The specificity of an empirical hypothesis, as we have found out, is that it is probabilistic knowledge and is descriptive in nature, that is, it contains an assumption about how an object behaves, but does not explain why. Example: the stronger the friction, the greater the amount of heat released; metals expand when heated.
Empirical law– this is already the most developed form of probabilistic empirical knowledge, using inductive methods to record quantitative and other dependencies obtained experimentally by comparing the facts of observation and experiment. This is what distinguishes it as a form of knowledge from theoretical law- reliable knowledge, which is formulated using mathematical abstractions, as well as as a result of theoretical reasoning, mainly as a consequence of a thought experiment on idealized objects.
Law is a necessary, stable, repeating relationship between processes and phenomena in nature and society. The most important task of scientific research is to raise experience to the level of universality, to find the laws of a given subject area, and to express them in concepts and 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 conformity of the world with laws, the fact that it is permeated by a set of objective laws.
The main function of science, scientific knowledge is 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 being studied in all its integrity and specificity as a unity of the diverse. The law is defined as a connection (relationship) between phenomena and processes, which is:
Objective, because it is inherent in the real world,
Significant, being a reflection of the relevant processes,
Internal, reflecting the deepest connections and dependencies of the subject area in the unity of all its moments,
Repeating, stable as an expression of the constancy of a certain process, the sameness of its action under similar conditions.
With changing conditions, the development of practice and knowledge, some laws disappear from the scene, others appear, and the forms of action of laws change. The cognizing subject cannot reflect the entire world; he can only get closer to it, formulating certain laws. Every law is narrow and incomplete, Hegel wrote. However, without them, science would grind to a halt.
Laws are classified according to the forms of motion of matter, 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 one-sidedly when:
The concept of law is absolutized,
When the objective nature of laws, their material source, is ignored,
When they are not considered systematically,
The law is understood as something unchangeable,
The boundaries within which certain laws are valid are violated,
A scientific law is a universal, necessary statement about the connection between phenomena. The general form of a scientific law is this: for any object from the field of phenomena being studied, 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 area, 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 current flows through a conductor, a magnetic field is formed 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 of 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 ordering 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 the only direct reflection of the fundamental characteristics of being.
One of the main functions of a scientific law is to explain why a particular phenomenon occurs. This is done by logical derivation of a given phenomenon from some general position and statements about the so-called initial conditions. This kind of explanation is usually called nomological, or explanation through a covering law. An explanation may rely not only on a scientific law, but also on an accidental general proposition, as well as on a statement of causality. An explanation through a scientific law has the advantage of giving the phenomenon a necessary character.
The concept of a 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. This is an example of celestial mechanics, such is the majority of social phenomena, especially economic ones. However, in political, historical sciences, and linguistics, there is an explanation that is not based on a scientific law, but a causal explanation or understanding that is 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 scientific laws of science, which are based on a system of absolute categories.
Scientific laws
A law is a theoretical conclusion that reflects the stable repeatability of certain phenomena. When establishing a law, we seem to 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, humans have intuition and the ability to think abstractly. This is how the first law-like conclusions arose, 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. Similarity in the minds of ancient thinkers concerned not only the external texture, but also the internal, deep content of things and concepts. In this sense, the division we establish exists only on the surface or physical layer, while analogy as a form of associative connection, on the contrary, unites existing things, but 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, no less important law that explains 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 studying the brain, came to the conclusion that the brain is a large hologram, where memory is contained not in neurons or 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. Physicist H. Zucarelli (2008) 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 ones, including mathematical structures. Moreover, if holography speaks of functional or informational similarity, then fractality confirms the same using the example of graphic and mathematical images.
The principle of hierarchy is the most important for understanding the world around us. The term "hierarchy" (from the Greek sacred and authority) was introduced to describe 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 the physical world is a roughened 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 it subsequently began to be successfully used in sociology, biology, physiology, cybernetics, general systems theory, and linguistics.
Any systems in their hierarchy exist fully as such only when they are considered the subjects of all their relationships. In all other cases they exist as objects with much less certainty. It must be borne in mind that there is a certain limiting number of elements of one level or another, the decrease or increase of which eliminates the level as such, where the philosophical law of the transition of quantity to 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 will 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. When the number of elements decreases below the Nth, this law simply ceases to apply. However, note that in this case other laws are updated, which seem to take the place of the lost ones. In nature, nothing can be acquired without losing, and, on the contrary, every loss is accompanied by new acquisitions, writes Schrödinger (E. Schrödinger. What is life? From the point of view of a physicist. - M.: Atomizdat, 1972. - 96 p.). 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 the personal properties inherent in them. Within the framework of catastrophe theory, the idea arose that with a small change in equilibrium (at bifurcation points), sharp upheavals in the system status can occur. After choosing one of the probable paths, a development trajectory, there is no turning back, unambiguous determinism operates, and the development of the system again becomes predictable until the next point.
The laws of science reflect regular, repeating connections or relationships between phenomena or processes in the real world. Until the second half of the 19th century, the true laws of science were considered to be universal statements that revealed regularly recurring, necessary and essential connections between phenomena. Meanwhile, regularity may not be universal, but existential in nature, i.e. do not apply to the entire 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 repeatable and stable nature of the regular connection between phenomena and processes of the objective world. For example, this is the law of thermal expansion of physical bodies, which can be expressed in qualitative language 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. Among the particular laws is 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 arises as a result of the interaction of random mass or repeated events, for example, throwing a die. Such processes are observed in demography, insurance, accident and disaster analysis, population statistics and economics. Since the mid-19th century, statistics began to be used to study the properties of macroscopic bodies consisting of a huge number of microparticles (molecules, atoms, electrons). 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 microworld are probabilistic and statistical in nature;
That the accuracy of measurement has a certain limit, which is established by the principle of uncertainty or inaccuracy of W. Heisenberg: two conjugate quantities of quantum systems, for example, the coordinate and momentum of a particle cannot be simultaneously determined with the same accuracy (which is why 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 them, which causes or generates another phenomenon, is called a cause. The second phenomenon, representing the result of the cause, is called a consequence (action). At the first empirical stage of research, the simplest causal relationships between phenomena are usually studied. However, in the future we have to turn to the analysis of other laws that reveal deeper functional relationships between phenomena. This functional approach is best realized through the discovery of theoretical laws, which are also called laws of unobservable objects. They play a decisive role in science, since with their help it is possible to explain empirical laws, and thereby 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 formulation 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 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 of universal gravitation was able to explain and even clarify the laws of Galileo and Kepler, which were empirical in origin.
Empirical and theoretical laws are interrelated and necessary stages in the study of 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, reason) 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 a particular language or symbolism, but is relative and dependent 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 allows one to obtain new truths from existing truths using pure reasoning, without resorting to experience or intuition.
Scientific proof
Since the times of the Greeks, to say “mathematics” means to say “proof,” so aphoristically Bourbaki defined his understanding of this issue. Here we point out that in mathematics there are the following types of proofs: direct, or by brute force; indirect evidence of existence; proof by contradiction: principles of greatest and least numbers and the method of infinite descent; proof by induction.
When we encounter a mathematical proof problem, we have to remove doubt 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 the integer is even and n is greater than 4, then n is the sum of two (odd) primes, i.e. Every number starting from 6 can be represented as the sum of three prime numbers. Anyone can check the validity of this statement for small numbers: 6=2+2+2; 7=2+2+3, 8=2+3+3. But it is, of course, impossible to test for all numbers, as required by the hypothesis. Some other proof than just verification is required. However, despite all efforts, such proof has not yet been found.
Holbach’s statement, writes D. Polya (Polya D. Mathematical discovery. - M.: Fizmatgiz, 1976. - 448 pp.) is formulated here in the most natural form for mathematical statements, since it consists of a condition and a conclusion: its first part, beginning the word “if” is a condition, the second part, beginning with the word “then” is a conclusion. When we need to prove or disprove a mathematical proposition stated in the most natural form, we call its condition (premise) and conclusion the main parts of the problem. To prove a proposition, you need to find a logical link connecting its main parts - the condition (premise) and the conclusion. To refute a proposition, you need to show (if possible, then with a 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 reasoned connection between them, and no one has been able to give an example that contradicts the hypothesis.
So, proof- a logical form of thought, which is a substantiation of the truth of a given position through other provisions, the truth of which is already substantiated, or self-evident. Since only one of the forms of thought we have already considered, namely 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 whose truth should be substantiated;
Arguments (or reasons) are provisions whose truth has already been established;
Demonstration, or method of proof, is a type of logical connection between the arguments themselves and the thesis. Arguments and thesis, since 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 conditionally categorical, divisive-categorical, conditionally divisive, purely conditional or purely disjunctive syllogisms.
Aristotle distinguished four types of evidence:
Scientific (apodictic, or didascal), substantiating the truth of the thesis strictly and correctly;
Dialectical, or polemical, i.e. those that substantiate the thesis through a series of questions and answers to them, clarifications;
Rhetorical, i.e. justifying the thesis only in a seemingly correct way, but in essence this justification is only probable;
Eristic, i.e. justifications that are only seemingly probabilistic, but are essentially false (or sophistical).
The subject of consideration in logic is 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 evidence is more common in applied, experimental and experimental sciences.
Based on 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 substantiated by arguments directly, directly, i.e. the arguments used serve as the premises of a simple categorical syllogism, where the conclusion from them will be the thesis of our proof. To emphasize the obvious advantage, direct evidence is sometimes called progressive evidence.
Let's use an example from V.I. Kobzar's textbook. (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 graduate student at a university” and the following: “All graduate students at universities are taking an exam in the history and philosophy of science.”
These arguments allow you to immediately obtain a conclusion that coincides with the thesis. In this case, we have a direct, progressive proof consisting of one inference, although the proof can consist of several inferences.
This same proof can be framed in a slightly different form, as a conditional 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 a conditional proposition, a general position is formulated, and in the second premise, in a categorical judgment, 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 substantiation of the proposition that the sum of the interior angles of a triangle on a plane is equal to two right angles. True, in this proof there is also clarity and evidence, since the proof is accompanied by drawings. The reasoning is as follows: let’s draw a straight line through the vertex of one of the corners of the triangle, parallel to its opposite side. In this case, we obtain equal angles, for example, No. 1 and No. 4, No. 2 and No. 5 lying crosswise. Angles No. 4 and No. 5, together with angle No. 3, form a straight line. And in the end, it becomes obvious that the sum of the internal angles of a triangle (No. 1, No. 2, No. 3) is equal to the sum of the angles of a straight line (No. 4, No. 3, No. 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. a position (judgment) that contradicts the thesis, or by excluding, according to the dividing-categorical syllogism, all members of the dividing judgment, except for our thesis, which is one of the members of this dividing judgment. 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 actually contradicts the thesis, and not the opposite of it, because contradiction does not allow for the simultaneous truth or falsity of these judgments, and the opposite allows for their simultaneous falsity.
In the event of a contradiction, the well-founded truth of the antithesis serves as a sufficient basis for the falsity of the thesis, and the well-founded falsity of the antithesis, on the contrary, indirectly substantiates the truth of the thesis. Justifying the falsity of a position opposite to a thesis is not a sufficient basis for the truth of the thesis itself, since opposing judgments can be false at the same time. Indirect evidence is usually used when there are no arguments for direct proof, when it is impossible for various reasons to substantiate the thesis directly.
For example, without having 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, it means that they will intersect somewhere and thus will have a common point for them. In this case, it turns out that through a point lying outside the third line, two lines parallel to it pass, which contradicts the previously substantiated position (through a point lying outside the line, only one line parallel to it can be drawn). Consequently, our assumption is incorrect, it leads us to absurdity, to a contradiction with an already known truth (previously proven position).
There are indirect evidence, when the substantiation of the fact that the desired object exists occurs without a direct indication of such an object.
V.L. Uspensky gives the following example. In a certain 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 reason as follows.
The movement of black pieces on the board occurs only after black moves. 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 have been castling. Therefore, the largest number of black pieces affected by moves is 15. 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 carries 380 passengers. Prove that some two of them celebrate their birthday on the same day of the year.
Let's think like this. There are a total of 366 possible dates for birthday celebrations. 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 the case that some date is common to two people. It is clear that this effect will definitely 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 all of them will have birthdays on different days of the year.
The logical technique used in the example with airplane passengers is named after the famous German mathematician Gustav Dirichlet. Here is a general formulation of this principle: if there are n boxes containing a total of at least n + 1 objects, then there will certainly be a box containing at least two objects.
You can offer direct proof of the existence of irrational numbers - for example, indicate “the number root of 2” and prove that it is irrational. But we can also offer such indirect evidence. The set of all rational numbers is countable, and the set of all real numbers is uncountable; This means that there are also numbers that are not rational, i.e. irrational. Of course, we still need to 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, you can explicitly indicate its recalculation. As for the uncountability of the set of real numbers, it can be deduced from the uncountability of all binary sequences by representing real numbers in the form of infinite decimal fractions.
Here it should be clarified that an uncountable set is called countable if it can be counted, i.e. name some of its elements first; some element different from the first - second; something different from the first two - the third and so on. Moreover, not a single element of the set should be skipped during recalculation. An infinite set that is not countable is called uncountable. The very fact of the existence of uncountable sets is very important, 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
We will illustrate this type of evidence with the following example. Let a triangle and its two unequal angles be given. We need to prove statement A: the larger side lies opposite the large angle.
Let's make the opposite assumption B: the side lying opposite the larger angle in our triangle is less than or equal to the side lying opposite the smaller angle. Assumption B contradicts the previously proven theorem that in any triangle, equal angles lie opposite equal sides, and if the sides are not equal, then a larger angle lies opposite the larger side. This means that assumption B is false, but statement A is true. It is interesting to note that the direct proof (that is, not by contradiction) of Theorem A turns out to be much more difficult.
Thus, the evidence by contradiction stands this way. make the assumption that statement B is true, the opposite, i.e. the opposite of the statement A that needs to be proven, and then, relying on this B, they come to a contradiction; then they conclude that it means that B is false, but A is true.
The principle of the greatest number
Scientific proofs include the principles of greatest and least numbers and the method of infinite descent. Let's look at them briefly.
The greatest number principle states that in any non-empty finite set of natural numbers there is a greatest number.
Least number principle: in any non-empty (not just finite) set of natural numbers there is a smallest number. There is a second formulation of the principle: there is no infinite decreasing (i.e. one in which each subsequent term is less than the previous) 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 not exist a smallest one. Now imagine that we managed to find a set of natural numbers in which the smallest number is missing; then for any element of this set there is another, smaller one, and for that there is an even smaller one, and so on, so that an infinite decreasing sequence of natural numbers arises. Let's look at examples.
You need to prove that any natural number greater than one has a prime factor. The number in question is divisible by one and itself. If there are no other divisors, then it is prime, which means it is the desired prime divisor. If there are other divisors, then we take the smallest of these others. If it were 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 their common divisors (i.e., such numbers, each of which is a divisor for both numbers under consideration) is even more finite. Having found the largest among them, we obtain what is required.
Or, suppose that the set of fractions does not have an irreducible one. 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 arise, which is impossible.
This version of the method by contradiction, when the contradiction that arises consists in the appearance of an infinite sequence of decreasing natural numbers (which cannot happen), is called the method of infinite (or limitless) descent.
Proofs by induction
The method of mathematical induction is used when one wants to prove that a certain statement holds for all natural numbers.
The proof by the method of induction begins with the formulation of two statements - the basis of induction and its step. There are no problems here. The problem is to prove both of these statements. If this fails, our hopes for using the method of mathematical induction are not justified. But if we are lucky, if we manage to prove both the basis and the step, then we obtain the proof of the universal formulation without any difficulty, using the following standard reasoning.
Statement A (1) is true because it is the basis of induction. Applying the induction transition to it, we find that statement A (2) is also true. Applying the induction transition to A (2), we find that A (3) is true. Applying the induction transition to A (3), we find that statement A (4) is also true. in this way we can go to each value of en and verify that A(en) is true. Consequently, for every en there is A (en), and this is the universal formulation that needed to be proved.
The principle of mathematical induction is essentially the permission not to carry out standard reasoning in each individual situation. Indeed, the standard reasoning has just been justified in general terms, and there is no need to repeat it each time in relation to this or that specific expression A(en). Therefore, the principle of mathematical induction allows one to draw 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., pp. 360-361)
Necessary explanations. Statements A (1), A (2), A (3), ... are called particular formulations. Statement: for every en there is A (en) - a universal formulation. The basis of induction is a particular formulation of A (1). The induction step, or inductive transition, is the statement: whatever en, the truth of the particular formulation A (en) implies the truth of the particular formulation A (ep + 1).
Refutation of evidence
The issue of refuting evidence is directly related to the problem of substantiating knowledge. The fact is that of the actions with evidence, only one of them is best known, namely, denial.
Negation of evidence is its refutation. A refutation is a substantiation of the falsity or inconsistency of one or another element of evidence, i.e. or a thesis, or an argument, or a demonstration, or 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 no different from a proof. Refuting a thesis, the refutation necessarily formulates an antithesis. Refuting the arguments, others are put forward. By refuting the demonstration of evidence, they discover a violation of the relationship between the arguments and thesis. At the same time, the refutation as a whole must also demonstrate in its structure strict adherence to the logical connections between its arguments and its thesis (i.e., antithesis).
Justification of the truth of the antithesis can be considered both as proof of the antithesis and as a refutation of the thesis. But substantiating the inconsistency of the arguments does not yet prove the falsity of the thesis itself, but only indicates the falsity or insufficiency of the given arguments 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, calling a refutation of arguments anti-evidence is not always correct.
The same goes for refuting the demonstration. By justifying the incorrectness (illogicality) of the connection between the thesis and the arguments, or the connection between the arguments in the proof, we only point out a violation of logic, but this does not deny either the thesis itself or the arguments that were given. Both may turn out to be quite acceptable - you just need to find more correct direct or indirect connections between them. Therefore, not every refutation can be called a refutation of the proof as a whole, or 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 demonstration), methods of refutation can also be indicated. Thus, a thesis can be refuted by proving the antithesis and by drawing consequences from the thesis that contradict obvious reality or a system of knowledge (principles and laws of theory). Arguments can be refuted both by justifying their falsity (the 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. It can also be refuted by justifying the fact that the arguments used themselves need justification.
It can also be refuted by establishing that the source of facts (grounds, 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 many rules of demonstration 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 between the arguments and the thesis itself, indicating a violation of the rules of figures of a categorical syllogism and their modes, indicating a violation of the rules of conditional and disjunctive syllogisms.
Is it useful to give falsification here??
1.2. Scientific law
Scientific law is the most important component of scientific knowledge. A scientific law represents knowledge in an extremely concentrated form. However, the goal of scientific activity in general should not be reduced only to the establishment of scientific laws, because there are also subject areas (primarily this concerns the humanities) where scientific knowledge is produced and recorded in other forms (for example, in the form of descriptions or classifications). Moreover, scientific explanation, as we will discuss further (§ 1.3), is possible not only on the basis of law: there is a whole range of different types of explanations. Nevertheless, it is the scientific law in its laconic formulation that makes the strongest impression both on the scientists themselves and on broad representatives of non-scientific activities. Therefore, scientific law often acts as a synonym for scientific knowledge in general.
The law is part of the theory, in the general theoretical context. This means that the law is formulated in the special language of a particular scientific discipline and is based on basic provisions in the form of a set of conditions under which the law is satisfied. That is, the law, despite its brief formulation, is part of the whole theory and cannot be torn out of its theoretical context. It cannot be applied to practice directly, without the theory surrounding it, and also, as is often the case, it requires for its applications the presence of certain intermediate theories, or “middle-level theories.” In other words, a scientific law is not an immediate product, always ready for use by any user.
Definition and characteristics of a scientific law
What is a scientific law? This is a scientific statement that is universal in nature and describes in a concentrated form the most important aspects of the subject area being studied.
A scientific law as a form of scientific knowledge can be characterized from two sides:
1) from the objective, ontological side. Here it is necessary to identify what features of reality are captured in the law;
2) from the operational and methodological side. Here it is necessary to identify how scientists come to understand the law, to formulate a law-like statement;
Let's move on to consider these two sides of the scientific law.
Objective (ontological) side of scientific law.
From the objective side, i.e. on the part of the referent of the theory, a scientific law is a stable, essential relationship between the elements of reality.
The stability of an attitude means that this attitude is stable, repeatable, reproducible under given unchangeable conditions.
The essence of the law means that the relationship described by the law does not reflect some random, randomly grasped properties of the described objects, but, on the contrary, the most important ones - those that determine either the structure of these objects, or the nature of their behavior (functioning) and, in general, that or explain the essence of the phenomenon being studied in a different way. The referent of a theory that includes laws is not a single object, but a certain (possibly infinite) set of objects, taken from the angle of universality; therefore, the law is formulated not for a single phenomenon, but refers to a whole class of similar objects, united in this class by certain properties.
Thus, the law fixes essential invariant relationships that are universal for a particular subject area.
What is universality of law
The universality of the law in itself is a rather complex quality. G.I. Ruzavin talks about three meanings of universality. The first meaning is universality, determined by the very nature of the concepts included in the law. Of course, there are different levels of generality of scientific concepts. Therefore, laws can be ordered on the basis of generality as more universal (fundamental) and less universal (derivative). The second sense of universality concerns spatiotemporal generality. A statement is universal in this sense if it applies to objects regardless of their spatial and temporal locations. Therefore, geological laws cannot be called universal in this sense, because characterize precisely earthly phenomena. In this case, we can talk about lower-level universality: regional and even local (or individual). Finally, the third meaning is associated with the logical form of law-like statements - with the use of a special logical operator in the formulation of the law, which allows one to speak about any “object in general.” Such an operator is called a quantifier. In universal statements, either a universal quantifier is used (for all objects of the type A there is...), or an existence quantifier (there is a certain object of the type A for which it is the case...). At the same time, laws of a lower level of universality use the existential quantifier, and fundamental laws use the universal quantifier.
In addition, the universality of a scientific law is expressed in the fact that, describing the essential aspects of a particular phenomenon, it relates directly not so much to existing phenomena, but to universal potential situations that can be realized if the appropriate conditions are met. In other words, the law, as it were, overcomes the sphere of what actually exists. Thus, K. Popper draws attention to this feature of scientific universal statements: they characterize the potential plane of reality, an objective predisposition to a particular phenomenon in the presence of appropriate conditions (such statements are called dispositions). Universal statements that play the role of scientific laws are, according to K. Popper, descriptions not so much of actually observed individual phenomena, but of potentials and predispositions.
Since it is the essential universality that must be fixed in the law, the question arises of how to distinguish genuine laws from random generalizations that only apparently have a law-like form. (For example, the statement “all the apples in this refrigerator are red” may be true without being a scientific law.) In general, this issue is not yet sufficiently clarified. But the important contribution of the American philosopher and logician N. Goodman should be noted. He also draws attention to the potential nature of laws. I. Goodman names this as a specific property of scientific laws. that conditional (or counterfactual) sentences can be derived from them, i.e. those that describe not the actual state of affairs, but what could or might happen in certain circumstances. For example, “if friction had not interfered, this stone would have continued to roll further” is a conditional statement based on the law of inertia. On the contrary, those judgments that reflect only the accidental properties of an object cannot serve as a basis for deriving counterfactual judgments from them."
Operational and methodological side of scientific law
From an operational perspective, a law can be viewed as a well-supported hypothesis. Indeed, we come to the recognition of a law after putting forward some kind of hypothesis that has a universal character, has the ability to explain a wide range of empirical data and captures the essential features of these individual facts. After carrying out some verification procedures, the scientific community accepts this hypothesis as confirmed and capable of appearing as a scientific law.
However, it should be noted that the property of the law, which is called universality, leads to certain difficulties, because universality assumes that we can apply the law to an unlimited class of homogeneous phenomena. But the very substantiation of the hypothesis is always based on a finite number of observations and empirical data. How does the transition from a finite empirical basis to a theoretical conclusion about an infinite number of applications occur? Further, where are the origins of categoricalness in the formulation of a scientific law? Do we have the right to say, for example, that “all bodies certainly expand when heated”?
This is a long-standing problem for the theory of knowledge and philosophy in general. D. Hume and I. Kant made a significant contribution to its clarification. Thus, D. Hume showed that from the observation of individual phenomena we cannot obtain a logically correct conclusion about the necessary connection of certain phenomena that underlies them. Ego means that in formulating a statement that is universal, we are doing more than simply describing an observed regularity. Moreover, this addition is not logically derived from a number of empirical data. In other words, we do not have reliable logical grounds for moving from single observations to postulating necessary connections between them.
Kant goes further than the negative results of D. Hume. I. Kant shows that the human mind, when putting forward certain universal provisions or laws, always “imposes” this or that law on nature, like a legislator, i.e. always takes an active position regarding the empirical basis. We are not simply registering a pattern that appears through empirical data, although sometimes this is exactly what it seems like, so naturally the work of a scientist looks like reading data and simply summarizing it. No, in fact, a scientist always puts forward a far-reaching judgment that fundamentally exceeds the capabilities of verification and is based on a number of prerequisite assumptions about the constancy of nature, etc. This judgment a priori anticipates an infinite series of cases, which obviously can never be fully investigated.
Of course, when putting forward a law-like hypothesis, the question arises about various kinds of necessities, but they are no longer of a general-logical nature, but of a more special, substantive nature. So, they talk about physical necessity, about causal (or causal) necessity; These shades of use of the term “necessity” are studied and clarified in modern modal logic.
Is the concept of a scientific law an anachronism?
Some modern philosophers of science argue that the very concept of law is currently not entirely successful. It refers us to the metaphysics of the 17th-18th centuries, when law was understood as something absolute, unconditional, inherent in nature with logical necessity. Today we have moved far away from such metaphysics. So, for example, says B. van Fraassen in the book “Laws and Symmetry” (1989). He raises a number of important issues concerning the status of laws in modern science. Nancy Cartwright's famous work How the Laws of Physics Lie (1983) reveals the complex context in which scientific laws operate. Thus, scientists, together with scientific laws, introduce strong idealizing assumptions and deliberately simplify the situation (including moving away from purely factual truth in itself). That is, the use of law in scientific activity is included in a rather complex practice.
It seems that it is still not worth abandoning the established concept of a scientific law in scientific practice. However, at the modern level of development of science, we really understand by laws not so much the unconditional laws of nature in the traditional metaphysical sense, but rather special theoretical constructs located in a complex context of abstract objects and abstract connections, idealizations, mental models, etc.
Scientific laws are effective theoretical constructs that perform a number of important functions in scientific knowledge.
Classification of laws
The classification of scientific laws can be carried out on various grounds. Let's indicate some ways. The simplest way is to group laws depending on the science (group of sciences) to which certain laws belong. In this regard, physical, biological, etc. laws can be distinguished.
There is, further, a division that goes back to the neo-positivist (§ 0.2) period. It is presented in a fairly clear form by R. Carnap. This is a distinction between empirical laws, in the formulation of which only observational terms are used (i.e., relating to objects that are fundamentally observable), and theoretical laws (including purely theoretical terms; such terms refer to fairly abstract objects). Despite the fact that, as we will see in § 1.4, the idea of the difference between the empirical and theoretical levels turns out to be quite complex upon closer examination, in general the division of laws into empirical and theoretical can be preserved, although today it no longer has such fundamental importance as it was in the neo-positivist period.
Finally, we note one more of the proposed classifications. It starts from the type of determinism that is expressed in certain laws. Thus, a distinction is made between deterministic (or dynamic) and statistical (or probabilistic) laws. Laws of the first type provide unambiguous characteristics of certain phenomena. Statistical laws give characteristics only in probabilistic terms: for example, in physics this applies either to mass, statistical phenomena, as, for example, in thermodynamics, or to objects of the microworld, where the probabilistic, uncertain nature of their properties also applies to individual objects, being their essential quality .
Functions of scientific laws
The most striking functions of scientific laws are explanation and prediction. Indeed, one of the most important features of theoretical thinking is the subsumption of certain phenomena under an established scientific law. Including, as we said above, it explains not only what actually takes place, but also what could happen in the presence of certain circumstances. Here the explanatory function turns into a predictive function. Further, the most important function of laws is the far-reaching unification of scientific knowledge. Thus, laws of a high degree of generality unite and systematize vast areas of knowledge.
In general, the functions of scientific laws are included in the functions of scientific theory, because the law always enters into the context of the theory, representing its fundamental provisions. We will talk about the functions of scientific theory in the appropriate place (§ 3.4).
Summary. So, a scientific law concentrates the essential, stable features of the phenomena being studied. A law is a universal statement that applies to an infinite number of individual cases corresponding to certain basic conditions. From an operational-methodological point of view, it is only a well-confirmed hypothesis, and not a logically necessary conclusion from a set of individual data. Any scientific law is a much stronger statement than those statements that would simply describe a finite collection of individual phenomena. Ultimately, theoretical reason itself “takes responsibility” for proposing a scientific law. The use of laws in scientific practice is immersed in a complex context of idealizations, assumptions, and abstract objects. Through scientific laws, descriptions, predictions, unification, etc. are carried out.
The goal of scientific knowledge is to establish the laws of science that adequately reflect reality. It is generally accepted that in nature there are objective patterns - stable, repeating connections between objects and phenomena. We will know laws - reflection of these objective laws in our consciousness. Laws are always objective in nature and express real processes that connect the phenomena of the objective world. Laws are stages of knowledge. It is customary to distinguish laws according to the degree of their generality: less general (concern a limited area of knowledge studied by specific sciences, for example, the law of natural selection); more general (affect several areas of knowledge, widespread in several related areas, for example, the law of conservation of energy); universal (fundamental laws of existence, for example, the principle of development and universal connection). The laws of functioning and the laws of development are also distinguished.
The signs of the law are universality and the necessary truth of propositions. Laws must relate to any object studied by a given science, and also adequately reflect objects and phenomena and their properties that are studied by the theory.
Development of scientific knowledge
The general course of development of science (and especially natural science, which will interest us in the future) includes the main stages of knowledge of nature and the world in general. It goes through several main steps:
1. Direct contemplation of nature as an undifferentiated whole - there is a correct embrace of the general picture of nature while neglecting particulars, which is characteristic of Greek natural philosophy;
2. Analysis of nature, dividing it into parts, isolating and studying individual things and phenomena, searching for individual causes and consequences, while the general picture of the universal connection of phenomena disappears behind the particulars - characteristic of the initial stage of the development of any specific sciences, in their historical development, for the late Middle Ages and early modern times;
3. Recreation of a complete picture on the basis of already known particulars by setting in motion what was stopped, reviving what was dead, linking what was previously isolated, that is, on the basis of combining analysis with synthesis - is characteristic of the mature period of development of specific sciences and of modern science in general.
So, it is obvious that scientific knowledge is not a once and for all given phenomenon, its volume and content are constantly changing, new hypotheses, theories are emerging and old ones are being abandoned. But what is the mechanism for the development of scientific knowledge, how do old and new relate in science, what models of the development of science exist?
Currently, three main models of historical reconstructions of science emerge most clearly:
1. The history of science as a cumulative, progressive, progressive process;
2. History of science as development through scientific revolutions;
3. The history of science as a set of individual, private situations (case studies).
All three models coexist in modern science, but they arose at different times, and this is associated with the dominance of individual models in specific periods of the development of science.
For a long time, the dominant model for the development of scientific knowledge was cumulative, closely connected with the philosophy of positivism. In science, more than in any other field of human activity, knowledge is accumulated. This circumstance became the basis for the formation of a cumulative model of the development of science. It is based on the idea that each subsequent step in science can be made only by relying on previous achievements, therefore new knowledge is always better, more perfect than the old, and more accurately reflects reality. Therefore, the previous development of science is only a preparation for its current state. Due to this circumstance, only those elements of knowledge that correspond to modern theories are significant; rejected ideas, recognized as erroneous, are nothing more than misunderstandings, delusions, deviations from the main path of development of science.
These ideas were most fully formulated in the works of E. Mach and P. Duhem at the end of the 19th century.
In connection with the general crisis of positivism - the methodological basis of the cumulative model - in the middle of the 20th century. ideas of discontinuity of development, peculiarities, and uniqueness of individual periods in the development of scientific knowledge penetrate into science. They are clearly formulated in the model of scientific revolutions.
It would be wrong to assume that before the appearance of this model in the history of science there were no ideas about scientific revolutions. Supporters of evolutionism recognized their existence, but they were either understood as accelerated evolutionary development, occurring in the same direction as the general course of development of knowledge, or they were pushed far into the past, as an absolute beginning, as a transition from pre-scientific concepts to scientific ones. In both cases, revolutions fit completely into the evolutionary movement.
The new interpretation of revolutions was based on the idea of absolute discontinuity in the development of scientific knowledge. It was assumed that the new theory emerging during the scientific revolution differed from the old one in the most fundamental way. After the revolution, the development of science begins anew and goes in a completely different direction.
This is precisely the point of view presented in the famous work of T. Kuhn “The Structure of Scientific Revolutions”. In this work, the author introduced the concept of “paradigm”, which is so often used today - scientific achievements recognized by all, which over a certain period of time provide the scientific community with a model for posing problems and solving them. Thus, Kuhn proposed a very fruitful idea that science is not a simple increment of knowledge, but a complex of knowledge of the corresponding era. Scientists whose scientific activities are based on the same paradigm are based on the same rules and standards of scientific practice. This is a prerequisite for normal science.
The transition from one paradigm to another occurs through a revolution; this is the usual model for the development of mature science (Kuhn believes that science can be considered mature since the time of Newton).
Before this, science was a cluster of small schools with different theoretical and methodological approaches. The identification of one of them led to the creation of a paradigm and marked the transition from prehistory to the history of science.
A paradigm is not just a model for blind copying, but an object for further development and concretization in new or more difficult conditions. The goal of science is to “squeeze” nature into a paradigm. It does not require the creation of new theories, but develops those with which its appearance is significantly connected. This explains the very deep study of a specific fragment of nature chosen by this paradigm.
The paradigm determines the setting up of experiments, the determination of universal constants and quantitative laws. Since during the revolution the paradigm emerges immediately as a whole, in its completed and perfect form, it does not require any significant modification; only concepts are clarified and experimental techniques are improved. On the one hand, this greatly limits the scientist’s field of vision and leads to stubborn resistance to any changes in the paradigm. Therefore, a change of paradigm is possible only with a change of generations of scientists - all supporters of the old paradigm must move away from scientific activity and give way to young ones. On the other hand, science is becoming more and more rigorous within those areas to which the paradigm directs researchers, and detailed information is accumulating. Only those who perfectly know their field of research form appropriate predictions and are able to recognize deviations from them and see anomalies against the background of the paradigm.
Only those anomalies that are evidence of a real crisis in science will lead to a new paradigm change. At the same time, awareness of the crisis situation and the exhaustion of all means presented by the old paradigm are not enough. Rejection occurs only if there is an alternative.
This approach to the scientific revolution assumes a constant division between the context of discovery and the context of confirmation of knowledge, with all efforts to invent something new, all creativity concentrated in revolutionary situations. Thus, scientific creativity is bright, exceptional flashes that determine the entire further development of science, during which previously acquired knowledge in the form of a paradigm is substantiated, expanded, and confirmed.
Activities during scientific revolutions are extraordinary (that is, extraordinary, unusual), while the work of scientists in the post-revolutionary period is normal, continuing most of the time.
As for scientific knowledge itself, the idea of scientific revolutions represented its development as absolutely discontinuous. All past history was viewed as a gradual, progressive movement towards modern theory, which today is the culmination, the pinnacle of all previous history. The next revolution comes, a new fundamental theory arises and a new radical breakdown of the past occurs, which is rebuilt as the prehistory of the new theory. Thus, every scientific theory entails the destruction of the past and the construction of history anew.
Subsequently, historians of science tried to combine models of evolutionary and revolutionary development of science. In scientific knowledge there is a pattern of unity of evolutionary and revolutionary transition from one stage of knowledge to another. During the period of evolutionary development of cognition, a process of improving knowledge occurs based on the accumulation of new facts, their systematization, the formation of laws, theories, and the development of new principles of cognition, its methods and means. Such an evolutionary process can lead to significant contradictions with the prevailing theory in science, to its replacement with a new theory, to the discovery of fundamentally new laws, and the use of new methods and means.
Methods for their discovery and justification
1. Laws and their role in scientific research.
The discovery and formulation of laws is the most important goal of scientific research: it is with the help of laws that the essential connections and relationships of objects and phenomena of the objective world are expressed.
All objects and phenomena of the real world are in an eternal process of change and movement. Where on the surface these changes seem random and unrelated to each other, science reveals deep, internal connections that reflect stable, repeating, invariant relationships between phenomena. Based on laws, science has the opportunity not only to explain existing facts and events, but also to predict new ones. Without this, conscious, purposeful practical activity is unthinkable.
The path to the law lies through a hypothesis. Indeed, to establish significant connections between phenomena, observations and experiments alone are not enough. With their help, we can only detect dependencies between empirically observed properties and characteristics of phenomena. In this way, only relatively simple, so-called empirical laws can be discovered. Deeper scientific or theoretical laws apply to unobservable objects. Such laws contain concepts that can neither be directly obtained from experience nor verified by experience. Therefore, the discovery of theoretical laws is inevitably associated with an appeal to a hypothesis, with the help of which they try to find the desired pattern. Having gone through many different hypotheses, a scientist can find one that is well confirmed by all the facts known to him. Therefore, in its most preliminary form, the law can be characterized as a well-supported hypothesis.
In his search for law, the researcher is guided by a certain strategy. He strives to find a theoretical scheme or an idealized situation with the help of which he could present in its pure form the pattern he has found. In other words, in order to formulate the law of science, it is necessary to abstract from all non-essential connections and relationships of the objective reality being studied and highlight only the connections that are significant, repeating, and necessary.
The process of comprehending the law, like the process of cognition in general, proceeds from incomplete, relative, limited truths to increasingly complete, concrete, absolute truths. This means that in the process of scientific knowledge, scientists identify increasingly deeper and more significant connections between reality.
The second significant point, which is associated with understanding the laws of science, relates to determining their place in the general system of theoretical knowledge. Laws form the core of any scientific theory. It is possible to correctly understand the role and significance of a law only within the framework of a certain scientific theory or system, where the logical connection between various laws, their application in constructing further conclusions of the theory, and the nature of the connection with empirical data are clearly visible. As a rule, scientists strive to include any newly discovered law in some system of theoretical knowledge, to connect it with other, already known laws. This forces the researcher to constantly analyze the laws in the context of a larger theoretical system.
The search for individual, isolated laws, at best, characterizes the undeveloped, pre-theoretical stage of the formation of science. In modern, developed science, law acts as an integral element of a scientific theory, reflecting, with the help of a system of concepts, principles, hypotheses and laws, a wider fragment of reality than a separate law. In turn, the system of scientific theories and disciplines strives to reflect the unity and connection that exists in the real picture of the world.
2. Logical-epistemological analysis of the concept of “scientific law”
Having clarified the objective content of the category of law, it is necessary to take a closer and more specific look at the content and form of the very concept of “scientific law”. We have previously defined a scientific law as a well-supported hypothesis. But not every well-confirmed hypothesis serves as a law. Emphasizing the close connection between hypothesis and law, we want first of all to point out the decisive role of hypothesis in the search and discovery of the laws of science.
In experimental sciences there is no other way to discover laws except by constantly putting forward and testing hypotheses. In the process of scientific research, hypotheses that contradict empirical data are discarded, and those that have a lower degree of confirmation are replaced by hypotheses that have a higher degree. Moreover, the increase in the degree of confirmation largely depends on whether the hypothesis can be included in the system of theoretical knowledge. Then the reliability of a hypothesis can be judged not only by those empirically verifiable consequences that directly follow from it, but also by the consequences of other hypotheses that are logically connected with it within the framework of the theory.
As an example, we can show how, using the hypothetico-deductive method, Galileo discovered the law of free fall of bodies. At first, like many of his predecessors, he proceeded from the more intuitively obvious hypothesis that the speed of the fall was proportional to the distance traveled. However, the consequences of this hypothesis contradicted empirical data, and therefore Galileo was forced to abandon it. It took him about three decades to find a hypothesis, the consequences of which were well confirmed by experiment. To arrive at the correct hypothesis, Kepler had to analyze nineteen different assumptions about the geometric orbit of Mars. At first, he proceeded from the simplest hypothesis, according to which this orbit has the shape of a circle, but such an assumption was not confirmed by astronomical observation data. In principle, this is the general way of discovering the law. A scientist rarely finds the right idea right away. Starting with the simplest hypotheses, he constantly makes adjustments to them and retests them experimentally. In sciences where mathematical processing of the results of observations and experiments is possible, such verification is carried out by comparing theoretically calculated values with actual measurement results. It was in this way that Galileo was able to verify the correctness of his hypothesis and finally formulate it in the form of the law of free fall of bodies. This law, like many other laws of theoretical natural science, is presented in mathematical form, which greatly facilitates its verification and makes the connection between quantities that it expresses easily visible. Therefore, we will use it to clarify the concept of law, which is at least used in the most developed branches of modern natural science.
As can be seen from the formula
,the law of free fall is expressed mathematically using the functional relationship between two variables quantities: time t and path S. We take the first of these quantities as an independent variable, or argument, and the second as a dependent variable, or function. In turn, these variables reflect the real relationship between such properties of the body as the path and time of fall. By choosing appropriate units of measurement, we can express these physical properties or quantities using numbers. In this way, it becomes possible to subject to mathematical analysis the relationship between physical or other properties of real objects and processes that are very different in their specific nature. The whole difficulty in this case will be not so much in finding a suitable mathematical function to display the relationship between properties, but in actually discovering such a connection. In other words, the task is to abstract from all the unimportant factors of the process under study and highlight the essential, basic properties and factors that determine the course of the process. Indeed, intuitively we can well assume that the distance traveled by a falling body depends on its mass, speed, and maybe even temperature. However, physical experience does not confirm these assumptions.
The question of which factors have a significant impact on the course of the process, and which can be abstracted from, is a very complex problem. Its solution is associated with the formulation of hypotheses and their subsequent testing. Reasoning abstractly, one can assume an infinite number of hypotheses that would take into account the influence of a variety of factors on the process. It is clear, however, that there is no practical possibility to test all of them experimentally. Returning to the law of free fall, we see that the movement of a falling body always occurs in a uniform way and depends primarily on time. But in the formula of the law there are also the initial path traversed by the body S0, and its initial speed V 0 , which represent fixed quantities, or options. They characterize the initial state of movement of any particular physical body. If these initial conditions are known, then we can accurately describe its behavior at any time, i.e., in this case, find the path traveled by the falling body during any period of time.
The possibility of abstracting the laws of motion from the chaotic multitude of phenomena occurring around us, notes the famous American physicist E. Wigner, is based on two circumstances. Firstly, in many cases it is possible to identify a set of initial conditions that contains all That, which is essential for the phenomena of interest to us. In the classic example of a freely falling body, almost all conditions can be neglected, except for the initial position and initial speed: its behavior will always be the same, regardless of the degree of illumination, the presence of other bodies near it, their temperature, etc. Equally important has the fact that under the same essential initial conditions, the result will be the same regardless of where and when we implement them. In other words, absolute position and time are never essential initial conditions. This statement, Wigner continues, became the first and perhaps the most important principle of invariance in physics. Without it, we would not be able to discover the laws of nature.
Classification of scientific laws.
empirical - referring to directly observable phenomena (for example, Ohm's, Boyle's - Mariotte's laws);
theoretical - related to unobservable phenomena.
dynamic - giving accurate, unambiguous predictions (Newtonian mechanics);
statistical - giving probabilistic predictions (the uncertainty principle, 1927).
By subject area. Laws of physical, chemical, etc.
By generality: general (fundamental) and particular. For example, Newton's laws and Kepler's laws, respectively.
By levels of scientific knowledge:
According to the predictive function:
The main functions of scientific law.
Explanation is revealing the essence of a phenomenon. In this case, the law acts as an argument. In the 1930s, Karl Popper and Karl Hempel proposed a deductive-nomological model of explanation. According to this model, in an explanation there is an explanandum - the phenomenon being explained - and an explanans - the explanatory phenomenon. The explanans includes provisions about the initial conditions in which the phenomenon occurs, and the laws from which the phenomenon necessarily follows. Popper and Hempel believed that their model was universal - applicable to any field. Canadian philosopher Dray objected, citing history as an example.
Prediction is going beyond the boundaries of the studied world (and not a breakthrough from the present to the future. For example, the prediction of the planet Neptune. It was before the prediction. Unlike an explanation, it predicts a phenomenon that may not have happened yet). There are predictions of similar phenomena, new phenomena and forecasts - predictions of a probabilistic type, based, as a rule, on trends rather than laws. A forecast differs from a prophecy - it is conditional, not fatal. Usually the fact of prediction does not affect the predicted phenomenon, but, for example, in sociology, forecasts can be self-fulfilling.
