The specifics of scientific laws. scientific law

1. The concept of scientific law: the laws of nature and the laws of science

Scientific knowledge acts as a complexly organized system that combines various forms of organization of scientific information: scientific concepts and scientific facts, laws, goals, principles, concepts, problems, hypotheses, science programs etc.

Scientific knowledge is a continuous process, i.e. a single developing system of a relatively complex structure, which formulates the unity of stable relationships between the elements of this system. Structure scientific knowledge can be depicted in various sections and therefore in the totality of its specific elements.

central link scientific knowledge is a theory. In the modern methodology of science, the following main elements of the theory are distinguished.

1. Initial principles - fundamental concepts, principles, laws, equations, axioms, etc.

2. Idealized objects - abstract models essential properties and connections of the studied subjects (for example, “absolute black body», « ideal gas" etc.).

3. The logic of the theory is a set of established rules and methods of proof aimed at clarifying the structure and changing knowledge.

4. Philosophical attitudes and value factors.

5. A set of laws and statements derived as consequences from the main provisions of this theory in accordance with specific principles.

A scientific law is a form of ordering scientific knowledge, consisting in the formulation of general statements about the properties and relationships of the studied subject area. Scientific laws are an internal, essential and stable connection of phenomena, causing their orderly change.

The concept of scientific law began to take shape in the 16th-17th centuries. during the creation of science in the modern sense of the word. For a long time it was believed that this concept is universal and applies to all areas of knowledge: each science is called upon to determine the laws and, on their basis, describe and explain the phenomena under study. The laws of history were discussed, in particular, by O. Comte, K. Marx, J.S. Mill, G. Spencer. At the end of the 90th century, W. Windelband and G. Rickert put forward the idea that, along with generalizing sciences, which have as their task the discovery of a scientific law, there are individualizing sciences that do not formulate any laws of their own, but represent the objects under study in their uniqueness and uniqueness.

The main features of scientific laws are:

Need,

universality,

repeatability,

Invariance.

In scientific knowledge, the law is presented as an expression of the necessary and general relationship between observed phenomena, for example, between charged particles of any nature (Coulomb's law) or any bodies that have mass (the law of gravity) in physics. In various currents of modern philosophy of science, the concept of law is compared with the concepts (categories) of essence, form, purpose, relationship, structure. As the discussions in the philosophy of science of the 20th century showed, the properties of necessity and generality (in the limit - universality) included in the definition of the law, as well as the correlation of classes of "logical" and "physical" laws, the objectivity of the latter are still among the most pressing and complex problems. research

A law of nature is a certain unconditional (often mathematically expressed) law natural phenomenon which is performed under familiar conditions always and everywhere with the same necessity. Such an idea of ​​the law of nature developed in the 17th-18th centuries. as a result of the progress of the exact sciences at the stage of development of classical science.

The universality of the law means that it applies to all objects in its field, acts at any time and at any point in space. Necessity as a property of a scientific law is determined not by the structure of thinking, but by the organization real world, although it also depends on the hierarchy of statements included in the scientific theory.

In the life of a scientific law, which captures a wide range of phenomena, three characteristic stages can be distinguished:

1) the era of formation, when the law functions as a hypothetical descriptive statement and is tested primarily empirically;

2) the era of maturity, when the law is fully confirmed empirically, acquired its systemic support and functions not only as an empirical generalization, but also as a rule for evaluating other, less reliable statements of the theory;

3) the era of old age, when it already enters the core of the theory, is used, first of all, as a rule for evaluating its other statements and can only be left together with the theory itself; the verification of such a law concerns, first of all, its effectiveness within the framework of the theory, although it still retains the old empirical support received during its formation.

At the second and third stages of its existence, a scientific law is a descriptive-evaluative statement and is verified like all such statements. For example, Newton's second law of motion for a long time was the actual truth.

It took many centuries of persistent empirical and theoretical research to give it a rigorous formulation. Now the scientific law of nature appears within the framework of Newton's classical mechanics as an analytically true statement that cannot be refuted by any observations.

Interpretation of the phenomena of nature around us and social life constitutes one of the most important tasks of natural science and social sciences. Long before the emergence of science, people tried in one way or another to explain the world around them, as well as their own mental characteristics and experiences. However, such explanations, as a rule, turned out to be unsatisfactory, since they were often based either on the animation of the forces of nature, or on belief in supernatural forces, God, fate, etc. Therefore, they, at best, could satisfy the psychological need of a person in search of some or answers to questions that tormented him, but did not at all give a true idea of ​​​​the world.

True explanations, which should be called truly scientific, arose with the advent of science itself. And this is quite understandable, since scientific explanations are based on precisely formulated laws, concepts and theories that are absent in everyday knowledge. Therefore, the adequacy and depth of explanation of the phenomena and events around us is largely determined by the degree of penetration of science into the objective laws governing these phenomena and events. In turn, the laws themselves can only be truly understood within the framework of an appropriate scientific theory, although they serve as the conceptual core around which the theory is built.

Of course, one should not deny the possibility and usefulness of explaining some everyday phenomena on the basis of an empirical generalization of observed facts.

Such explanations are also considered real, but they are limited only to ordinary, spontaneous-empirical knowledge, in reasoning based on the so-called common sense. In science, not only simple generalizations, but also empirical laws are tried to be explained with the help of perfect theoretical laws. Although real explanations can be very diverse in their depth or strength, nevertheless, they must all satisfy two essential requirements.

Firstly, any true interpretation must be based in such a way that its arguments, argumentation and specific characteristics have a direct relationship to those objects, phenomena and events that they explain. The fulfillment of this request is the necessary prerequisite for considering the explanation adequate, but this circumstance alone is not enough for the fidelity of the interpretation.

Secondly, any interpretation must be fundamentally verifiable. This request has an extremely important meaning in natural science and experimental sciences, as it makes it possible to sort out truly scientific explanations from all kinds of purely speculative and natural philosophical constructions that also claim to explain real phenomena. The fundamental verifiability of an explanation does not at all preclude the use as arguments of such theoretical principles, postulates, and laws that cannot be verified directly empirically.

It is only necessary that the clarification provides the potential for deriving individual results that allow experimental testing.

Based on knowledge of the law, a reliable prediction of the course of the process is likely. "To know the law" means to reveal one or another side of the essence of the object under study, the phenomenon. Knowledge of the laws of organization is the main task of the theory of organization. In relation to the organization, law is a necessary, significant and permanent connection between the elements of internal and external environment, which determines their ordered change.

The concept of law is close to the concept of regularity, which can be considered as some kind of "extension of the law" or "a set of laws interrelated in content that provide a stable trend or aspiration for changes in the system."

Laws differ in degree of generality and scope. Universal laws reveal the relationship between the most universal properties and phenomena of nature, society and human thinking.

A scientific law is a formulation of the objective connection of phenomena and is called scientific because this objective connection is known by science and can be used in the interests of the development of society.

A scientific law formulates a constant, repetitive and necessary connection between phenomena and, therefore, we are not talking about a simple coincidence of two series of phenomena, not about randomly discovered connections, but about such a causal interdependence when one group of phenomena inevitably gives rise to another, being their cause.

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“A scientific law is a statement (statement, judgment, proposition) that has the following characteristics:

1) it is true only under certain conditions;

2) under these conditions, it is true always and everywhere without any exceptions (an exception to the law that confirms the law is dialectical nonsense);

3) the conditions under which such a statement is true are never fully realized in reality, but only partially and approximately.

Therefore, one cannot literally say that scientific laws are found in the reality being studied (discovered). They are invented (invented) on the basis of the study of experimental data in such a way that they can then be used in obtaining new judgments from these judgments about reality (including for predictions) in a purely logical way. By themselves, scientific laws cannot be confirmed and cannot be refuted empirically. They can be justified or not, depending on how well or poorly they fulfill the above role.

Take, for example, the following statement: “If in one institution a person is paid more for the same work than in another institution, then the person will go to work in the first of them, provided that for him work in these institutions does not differ in anything except salary ". The part of the phrase after the words "on that condition" fixes the condition of the law. Obviously, there are no jobs that are the same in everything except the salary. There is only some approximation to this ideal from the point of view of this or that person. If there are cases when a person goes to work in an institution where the salary is lower, then they do not refute the statement in question. In such cases, obviously, the condition of the law is not fulfilled. It may even be that, in observed reality, people always choose to work in institutions with lower pay. And this should not be interpreted as an indicator of the fallacy of our assertion. This may be due to the fact that in such institutions other circumstances of work are more acceptable (for example, shorter working hours, less workload, there is an opportunity to do some of their own business). In such a situation, the statement in question can be excluded from the number of scientific laws as inoperative , unnecessary.

From what has been said, it should be clear that a statement that simply generalizes the results of observations cannot be considered a scientific law.

For example, a person who had to walk through the chain of command and observe different types of bosses can conclude: "All bosses are grabbers and careerists." This statement may or may not be true. But it is not a scientific law, because the conditions are not specified. If the conditions are any or indifferent, this is a special case of the conditions, and this must be indicated. But if the conditions are indifferent, then any situation will give an example of completely realizable conditions of this kind, and the concept of a scientific law cannot be applied to this case.

Usually, as conditions, those conditions are fixed in the sense mentioned above, but only some specific phenomena that can actually be observed. Take, for example, the following statement: "In the case of mass production of products, its quality is reduced, provided that there is mediocre management of this branch of production, there is no personal responsibility for quality and personal interest in maintaining quality." Here the condition is formulated in such a way that examples of such conditions can be given in reality. And the possibility of cases when mass production of products is associated with an increase in its quality is not ruled out, because there are some other strong reasons that are not indicated in the condition. Such statements are not scientific laws. These are simply general statements which may be true or false, may be supported by examples and refuted by them.

Speaking of scientific laws, we must distinguish between what are called the laws of things themselves, and the statements of people about these laws.

The subtlety of this distinction lies in the fact that we know about the laws of things only by formulating some statements, while we perceive the laws of science as a description of the laws of things. However, the distinction here can be made quite simply and clearly. The laws of things can be written in a variety of ways language means, including statements like "All men are deceivers", "Click a mare on the nose, she will wave her tail", etc., which are not scientific laws. If in a scientific law we separate its main part from the description of conditions, then this main part can be interpreted as fixing the law of things. And in this sense, scientific laws are statements about the laws of things.

But singling out scientific laws as special linguistic forms is a completely different orientation of attention compared to the question of the laws of things and their reflection. The similarity of phraseology and the apparent coincidence of problems create here difficulties that are completely inadequate to the banality of the very essence of the matter.

Distinguishing between scientific laws and the laws of things, one must obviously distinguish between the consequences of both. The consequences of the former are statements deduced from them according to general or special (accepted only in a given science) rules. And they are also scientific laws (though derivative of those from which they are derived). For example, it is possible to build a sociological theory in which, from certain postulates about the individual's desire for irresponsibility for his actions to other individuals who are with him in relation to the commonwealth, statements about the tendency of individuals to be unreliable (not to keep given word, do not keep someone else's secret, squander someone else's time).

The consequences of the laws of things, fixed by the laws of science, are not the laws of things, but certain facts of reality itself, to which scientific laws refer. Let us take, for example, the law according to which there is a tendency to appoint not the most intelligent and talented people, but the most mediocre and averagely stupid people, but who are pleasing to the authorities in other respects and who have suitable connections, to leadership positions. Its consequence is that in a certain field of activity (for example, in research institutions, in educational institutions, in management art organizations, etc.) leading positions in most cases (or at least often) are occupied by people who are stupid and mediocre from the point of view of business interests, but cunning and dodgy from the point of view of career interests.

People at every step face the consequences of social laws. Some of them are subjectively perceived as accidents (although strictly logically the concept of randomness is not applicable here at all), some are surprising, although they occur regularly. Who has not heard and even spoken about the appointment of a certain person to a leading position: how could such a scoundrel be appointed to such a responsible post, how could such a cretin be entrusted with such a thing, etc. But one should be surprised not by these facts, but by those when smart, honest and talented people get to leadership positions. This is indeed a departure from the law. But it's not a coincidence either. Not randomness, not in the sense that it is natural, but in the sense that the concept of randomness is again inapplicable here. By the way, the expression "responsible post" is absurd, because all posts are irresponsible, or only an indication of the high rank of the post makes sense.

Zinoviev A.A., Yawning heights / Collected works in 10 volumes, Volume 1, M., "Tsentrpoligraf", 2000, p. 42-45.

The specificity of the empirical hypothesis, as we found out, is that it is probabilistic knowledge, is descriptive, that is, it contains an assumption about how the object behaves, but does not explain why. Example: the stronger the friction, the more heat is released; metals expand when heated.

empirical law- this is already the most developed form of probabilistic empirical knowledge, with the help of inductive methods, fixing quantitative and other dependences obtained empirically, when comparing the facts of observation and experiment. This is its difference as a form of knowledge from theoretical law- reliable knowledge, which is formulated with the help of mathematical abstractions, as well as as a result of theoretical reasoning, mainly as a result of a thought experiment on idealized objects.

Law is a necessary, stable, recurring relationship between processes and phenomena in nature and society. The most important task scientific research- to raise experience to the universal, to find the laws of a given subject area, to express them in concepts, theories. The solution to this problem is possible if the scientist proceeds from two premises:

Recognition of the reality of the world in its integrity and development,

Recognition of the lawfulness of the world, that it is permeated with a set of objective laws.

Main function science, scientific knowledge - the discovery of the laws of the studied area of ​​reality. Without establishing laws, without expressing them in a system of concepts, there is no science, and there can be no scientific theory.

The law is a key element of the theory, expressing the essence, deep connections of the object under study in all its integrity and concreteness as a unity of the manifold. The law is defined as a connection (relationship) between phenomena, processes, which is:

Objective, because it is inherent in the real world,

Essential, being a reflection of relevant processes,

Internal, reflecting the deepest connections and dependencies of the subject area in the unity of all its moments,

Repetitive, stable as an expression of the constancy of a certain process, the sameness of its action in similar conditions.

With changing conditions, the development of practice and knowledge, some laws leave the stage, others appear, and the forms of operation of laws change. The cognizing subject cannot reflect the whole world as a whole, he can only approach this by formulating certain laws. Every law is narrow, incomplete, wrote Hegel. However, without them, science would stop.

Laws are classified according to the forms of matter movement, according to the main spheres of reality, according to the degree of generality, according to the mechanism of determination, according to their significance and role, they are empirical and theoretical.

Laws are interpreted unilaterally when:

The concept of law is absolutized,

When the objective character of laws is ignored, their material source,

When they are not considered systematically,

The law is understood as something unchanging,

The boundaries within which certain laws are valid are violated,

A scientific law is a universal, necessary statement about the connection of phenomena. General form The scientific law is as follows: for any object from the investigated field of phenomena, it is true that if it has property A, then it necessarily also has property B.

The universality of the law means that it applies to all objects in its field, acting at any time and at any point in space. The necessity inherent in scientific law is not logical but ontological. It is determined not by the structure of thinking, but by the structure of the real world itself, although it also depends on the hierarchy of statements included in the scientific theory. (Ivin A.A. Fundamentals of social philosophy, pp. 412 - 416).

Scientific laws are, for example, the following statements:

If a current flows through a conductor, a magnetic field forms around the conductor;

If a country does not have a developed civil society, it does not have a stable democracy.

Scientific laws are divided into:

Dynamic laws, or patterns of rigid determination, which fix unambiguous connections and dependencies;

Statistical laws, in the formulation of which the methods of probability theory play a decisive role.

Scientific laws related to broad areas phenomena, have a clearly expressed dual, descriptive-prescriptive character, they describe and explain a certain set of facts. As descriptions, they must correspond to empirical data and empirical generalizations. At the same time, such scientific laws are also standards for evaluating both other statements of the theory and the facts themselves.

If the role of the value component in scientific laws is exaggerated, they become only a means for streamlining the results of observation, and the question of their correspondence to reality (their truth) turns out to be incorrect. And if the moment of description is absolutized, scientific laws appear as a direct and only possible reflection of the fundamental characteristics of being.

One of the main functions of scientific law is to explain why a particular phenomenon occurs. This is done by logically deriving the given phenomenon from some general position and asserting the so-called initial conditions. This kind of explanation is usually called nomological, or explanation through the enveloping law. The explanation can be based not only on a scientific law, but also on a random general position, as well as the assertion of a causal connection. Explanation through scientific law has the advantage of giving the phenomenon a necessary character.

The concept of scientific law arises in the 16th - 17th centuries, during the formation of science. Science exists where there are patterns that can be studied and predicted. Such is an example of celestial mechanics, such is most of social phenomena, especially economic ones. However, in the political, historical sciences, linguistics, there is an explanation based not on a scientific law, but a causal explanation or understanding based not on descriptive, but on evaluative statements.

Scientific laws are formulated by those sciences that use comparative categories as their coordinate system. They do not establish the scientific laws of science, which are based on a system of absolute categories.

scientific laws

A law is a theoretical conclusion that reflects the steady repetition of certain phenomena. When approving a law, we, as it were, arbitrarily separate some part of the set accessible to us, study it thoroughly and draw some general conclusions on the basis of this. It turns out that our conclusions are based on insufficient information. However, a person has intuition and the ability to abstract thinking. Thus arose the first legal conclusions attributed to Hermes Trismegistus: what is below corresponds to what is above; and that which is above corresponds to that which is below, to work the wonders of the one thing. The similarity in the view of the ancient thinkers concerned not only the external texture, but also the internal, deep content of things and concepts. In this sense, the division established by us exists only on the surface or physical layer, while analogy as a form of associative connection, on the contrary, unites the existent, but already from a multidimensional position. Moreover, this law-like principle asserts not only structural similarity, or isomorphism, but also spiritual affinity, which today is still outside the sphere of interest of academic science.

Another equally important law explaining the interaction of a system and an element is the principle of holography, the discovery of which is associated with the names of D. Gabor (1948), D. Bohm and K. Pribram (1975). The latter, while doing research on the brain, came to the conclusion that the brain is a large hologram, where memory is contained not in neurons and not in groups of neurons, but in nerve impulses circulating throughout the brain, and just like a piece of a hologram contains everything the entire image without significant loss of information quality. The physicist J. Zucarelli (2008) also came to similar conclusions, who transferred the principle of holography to the field of acoustic phenomena. Numerous studies have established that holography is inherent in all structures and phenomena of the physical world without exception.

A further development of the relationship between part and whole is the principle of fractality, discovered by B. Maldenbrot in 1975 to designate irregular self-similar sets: a fractal is a structure consisting of parts that are in some sense similar to the whole. Thus, as in holography, the main property of a fractal is self-similarity. Fractality is inherent in all natural phenomena, as well as artificial, including mathematical structures. Moreover, if holography speaks of a functional or informational similarity, then fractality confirms the same on the example of graphic and mathematical images.

Critical importance for knowledge of the world is the principle of hierarchy. The term "hierarchy" (from the Greek sacred and power) was introduced to characterize the organization of the Christian church. Later, in the 5th century, Dionysius the Areopagite expands his interpretation in relation to the structure of the universe. He believed, not without reason, that physical world is a coarsened analogue of the heavenly world, where there are also levels or layers that obey general laws. The term “hierarchy”, as well as “hierarchical levels”, turned out to be so successful that later it began to be successfully used in sociology, biology, physiology, cybernetics, general theory systems, linguistics.

Any systems in their hierarchy exist fully as such only when they rely on the subjects of all their relations. In all other cases, they are available as objects with much less certainty. It must be borne in mind that there is a certain limiting number of elements of a particular level, the decrease or increase of which eliminates the level as such, where the philosophical law of the transition of quantity into quality operates, which is the most common reason for the formation of other levels of the hierarchy.

Below we will consider statistical laws in more detail, but here we point out that E. Schrödinger believed that all physical and chemical laws occurring inside organisms are statistical and manifest themselves with a large number of interacting elements. With a decrease in the number of elements below the Nth, this law simply ceases to operate. However - note - in this case, other laws are updated, which, as it were, take the place of the lost ones. In nature, nothing can be acquired without losing, and, on the contrary, any loss is accompanied by new acquisitions, writes Schrödinger (Schrödinger E. What is life? From the point of view of a physicist. - M .: Atomizdat, 1972. - 96p.). Violation of statistical reliability with a small number of elements leads to an increase in the individual role of each of them with a corresponding actualization of their inherent personal properties. Within the framework of the theory of catastrophes, the idea arose that with a small change in equilibrium (at bifurcation points), sharp reversals of the systemic status can occur. After choosing one of the possible paths, the trajectory of development, there is no way back, unambiguous determinism operates, and the development of the system again becomes predictable until the next point.

The laws of science display regular, recurring connections or relationships between phenomena or processes in the real world. Until the second half of the 19th century, universal statements were considered to be the true laws of science, revealing regularly repeated, necessary and essential connections between phenomena. Meanwhile, regularity may not be universal, but existential in nature, i.e. does not apply to the whole class, but only to a certain part of it. Hence, all laws are divided into the following types:

Universal and particular laws;

Deterministic and stochastic (statistical) laws;

Empirical and theoretical laws.

It is customary to call universal laws that reflect the universal, necessary, strictly repeating and stable nature of a regular connection between phenomena and processes of the objective world. For example, this is the law thermal expansion physical bodies, which is on quality language can be expressed using the sentence: all bodies expand when heated. More precisely, it is expressed in quantitative language through the functional relationship between temperature and increase in body size.

Particular or existential laws are either laws derived from universal laws or laws reflecting the regularities of random mass events. Particular laws include the law of thermal expansion of metals, which is secondary or derivative in relation to the universal law of expansion of all physical bodies.

Deterministic and stochastic laws are distinguished by the accuracy of their predictions. Stochastic laws reflect a certain regularity that occurs as a result of the interaction of random massive or repetitive events, such as throwing a dice. Such processes are observed in demography, insurance business, analysis of accidents and catastrophes, population statistics and economics. Since the middle of the 19th century, statistical methods have been used to study the properties of macroscopic bodies consisting of a huge number of microparticles (molecules, atoms, electrons). At the same time, it was believed that statistical laws could, in principle, be reduced to deterministic laws inherent in the interaction of microparticles. However, these hopes were dashed with the advent of quantum mechanics which proved:

That the laws of the microcosm have a probabilistic-statistical character;

That the accuracy of measurement has a certain limit, which is set by the principle of uncertainties or inaccuracies of W. Heisenberg: two conjugate quantities of quantum systems, for example, the position and momentum of a particle, cannot be simultaneously determined with the same accuracy (in connection with which Planck's constant was introduced).

So, among the laws, the most common are causal, or causal, which characterize the necessary relationship between two directly related phenomena. The first of these, which causes or gives rise to another phenomenon, is called the cause. The second phenomenon, representing the result of the action of the cause, is called the effect (action). At the first empirical stage of research, the simplest causal relationships between phenomena are usually studied. However, in the future, one has to turn to the analysis of other laws that reveal deeper functional relationships between phenomena. This functional approach is best realized when discovering theoretical laws, which are also called the laws of unobservable objects. It is they that play a decisive role in science, since with their help it is possible to explain empirical laws, and thus the numerous individual facts that they generalize. The discovery of theoretical laws is an incomparably more difficult task than the establishment of empirical laws.

The path to theoretical laws lies through the advancement and systematic testing of hypotheses. If, as a result of numerous attempts, it becomes possible to deduce an empirical law from a hypothesis, then there is a hope that the hypothesis may turn out to be a theoretical law. Even greater confidence arises if, with the help of a hypothesis, it is possible to predict and discover not only new important, previously unknown facts, but also previously unknown empirical laws: the universal law gravity was able to explain and even clarify the laws of Galileo and Kepler, empirical in their origin.

Empirical and theoretical laws are interrelated and necessary stages in the study of the processes and phenomena of reality. Without facts and empirical laws it would be impossible to discover theoretical laws, and without them to explain empirical laws.

Laws of logic

Logic (from the Greek word, concept, reasoning, mind) is the science of the laws and operations of correct thinking. According to the basic principle of logic, the correctness of reasoning (conclusion) is determined only by its logical form, or structure, and does not depend on the specific content of the statements included in it. The distinction between form and content can be made explicit by means of a particular language or symbolism, and is relative and depends on the choice of language. A distinctive feature of a correct conclusion is that it always leads from true premises to a true conclusion. Such a conclusion makes it possible to obtain new truths from existing truths with the help of pure reasoning, without resorting to experience, intuition.

scientific proof

Since the time of the Greeks, to say "mathematics" means to say "proof", so aphoristically Bourbaki defined his understanding of this issue. Immediately, we point out that the following types of evidence are distinguished in mathematics: direct, or by enumeration; indirect evidence of existence; proof by contradiction: the principles of the largest and smallest number and the method of infinite descent; proof by induction.

When we meet mathematical problem to prove, we have to remove doubts about the correctness of a clearly formulated mathematical statement A - we must prove or disprove A. One of the most entertaining problems of this kind is to prove or disprove the hypothesis of the German mathematician Christian Goldbach (1690 - 1764): if an integer is even and n greater than 4, then n is the sum of two (odd) prime numbers, i.e. every number starting from 6 can be represented as the sum of three prime numbers. The validity of this statement for small numbers can be checked by everyone: 6=2+2+2; 7=2+2+3, 8=2+3+3. But to check for all numbers, as required by the hypothesis, of course, is impossible. Some other proof than just verification is required. However, despite all efforts, no such proof has yet been found.

Holbach's statement, writes D.Poya (Poya D. Mathematical discovery. - M .: Fizmatgiz, 1976. - 448s.) is formulated here in the most natural form for mathematical statements, since it consists of a condition and a conclusion: its first part, starting the word "if", is a condition, the second part, beginning with the word "then", is the conclusion. When we need to prove or disprove a mathematical proposition formulated in the most natural form, we call its condition (premise) and conclusion the main parts of the problem. To prove a sentence, it is necessary to find a logical link connecting its main parts - a condition (premise) and a conclusion. To refute the proposition, one must show (if possible, then by counterexample) that one of the main parts, the condition, does not lead to the other, the conclusion. Many mathematicians have tried to remove the veil of obscurity from Goldbach's conjecture, but to no avail. Despite the fact that very little knowledge is required to understand the meaning of the condition and the conclusion, no one has yet been able to establish a strictly argued connection between them, and no one has been able to give an example that contradicts the hypothesis.

So, proof – logical form thought, which is the justification of the truth of a given proposition by means of other propositions, the truth of which is already substantiated, or self-evident. Since only one of the forms of thought we have already considered, namely, a judgment, has the property of being true or false, then the definition of proof is about it.

Proof is a truly rational, thought-mediated form of reflection of reality. Logical connections between thoughts are much easier to detect than between the objects themselves, about which these thoughts speak. Logical connections are more convenient to use.

Structurally, the proof consists of three elements:

Thesis is a position, the truth of which should be substantiated;

Arguments (or grounds) - positions, the truth of which has already been established;

A demonstration, or method of proof, is a kind of logical connection between the arguments themselves and the thesis. Arguments and thesis, insofar as they are judgments, can be correctly connected with each other either according to the figures of a categorical syllogism, or according to the correct modes of a conditionally categorical, divisive-categorical, conditionally dividing, purely conditional, or purely divisive syllogisms.

Aristotle distinguished four types of evidence:

Scientific (apodictic, or didascal), substantiating the truth of the thesis strictly, correctly;

Dialectical, or polemical, i.e. those who substantiate the thesis in the process of a series of questions and answers to them, clarifications;

Rhetorical, i.e. substantiating the thesis only by seeming the right way, in essence, this justification is only probable;

Eristic, i.e. justifications that only seem probabilistic, but in essence are false (or sophistical).

The subject of consideration in logic are only scientific, i.e. correct evidence regulated by this science.

Deductive proofs are common in mathematics, theoretical physics, philosophy and other sciences that deal with objects that are not directly perceived.

Inductive proofs are more common in the sciences of applied, experimental and experimental character.

According to the type of connections between arguments and thesis, evidence is divided into direct, or progressive, and indirect, or regressive.

Direct evidence- those in which the thesis is justified by arguments directly, directly, i.e. the arguments used play the role of premises of a simple categorical syllogism, where the conclusion from them will be the thesis of our proof. To emphasize the obvious advantage, sometimes direct evidence is called progressive.

Let's use an example from study guide V.I. Kobzar. (Kobzar V.I. Logic in questions and answers, 2009), replacing the heroes.

To prove the thesis: “My friend is taking an exam in the history and philosophy of science”, the following arguments should be given: “My friend is a university graduate student” and the following: “All university graduate students take an exam in the history and philosophy of science.”

These arguments allow you to immediately get a conclusion that coincides with the thesis. In this case, we have a direct, progressive proof, consisting of one inference, although the proof may consist of several conclusions.

This same proof can also be formulated in a slightly different form, as a conditionally categorical syllogism: "If all university graduate students pass the exam in the history and philosophy of science, then my friend also passes the exam, because he is a graduate student." Here, in the conditional proposition, the general proposition is formulated, and in the second premise, in the categorical proposition, it is established that the basis of this conditional proposition is true. According to the logical norm: if the basis of a conditional proposition is true, its consequence will necessarily be true, i.e. we get our thesis as a conclusion.

An example of a direct proof is the justification of the proposition that the sum of the interior angles of a triangle in a plane is equal to two right angles. True, in this proof there is also visibility, obviousness, since the proof is accompanied by drawings. The reasoning is as follows: let us draw a straight line through the vertex of one of the angles of the triangle, parallel to its opposite side. In this case, we get equal angles, for example, No. 1 and No. 4, No. 2 and No. 5 as lying crosswise. Angles #4 and #5, together with angle #3, form a straight line. And in the end it becomes obvious that the sum of the interior angles of the triangle (#1, #2, #3) is equal to the sum of the angles of a straight line (#4, #3, #5), or two right angles.

Another thing - circumstantial evidence, analytical, or regressive. In it, the truth of the thesis is substantiated indirectly, by substantiating the falsity of the antithesis, i.e. position (judgment) that contradicts the thesis, or by excluding all members of the disjunctive judgment, except for our thesis, which is one of the members of this disjunctive judgment, according to the dividing-categorical syllogism. In both cases, it is necessary to rely on the requirements of logic for these forms of thought, on the laws and rules of logic.

Thus, when formulating an antithesis, care must be taken to ensure that it is really contrary to the thesis, and not opposite to it, because the contradiction does not allow the simultaneous truth or falsity of these judgments, and the opposite allows their simultaneous falsity.

In case of contradiction, the justified truth of the antithesis acts as a sufficient reason for the falsity of the thesis, and the justified falsity of the antithesis, on the contrary, indirectly justifies the truth of the thesis. The justification of the falsity of the position opposite to the thesis is not a sufficient basis for the truth of the thesis itself, since the opposite judgments can be false at the same time. Indirect evidence is usually used when there are no arguments for direct evidence, when it is impossible to different reasons justify the thesis directly.

For example, having no arguments to directly substantiate the thesis that two lines parallel to a third are parallel to each other, they admit the opposite, namely, that these lines are not parallel to each other. If this is so, then they will intersect somewhere and thus will have a common point for them. In this case, it turns out that two lines parallel to it pass through a point lying outside the third line, which contradicts the previously justified position (only one line parallel to it can be drawn through a point lying outside the line). Consequently, our assumption is wrong, it leads us to absurdity, to a contradiction with the already known truth (previously proven position).

There are indirect proofs when the substantiation of the fact that the desired object exists occurs without a direct indication of such an object.

VL Uspensky gives the following example. In some chess game, the opponents agreed to a draw after White's 15th move. Prove that one of the black pieces has never moved from one square of the board to another. We argue as follows.

The movement of black pieces on the board occurs only after black's move. If such a move is not castling, one piece moves. If the move is castling, two pieces move. Black managed to make 14 moves, and only one of them could be castling. Therefore, the most a large number of There are 15 black pieces affected by the moves. But there are only 16 black pieces. This means that at least one of them did not participate in any of Black's moves. Here we do not specifically indicate such a figure, but only prove that it exists.

Second example. The plane is carrying 380 passengers. Prove that some two of them celebrate their birthday on the same day of the year.

We reason like this. In total there are 366 possible dates for celebrating a birthday. And there are more passengers. This means that it cannot be that all of them have birthdays on different dates, and it must certainly be that some date is common to two people. It is clear that this effect will necessarily be observed starting with the number of passengers equal to 367. But if the number is 366, it is possible that the dates and months of their birthdays will be different for everyone, although this is unlikely. By the way, probability theory teaches that if a randomly selected group of people consists of more than 22 people, then it is more likely that some of them will have the same birthday than that they all have birthdays on different days of the year.

The logical device used in the example with the passengers of the plane is named after the famous German mathematician Gustav Dirichlet. Here is the general formulation of this principle: if there are en boxes containing a total of at least en + 1 items, then there is bound to be a box containing at least two items.

Can be offered direct evidence existence of ir rational numbers- for example, indicate "the number is the root of 2", and prove that it is irrational. But it is possible to offer such indirect evidence. The set of all rational numbers is countable, but the set of all real numbers is uncountable; this means that there are also numbers that are not rational, i.e. irrational. Of course, one must also prove that one set is countable and the other is uncountable, but this is relatively easy to do. As for the set of rational numbers, one can explicitly indicate its recalculation. As for the uncountability of the set of real numbers, then it - with the help of representing real numbers in the form of infinite decimal fractions- can be deduced from an uncountable set of all binary sequences.

Here it should be clarified that an uncountable set is called countable if it can be recalculated, i.e. name some element of it first; some element that is different from the first - the second; some different from the first two - the third and so on. Moreover, not a single element of the set should be omitted during the recalculation. An infinite set that is not countable is called uncountable. The very fact of the existence of uncountable sets is very fundamental, since it shows that there are infinite sets, the number of elements in which is different from the number of elements of the natural series. This fact was established in the 19th century and is one of the greatest achievements of mathematics. Note also that the set of all real numbers is uncountable.

Evidence by contradiction

This type of evidence will be illustrated by the following example. Let a triangle and its two unequal angles be given. It is required to prove statement A: a large side lies opposite a large angle.

Let us make the opposite assumption B: the side lying in our triangle opposite the larger angle is less than or equal to the side lying opposite the smaller angle. Assumption B conflicts with the previously proven theorem that in any triangle there are equal angles opposite equal sides, and if the sides are not equal, then opposite larger side lies a larger angle. Hence, assumption B is false, but statement A is true. It is interesting to note here that the direct proof (that is, not by contradiction) of Theorem A turns out to be much more complicated.

So the evidence to the contrary stands in this way. make the assumption that statement B is true, the opposite, i.e. the opposite of the assertion A that needs to be proved, and further, relying on this B, come to a contradiction; then they conclude that it means that B is wrong, but A is right.

principle of greatest number

To scientific evidence include the principles of the largest and smallest number and the method of infinite descent. Let's consider them briefly.

The principle of the greatest number states that in any non-empty finite set natural numbers find the largest number.

The principle of least number: in any non-empty (and not only finite) set of natural numbers, there is a smallest number. There is also a second formulation of the principle: there is no infinite decreasing (that is, one in which each subsequent term is less than the previous one) sequence of a natural number. Both formulations are equivalent. If there were an infinite decreasing sequence of natural numbers, then among the members of this sequence there would be no smallest. Now imagine that we managed to find a set of natural numbers in which the smallest number is absent; then for any element of this set there is another, smaller one, and for it, an even smaller one, and so on, so that an infinite decreasing sequence of natural numbers arises. Consider examples.

It is required to prove that any natural number greater than one has a prime divisor. The number in question is divisible by one and itself. If there are no other divisors, then it is prime, which means that it is the desired prime divisor. If there are other divisors, then we take the smallest of these others. If it is divisible by something other than one and itself, then this something would be an even smaller divisor of the original number, which is impossible.

In the second example, we need to prove that for any two natural numbers there is a greatest common divisor. Since we agreed to start the natural series from one (and not from zero), then all divisors of any natural number do not exceed this number itself and, therefore, form a finite set. For two numbers, the set of them common divisors(i.e. such numbers, each of which is a divisor for both numbers under consideration) is all the more finite. Finding the largest among them, we obtain the desired.

Or, suppose that in the set of fractions there is no irreducible. Let's take an arbitrary fraction from this set and reduce it. We will also reduce the resulting one, and so on. The denominators of these fractions will become smaller and smaller, and an infinite decreasing sequence of natural numbers will appear, which is impossible.

This variant of the method by contradiction, when the emerging contradiction consists in the appearance of an infinite sequence of decreasing natural numbers (which cannot be), is called the method of infinite (or unlimited) descent.

Proofs by induction

Method mathematical induction is used when one wants to prove that a certain statement holds for all natural numbers.

The proof by induction begins with the fact that two statements are formulated - the basis of induction and its step. There are no problems here. The problem is to prove both these statements. If this fails, our hopes for the application of the method of mathematical induction are not justified. But if we are lucky, if we succeed in proving both the basis and the step, then we obtain the proof of the universal formulation without any difficulty, applying the following standard argument.

Statement A(1) is true because it is the basis of the induction. Applying the inductive step to it, we obtain that assertion A (2) is also true. applying the inductive step to A (2), we find that A (3) is true. Applying the inductive step to A (3), we obtain that assertion A (4) is also true. in this way we can go to each value of en and verify that A(en) is true. Therefore, for every en, A(en) holds, and this is the universal formulation that was required to be proved.

The principle of mathematical induction is, in essence, the permission not to carry out standard reasoning in each individual situation. indeed, the standard reasoning has just been justified in general view, and there is no need to repeat it every time in relation to one or another specific expression A (en). Therefore, the principle of mathematical induction allows one to make a conclusion about the truth of the universal formulation, as soon as the truth of the basis of induction and the inductive transition are established. (V.L. Uspensky, op. cit., p. 360-361)

Necessary explanations. Statements A (1), A (2), A (3), ... are called particular formulations. Statement: for any en, A (en) takes place - a universal formulation. The induction basis is a particular formulation of A (1). The step of induction, or the induction step, is the assertion: whatever en is, the truth of the particular statement A (en + 1) follows from the truth of the particular statement A(en).

Refutation of evidence

The problem of substantiating knowledge is directly related to the question of refutation of evidence. The fact is that of the actions with proof, only one of them is best known, namely, negation.

The denial of a proof is its refutation. A refutation is a substantiation of the falsity or inconsistency of one or another element of evidence, i.e. or thesis, or arguments, or demonstration, and sometimes all of them together. This topic is also well covered in the manual by V.I. Kobzar.

Many properties of a refutation are determined by the properties of a proof, because a refutation is structurally almost the same as a proof. In refuting the thesis, the refutation necessarily formulates the antithesis. Refuting arguments, others are put forward. Refuting the demonstration of evidence, they reveal a violation in it of the relationship between the arguments and the thesis. At the same time, the refutation as a whole must also demonstrate by its structure the strict observance of logical connections between its arguments and its thesis (i.e., antithesis).

Justification of the truth of the antithesis can be considered both as a proof of the antithesis and as a refutation of the thesis. On the other hand, the justification of the inconsistency of the arguments does not yet prove the falsity of the thesis itself, but only indicates the falsity or insufficiency of the arguments presented to substantiate the thesis, only rejects them, although it is quite possible that there are arguments in favor of the thesis, and there are even many of them, but for various reasons they are in evidence was not used. Thus, it is not always correct to call refutation of arguments anti-proof.

So it is with the refutation of the demonstration. Justifying the incorrectness (illogicality) of the connection of the thesis with the arguments, or the connection between the arguments in the proof, we only point out the violation of logic, but this does not negate either the thesis itself or the arguments that were given. Both that, and another can appear quite comprehensible - it is only necessary to find more correct direct or mediated connections between them. Therefore, not every refutation can be called a refutation of the proof as a whole, more precisely, not every refutation rejects the proof as a whole.

According to the types of refutation (refutation of the thesis, refutation of arguments and refutation of the demonstration), one can also indicate the methods of refutation. So, the thesis can be refuted by proving the antithesis and by deriving consequences from the thesis that contradict the obvious reality, or the system of knowledge (principles and laws of the theory). Arguments can be refuted both by justifying their falsity (arguments only seem to be true, or are uncritically accepted as true), and by justifying that the arguments given are not enough to prove the thesis. You can also refute by justifying the fact that the arguments used themselves need to be justified.

You can also refute by establishing that the source of facts (reasons, arguments) to substantiate the thesis is unreliable: the effect of forged documents.

There are quite a lot of ways to refute a demonstration due to the multitude of the demonstration rules themselves. A refutation may indicate a violation of any rule of inference, if the arguments of the proof are not connected according to the rules, either premises or terms. A refutation can reveal a violation of the connection of arguments with the thesis itself, pointing to a violation of the rules of categorical syllogism figures and their modes, indicating a violation of the rules of conditional and disjunctive syllogisms.

Here it is useful to give falsification??