Realism and Antirealism in Mathematics
The history of realism and antirealism has its origins in medieval philosophy. It seemed that the confrontation between realism and antirealism based on the disagreement on the problem of universals lost its urgency a long time ago. However, this opposition was resumed in the twentieth century in the context of reasoning on the basis of logic and mathematics in fairly acute forms within almost two hundred years. Certainly, the problem of status and relations of the notion of general and individual in logic and mathematics is still relevant today as the debate between realists and antirealists is actively continuing. Besides, there is no good reason to hope that the debates will stop because of the victory of one of the parties in the foreseeable future. On the contrary, each of the parties is finding more and more sophisticated and compelling arguments in favor of the validity of own position, and finds weaknesses in the arguments of the opponent. Analysis and subject assessment of this discussion imply that it is more promising to find the third, middle line, which could be able to avoid both realism and antirealism weaknesses.
The key aspect that differentiates realism from antirealism is the mode of existence of logical and mathematical objects, which, despite its apparent simplicity, proved to be very complicated. The bond between the requirement of existence with some a priori necessary and sufficient conditions for the recognition of the existence of objects can lead to “metaphysical nihilism” (Azzouni, 2010). Meanwhile, a statement of the mode of “existence” has always been firmly inscribed in a specific context, namely ontological, sociological, mereological, etc. It directly depends on the cognitive abilities of the subject as well as its language, mental and other characteristics because every time it relates to abstract objects.
The strong version of realism, which, in fact, is identical to the traditional Platonism, affirmatively replies to the question of the mode of existence of mathematical objects. It refers to the actual existence regardless of the language that describes these objects or the mental state of a person who comprehends the nature or his/her activity. Similarly, it approves the availability of truth values of certain mathematical propositions regardless of the subject of mathematical activity (the semantic version of realism). In other words, it is an actual hyper-sensual, transcendental reality in which these objects are submerged. In terms of its perception, this reality has the same status for the subject as the reality of its own feelings.
The comparison of objects of supersensible reality and Platonic ideas is permissible only in limited terms and is not always perfect in this case. However, the term “Platonism” primarily points to the remote origins of modern realism. Logical and mathematical research from the standpoint of Platonism is “the opening” of the relevant objects and their properties. Moreover, the methods of proof that can evidently point to the existence of these objects (or properties) without determining the way of their construction (e.g., the so-called theorem of pure existence) are permitted. Objects are considered to be known if the rules of action regarding them are defined.
More specifically, this position requires:
- Objective existence of mathematical reality;
- Objective existence of the elements of this reality;
- Independence of these elements from the cognitive activity of the subject and its abilities.
- Values of mathematical propositions are given by the truth conditions that are specified by the features of mathematical reality.
In this sense, mathematics is considered to be a science concerning structures, procedures, and relations that arose in the process of computing practices, measurements, and descriptions of forms of real and abstract objects as well as relations between them, and which is based on logical proofs and numerical calculations.
Despite the appeal of realism to the supersensible reality, its supporters are actively searching for empirical support and perceive its presence in the indispensability argument, which connects the existence of this kind of reality with the fact of indispensability (super efficiency) of mathematics in science (Batterman, 2010). However, for many logicians and mathematicians who are distant from philosophical meditations concerning science base, but are related to the main keynote of Platonism, the indispensability argument is associated with a set of mathematical principles and methods, which only allowed for qualified mathematical research.
It means the use of the following:
- Classical primary formal languages;
- Classical logic;
- Non-structural methods and non-constructive axioms;
- Impredicative definitions.
This group of logical-mathematical community adheres to the natural line of epistemological optimism, which is expressed in the belief that every problem is eventually solvable. It refers to the Platonism of the “ordinary” consciousness of mathematicians that is adopted by them not due to some ideological reasons but based on utilitarian considerations.
According to the traditional Platonism, abstract logical-mathematical objects do not have space-time characteristics and cannot be comprehended in terms of cause-and-effect relationships that permeate objects of the physical origin. If a subject is treated in mathematics as a study of some abstract entities and logical-mathematical reality as a system of abstract objects (numbers, functions, sets, categories, etc.), then this type of realism is usually called objective Platonism. In this case, it is believed that the set theory and the number theory only describe these objects (Balaguer, 2009).
The fact that the supersensible Platonic reality exists is difficult to believe as it does not find itself in any of the physical effects. Its presence can be only postulated as, indeed, consistent realists do. The appeal to the argument of indispensability in an empirical study to a rather artificial argument does not add confidence in the sense of ascertaining the existence of the super-sensuous reality.
The mathematical reality is objective and independent in relation to a person as a component of culture, but it is historically conditioned and predetermined as a component of a person’s inner world, which defines its cognitive abilities and activity vector. It is not necessary to search for the philosophical foundations of mathematics, as the traditional mathematical realism does, or postulate the existence of a supersensible reality that opens a priori intuition. Moreover, from the perspective of social constructivism, mathematics naturally appears as a typical result of the creative activity of a person as well as a historically predetermined mathematical practice (Cole, 2008). Similarly, search and improvement motives of evidence in logic and mathematics can be discerned not in a “search for truth” factor but in an appeal to the scientific community in order to test reliability and validity of the path connected with the thesis formulation of proof (for example, the theorem).
Modern antirealism includes a range of different approaches and directions. Usually, it consists of conventionalism, which interprets the expression of logical-mathematical language as analytically true propositions, Frege’s formalism, metamathematics, Curry’s formalism, and fictionalism of Field, who considers the logical-mathematical expressions as non-relative to reality (Balaguer, 2009).
Nominalism as a major kind of antirealism in its naturalistic expression has two mutually independent forms: the denial of the possibility of the existence of some (or all) abstract objects and the denial of the possibility of the existence of universals (general concepts). It may even mean that any logical-mathematical objects, such as numbers, functions or sets are deprived of any mode of existence. In fact, according to Musgrave, the main aspect does not concern objects but relationships between statements (Musgrave, 1999).
It is accepted to allocate the following types of nominalism:
- Predicate nominalism (things are combined by the presence of a certain property);
- Conceptual nominalism (things are under a single concept);
- Similarity nominalism (things are alike in certain respects);
- Mereological nominalism (includes the presence of some property as a part, which excels in a holistic formation);
- Instrumental nominalism (constants are applied only to concrete objects and quantifiers “run” only on sets of object data and their properties) (Rosen 2011).
In the framework of modern nominalism, the nature and the main purpose of mathematical knowledge is associated with its communicative function, whereas the logical-mathematical language is considered to be saturated with a kind of metaphors that enable efficient communication of the corresponding community about some abstract designs. It is so-called “abstract expressionism,” which seeks to avoid a number of conceptual difficulties inherent to nominalism. Since the implementation of this communication function is possible only because the mathematical community believes in reliability, connectivity, and the order of conclusions via and in the context of the abstract language, the nominalism position can be more precisely called “belief expressionism” (Liggins, 2013).
Reflection on realism and antirealism leads to the conclusion that, in a certain sense, it is permissible to talk about the need to introduce the concept of meta-ontology, which refers to a set of questions relating to the actual ontology, such as their meaningfulness, objectivity, and complexity levels of its analysis. Only in the context of meta-ontology, mathematics can be naturally represented as a set of metaphorical proposals relating to a “reality” that allows a range of different interpretations.
The Third Line
The combination of provisions of structuralism and antirealism (nominalism) allows designating a median – the third line between realism and nominalism in the grounds logic and mathematics, withdrawing their opposition, and shedding new light on the real mode of existence of mathematical objects. It must be mentioned that such attempts have already been made, but it is usually estimated as not entirely successful as it proved that it was basically a separate improvement of reasoning of realism or nominalism. The cause of these failures was due to these attempts that implicitly meant only double (internal and external) determination of mathematics. The idea of triple determination of mathematics suggests further progress of neurobiological and psychological studies of cognitive processes. According to the idea of the third line, patterns are basic, elementary formations that form mathematical reality. Meanwhile, in this reality, the human intellect cuts what is presented by nature of its activities and previous experience. In this sense, it should be mentioned the normativity of abstract objects of mathematics and an a priori process of mathematical creativity is due to the activity of the subject logical-mathematical cognition. However, the cognitive abilities of the subject of logical-mathematical cognition are historically conditioned and mathematics can be regarded as an essential element of human culture and human activity. This element is as versatile and defined as the other elements in a specific historical period and a particular social environment. Cognitive abilities of the subject are determined not only by the current and previous activities but also by its biological predetermination, double (internal and external) determination of the mind, and perceptual components. In this case, there is a direct crossing of the philosophy of science (and its socio-cultural dimension) and the philosophy of mind.
Indeed, the specificity of human perception sense is that, in the perceptual space, the segment is more of a simple object rather than a point. Besides, in the process of object perception, there are integration and synthesis of individual neurons, and data integral configuration are formed via which the brain operates and through the “prism” of which it analyzes the reality. It is understandable because in the process of evolution distinguishing large visual configurations that are primary in terms of the formation of an image of the external object was important for survival. Thus, Cantor’s theorem on embedded intervals that is underlying the theory of real numbers (any sequence of decreasing bounded sets of real numbers has a non-empty intersection) is compelled by features of (perceptual) space. In its turn, Cauchy’s theorem on intermediate values of a continuous function on the interval corresponds to people’s innate representation of continuity. Analysis of perceptual features of infants indicates that even when they have not acquired any visual experience, they still choose some form of another. Hence the selection of forms occurs at the level of the deep, inborn brain structures. The subsequent evolution of intelligence certainly corrects selection mechanisms. If to consider the existence of such structures, the logical-mathematical languages are like the second stage of the neurobiological structures that as a result of data synthesis of these perceptions and external configurations will ultimately determine the components of mathematical reality. The very nature of human activities through their regulatory components contributes to the formation of objects of this reality. In this case, the meaning of objective mathematics may be concluded as a constantly expandable set of ideas that will never be exhausted by a human mind, while a subjective as a product of human intelligence, its ingenious design, which it continually creates and improves.
From the enactivism perspective, mathematics is not only the process of representation of quantitative and spatial relationships of the outside world but also the process of creating a particular reality that bears the stamp of the specifics of the body organization of man. It is a process, in which representation mechanisms (external stimuli), features of its physical organization (internal stimulus), and the actual activity (with its historical conditioning and socio-cultural predetermination that forms action standards with abstract objects) are closely intertwined.
There is a triple determination of mathematical reality (as a system of logical-mathematical objects): internal, external, and activity-related (normative). It seems that this approach allows outlining the direction that is capable of overcoming the opposition of mathematical realism and antirealism in the form of a “third” path, which is smoothing “corners” of both realism and antirealism as well as removes a number of epistemological difficulties that are characteristic of the philosophy of mathematics of the last decades.