The axiomatic conception

In similar fashion, contemporary philosophy of science is moving beyond the question of the structure of scientific theories. For a variety of reasons, that question was of enormous importance to the logical positivists and to the logical empiricists. Mathematical logic supplied a clear conception: a theory is a collection of statements (the axioms of the theory) and their deductive consequences. The logical positivists showed how this conception could be applied in scientific cases—one could axiomatize the theory of relativity, for example. Nor was the work of axiomatization an idle exercise, for the difficulties of formulating a precise criterion of cognitive significance (intended to separate good science from meaningless philosophical discussion) raised questions about the legitimacy of the special vocabulary that figures in scientific theories. Convinced that the sound and fury of German metaphysics—references to “Absolute Spirit” by Georg Wilhelm Friedrich Hegel (1770–1831) and talk of “the Nothing” by Martin Heidegger (1889–1976)—signified, indeed, nothing, logical positivists (and logical empiricists) recognized that they needed to show how terms such as electron and covalent bond were different.

They began from a distinction between two types of language. Observational language comprises all the terms that can be acquired by presentation of observable samples. Although they were skeptical about mixing psychology and philosophy, logical empiricists tacitly adopted a simple theory of learning: children can learn terms such as red by being shown appropriate swatches, hot by holding their hands under the right taps, and so forth. Logical empiricists denied that this observational vocabulary would suffice to define the special terms of theoretical science, the theoretical language that seemed to pick out unobservable entities and properties. Conceiving of theories as axiomatic systems, however, they drew a distinction between two types of axioms. Some axioms contain only theoretical vocabulary, while others contain both theoretical and observational terms. The latter, variously characterized as “correspondence rules” or “coordinating definitions,” relate the theoretical and observational vocabularies, and it is through them that theoretical terms acquire what meaning they have.

The last formulation blurs an important difference between two schools within logical empiricism. According to one school, the theoretical terms are “partially interpreted” by the correspondence rules, so, for example, if one such rule is that an electron produces a particular kind of track in a cloud chamber, then many possibilities for the meaning of the previously unfamiliar term electron are ruled out. A more radical school, instrumentalism, held that, strictly speaking, the theoretical vocabulary remains meaningless. Instrumentalists took scientific theories to be axiomatic systems only part of whose vocabulary—the observational language—is interpreted; the rest is a formal calculus whose purpose is to yield predictions couched in the observational vocabulary. Even instrumentalists, however, were able to maintain a distinction between serious theoretical science and the much-derided metaphysics, for their reconstructions of scientific theories would reveal the uninterpreted vocabulary as playing an important functional role (a result not to be expected in the metaphysical case).

Logical empiricists debated the merits of the two stances, exploring the difficulties of making precise the notion of partial interpretation and the possibility of finding axiomatic systems that would generate all the observational consequences without employing any theoretical vocabulary. Their exchanges were effectively undercut by the American philosopher Hilary Putnam, who recognized that the initial motivation for the approach to theories was deeply problematic. In their brief sketches of the differences between the two languages, logical empiricists had conflated two distinctions. On the one hand there is a contrast between things that can be observed and things that cannot—the observable-unobservable distinction; on the other hand, there is the difference between terms whose meanings can be acquired through demonstration and those whose meanings cannot be acquired in this way—the observational-theoretical distinction. It is a mistake to believe that the distinctions are congruent, that observational terms apply to observable things and theoretical terms to unobservable things. In the first place, many theoretical terms apply to observables (spectroscope is an example). More important, many terms learnable through demonstration apply to unobservables—in Putnam’s telling example, even small children learn to talk of “people too little to see.”

Once the second point was appreciated, the way was open for introducing theoretical vocabulary that logical empiricism had never taken seriously (even though many eminent scientists and gifted science teachers had often developed such modes of conveying meaning). One can see that the term part might be learned in connection with pieces of observable objects and that its use might cover unobservable things as well, so the specification of atoms as “parts of all matter that themselves have no parts” (whatever its merits today) might have served the contemporaries of John Dalton (1766–1844), an early developer of atomic theory, as a means of appreciating what he was claiming.

Logical empiricism lavished great attention on the problem of exposing the structure of scientific theories because solving that problem seemed crucial to the vindication of the theoretical vocabulary employed by the sciences. Putnam showed, in effect, that no such strenuous efforts were required.