation of a phenomenon, but is furthermore the theoretical interpretation of it. And this theoretical interpretation is not just a technical language, but one that makes possible the use of instruments.
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