True knowledge at all times was based on establishing the law and proving its veracity in certain circumstances. For such a long period of existence of logical reasoning, the wording of the rules was given, and Aristotle even compiled a list of “correct reasoning”. Historically, it is customary to divide all inferences into two types - from the concrete to the plural (induction) and vice versa (deduction). It should be noted that the types of evidence from the particular to the general and from the general to the particular exist only in interconnection and cannot be interchangeable.
Induction in mathematics
The term induction has Latin roots and is literally translated as “guidance”. When closely studied, you can distinguish the structure of the word, namely the Latin prefix - in- (denotes a directed action inside or inside) and -duction - introduction. It is worth noting that there are two types - full and incomplete induction. The full form is characterized by conclusions drawn from the study of all objects of a certain class.
Incomplete - conclusions applied to all subjects of the class, but made on the basis of the study of only some units.
Complete mathematical induction is an inference based on a general conclusion about the entire class of any objects that are functionally related by the relations of a natural series of numbers on the basis of knowledge of this functional connection. The proof process takes place in three stages:
- the first proves the correctness of the position of mathematical induction. Example: f = 1, this is the basis of induction;
- the next stage is based on the assumption of the validity of the situation for all natural numbers. That is, f = h, this is an induction hypothesis;
- at the third stage, the validity of the position for the number f = h + 1 is proved; on the basis of the correctness of the position of the previous paragraph, this is an induction transition, or a step of mathematical induction. An example is the so-called "domino principle": if the first bone in a row falls (basis), then all the bones in a row fall (transition).
And as a joke, and seriously
For ease of perception, examples of solutions by mathematical induction are exposed in the form of joke problems. Such is the “Polite Queue” task:
- Rules of conduct prohibit a man from taking a line in front of a woman (in such a situation, she is let forward). Based on this statement, if the last in line is a man, then all the rest are men.
A striking example of the method of mathematical induction is the problem “Dimensionless flight”:
- It is required to prove that any number of people fits in the minibus. It is true that one person can be accommodated inside a transport without difficulty (basis). But no matter how full the minibus is, 1 passenger will always fit in it (induction step).
Familiar circles
Examples of solving the method of mathematical induction of problems and equations are quite common. As an illustration of this approach, we can consider the following problem.
Condition : h circles are placed on the plane. It is required to prove that for any arrangement of figures the map formed by them can be correctly painted with two colors.
Solution : for h = 1, the truth of the statement is obvious, so the proof will be constructed for the number of circles h + 1.
We assume that the statement is valid for any map, and h + 1 circles are given on the plane. Removing one of the circles from the total number, you can get a card correctly painted with two colors (black and white).
When restoring a deleted circle, the color of each area changes to the opposite (in the indicated case, inside the circle). It turns out the card, correctly painted in two colors, which was required to prove.
Examples with natural numbers
The application of the method of mathematical induction is clearly shown below.
Examples of solutions:
Prove that for any h the equality will be correct:
1 2 +2 2 +3 2 + ... + h 2 = h (h + 1) (2h + 1) / 6.
Decision:
1. Let h = 1, then:
R 1 = 1 2 = 1 (1 + 1) (2 + 1) / 6 = 1
It follows from this that for h = 1 the statement is correct.
2. Assuming that h = d, the equation is obtained:
R 1 = d 2 = d (d + 1) (2d + 1) / 6 = 1
3. Assuming that h = d + 1, it turns out:
R d + 1 = (d + 1) (d + 2) (2d + 3) / 6
R d + 1 = 1 2 +2 2 +3 2 + ... + d 2 + (d + 1) 2 = d (d + 1) (2d + 1) / 6 + (d + 1) 2 = (d ( d + 1) (2d + 1) +6 (d + 1) 2 ) / 6 = (d + 1) (d (2d + 1) +6 (k + 1)) / 6 =
(d + 1) (2d 2 + 7d + 6) / 6 = (d + 1) (2 (d + 3/2) (d + 2)) / 6 = (d + 1) (d + 2) ( 2d + 3) / 6.
Thus, the equality is proved for h = d + 1; therefore, the statement is true for any natural number, as shown in the example of a solution by mathematical induction.
Task
Condition : proof is required that for any value of h the expression 7 h -1 is divisible by 6 without a remainder.
Solution :
1. Suppose h = 1, in this case:
R 1 = 7 1 -1 = 6 (i.e. divided by 6 without remainder)
Therefore, for h = 1 the statement is true;
2. Let h = d and 7 d -1 be divided by 6 without remainder;
3. The proof of the validity of the statement for h = d + 1 is the formula:
R d +1 = 7 d +1 -1 = 7 ∙ 7 d -7 + 6 = 7 (7 d -1) +6
In this case, the first term is divided by 6 under the assumption of the first paragraph, and the second term is 6. The statement that 7 h -1 is divisible by 6 without remainder for any positive integer h is true.
Fallacy of judgment
Often, in the evidence, incorrect reasoning is used, due to the inaccuracy of the used logical constructions. This mainly occurs when the structure and logic of the proof is violated. An example of wrong reasoning is such an illustration.
Task
Condition : proof is required that any pile of stones is not a pile.
Solution :
1. Suppose h = 1, in this case in a bunch of 1 stone and the statement is true (basis);
2. Let it be true for h = d that a heap of stones is not a heap (assumption);
3. Let h = d + 1, which implies that when adding another stone, the set will not be a heap. The conclusion suggests itself that the assumption is valid for all natural h.
The mistake is that there is no definition of how many stones form a heap. Such an omission is called a hasty generalization in the method of mathematical induction. An example of this clearly shows.
Induction and the laws of logic
Historically, examples of induction and deduction always "go hand in hand." Such scientific disciplines as logic, philosophy describe them in the form of opposites.
From the point of view of the law of logic, inductive definitions are based on facts, and the truthfulness of the premises does not determine the correctness of the resulting statement. Often, inferences are obtained with a certain degree of probability and credibility, which, of course, must be verified and confirmed by additional studies. An example of induction in logic can be the statement:
In Estonia - drought, in Latvia - drought, in Lithuania - drought.
Estonia, Latvia and Lithuania are the Baltic states. In all Baltic states drought.
From the example we can conclude that new information or truth cannot be obtained using the induction method. All you can count on is some possible truthfulness of the conclusions. Moreover, the truth of the premises does not guarantee the same conclusions. However, this fact does not mean that induction vegetates on the margins of deduction: a huge number of provisions and scientific laws are substantiated using the induction method. An example is the same mathematics, biology, and other sciences. This is mainly due to the method of complete induction, but in some cases partial is applicable.
The venerable age of induction allowed it to penetrate into almost all spheres of human activity - this is science, economics, and everyday conclusions.
Scientific Induction
The induction method requires a scrupulous relationship, since too much depends on the number of studied particulars of the whole: the larger the number studied, the more reliable the result. Based on this feature, the scientific laws obtained by the method of induction are checked for a long time at the level of probabilistic assumptions for isolating and studying all possible structural elements, relationships, and effects.
In science, the induction conclusion is based on significant features, with the exception of random provisions. This fact is important in connection with the specifics of scientific knowledge. This is clearly seen in the examples of induction in science.
There are two types of induction in the scientific world (in connection with the method of study):
- induction selection (or selection);
- induction is an exception (elimination).
The first type is distinguished by a methodical (scrupulous) selection of class samples (subclasses) from its different areas.
An example of this type of induction is as follows: silver (or silver salts) purifies water. The conclusion is based on long-term observations (a peculiar selection of confirmations and refutations - selection).
The second type of induction is based on conclusions establishing causal relationships and excluding circumstances that do not meet its properties, namely universality, observance of the time sequence, necessity and uniqueness.
Induction and deduction from the perspective of philosophy
If you look at a historical retrospective, the term "induction" was first mentioned by Socrates. Aristotle described examples of induction in philosophy in a more approximate terminological dictionary, but the question of incomplete induction remains open. After the persecution of the Aristotelian syllogism, the inductive method began to be recognized as fruitful and the only possible in natural science. Bacon is considered the father of induction as an independent special method, but he failed to separate, as contemporaries demanded, induction from the deductive method.
Further development of induction was carried out by J. Mill, who considered the induction theory from the perspective of four basic methods: agreement, difference, residuals, and corresponding changes. It is not surprising that today the listed methods, when examined in detail, are deductive.
Awareness of the failure of the theories of Bacon and Mill led scientists to study the probabilistic basis of induction. However, there were some extremes: attempts were made to reduce induction to probability theory with all the ensuing consequences.
Induction receives a vote of confidence in practical application in certain subject areas and due to the metric accuracy of the inductive basis. An example of induction and deduction in philosophy can be considered the Law of universal gravitation. On the date the law was opened, Newton was able to verify it with an accuracy of 4 percent. And when checking after more than two hundred years, the correctness was confirmed with an accuracy of 0.0001 percent, although the verification was carried out by the same inductive generalizations.
Modern philosophy pays more attention to deduction, which is dictated by a logical desire to derive new knowledge (or truths) from an already known one, without resorting to experience, intuition, but operating with “pure” reasoning. When referring to the true premises in the deductive method in all cases, the output is a true statement.
This very important characteristic should not overshadow the value of the inductive method. Since induction, relying on the achievements of experience, also becomes a means of processing it (including generalization and systematization).
The use of induction in economics
Induction and deduction have long been used as methods of researching the economy and predicting its development.
The range of use of the induction method is wide enough: the study of the implementation of forecast indicators (profit, depreciation, etc.) and a general assessment of the state of the enterprise; formation of an effective enterprise promotion policy based on facts and their relationships.
The same induction method is used in the “Shekhart maps”, where, under the assumption of the separation of processes into controlled and uncontrolled, it is stated that the framework of the controlled process is inactive.
It should be noted that scientific laws are justified and confirmed using the induction method, and since economics is a science that often uses mathematical analysis, risk theory and statistical data, it is not surprising that induction is on the list of basic methods.
The following situation can serve as an example of induction and deduction in economics. The increase in the price of food (from the consumer basket) and essential goods push the consumer to think about the emerging high cost in the state (induction). At the same time, from the fact of high cost using mathematical methods, it is possible to derive indicators of price increases for individual goods or categories of goods (deduction).
Most often, management personnel, managers, and economists turn to the induction method. In order to be able to predict with sufficient truthfulness the development of the enterprise, market behavior, the consequences of competition, an induction-deductive approach to the analysis and processing of information is necessary.
A good example of induction in economics related to erroneous judgments:
- company profit decreased by 30%;
a competing company has expanded its product line;
nothing else has changed; - a competing company’s production policy caused a 30% reduction in profits;
- therefore, the same production policy is required.
The example is a colorful illustration of how inept use of the induction method contributes to the ruin of an enterprise.
Deduction and induction in psychology
Since there is a method, then, according to the logic of things, there is a properly organized thinking (for using the method). Psychology as a science that studies mental processes, their formation, development, relationships, interactions, pays attention to "deductive" thinking, as one of the manifestations of deduction and induction. Unfortunately, on the pages of psychology on the Internet there is practically no justification for the integrity of the deductive-inductive method. Although professional psychologists are more likely to encounter manifestations of induction, or rather, erroneous conclusions.
An example of induction in psychology, as an illustration of erroneous judgments, is the statement: my mother is deceiving, therefore, all women are deceivers. Even more can be learned from the “erroneous” examples of induction from life:
- the student is not capable of anything if he received a deuce in mathematics;
- he is a fool;
- he is smart;
- I can do everything;
- and many other value judgments, derived on completely random and, at times, insignificant messages.
It should be noted: when the fallacy of a person’s judgment reaches the point of absurdity, the front of work for the therapist appears. One example of induction at a specialist appointment:
“The patient is absolutely sure that the red color carries only danger for him in any manifestations. As a result, a person excluded from this life this color scheme - as much as possible. At home, there are many opportunities for a comfortable stay. You can refuse all objects of red color or replace them with analogues made in a different color scheme. But in public places, at work, in the store - it is impossible. In a situation of stress, the patient each time experiences a “rush” of completely different emotional states, which can be dangerous for others. "
This example of induction, and unconsciously, is called "fixed ideas." If this happens to a mentally healthy person, we can talk about the lack of organization of mental activity. A way to get rid of obsessive states can be the elementary development of deductive thinking. In other cases, psychiatrists work with such patients.
The above examples of induction indicate that "ignorance of the law does not exempt from the consequences (erroneous judgments)."
Psychologists, working on the topic of deductive thinking, have compiled a list of recommendations designed to help people learn this method.
The first paragraph is the solution of problems. As one could see, that form of induction, which is used in mathematics, can be considered "classical", and the use of this method contributes to the "discipline" of the mind.
The next condition for the development of deductive thinking is broadening one's horizons (he who thinks clearly, he clearly states). This recommendation directs the “afflicted” to the compartments of sciences and information (libraries, websites, educational initiatives, travels, etc.).
Accuracy is the next recommendation. Indeed, from examples of the use of induction methods, it is clearly seen that it is it that is in many respects the key to the truth of statements.
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Separately, mention should be made of the so-called "psychological induction." This term, although infrequently, can be found on the Internet. All sources do not give at least a brief wording of the definition of this term, but refer to “life examples”, while pretending to be suggestion, some forms of mental illness, or extreme conditions of the human psyche as a new type of induction. From all of the above, it is clear that an attempt to derive a "new term", based on false (often not true) premises, dooms the experimenter to receive an erroneous (or hasty) statement.
It should be noted that a reference to the 1960 experiments (without indicating the venue, the names of the experimenters, the sample of subjects and, most importantly, the purpose of the experiment) looks, to put it mildly, unconvincing, and the statement that the brain perceives information bypassing all organs of perception (phrase “Experiencing an impact” in this case would fit in more organically), makes you think about the credulity and uncriticality of the author of the statement.
Instead of a conclusion
The queen of sciences is mathematics; it is not in vain that she uses all possible reserves of the method of induction and deduction. The considered examples allow us to conclude that the superficial and inept (thoughtless, as they say) application of even the most accurate and reliable methods always leads to erroneous results.
In the mass consciousness, the deduction method is associated with the famous Sherlock Holmes, who often uses examples of induction in his logical constructions, using deduction in necessary situations.
The article examined examples of the application of these methods in various sciences and spheres of human life.