The Construction of Theories by Moritz Schlick

Theoretical science, as is obvious from its name, consists of theories -- that is, of systems of propositions. Propositions constitute a system when they are related to one another through being concerned with the same objects; or even when they can be deduced from one another. The process of formulating a law of nature is, fundamentally, always the same. It consists, in the first place, of recording the observations of a natural process in a table which always contains the relevant measured values of those variable magnitudes which characterize the process. The next step is to discover a function which will represent in a single formula the distribution of values in this table. This formula is then considered to be the law describing the process as long as all new observations are in agreement with it. Inasmuch as the formula always contains more than what is actually observed, and also because it must hold for all processes of a similar kind, the formulation of any law involves a generalization, or a so-called induction. There is no such thing as a logically valid deduction going from the particular to the general: the latter can only be conjectured, but never logically inferred. Thus, the universal validity, or truth, of laws must always remain hypothetical. All laws of nature have the character of hypotheses: their truth is never absolutely certain. Hence, natural science consists of a combination of brilliant guesses and exact measurements. . . .

In the same way as a special law is the result of a series of single observations, a general law is the consequence of the inductive combination of several individual laws, until finally a relatively small number of general propositions which include the totality of natural laws is obtained. Thus today, for instance, all chemical laws can, in principle, be reduced to physical laws; and the dividing line between the different domains of physics which used to be externally related to one another (mechanics, acoustics, optics, theory of heat, etc.) has long since completely disappeared. At the present time, only mechanics and electrodynamics are left; and these are nowise independent of each other, hut interpenetrate everywhere. Whether biology will continue to remain a special province, or whether it also will become incorporated in the domain of physics, is a question that will be discussed in due course.

In order to obtain a concrete description of nature (i. e., of nature as it really is), it is not sufficient to formulate laws: the abstract laws must, as it were, be given content. And in addition to these abstract laws, the constellation of reality (at the time of consideration), to which the formulas can be applied, must be stated. Such constellations are called by physicists boundary or initial conditions; and mathematically, they are expressed by the introduction of constants.

Here, we are considering the system of laws in itself, independently of all applications -- that is to say, we are only studying general, and not particular, propositions. We can thus select out of this system, a group of the most general propositions from which all the others are derivable. This derivation is a purely logical deduction which can be undertaken without knowledge of the meaning of the symbols which occur in the laws. Hence, we will disregard, not only all application to individual cases, but also the meaning of all words and symbols -- until the system is reduced to a purely formal structure, or empty framework which does not consist of actual propositions, but only of their forms (in logic, these are known as prepositional functions). A system of this kind, which does not represent nature in actuality, but all the possibilities in nature, or in other words, its most general form -- is known as a hypothetico-deductive system (Fieri). The propositions forming a group at the apex of this system, are called axioms; and the choice as to which propositions shall be taken as axioms is, to a certain extent, arbitrary. We may regard any proposition as an axiom, so long as we fulfil one condition, which is that all the other propositions in the system be derivable from the chosen group of axioms. Thus, the quality of being an axiom is not only in any sense a natural, intrinsic attribute or characteristic of a law; the only reason for choosing certain propositions as axioms, are those of their expediency or convenience. In the propositions derived from these axioms, further symbols, other than those used in the axioms, are introduced by definition. A definition consists of the introduction of new symbols, or signs, for the purpose of abbreviation. The choice as to which of these signs shall be regarded as fundamental symbols and which as derived from the latter by definition, is likewise arbitrary.

Examples:

E = 1/2mv2 M = mv Definition of Energy Definition of Momentum But instead of mass and velocity, we can also write:

Energy



Momentum : v = 2E



M

Thus, it is immaterial which magnitudes or quantities occur in the axioms.

Hence, the structure of a theory consists of: 1) axioms; 2) derived propositions and 3) definitions. In the symbolic representation of natural science, whether by means of words or of mathematical symbols, the three structural elements cannot be outwardly distinguished from one another.

The symbolic representation of a theory consists of sentences which in their turn are constituted of certain series of spoken or written signs: the theory itself consists primarily of "propositions." The question as to whether a sentence represents a true proposition or only a definition for example depends on the interpretations which explain it and give it its meaning. These do not form part of the symbolic representation itself, but are added to it -- that is, they are added to a hypothetico-deductive system -- from outside as it were, for example, in the form of ostensive definitions. They constitute the rules of the application of the sentences and are conclusive for the philosophical interpretation of the latter. It is, after all, necessary to refer to a reality which is described by the system of signs or symbols since, at some time or another, we must break out from their system. Only those sentences which, by virtue of their interpretation, represent genuine propositions, can communicate something about nature; the others are merely internal rules for signs and consequently are definitions.

Excerpted from Philosophy of Nature, by Moritz Schlick