Axiomatic Method

means of constructing a scientific theory, in which this theory has as its basis certain points of departure (hypotheses)—axioms or postulates, from which all the remaining assertions of this discipline (theorems) must be derived through a purely logical method by means of proofs.

The aim of the axiomatic method is a limitation of the arbitrariness of scientific proposition as truths for a given theory. Scientific constructions based on the axiomatic method are usually called deductive. All the concepts of a deductive theory, except for a fixed number of initial ones, are introduced by means of definitions which express (or explicate) the concepts through previously introduced concepts. In varying degrees, deductive proofs characteristic of the axiomatic method have been applied in many disciplines. But despite the efforts to systematically apply the axiomatic method to the exposition of philosophy (B. Spinoza), sociology (G. Vico), political economy (K. Rodbertus-Jagetzow), biology (J. Woodger), and other scientific disciplines, the principal sphere of its application up to the present time has been mathematics and symbolic logic, as well as certain branches of physics (mechanics, thermodynamics, electrodynamics, and others).

In its historical development, the axiomatic method has passed through three stages. The first of these is connected with the formulation of geometry in ancient Greece. The basic work of this period is Euclid’s Elements, although, evidently, even before him, Pythagoras, who is credited with the origin of the axiomatic method, and subsequently Plato, with his disciples, accomplished a great deal in developing geometry on the basis of the axiomatic method. At that time it was considered that the only statements chosen as axioms should be those whose truth was “self-evident,” so that the truth of the theorems was considered to be guaranteed by the infallibility of logic itself. But Euclid did not succeed in limiting himself to purely logical means in constructing geometry on the basis of axioms. He willingly resorted to intuition in problems of continuity, mutual position, and congruence of geometric objects. However, in Euclid’s time such recourse to intuition may not have been considered to be going beyond the bounds of logic, primarily because logic itself had not yet been axiomatized, although a partial formalization of logic, accomplished by Aristotle and his followers, was an approach to axiomatization. Nor was there sufficient precision in introducing initial concepts and defining new concepts.

The beginning of the second stage in the history of the axiomatic method is usually linked with the discovery by N. I. Lobachevskii, J. Bolyai, and K. F. Gauss of the possibility of consistently (without contradiction) constructing a geometry proceeding from systems of axioms different from those of Euclid. This discovery destroyed belief in the absolute (obvious or a priori) truth of the axioms and of the scientific theories based upon them. Now axioms began to be conceived simply as points of departure for a given theory; moreover, the problem of their validity in one sense or another, as well as the choice of axioms, went beyond the axiomatic theory as such and had a bearing on its relationship with facts lying outside of it. Many, and varied geometric, arithmetic, and algebraic theories appeared which were constructed by means of the axiomatic method (the works of J. W. R. Dedekind, H. Grassmann, and others). This stage of development of the axiomatic method culminated in the creation of axiomatic systems of arithmetic (G. Peano, 1891), geometry (D. Hubert, 1899), statement and predicate calculations (A. N. Whitehead and B. Russell, England, 1910), and the axiomatic theory of sets (E. Zermelo, 1908).

Hubert’s axiomatization of geometry permitted F. Klein and H. Poincaré to prove the noncontradictory nature of Lobachevskii’s geometry relative to Euclid’s geometry by means of showing the interpretation of concepts and propositions of non-Euclidean geometry in terms of Euclidean geometry or, as it is said, by constructing a model of the former by means of the latter. From that time on, the method of models (interpretations) has become the most important method for establishing the relative consistency of axiomatic theories. At the same time, it became very clear that besides its “natural” interpretation—that is, that interpretation from whose refinement and development a given theory was constructed—axiomatic theory could have other interpretations as well. Moreover, it could equally be considered to “have something to say” about each of them.

The subsequent development of this idea and the striving to describe exactly the logical means of deriving theorems from axioms led Hubert to the conception of a formal axiomatic method, characteristic of its third and contemporary stage of development. Hilbert’s fundamental idea was the complete formalization of scientific language, within which its statements are regarded simply as sequences of signs (formulas) which in themselves have no meaning whatsoever; they would acquire meaning only in a certain specific interpretation. This idea also had a bearing on axioms—those of general logic as well as those specific axioms used in a given theory. In order to derive theorems from axioms and, in general, certain formulas from others, special rules of derivation were formulated; for example, the so-called rule of modus ponens (the “means of assertion”) which allows B to be obtained from A and “A implies B.” Proof in such a theory (calculation, or formal system) is simply a sequence of formulas, each of which is either an axiom or is obtained from previous sequential formulas in accordance with some rules of derivation. In contrast to such formal proofs, the properties of the formal system itself are judged as a whole, and sometimes even successfully proved, by their own content of the so-called metatheory—that is, the theory that considers the given (“object”) theory as the object of study. In the language of metatheory (metalanguage), rules have also been formulated for deriving an object theory. According to Hilbert’s conception, within the framework of the theory of proofs created by him—that is, allowing metatheory only the so-called finite means of reasoning (those which do not employ references to any object whatsoever not having a finite construction)—one could prove the consistency and completeness of all classical mathematics (that is, the demon-strability of every formula as truth in a certain, defined interpretation). Despite a number of significant results along these lines, Hilbert’s program, which is usually called formalism, cannot be carried out in its entirety because, according to K. Gödel’s most important result (1931), every sufficiently rich and consistent formal system is necessarily incomplete (the so-called incompleteness theorem). Gödel’s theorem bears witness to the limited nature of the axiomatic method, although well-defined extensions of allowable metatheoretical means have made it possible for the German mathematician G. Gentzen, as well as P. S. Novikov and others, to prove the consistency of formalized arithmetic.

The axiomatic method has also been subject to criticism proceeding from various semantic criteria. Thus, the intuitionists (L. E. J. Brouwer, H. Weyl, and others) do not acknowledge the validity of applying the principle of the excluded middle to infinite sets. This principle, however, is not only accepted as a logical axiom in the majority of formal theories, but is also used (although not obviously) in the fundamental premises of Hilbert’s program, according to which consistency of a theory is a sufficient condition for its “truthfulness.” Like intuitionism, the constructivist school in mathematics (represented in the USSR by A. A. Markov and N. A. Shanin) considers mathematics as the study not of arbitrary models of consistent formal systems but merely of the aggregate of objects which allow effective construction in a specific sense.

Even more hostile views toward the axiomatic method have been put forth by the ultraintuitionist critics, who express doubt about the uniqueness of the natural series of numbers and who, in the same vein, question the simple conceptual definitiveness embodied in the theorem of a formal system. According to this school of criticism, the axiomatic method is based on the “principle of locality for proofs,” which assumes that, if the axioms are true and the rules of derivation remain a truth, then the theorems must also necessarily be true. Thus, the intuitive basis for the generally accepted principle of mathematical induction contains, in the opinion of the ultraintuitionist critics, an irremovable vicious circle. Ultraintuitionism does not, however, limit itself to negative criticism; it also proposes a positive program to overcome the difficulties mentioned above.