Axiomatic systems

An axiomatic system is a logical system consisting of the following elements: (i) a list of primitive terms, which can be shared with other axiomatic systems; (ii) a list of axioms, which are general statements –sometimes called postulates– that are accepted as valid throughout the scope of the axiomatic system; (iii) definitions of new terms based on previous terms; and (iv) a list of proven theorems that refer to the theory as a whole or to special cases.

The term "axiom" derives from the Ancient Greek "ἀξίωμα" which means "a requisite" or "something self-evident". Each of the classical subjects of physics was originally developed by direct induction from experimental observation. Inductive developments in mechanics later led to axiomatic reformulations that were more elegant, abstract, and concise. The Newtonian formulation of mechanics led to the Lagrangian and Hamiltonian formulations. Other examples of axiomatized scientific theories include macroscopic thermodynamics of equilibrium –based on axioms of László Tisza and Herbert Callen [1]– and Copenhagen quantum mechanics –based on axioms of Niels Bohr, Werner Heisenberg, Max Born, Erwing Schrödinger, and John von Neumann [2]–.

The axiomatic systems used in science would be cumulative, and the new axiomatic systems would not contradict the old ones, but embrace them as a particular case.

The list of axioms must be internally consistent –that is, one axiom cannot contradict another– and cannot lead to contradictory theorems; if a list of axioms is used to derive a consequence and its contrary, this implies that the list is inconsistent and must be abandoned. For instance, intuitionistic type theory, was based on a strong impredicative axiom that allowed a type of all types to be both a type and an object of that type; it was abandoned after Jean Yves Girard showed it led to a contradiction [3].

Axioms are not unique; in general, it is possible to find an equivalent formulation of a given axiomatic system, using another set of axioms. The criteria for selecting the most adequate list of axioms include, among others, simplicity, generality, intuitive scientific meaning –for scientific theories–, and economy in number of axioms. Moreover, axioms would be independent –one axiom would not be derived from others– and would not be based on operational definitions or procedures.

The number and the nature of the axioms depend on the scientific discipline considered. However, three different types of axioms acquire special relevance from the point of view of a fundamental theory: the descriptor, the evolutor, and the state. The descriptor is the axiom that defines the system under study. The axiom of state unambiguously identifies how the states of this system are identified. Finally, the evolutor is the axiom that establishes how a given system evolves in time from a given state to another state. For ordinary quantum mechanics, the descriptor is the number and type of particles in the system under study, the state is given by a wave function in a Hilbert space, and the evolution is given by the collapse postulate for measurements and by the Schrödinger equation in the rest of the cases.

ADVANTAGES OF AXIOMATIZATION

The advantages of axiomatic formulations of scientific theories are numerous. First, the concise axioms exhibit the internal consistency of the logical structure. Furthermore, after sufficient experience with the axioms, it actually becomes possible to develop a deeper insight and intuition than on the basis of the elementary theory. And finally, important extensions and generalizations of the theory frequently are suggested and put into practice by abstract formulations, just as the development of quantum mechanics was accelerated by the Hamiltonian formulation of classical mechanics.

CRITICISM

Some authors define axioms as self-evident and unprovable propositions. The first assumption is difficult to accept because "self-evident" is a subjective term, while the second assumption needs to be clarified. It is true that the axioms of a given axiomatic system cannot be proved within the system itself, but they can usually be demonstrated from a more fundamental axiomatic system. For instance, the axioms of classical equilibrium thermodynamics can be obtained from the axioms of extended thermodynamics.

REFERENCES AND NOTES

  1. Thermodynamics and an introduction to thermostatistics; Second edition 1985: John Wiley & Sons, Inc.; New York. Callen, Herbert B.
  2. The General Formulation of Quantum Mechanics 2003: In Handbook of Molecular Physics and Quantum Chemistry, Volume 1 Fundamentals; John Wiley & Sons Ltd.; Wilson, Stephen (Editor-in-chief); Bernath, Peter F. (Associate Editor); McWeeny; Roy (Associate Editor). McWeeny, R.
  3. Information and Knowledge, A Constructive Type-theoretical Approach 2008: In Logic, Epistemology, And The Unity Of Science 10; Springer, Dordrecht; Rahman, Shahid (Editor); Symons, John (Editor). Primiero, Giuseppe.