The connection of mathematics and logic in the structure of axiomatized and formalized theories

Chechetkina Irina Igorevna PhD in Chemistry Associate professor, Department of Philosophy and History of Science, Kazan National Research Technological University 420097, Russia, respublika Tatarstan, g. Kazan', ul. Dostoevskogo, 74 A, of. 1 iralena@mail.ru Other publications by this author

Introduction: the axiomatic method, its relevance, history, and prospects in natural science.

Most scientists and engineers working in fundamental and applied natural sciences, in technology, use mostly informal and non-axiomatized mathematical models and theories. By non-formalized models and theories, we will understand those that are not related to set theory, state space, or model theory.

 These are, for example, heuristic models in biology (Watson and Crick's DNA double helix model), sign-symbolic models of structural chemistry, analog models in microphysics (Rutherford's atomic structure model), mathematical models and particular theories of physico-chemical processes in the field of kinetics and catalysis, thermodynamics describing the state using a system of differential equations process at any given time, simulation models in computer science ^[1].  All these models require insight and logic of model construction (analogy, simplification, use of approximations), search for mathematical analogues of a natural science object, and ways to study the iconic patterns of model parameters. Unformalized and non-axiomatized models and theories can be contradictory, describe the same reality using different representations (the model of the structure of the Boron atom), their inconsistency is revealed when their obvious contradictory positions are revealed, which leads to further detailed analysis of their mathematical and logical structures ^[2].

For a better understanding of physical reality and the essence of its laws, it is necessary to come to an axiomatized and formalized version of the theory. Axioms allow not only to systematize theoretical knowledge and carry out formal routine verification of evidence, but also to come to new laws, better understand and establish their connection with physical reality. This is the role of axioms in cognition. Axiomatization allows you to get rid of confusion in the origin of concepts and their description of phenomena, replace intuitive or semi-empirical concepts of scientific theory with strict mathematical ones, and do away with thought experiments.

Initially, physics was concerned with the axiomatization of specific informal theories, which meant the explicit presentation of primitive concepts and axioms. An example is Newton's mechanics, where logic was applied intuitively and the axioms were meaningful.

In physics, Newton was close to the axiomatic method. He believed that two or three general principles can be deduced from natural phenomena and then explain how the actions of things follow from these principles ^[3]. Although Newton calls his laws of motion axioms, they are derived from his own experiments or Galileo's experiments, while he does not present a method for deducing these laws.

According to Newton, the essence of science is to establish cause-and-effect relationships, which are expressed in the fundamental law (the "beginning"). The laws (the three laws of motion and the law of gravity) are mathematical formulas, which he calls axioms, from which simple consequences are derived, serving to solve particular problems of mechanics and explain the phenomenological third Kepler's law.

However, mathematically developed theories of physics of the 19th century, such as thermodynamics and electrodynamics, required a logical justification of their knowledge, which meant the formalization of physical theory and an explicit presentation of the underlying logical apparatus. It was necessary to improve the logical apparatus and create reliable logical languages.

At the beginning of the 20th century, Hilbert put forward a project to rebuild the foundations of classical mathematics, his attention was directed to the problem of consistency of mathematics. To do this, Hilbert proposed to solve it by formalizing all its theories and reducing them sequentially to set theory. He wanted to create an axiomatic system that would serve as the foundation of all mathematical theories. Hilbert realized that axiomatization requires well-designed symbolic formal systems with which to formalize mathematical theories and proofs. The point of the program was that it was possible to neglect the semantic meanings of mathematical expressions, replacing them with symbols or strings of algebraic symbols. As a result, mathematical theories had to be replaced by formal systems, and proofs – by sequences of formulas ^{[4, pp. 464 480]}. Thus mathematical theories acquired a logical structure, and soon mathematics opened the way of reduction to logic (Russell, Whitehead, Wittgenstein, logical empiricism).

Hilbert's project was also aimed at using the formal axiomatic method in mathematical natural science, where it was to become the method of all theoretical research.

Hilbert's program influenced the construction of a system of axioms of set theory (Zermelo, von Neumann, Frenkel) and the establishment of its consistency. This required a deeper study of the axiom system, which entailed an analysis of the structure of a scientific statement, truth, expressiveness, provability, and the study of mathematical models. Together with the discovery of formal languages and their systems, all these new concepts led to the creation of mathematical logic ^[5].

Mathematical logic opened up new perspectives for natural science. The theories of natural sciences were presented as axiomatically formalized systems. The question was to correlate the natural language of theory, in which empirical facts were described, with their mathematical and logical structures. This issue is discussed in syntactic and semantic concepts of science in connection with the problem of reconstruction of scientific theories so far.

Syntactic view: an example from the gas molecular kinetic theory.

The syntactic view (Vienna Circle, logical empiricism) on the subject area of science is that its language, including, for example, physical concepts and natural language, can be expressed using an axiomatic method as a set of sentences in a symbolic logical language for a given subject area ^[6]. The syntactic interpretation of the natural sciences required their formal axiomatic construction according to Hilbert's recipe and the reduction of mathematics to symbolic logic.

The theory was a partially interpreted axiomatized system, its language was divided into a theoretical language, a language of correspondence, and a language of observation. Theoretical propositions are, for example, the laws of thermodynamics, which are composed of theoretical concepts (temperature, pressure, volume), this also includes the calculus of an accepted axiomatic system, for example, algebra. The propositions of correspondence connected the propositions of theory with the propositions of observation, the latter setting the semantics of the propositions of theory.

 Thus, the language of molecular kinetic theory for gases included the observation sentences following from Boyle's particular law, PV = const. This law was based on experiments with a real gas, a glass cube with specified geometric characteristics – length, width and height - was used to measure the volume of the released gas. The observation proposals recorded the volume of the cube, the pressure and the temperature of the gas, for example, the gas pressure was such and such, a glass cube was used in the laboratory to measure the volume of gas. The empirical content of the concepts of volume as a geometric characteristic and pressure as a force pressing on a gas, which was fixed in the correspondence sentences, which, in turn, gave meaning to the ideal concepts of pressure, temperature and volume in the axiomatized equation of the ideal gas state PV = nRT ^[7]. Carnap described similar sentences and believed that they were all described in logical terms ^[8]. Mathematical concepts in physical laws, as well as the relationship between them in sentences, were expressed in the language of first-order predicate logic with equality. The axioms of logic were connected with the formalized propositions of scientific theory, and by deduction, particular concepts (consequences) of analytical judgments were derived from them, which were then correlated with the propositions of observation. Mathematics was reduced to logic.

Thus, a scientific theory was understood as a syntactically formulated set of theoretical propositions (axioms, theorems and laws) together with their interpretation using appropriate sentences.

Further analysis of the logical and methodological problems of specific scientific theories by proponents of the semantic approach (van Fraasen, Suppes) revealed the shortcomings of the syntactic approach. He was reproached for the gap between theoretical description and empirical statements, leading to the identification of many theoretical statements with many empirically verifiable consequences, the more axioms and postulates were introduced, the more empirically verifiable consequences were required. Criticism was also conducted for recognizing only the only language of formalization and axiomatization, which was the logic of first-order predicates with equality, and the theory using this logic was declared the standard of scientific knowledge.

Nevertheless, the syntactic approach is of lasting importance for the philosophy of science, since it was an attempt to find the elementary foundations of scientific knowledge, which were analytical judgments, and to reduce complex theoretical descriptions to them. The methods of analyzing scientific knowledge were formal logical methods using symbolic languages and the axiomatic method, which, using a system of axioms, provided a link between formalized concepts and statements of theory. Note that interest in the syntactic approach has not faded to this day, for example, Friedman's work, where he attempts to axiomatize the special theory of relativity ^[9], or his reconstruction of Carnap's work ^[10].

A semantic view.

Semantic view (Suppes, van Fraasen) the reconstruction of the structure of scientific theory was due to the fact that in the middle of the XX century.  Most logicians and mathematicians have recognized that there is an alternative way to axiomatize theory – the construction of a set-theoretic predicate, unlike logical empiricists who used symbolic algebra in logic, while maintaining the axiomatic architecture of the theory in the style of Hilbert.

Suppes' proposal was to apply the set-theoretic predicate to empirical sciences, to present it as a formula of the language of set theory specific to each science. There are differences in the approaches of Suppes and van Fraasen. Suppes focuses on describing mathematical structures satisfying the set-theoretic predicate, while van Fraassen focuses on the models themselves, trying to minimize the role of the axioms of the theory. Both of them agree that reconstructing the structure of a theory means presenting it as a class of models ^{[11, p. 64]}.

The concept of a model was borrowed from Tarski's logic, it is a logical construction expressed in the language of mathematics. The model is constructed in various ways, including using the set-theoretic predicate of Suppes.

Suppes believes that mathematics is the main tool for the philosophy of science, and not the metamathematics of logical empiricists ^[12]. The construction of a set-theoretic predicate means the inclusion in set theory of both the entire mathematical apparatus of natural science theory and logic. Let's say general relativity depends on Riemann geometry, differential calculus, tensor calculus, and real numbers. All these theories do not need to be formalized step by step and axiomatized step by step (this was what logical empiricists used to do), theories are set by set-theoretic resources. Let U be a mathematical structure that can be constructed from basic sets, then the predicate P consists of two parts, the first is U from the basic sets, and the second is a union of the axioms of the mathematical structure. Let's denote a?, . . ., a? by sets of U, then we can write the predicate P as P(U; a?, . . ., a?). All kinds of structures corresponding to A are predicate P by definition. Finally, the Suppes predicate is written as the formula P(x) xx?. . . ?x?P(x?; x?, . . . , x?) ^{[13, 120 ? 126 p.]}.

The Suppes predicate and non-relativistic quantum mechanics.

What is the structure of a physical theory constructed using the Suppes predicate? We will try to answer this question. As an example, we can describe the Suppes predicate for nonrelativistic quantum mechanics using Zermelo–Frenkel set theory (ZF) without the axiom of choice. Krause ^[14] formulated a set-theoretic predicate and identified five axioms for this field of knowledge. Our strategy will be as follows: first we will identify the axioms, and then we will interpret them and show how the mathematical structures of physics (representations of Hilbert space) are "translated" into the language of non-classical logic using set theory. Next, we will highlight the stages of building a scientific theory.

From Krause's point of view, the mathematical formula for non-relativistic quantum mechanics in set theory would be:

QMNR = <S, {Hi}, {Ai j}, {Tik}> i?I, j?J, k?K  (1)

In this formula, we denote nonrelativistic quantum mechanics as QMNR. S there are many physical systems, {Hi} a collection of Hilbert spaces, {Aij} the set of Hermitian operators in the space Hi, and {Tik} the set of unitary operators in Hi, {Tik} ? {Aij}.

Now we formulate the axioms (1 5) for this formula:

(1). For each physical system s ? S. This means that the Hilbert space for a given physical system is associated with a set of Hilbert spaces: Hs ? {Hi}. The vectors |?I of this space represent the states of a physical system. The state vectors must be normalized, i.e. multiplied by complex numbers, all normalized vectors must represent the same state of a quantum mechanical system. So, k.|?I (for any complex number k) represents the same state as |?I. When there is a system consisting of several elements of S, it is associated with the tensor product of the Hilbert spaces of the constituent elements of the system. If the cardinal of a subset of systems is equal to n (let's call them s?, . . ., s?), then the Hilbert space H is equal to H = Hsi ? . . . ? Hs?. A typical vector of this space is written as |?I > ? . . . ?|?N >, or just |?I > . . . |?n>. The last formula is an axiom.

(2). Let |?(t) represent the state of a quantum mechanical system at time t. For each |?I, you need to associate the unitary operator Ts in such a way that for any moment of time t is executed:

|?(t)> = Ts(t).|?(0)>, where |?(0)> (2)

This equation represents the unitary evolution in time of the vector of state, which is called the Schrodinger equation, which is an axiom.

(3). The Hermitian operator A corresponds to the scalars ?i, which is written using the formula:

  A|?i> = ?i.|?i>  (3)

The formula is an axiom. It is assumed that Hermitian operators represent observable physical quantities that can be measured in a physical system in a certain state.

(4). It is known that any Hermite operator A is diagonalizable, which means that we can find a basis {|ai>} for the Hilbert space under consideration formed by engineering vectors A. So, for any state |?I, we can write:

|?i> = ?ci|?i>  (4),

 where ?ci = < ai|? are the Fourier coefficients. Thus, |Ci|2= Pi represents the probability that dimension A will get the value ?i. This postulate is known as Born's rule, it is also an axiom.

 (5). The Dirac equation (equivalent to the Schrodinger equation) is an axiom.

Construction of non-relativistic quantum mechanics QM NR using the Suppes predicate.

The construction of the theory from the point of view of the semantic view is reduced to the following procedures:

1. A mathematical structure is being formed, in this case, the set-theoretic predicate of Suppes, representing sets of physical systems, Hilbert spaces, Hermitian and unitary operators, acts as the formula of nonrelativistic quantum mechanics QMNR. This mathematical formula is subject to further axiomatization (in a logical sense).

2. The language of mathematical structure is being sought, with the help of which the axioms of logic are formulated, giving symbolic representations. This language is the theory of Zermelo–Frenkel sets without the axiom of choice.

3. Five axioms are formulated. They represent the postulates of non-relativistic quantum mechanics, in this case they are mathematical formulas of Hilbert space, the vector of state, its evolution over time in the notation of Schrodinger or Dirac, Hermitian operators.  Now mathematical formulas need to be "translated" into the language of logic of set theory ZF, for example, the state vector of a quantum mechanical system, which is described using the mathematical apparatus of Hilbert space, is written in symbolic form in set theory as | >. . . |?n>.

4.                  The axioms of logic act as descriptions of mathematical models or set-theoretic predicate. Note that many logical concepts have an interpretation not only in mathematics (formal), but also in physical theory (meaningful). Thus, the vectors |?I of the Hilbert space denote everything that we know about the state of a physical system (for example, the quantum numbers of an electron), and Hermitian operators are associated with observable quantities.

5.                  The mathematical structures of nonrelativistic quantum mechanics (mathematical representations of Hilbert space) become models of axioms, and the theory itself becomes a family of models. The structure of the theory appears as a hierarchy of models.

The study of mathematical structures, which can be models (in a logical sense) of scientific theories, can be viewed from several points of view. Theoretical structures can be described in terms of state spaces (Lloyd, van Fraasen, Weisberg ^[15], understood as relational systems ^[16], or considered as set-theoretic predicates (Suppes and Sneed) ^[17]. The most common version is the third one ^[18].

The semantic approach of Suppes understands a specific theory as a class of models (set-theoretic structures in logic), as they are given in some set theory, satisfying the axioms of the theory and serving to identify the theory.

The structure of non-relativistic quantum mechanics QM NR in a semantic view.

The structure of the theory can be considered as a hierarchy of models ^[19]. It highlights various levels of axiomatization in it: from the highest to the lowest – axioms, theory models, then experimental models and data models. Let's give an interpretation to the structural levels of the theory:

1.                  The axioms of the theory. They define the structure of a scientific theory, on the one hand, they are related to its postulates, and on the other hand, they define set-theoretic predicates and its models. The choice of an axiom is related to intuition, da Costa says about this: intellectual knowledge always requires intuition, explains: mathematicians do not have a clear "vision" of either transfinite cardinals or a set of real numbers, but only the intuition of a system of relations that implicitly define these concepts through axiomatic systems ^{[20, p. 54]}. Further, he argues in the style of constructivism, believing that this intuition has a constructive nature, all formal representations begin with intuition, but then they go beyond the intuitive core and begin to become more complicated, leading to the construction of increasingly complex concepts, and then the axiomatic method becomes our "autopilot".

2.                  Models of the theory. The identification of many models of the theory is connected with the mathematical theory of measurement, which establishes a way to move from qualitative empirical concepts of theory to quantitative representations. To do this, experimental procedures must be axiomatized using special sections of algebra and put forward a mathematical model that includes a set of numbers, relations and operations. The correspondence of the experimental model to the numerical model is proved using the representation theorem. It turns out that between the postulates of the theory and its subject area lies a whole class of models, which was ignored by supporters of the syntactic approach. The proof of the representation theorem for various classes of models is associated with the construction of a set-theoretic predicate ^[21].

3.                  Experimental models and data models. The theory of experiment planning and its criteria are used to set goals and objectives and analyze the experiment. Statistical compliance tests and parameter estimation in the context of theoretical models are used to build data models. There are complex correspondences (homophism, isomorphism) between experimental models and theory models, measurement theory and data models. This correspondence is based on an extension of the methodology of the representation theorem, which extends to the entire structure of scientific theory, covering theory models, experimental models and data models ^[22]. However, the most difficult thing is to correlate data models with reality itself ^[23]. Therefore, representatives of the semantic approach, such as Lediman and van Fraasen ^[24] believe that the semantic approach alone cannot solve this problem.

Thus, in order to explicate the structure of non-relativistic quantum mechanics, its axioms were defined and interpreted using non-classical logic. Highlighting the stages of theory construction made it possible to better understand the relationship between logic and mathematics between mathematical structures and axioms of logic, which consists in the fact that mathematical structures become models of axioms, and the theory itself becomes a family of models. Mathematics includes non-classical logic. Further consideration of the structure of the theory as a hierarchy of models reveals the relationship between the axioms, postulates of the theory, its models, experimental models, and data models. The whole hierarchical structure is connected by the methodology of the representation theorem.

Conclusion.

The focus was on how the basic structures of logic and mathematics are involved in the construction of scientific theories and how they relate to the natural language of theory. For this purpose, a comparative analysis was made between the syntactic and semantic approaches in the philosophy of science. The focus of the research was also on the structure of scientific knowledge, which was understood in different ways by representatives of these approaches. In the syntactic view, the structure of the theory was understood as a linguistic construct, and the connection between logic and mathematics was that the theoretical level of knowledge included mathematical descriptions of theoretical concepts and laws, which were expressed in the language of logic of predicates of the first order with equality. The connection between mathematical and logical concepts was provided by the axiomatic method, which acted as a tool of cognition, a means of formalizing mathematical structures. Mathematics was "translated" into the language of symbolic logic. Mathematics was reduced to logic. The theoretical language was connected by means of correspondence sentences with observation sentences, which were analytical judgments S is P, expressed in natural language. This structure of the theory was too simple and did not take into account the variety of model representations of the theory, which gave rise to many linguistic constructs.

  The semantic view tries to overcome this disadvantage. He puts forward two approaches to understanding the structure of the theory: state space and model theory, the latter was developed using the set-theoretic predicate of Suppes, which is the focus of this study to illustrate the connection between logic, mathematics and natural language. For this purpose, the reconstruction of non-relativistic wave mechanics was carried out and it was shown how the construction of a scientific theory takes place and its structure was highlighted.

The Suppes predicate implies an even greater use of formal languages to describe the mathematical structures of the theory, usually the resources of set theory as sets of abstract objects (relations or functions) are used to formalize mathematical structures. Logic and mathematics are related to each other in such a way that metamathematics or linguistics is a part of mathematics. Mathematics includes set theory and model theory, i.e. mathematical logic. A set-theoretic predicate is a single generative mathematical system in which mathematical structures are associated with the axioms of logic. The axioms of logic play a twofold role in cognition: first, they act as descriptions of mathematical models, and secondly, they describe the postulates of a theory, as which its basic laws are used. Axioms are chosen intuitively, and this gives rise to further philosophical discussions. As soon as the axioms become explicit, their models can be defined, and they, in turn, can be applied to real systems, thereby providing the semantics of the axioms.  A specific system satisfying these five axioms is the physical system of non-relativistic quantum mechanics. The structure of the theory is interpreted as a hierarchy of models. The connection between theory models, experimental models, and data models is provided by the representation theorem. The connection of the model with phenomena and their expression in natural language remains problematic. There is also a disadvantage in the syntactic approach. In our opinion, this problem is basically insoluble, since it is impossible to reduce a flexible and imaginative natural language to mathematical formalisms and axioms.