Naturalism and the philosophy of mathematics
The term naturalism is as popular in modern philosophy as it is multi-valued and elusive. Perhaps the easiest way to get close to its main feature, pointing at its opposite. Naturalism is the opposite of supranaturalism. The latter means an appeal in philosophical reasoning to the supernatural, supernatural, ie. to the sphere of religious, and metaphysical representations in their traditional understanding.
The classic formulation of the naturalistic position in modern epistemology and the philosophy of science is the article by W. Quine Naturalized epistemology (1969). The starting point for Quine's reasoning is the failure of Logicism projects (B. Russell) and logical positivism (R. Carnap): he is convinced of the impossibility of reducing scientific proposals to the language of observation, logic and set theory. The reason for this impossibility is rooted in the validity of the pragmatist view of cognition (C. Pearce, see paragraph 4.3) and the holic nature of the verification of scientific theories (P. Duhem). In this situation, we have no grounds for recognizing a higher status for epistemology (and the philosophy of science) in comparison with concrete natural theories.
However, depriving the epistemology of the status of the first philosophy does not mean its death, on the contrary - only now it gets certain outlines and a clear place in the system of human knowledge [78, p. 87-88; 22, p. 383]. Once on a level with such spiders as psychology, linguistics, biology, she gets the legal right to mutually beneficial exchange with them. Quine, however, somewhat narrowing the opening perspective, preferring to treat epistemology as a section of psychology. He writes: "The old epistemology sought, in a sense, to include (in contain ) the natural science & lt; ... & gt ;. On the contrary, epistemology in its new form is itself included in natural science as one of the chapters of psychology. At the same time, the previous claim to the inclusion of natural science within the framework of epistemology, in its own way, retains its force. & lt; ... & gt; Thus, there is mutual inclusion, although in different senses: both epistemology in natural science and natural science in epistemology. & lt; ... & gt; We are looking for an understanding of science as an institution or process taking place in the world, and we do not suppose that this understanding should be any better than the science itself, which is its object [78, p. 83-84; 22, p. 379-380].
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For naturalism, mathematics is part of human culture. The very same culture is the top floor of a three-story fundamental naturalistic pyramid: biological - social - cultural. Every next floor in it is the product of the previous one; he can not be imagined without the previous one, although we do not reduce it to him . Within the framework of such a broad scheme, different constructions are possible.
In practice, most often there is either a version of the naturalistic philosophy of mathematics, which makes the main bet on the upper, socio-cultural floors of the fundamental naturalistic pyramid (in two versions - culturological and sociological), or on its lower, biological floor (again in two versions - cognitive and evolutionary). However, in both cases, we are talking about reasoning unfolding in principle in the post-Darwinian intellectual space.
Remembering Quine, one can say that the versions of naturalism in the philosophy of mathematics differ in which area of the natural or social sciences they are oriented first: cultural anthropology, sociology, cognitive psychology or biology.
The naturalistic position is a kind of realism (see Chapter 8). However, this is a realistic, essentially anti-platonic type. It is often described as "hypothetical realism" [54, p. 156]. According to this approach, our concept of the world, both in general and in particular, is a set of hypotheses, but not accidental, but formed and tested in the course of biological and cultural evolution. This set can not be completely devoid of adequacy. As the American paleontologist D. Simpson put it, "a monkey who did not have a realistic perception of the tree branch, on which she jumped, was soon a dead monkey and thus did not become one of our ancestors. Our perceptions really give us a true, though not complete, idea of the external world, because it was and is a biological necessity, built into us by natural selection. If we were not like that, we would not be here! " [87, p. 84].
The Austrian biologist K. Lorenz wrote that the evolutionary origin of our cognitive apparatus makes it, however, in its own way one-sided and limited: "we have developed" organs "only for those aspects of Jehovah-in-themselves, what is important was taken into account to preserve the species, i.e. in those cases when selection pressure was sufficient to create this special apparatus of cognition [26, p. 249]. But in regard to this sphere, we are like a primitive sealer or whaler, who notices "only that which is of practical interest to him" [26, p. 249]. Moreover: from the biological point of view, we need to use so many mental images that we use, rather than a specific connection between the stimuli received and our responses (remember the pragmatist principle of Pierce).
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However, modern physics and mathematics represent the fruits of thinking of a high level of abstraction that lead us far beyond the network of everyday stimuli and reactions. How can evolutionary thinking with all its limitations create such theories? Abstract thinking in something like the games of young animals. The game prepares them for adulthood through pre-playing vital relationships in the "not real" of the situation. The main value of theoretical thinking from the biological point of view also consists in the ability to "lose" mentally, in the imagination, all sorts of situations before we have to face them in the "real" life. Such an anticipation substantially minimizes the risks. As you progress, mental playback can become "high-rise", generating higher and higher levels of abstraction.
However, the main principle is to try to apply in the new conditions those methods and techniques that have proved themselves well in the old. Therefore, even the highly abstract mindset of a person does not lose its connection with its basic perceptual (pertaining to sensory perception) and kinesthetic ( muscular feeling associated with movement) experience.
This topic was developed in the studies of modern cognitive psychologists. As an example, we refer to the concept of the inseparable connection between thinking and the body ( embodied knowledge) and the problems of the conceptual metaphors of the American cognitive linguist Lakoff. Together with the Chilean-Swiss-American psychologist, a specialist in problems of mathematical knowledge, R. Nunez he published a book "Where does mathematics come from?" (2000) .
Their central thesis is as follows: "Mathematics as we know it ( as we know it) is created and used by people: mathematicians, physicists, computer scientists, economists - they all are representatives of the form Homo sapiens . Perhaps this is an obvious fact, but it has one important consequence. Mathematics (as we know it) is limited and structured by the properties of the human brain and the human mind. The only mathematics that we know or can know is mathematics based on our brain and mind ( and brain-and-mind-based mathematics) [71, p. 1; 23, p. 29].
In this connection, Lakoff and Nunez formulate two main questions.
1) What exactly are the mechanisms of the human brain and consciousness that allow people to formulate mathematical ideas and build mathematical reasoning?
2) Is there a mathematics based on the brain and consciousness - everything that mathematics itself represents? Or so: is there, as Platonists supposed, free from the incarnation ( disembodied ) mathematics that transcends all bodies and consciousnesses and structures the universe (like this universe, and every possible universe)? [71, p. 1]
The first of these questions is called upon to answer cognitive science as an interdisciplinary study of consciousness, the brain, and their interrelationship. The second, according to the authors, lies outside science. The answer to it concerns the area of faith, which is akin to faith in God. Human mathematics can not be a part of platonic mathematics, because it is based on a conceptual metaphor at all levels and at all levels, the latter is specific for living beings.
Our consciousness is not a universal consciousness, it is bodily rooted ( embodied ) and is specific for a person and, perhaps, his closest biological relatives. "The specific nature of our bodies, the brain and our everyday functioning in the world determines the structure of human concepts and reasonings, including mathematical concepts and reasoning." [71, p. 5; 23, p. 31]. In addition, most of our thinking is not available for direct introspection, which means that we are not aware of it. The abstract, as a rule, is understood by us through a concrete, ie. with the use of ideas and methods of reasoning, which are rooted in the sensorimotor system. This cognitive mechanism is called a conceptual metaphor.
Conceptual metaphors permeate all mathematics. The simplest examples of such metaphors are: numbers are points on the line or numbers are sets & quot ;. In general, mathematics "piles up a metaphor for metaphor", and the task of a cognitive psychologist is to unravel their cobwebs and show how they are ultimately guided by the cognitive apparatus used in everyday thinking. For example, mathematical thinking makes possible everyday concepts such as the "set of objects in a limited area of space", "repetitive action", "movement", "rotation", "approaching the border" etc. Next to them are visual diagrams, such as the schema container (inside - border - outside). Our ability to manage our movements significantly affects the design of our conceptual system, primarily through the process scheme. The reason is already known to us - the same nervous control system is responsible for both complex bodily movement and rational output.
Mathematical knowledge is passed down from generation to generation, but - how? From the naturalistic point of view, there are two main ways of such transfer - the biological mechanisms of the transfer of genetic information and socio-cultural mechanisms (transmission through imitation and learning). At what level is math transmitted? Biological or social?
"The central element, the deep structure of mathematics," wrote an American-French mathematician and philosopher of mathematics of Israeli descent on this subject. And. In the article "The Philosophical Problems of Mathematics in the Light of Evolutionary Epistemology" (1989), it embraces the cognitive mechanisms that, like other biological mechanisms, developed as they collided with reality, and found themselves genetically anchored in the course of evolution. I will call this central structure of the logic-operational component of mathematics. On its basis, the thematic component of mathematics has grown and continues to grow, which consists of the specific content of mathematics. This second level is culturally conditioned [79, p. 58].
The first level includes innate ability of counting and orientation in space, which to some extent manifest themselves not only in humans, but also in other biological species, for example in birds. In all humans, they are the same because of the genetic identity of the biological species. Relating to this first level is, as far back as in 1941, K. Lorenz noted, the biological version of Kant's a priori : what is a priori for the individual is a posteriori for the species .
However, actually the mathematics (the second level) is an element of human culture, and consequently, it must obey the main features of socially transmitted knowledge and skills. A mathematician deals with something very real and even objective, but this reality and objectivity is not natural, but cultural, which does not deprive her of the ability to decisively determine the human individual. Along with the biological a priori there is a "historical a priori" (Michel Foucault, see Chapter 7) or a priori of culture (neo-Kantians, see paragraph 3.7).Sometimes in this connection they talk about social constructivism, implying a position according to which everything that a person is inclined to perceive as an objective reality is in fact a social construct. In fact, even the natural world, we see in many respects through the prism of our culture, which is socially conditioned; the very same cultural prism in the usual situation remains invisible to us. However, as the advocate of this approach writes, the American sociologist R. Collins: "The theory of social constructivism in relation to intellectual life is far from being antirealistic, and provides us with a whole abundance of realities. Social networks exist; there are also their material foundations-churches and schools, audiences and patrons who fed and clothed intellectuals; in addition, there are economic, political and geopolitical processes that make up the outer sphere of causality [21, p. 1114].
One of the most important features of human culture is its diversity. As Spengler noted during the First World War, there is no single human culture, but many different cultures, and therefore, there should not be one common for all people of mathematics, but a mathematician should be as much as these cultures [44, p. 151].
So, we must have many alternative mathematicians instead of one universal, just like we have, for example, many alternative religions or ethical systems. However, where are similar alternatives in mathematics? We must either present examples of real alternatives in the field of mathematics, or give a sociocultural explanation of their absence or our "blindness" to them.
In formulating this problem, the British sociologist D. Bloor tried to imagine what an alternative would look like to the mathematics we are familiar with [51, p. 95-97]: it must seem to us a mistake or an inadequacy, but to be legally rooted in its own holistic cultural context. Do we know something similar? Bloor tries to give a number of examples. True, these are not so much full-fledged alternatives as historical and cultural variations.
One of these characteristic examples is the difference in understanding numbers in Antiquity and in Modernity. For Antiquity, the number (αριθμός) is a collection of units. The unit itself is not a number, it is the principle of unity and the beginning of numbers. A characteristic representation of such a number is the collection of counting stones (ψήφοι). For this approach, a number is always a natural number. The alternative representation of the number as the length of the segment, rather than the set of counting stones, allows one to recognize the number, fractions (more precisely, different fractions representing the same length, are identified, allowing to introduce the concept of a rational number) and irrational numbers, resulting in the concept real number.
The change in the fundamental representation in arithmetic is explained, according to Bloor, by a different understanding of the relation between arithmetic and geometry, which was rooted in "past experience and current goals" [51, p. 104]. The first understanding was connected with the Pythagorean-platonic worldview. The second point of view, advocated by S. Stevin, a Flemish mechanic and engineer of the late 16th century, was a point of view of practice for which the number is a tool for measuring and calculating in the daily affairs of the "sublunary" of the world, it represented the "leveling and secularization of the number" [51, p. 107].
In the past three decades, a special direction has emerged - ethnomathematics, which considers it necessary to talk seriously about forcibly planted and implanted Eurocentrism and discrimination of other cultural traditions in mathematics. The term ethnomathematics introduced in 1984 one of the main enthusiasts of this idea, the Brazilian teacher and historian of mathematics U. D'Ambrosio . For its supporters, modern academic mathematics is just one of many traditions that originated in the Mediterranean area and, for a number of historical reasons, went far beyond its limits and imposed itself upon all mankind. They see their task in restoring the rights of other mathematicians to human mathematicians.
You can go in the solution of the problem of alternative mathematicians and, in other ways (in comparison with ethnomathematics), recognize unification and universalization in the field of mathematics as inevitable for human culture, but also raise questions about the naturalistic mechanisms for achieving it and the naturalistic reasons for its inevitability. In our opinion, in the history of European mathematics, there were regular situations that in another cultural field (such as religion or ethics) would surely lead to a split and the formation of competing alternatives. However, each time there was one of two things: 1) or the alternative did not receive sufficient development and disappeared; 2) or a new, more general point of view was developed that absorbed all the alternatives and legitimized their coexistence. An example of the first type is atomistic geometry as an alternative to the continuum geometry in the V-IV centuries. BC. . Examples of the second type are situations with non-Euclidean geometries in the 19th century. and with competing approaches in the foundations of mathematics in the early 20th century. The reasons for this tendency to get rid of alternatives are not completely clear, and it itself can serve as a serious argument against the naturalistic interpretation of mathematics.
In conclusion, it remains to note that the philosophy of mathematics is a living field of research, where different directions, schools and points of view collide. In the presented review, we tried to talk about the most notable phenomena in this area, although there are still many interesting approaches beyond the scope of the review.
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