True knowledge at all times was based on establishing a pattern and proving its veracity in certain circumstances. For such a long period of existence of logical reasoning, the formulations of the rules were 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 particular to general and from general to particular exist only in relation and cannot be interchanged.
Induction in mathematics
The term "induction" (induction) has Latin roots and literally translates as "guidance". Upon closer study, one can distinguish the structure of the word, namely the Latin prefix - in- (denotes directed action inward or being inside) and -duction - introduction. It is worth noting that there are two types - complete and incomplete induction. The full form is characterized by conclusions drawn from the study of all subjects of a certain class.
Incomplete - conclusions,applied to all items of the class, but based on the study of only some units.
Full mathematical induction is a conclusion based on a general conclusion about the entire class of any objects that are functionally related by relations of the natural series of numbers based on the knowledge of this functional connection. In this case, the proof process takes place in three stages:
- on the first one, the correctness of the statement of mathematical induction is proved. Example: f=1, this is the basis of induction;
- The next stage is based on the assumption that the position is valid for all natural numbers. That is, f=h, this is the induction hypothesis;
- at the third stage, the validity of the position for the number f=h+1 is proved, based on 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 stones in the row fall (transition).
Joking and serious
For ease of perception, examples of solutions by the method of mathematical induction are denounced as joke problems. This is the Polite Queue task:
Rules of conduct forbid a man to take a turn in front of a woman (in such a situation she is let in front). Based on this statement, if the last one 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 inthe minibus fits any number of people. It is true that one person can fit inside the 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 problems and equations by mathematical induction are quite common. As an illustration of this approach, consider the following problem.
Condition: there are h circles on the plane. It is required to prove that for any arrangement of the figures, the map formed by them can be correctly colored with two colors.
Decision: for h=1 the truth of the statement is obvious, so the proof will be built for the number of circles h+1.
Let's assume that the statement is true for any map, and h+1 circles are given on the plane. By removing one of the circles from the total, you can get a map correctly colored with two colors (black and white).
When restoring a deleted circle, the color of each area changes to the opposite (in this case, inside the circle). The result is a map correctly colored with two colors, which was required to be proved.
Examples with natural numbers
The application of the method of mathematical induction is illustrated below.
Examples of solution:
Prove that for any h the equality will be correct:
12+22+32+…+h 2=h(h+1)(2h+1)/6.
Solution:
1. Let h=1, then:
R1=12=1(1+1)(2+1)/6=1
It follows that for h=1 the statement is correct.
2. Assuming h=d, the equation is:
R1=d2=d(d+1)(2d+1)/6=1
3. Assuming that h=d+1, it turns out:
Rd+1=(d+1) (d+2) (2d+3)/6
Rd+1=12+22+3 2+…+d2+(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)(2d2+7d+6)/6=(d+1)(2(d+3/2)(d+2))/6=(d+1)(d+2)(2d+3)/6.
Thus, the validity of equality for h=d+1 has been proved, therefore the statement is true for any natural number, which is shown in the example of the solution by mathematical induction.
Task
Condition: proof is required that for any value of h, the expression 7h-1 is divisible by 6 without a remainder.
Solution:
1. Let's say h=1, in this case:
R1=71-1=6 (i.e. divisible by 6 without a remainder)
Hence, for h=1 the statement is true;
2. Let h=d and 7d-1 is divisible by 6 without a remainder;
3. The proof of the validity of the statement for h=d+1 is the formula:
Rd+1=7d+1 -1=7∙7d-7+6=7(7d-1)+6
In this case, the first term is divisible by 6 according to the assumption of the first paragraph, and the secondthe term is 6. The statement that 7h-1 is divisible by 6 without a remainder for any natural h is true.
False Judgment
Often, incorrect reasoning is used in proofs, due to the inaccuracy of the logical constructions used. Basically, this happens when the structure and logic of the proof are violated. An example of incorrect reasoning is the following illustration.
Task
Condition: proof is required that any pile of stones is not a pile.
Solution:
1. Let's say h=1, in this case there is 1 stone in the pile and the statement is true (basis);
2. Let it be true for h=d that a pile of stones is not a pile (assumption);
3. Let h=d+1, from which it follows that when one more stone is added, the set will not be a heap. The conclusion suggests itself that the assumption is valid for all natural h.
The error lies in the fact that there is no definition of how many stones form a pile. Such an omission is called hasty generalization in the method of mathematical induction. An example shows this clearly.
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 as opposites.
From the point of view of the law of logic, inductive definitions are based on facts, and the veracity of the premises does not determine the correctness of the resulting statement. Often obtainedconclusions with a certain degree of probability and plausibility, which, of course, must be verified and confirmed by additional research. An example of induction in logic would be the statement:
Drought in Estonia, dry in Latvia, dry in Lithuania.
Estonia, Latvia and Lithuania are the B altic states. Drought in all B altic states.
From the example, we can conclude that new information or truth cannot be obtained using the method of induction. All that can be counted on is some possible veracity of the conclusions. Moreover, the truth of the premises does not guarantee the same conclusions. However, this fact does not mean that induction vegetates in the backyard of deduction: a huge number of provisions and scientific laws are substantiated using the method of induction. Mathematics, biology and other sciences can serve as an example. This is due for the most part to the full induction method, but in some cases partial is also applicable.
The venerable age of induction allowed it to penetrate into almost all areas of human activity - this is science, economics, and everyday conclusions.
Induction in the scientific environment
The method of induction requires a scrupulous attitude, 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, scientific laws obtained by induction are tested for a long time at the level of probabilistic assumptions in order to isolate and study all possiblestructural elements, connections and influences.
In science, the inductive 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 way of studying):
- induction-selection (or selection);
- induction - exclusion (elimination).
The first type is characterized by methodical (scrutinous) sampling of a class (subclasses) from its different areas.
An example of this type of induction is as follows: silver (or silver s alts) purifies water. The conclusion is based on long-term observations (a kind of selection of confirmations and refutations - selection).
The second type of induction is based on conclusions that establish causal relationships and exclude circumstances that do not meet its properties, namely, universality, observance of the temporal sequence, necessity and unambiguity.
Induction and deduction from the standpoint of philosophy
If you look at the 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 one 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 position of four main 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 investigate the probabilistic basis of induction. However, even here there were some extremes: attempts were made to reduce the induction to the theory of probability 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 is the law of universal gravitation. At the date of discovery of the law, Newton was able to verify it with an accuracy of 4 percent. And when tested after more than two hundred years, the correctness was confirmed with an accuracy of 0.0001 percent, although the test was carried out with the same inductive generalizations.
Modern philosophy pays more attention to deduction, which is dictated by a logical desire to derive new knowledge (or truth) from what is already known, without resorting to experience, intuition, but using "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).
Application of induction in economics
Induction and deduction have long been used as methods of studying the economy and predicting its development.
The range of use of the induction method is quite wide: the study of the fulfillment 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 method of induction is used in Shewhart's charts, where, under the assumption that processes are divided into controlled and unmanaged, 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 method of induction, and since economics is a science that often uses mathematical analysis, risk theory and statistical data, it is not surprising that induction is included in the list of main methods.
The following situation can serve as an example of induction and deduction in economics. An increase in the price of food (from the consumer basket) and essential goods pushes 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 toit was possible to predict with sufficient truthfulness the development of the enterprise, the behavior of the market, the consequences of competition, an inductive-deductive approach to the analysis and processing of information is needed.
An illustrative example of induction in economics relating to fallacious judgments:
-
company's profit down 30%;
competitor expands product line;
nothing else has changed;
- competitor's production policy caused a 30% profit cut;
- hence the need to implement the same production policy.
The example is a colorful illustration of how the inept use of the method of induction contributes to the ruin of the enterprise.
Deduction and induction in psychology
Since there is a method, then, logically, there is also a properly organized thinking (to use the method). Psychology as a science that studies mental processes, their formation, development, relationships, interactions, pays attention to "deductive" thinking as one of the forms of manifestation 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 a deceiver, therefore, all women are deceivers. You can learn even more "erroneous" examples of induction from life:
- a student is not capable of anything if he received a deuce in mathematics;
- he is a fool;
- he is smart;
- I can do anything;
- and many other value judgments based on absolutely random and sometimes insignificant messages.
It should be noted: when the fallacy of a person's judgments reaches the point of absurdity, there is a front of work for the psychotherapist. 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 has excluded this color scheme from his life - as far as possible. In the home environment, there are many opportunities for comfortable living. You can refuse all red items or replace them with analogues made in a different color scheme. But in public places, at work, in the store - it is impossible. Getting into a situation of stress, the patient each time experiences a “tide” 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 he althy person, we can talk about a lack of organization of mental activity. The elementary development of deductive thinking can become a way to get rid of obsessive states. In other cases, psychiatrists work with such patients.
The above examples of induction show that “ignorance of the law does notliberates from consequences (erroneous judgments).”
Psychologists, working on the topic of deductive reasoning, have compiled a list of recommendations designed to help people master this method.
The first item is problem solving. As can be seen, the form of induction 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 the expansion of horizons (those who think clearly, clearly state). This recommendation directs the "afflicted" to the treasuries of science and information (libraries, websites, educational initiatives, travel, etc.).
Accuracy is the next recommendation. After all, it is clearly seen from examples of using induction methods that it is in many respects the guarantee of the truth of statements.
They did not bypass the flexibility of the mind, implying the possibility of using different ways and approaches in solving the problem, as well as taking into account the variability of the development of events.
And, of course, observation, which is the main source of empirical experience.
Special 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 formulation of the definition of this term, but refer to "examples from life", while presenting either suggestion or some forms of mental illness as a new type of induction,These are the extreme states of the human psyche. From all of the above, it is clear that an attempt to derive a “new term” based on false (often untrue) premises dooms the experimenter to receive an erroneous (or hasty) statement.
It should be noted that the reference to the 1960 experiments (without specifying 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 assertion that the brain perceives information bypassing all organs of perception (the phrase “experienced by” in this case would fit in more organically), makes one think about the gullibility and uncriticality of the author of the statement.
Instead of a conclusion
Queen of sciences - mathematics, knowingly 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 in his logical constructions often uses examples of induction, using deduction in necessary situations.
The article examined examples of the application of these methods in various sciences and spheres of human life.
