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A bit of the history of sophism The term "sophism" was first introduced by Aristotle, comes from the ancient Greek word sophisma - "skill, cunning trick, invention, imaginary wisdom."

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Examples of sophisms, famous in antiquity “What you have not lost, you have; you did not lose your horns; it means you have horns.” “He who sits stood up; who got up, he stands; therefore, the one sitting is standing.” “This dog is yours; he is a father; so he is your father.” “Do you know what I want to ask you now? - Not. "Don't you know it's wrong to lie?" - Of course I know. “But that's exactly what I was going to ask you, and you said you didn't know; so you know what you don't know"

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Sophistry has been around for more than two millennia. Their emergence is usually associated with the philosophical activity of the sophists (Ancient Greece of the 5th-4th centuries BC) - paid teachers of wisdom who taught everyone philosophy, logic and, especially, rhetoric (the science and art of eloquence). The most famous representatives of the direction of sophistry in Ancient Greece are Protagoras, Gorgias, Prodik.

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Classification of sophisms Medicines “The medicine taken by the sick is good. The more you do good, the better. So, you need to take as many medicines as possible.” Thief “The thief does not want to acquire anything bad. The acquisition of good things is a good thing. Therefore, the thief desires good things." logical algebraic Unit is equal to zero Take the equation x-a=0, divide both sides of the equation by (x-a), we get (x-a)/(x-a)=0/(x-a) and hence 1=0. Error: The error is that x-a is zero, and you cannot divide by zero.

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terminological "All angles of a triangle = π" in the sense of "The sum of the angles of a triangle = π" "how much five plus two times two?" Here it is difficult to decide whether 9 (i.e. 5 + (2*2)) or 14 (i.e. (5 + 2) * 2) is meant. . arithmetic One ruble is not equal to one hundred kopecks. 1 rub. = 100 kopecks 10 rubles = 1000 kopecks We multiply both parts of these correct equalities, we get: 10 rubles = 100,000 kopecks, from which it follows: 1 p. = 10,000 kopecks, i.e. 1 p. not equal to 100 kopecks. Mistake: The mistake made in this sophism is the violation of the rules of action with named quantities: all operations performed on quantities must also be performed on their dimensions.

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geometric “from a point on a line, two perpendiculars can be lowered” Let's try to “prove” that through a point lying outside the line, two perpendiculars can be drawn to this line. For this purpose, take the triangle ABC. On the sides AB and BC of this triangle, as on diameters, we construct semicircles. Let these semicircles intersect with the side AC at points E and D. Let us connect the points E and D with straight lines to the point B. The angle AEB is straight, as inscribed, based on the diameter; angle BDC is also a right angle. Therefore, BE is perpendicular to AC and B D is perpendicular to AC. Two perpendiculars to line AC pass through point B.

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Why are sophisms useful for students of physics? What can they give? Analysis of sophisms, first of all, develops logical thinking, that is, instills the skills of correct thinking. What is especially important, the analysis of sophisms helps the conscious assimilation of the material being studied, develops observation, thoughtfulness and a critical attitude towards what is being studied. Finally, the analysis of sophisms is fascinating. The more difficult the sophism, the more satisfying is its analysis. It is valuable, not that he did not make mistakes, but that he found the cause of the error and eliminated it.

Danilov Dmitry, 8th grade student

Research work. A definition of sophism is given, historical information is described, various sophisms are analyzed: arithmetic, algebraic, geometric and others.

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MOU "OOSH village of Mavrinka, Pugachevsky district, Saratov region" Research work at the municipal scientific and practical conference "Step into the Future"

The purpose of my work is to prove that sophisms are not just an intellectual fraud, but an important engine of human thought. Show practical application, their relevance in our time. Tasks: Consider mathematical, algebraic, geometric sophisms in terms of their importance for the study of mathematics. Try to find errors in the presented sophisms. Show sophisms from life and modern practice.

Introduction. Brains are obliged to work Sophisms are usually called statements, in the evidence of which imperceptible, and sometimes quite subtle errors are hidden. Every branch of mathematics, from simple arithmetic to modern, more complex areas, has its sophistry. In the best of them, reasoning with carefully disguised error leads to the most incredible conclusions. Euclid devoted a whole book to errors in geometric proofs, but it has not reached our days, and we can only guess about what an irreparable loss elementary mathematics suffered because of this. Analysis of sophisms, first of all, develops logical thinking, i.e. instills the skills of correct thinking. To discover an error in sophism means to recognize it, and awareness of an error prevents it from being repeated in other mathematical reasoning. The development of critical thinking will allow not only to successfully master the exact sciences, but also not to be a victim of scammers in life. For example, when applying for a loan at a bank, you will not be in debtor for life. I think many at least once in their lives have heard such statements: "All numbers are equal" or "two equals three." There can be many such examples, but what does it mean? Who came up with this? Is it possible to somehow explain these statements or is it all fiction? I want to answer these questions and many others in my work. There are various sophisms: logical, terminological, psychological, mathematical, etc.

THE CONCEPT "SOPHISM" Sophism - (from the Greek sophisma, "skill, skill, cunning invention, trick") - a conclusion or reasoning that justifies some deliberate absurdity, absurdity or paradoxical statement that contradicts generally accepted ideas. Sophism, unlike paralogism, is based on a deliberate, conscious violation of the rules of logic. Whatever the sophism, it always contains one or more disguised errors. Mathematical sophism is an amazing statement, the proof of which hides imperceptible, and sometimes quite subtle errors. The history of mathematics is full of unexpected and interesting sophisms, the resolution of which sometimes served as an impetus for new discoveries. Mathematical sophisms teach us to move forward attentively and cautiously, to carefully monitor the accuracy of formulations, the correctness of drawing drawings, and the legality of mathematical operations. Very often, understanding errors in sophistry leads to an understanding of mathematics in general, helps to develop logic and skills of correct thinking. If you find an error in sophism, then you have realized it, and the awareness of the error warns against its repetition in further mathematical reasoning. Sophisms are of no use if they are not understood.

EXCURSION TO HISTORY The sophists were a group of ancient Greek philosophers of the 4th-5th centuries BC, who achieved great skill in logic. The most famous activities of senior sophists, which include Protagoras from Abdera, Gorgias from Leontip, Hippias from Elis and Prodice from Keos. . Aristotle called sophism "imaginary evidence", in which the validity of the conclusion is apparent and is due to a purely subjective impression caused by a lack of logical analysis. . The persuasiveness at first glance of many sophisms, their “logicality” is usually associated with a well-disguised error: substitution of the main idea (thesis) of the proof, acceptance of false premises as true, non-observance of acceptable methods of reasoning (rules of logical inference), use of “unresolved” or even “forbidden » rules or actions, such as division by zero in mathematical sophistry.

ARITHMETIC SOPHISMS Arithmetic - (Greek arithmetika, from arithmys - number), the science of numbers, primarily about natural (positive integer) numbers and (rational) fractions, and operations on them. So what are arithmetic sophisms? Arithmetic sophisms are numerical expressions that have an inaccuracy or error that is not noticeable at first glance. 1. “If A is greater than B, then A is always greater than 2B.” Take two arbitrary positive numbers A and B, such that A>B. Multiplying this inequality by B, we obtain a new inequality AB>B*B, and subtracting A*A from both of its parts, we obtain the inequality AB-A*A>B*B-A*A, which is equivalent to the following: A(B-A )>(B+A)(B-A). (1) After dividing both parts of the inequality (1) by B-A, we get that A>B+A (2), And adding to this inequality the original inequality A>B term by term, we have 2A>2B+A, whence A>2B . So, if A>B, then A>2B. This means, for example, that from the inequality 6>5 it follows that 6>10. Where is the error???

2. "A number equal to another number is both greater and less than it." Let's take two arbitrary positive equal numbers A and B and write the following obvious inequalities for them: A>-B and B>-B. (1) Multiplying both of these inequalities term by term, we get the inequality A*B>B*B, and after dividing it by B, which is quite legal, because B>0, we come to the conclusion that A>B. (2) Having written two other equally indisputable inequalities B>-A and A>-A, (3) Similarly to the previous one, we obtain that B*A>A*A, and dividing by A>0, we arrive at the inequality A>B . (4) So, the number A, equal to the number B, is both greater and less than it. Where is the mistake???

3. "2+2=5" To prove that 2+2=5 , you can only prove that 4=5 Let's start with equality: 16-36=25-45 Add 20.25 to both parts, we get: 16 -36+20.25=25-45+20.25 Note that in both parts of the equality, a full square can be displayed: 4²-2*4*4.5+4.5²=5²-2*5*4.5+ 4.5² We get: (4-4.5)²=(5-4.5)² We extract the root of both sides of the equality, we get: 4-4.5=5-4.5 4=5 which was required to prove .

4. "Twice two equals five" Denote 4=a, 5=b, (a+b)/2=d. We have: a+b=2d, a=2d-b, 2d-a=b. Let's multiply the last two equalities by parts. We get: 2da-a 2 =2db-b 2 . Multiply both sides of the resulting equality by –1 and add d 2 to the results. We will have: a 2 -2da+d 2 =b 2 -2bd+d 2 , or (a-d)(a-d)=(b-d)(b-d), whence a-d=b-d and a=b , i.e. 2*2=5 Where is the error???

5. "Missing ruble" Three friends went to a cafe to drink a cup of coffee. We drank. The waiter brought them a bill for 30 rubles. Girlfriends paid 10 rubles each and left. However, the owner of the cafe for some reason decided that the coffee served at this table costs 25 rubles, and ordered the visitors to return 5 rubles. The waiter took the money and ran to catch up with his friends, but while he was running, he thought that it would be difficult for them to divide 5 rubles into three, and therefore decided to give them 1 ruble each, and keep two rubles for himself. And so he did. What happened? Friends paid 9 rubles each. 9 * 3 = 27 rubles, but the waiter had two rubles left. And where is another 1 ruble?

ALGEBRAIC SOPHISMS Algebra is one of the major branches of mathematics, which, along with arithmetic and geometry, belongs to the oldest branches of this science. Problems, as well as methods of algebra, which distinguish it from other branches of mathematics, were created gradually, starting from antiquity. Algebra arose under the influence of the needs of social practice, as a result of the search for common methods for solving the same type of arithmetic problems. These techniques usually consist in compiling and solving equations. Those. algebraic sophisms - deliberately hidden errors in equations and numerical expressions.

1. “Two unequal natural numbers are equal to each other” We solve a system of two equations: x + 2y \u003d 6, (1) y \u003d 4- x / 2 (2) Let's do this by substituting y from the 2nd equation into 1, we get x + 8- x=6, whence 8=6 Where is the mistake???

2. "A negative number is greater than a positive number." Take two positive numbers a and c. Let's compare two ratios: a/- c and -a/ c They are equal, since each of them is equal to -(a/c). You can make a proportion: a / - c = - a / c But if in the proportion the previous member of the first relation is greater than the next, then the previous member of the second relation is also greater than its subsequent one. In our case, a>-c, therefore, it should be -a>c, i.e. a negative number is greater than a positive number. Where is the mistake???

3. Any number a is equal to a smaller number b Let's start with equality: a=b+c Multiply both of its parts by a-b , we get: a²-ab = ab+ac-b²-bc Move ac to the left side: a²-ab-ac = ab-b²-bc and factorize: a (a-b-c) =b (a-b-c) Dividing both sides of the equality by a-b-c , we find a=b which was required to be proved.

4. The equation x-a=0 has no roots Given the equation: x-a=0 Divide everything by x-a, we get: 1=0 This equality is incorrect, therefore the original equation has no roots.

5. The weight of an elephant is equal to the weight of a mosquito. Let x be the weight of the elephant and y be the weight of the mosquito. Let's denote the sum of these weights as 2n, we get x+y=2n. From this equality, you can get two more: x - 2p \u003d -y and x \u003d -y + 2p. We multiply these two equalities term by term: x 2 - 2px + p 2 \u003d y 2 - 2pu + p 2 or (x - p) 2 \u003d (y - p) 2. Extracting the square root of both parts of the last equality, we get: x - n \u003d y - n or x \u003d y, i.e. The weight of an elephant is equal to the weight of a mosquito! What's the matter here?

GEOMETRIC SOPHISMS Geometric sophisms are conclusions or reasonings substantiating some deliberate absurdity, absurdity or paradoxical statement connected with geometric figures and actions on them. 1. "A match is twice as long as a telegraph pole" Let a dm be the length of the match and b dm be the length of the pole. The difference between b and a will be denoted by c . We have b - a = c , b = a + c . We multiply these two equalities by parts, we find: b 2 - ab = ca + c 2. Subtract bc from both parts. We get: b 2 - ab - bc \u003d ca + c 2 - bc, or b (b - a - c) \u003d - c (b - a - c), from where b \u003d - c, but c \u003d b - a, therefore b = a - b, or a = 2b. Where is the mistake???

2.Triangle problem Given a right triangle 13×5 cells, composed of 4 parts. After rearranging the parts while visually maintaining the original proportions, an additional cell appears, not occupied by any part. Where does it come from?

The statement is easy to check by calculations.

3. Disappearing Square A large square is made up of four identical quadrilaterals and a small square. If the quadrilaterals are expanded, they will fill the area occupied by the small square, although the area of ​​the large square will not visually change.

Sophism of Aristotle All circles have the same length. Indeed, when wrapping two circles with different diameters OA 1 and OA 2, each of them is straightened in one revolution to the same segment OO 1

To identify the error, a drawing was constructed showing which trajectory the various points of the circle actually go through, and the error in the proof becomes obvious. Points A 1 and A 2 during the movement of the wheel describe curves of different lengths, they are called cycloidal curves.

OTHER SOPHISMS In addition to mathematical sophisms, there are many others, for example: logical, terminological, psychological, etc. It is easier to understand the absurdity of such statements, but this does not make them any less interesting. So many sophisms look like a game with language devoid of meaning and purpose; a game based on the ambiguity of linguistic expressions, their incompleteness, understatement, the dependence of their meanings on the context, etc. These sophisms seem especially naive and frivolous. “Half-empty and half-full” “Half-empty is the same as half-full. If the halves are equal, then the whole ones are equal. Therefore, the empty is the same as the full. “Even and odd” “5 is 2 + 3 (“two and three”). Two is an even number, three is an odd number, it turns out that five is a number both even and odd. Five is not divisible by two, nor is 2 + 3, which means both numbers are not even! “Medicines” “Medicine taken by the sick is good. The more you do good, the better. So, you need to take as many medicines as possible.”

"The fastest creature cannot catch up with the slowest" The swift-footed Achilles will never overtake the slowest tortoise. By the time Achilles reaches the tortoise, it will move forward a little. He will quickly overcome this distance, but the turtle will go a little further ahead. And so on ad infinitum. Whenever Achilles reaches the place where the tortoise was before, it will be at least a little, but in front. "No End" A moving object must reach half of its path before it reaches its end. Then he must go through half of the remaining half, then half of this fourth part, and so on. to infinity. The object will constantly approach the end point, but will never reach it.

"Pile" One grain of sand is not a pile of sand. If n grains of sand are not a heap of sand, then n + 1 grains of sand are not a heap either. Therefore, no number of grains of sand form a pile of sand. "Can an all-powerful magician create a stone that he cannot lift?" If he cannot, then he is not omnipotent. If he can, then he is still not omnipotent, because. he cannot lift this stone. Is a full glass equal to an empty one? Yes. Let's have a discussion. Suppose there is a glass filled with water up to half. Then we can say that a glass half full is equal to a glass half empty. By doubling both sides of the equation, we get that a full glass is equal to an empty glass.

"Evatl's sophism" Euathl took lessons in sophistry from the sophist Protagoras on the condition that he would pay the fee only if he won the first trial. After the training, the student did not take over the conduct of any process and therefore considered himself entitled not to pay the fee. The teacher threatened to file a complaint with the court, telling him the following: "The judges will either order you to pay the fee or not. In both cases, you will have to pay. In the first case, by virtue of the judge's verdict, in the second case, by virtue of our contract." To this Euathlus answered: “In neither case will I pay. If I am adjudged to pay, then, having lost the first trial, I will not pay by virtue of our contract, but if I am not adjudged to pay a fee, then I will not I will pay by virtue of the verdict of the court." (The mistake becomes clear if we separately pose two questions: 1) whether Euathlus should pay or not, and 2) whether the conditions of the contract are fulfilled or not. and the same river (an image of nature) cannot be entered twice, for the next time the one enters, another water will flow on him. His student Cratyl drew other conclusions from the teacher's statement: one and the same river cannot be entered even once, because by the time you enter, it will already change.

Conclusion. One can talk endlessly about mathematical sophisms, as well as about mathematics in general. Every day new paradoxes are born, some of them will remain in history, and some will last one day. Sophisms are a mixture of philosophy and mathematics, which not only helps to develop logic and look for errors in reasoning. Literally remembering who the sophists were, one can understand that the main task was to comprehend philosophy. But, nevertheless, in our modern world, if there are people who are interested in sophisms, especially mathematical ones, they study them as a phenomenon only from the side of mathematics in order to improve the skills of correctness and logical reasoning.

To understand sophism as such (to solve it and find a mistake) is not immediately obtained. It takes some skill and ingenuity. A developed logic of thinking can be useful in life. Sophistics is a whole science, namely, mathematical sophisms are only part of one big trend. It is indeed very interesting and unusual to explore sophisms. Sometimes their reasoning seems irreproachable! Thanks to sophisms, you can learn to look for errors in the reasoning of others, learn to correctly build your own reasoning and logical explanations.

mathematic teacher

Livadia UVK

Posternakova Olga Glebovna

THE CONCEPT OF SOPHISM

Sophism - (from the Greek sophisma - trick, trick, fiction, puzzle), a conclusion or reasoning that justifies some deliberate absurdity, absurdity or paradoxical statement that contradicts generally accepted ideas.

• Sophists were a group of ancient Greek philosophers of the 4th-5th century BC who achieved great skill in logic. During the period of the decline of the morals of ancient Greek society (5th century), the so-called teachers of eloquence appeared, who considered and called the acquisition and dissemination of wisdom the goal of their activity, as a result of which they called themselves sophists.

• The most famous are the activities of the senior sophists, which include Protagoras from Abdera, Gorgias from Leontip, Hippias from Elis and Prodice from Ceos.

• The most famous scientist and philosopher Socrates was at first a sophist, actively participated in the disputes and discussions of the sophists, but soon began to criticize the teachings of the sophists and sophistry in general. The philosophy of Socrates was based on the fact that wisdom is acquired with communication, in the process of conversation.

• Prohibited actions;
• neglect of the conditions of theorems; formulas and rules;
• erroneous drawing;
• reliance on erroneous assumptions.

FORMULA OF SUCCESS OF SOPHISM

• The success of sophism is determined by the following formula:

a + b + c + d + e + f ,

where (a + c + e) ​​is the indicator of the strength of the dialectician, (b + d + f) is the indicator of the weakness of his victim.

• a - negative qualities of the face (lack of development of the ability to control attention). b - positive qualities of a person (the ability to think actively) c - an affective element in the soul of a skilled dialectician d - qualities that awaken in the soul of the sophist's victim and cloud the clarity of thinking in it e - categorical tone that does not allow objection, a certain facial expression f - passivity of the listener
• a - negative qualities of the face (lack of development of the ability to control attention).
• b - positive qualities of a person (ability to think actively)
• c - affective element in the soul of a skilled dialectician
• d - qualities that awaken in the soul of the sophist's victim and cloud the clarity of thinking in her
• e - categorical tone that does not allow objection, a certain facial expression
• f - listener passivity

• The sum of any two identical numbers is zero.
• Take an arbitrary non-zero number a and write the equation x = a. Multiplying both its parts by (-4a), we get -4ax \u003d -4a 2. Adding to both sides of the last equality X 2 and moving the term -4a 2 to the left with the opposite sign, we get x 2 -4ax + 4a 2 \u003d x 2, from where, noticing that there is a full square on the left, we have
• (x-2a) 2 \u003d x 2, x-2a = x.
• Replacing in the last equality X by the number a equal to it, we get a-2a = a, or -a = a, whence 0 = a + a,
• i.e. the sum of two arbitrary identical numbers a equals 0.

• All numbers are equal
• Let's prove that 5=6.
• Let's write the equation:
• 35+10-45=42+12-54
• Let's bracket the general
• multipliers: 5∙(7+2-9)=6∙(7+2-9).
• We divide both sides of this equality by
• common factor (it is enclosed in brackets):
• 5∙(7+2-9)=6∙(7+2-9).
• Means, 5=6 .

• "Two times two equals five."
• Denote 4=a, 5=b, (a+b)/2=d. We have: a+b=2d, a=2d-b, 2d-a=b. Let's multiply the last two equalities by parts. We get: 2da-a*a=2db-b*b. Multiply both sides of the resulting equality by -1 and add d * d to the results. We will have: a 2-2da+d2=b2 -2bd+d2, or (a-d)(a-d)=(b-d)(b-d), whence a-d=b-d and a=b, i.e. 2*2=5

• « A match is twice as long as a telegraph pole.
• Let a dm- match length and b dm - column length. The difference between b and a will be denoted by c .
• We have b - a = c, b = a + c. We multiply these two equalities by parts, we find: b 2 - ab = ca + c 2. Subtract bc from both parts. We get: b 2 - ab - bc \u003d ca + c 2 - bc, or b (b - a - c) \u003d - c (b - a - c), from where: b \u003d - c, but c \u003d b - a, so b = a - b, or a = 2b.

TRIGONOMETRIC SOPHIS m

• An infinite large number is zero
• If the acute angle increases. Approaching 900 as a limit, its tangent, as is known, grows indefinitely in absolute value, remaining positive: tg90 0 = +∞.
• But if we take an obtuse angle and reduce it, bringing it closer to 900 as a limit, then its tangent, remaining negative, also grows indefinitely in absolute value: tg90 0 = - ∞.
• Let's compare formulas (1) and (2): - ∞ = +∞

• "The fastest being cannot catch up with the slowest"
• Swift-footed Achilles will never overtake a slow-moving tortoise. By the time Achilles reaches the tortoise, it will move forward a little. He will quickly overcome this distance, but the turtle will go a little further ahead. And so on ad infinitum. Whenever Achilles reaches the place where the tortoise was before, it will be at least a little, but in front.

• "The Sophism of Cratylus"
• The dialectician Heraclitus, proclaiming the thesis "everything flows", explained that one and the same river (an image of nature) cannot be entered twice, because when the one enters the next time, another water will flow on him. His student Cratyl drew other conclusions from the teacher's statement: one and the same river cannot be entered even once, because by the time you enter, it will already change.

• “He who sits has risen; who got up, he stands; therefore the one who sits is standing.
• “Socrates is a man; man is not the same as Socrates; So Socrates is something other than Socrates.”
• “In order to see, it is not at all necessary to have eyes, because without the right eye we see, without the left we also see; apart from the right and left, we have no other eyes; therefore it is clear that the eyes are not necessary for sight.”
• “He who lies speaks of the matter in question, or does not speak of it; if he talks about business, he does not lie; if he does not talk about the deed, he talks about something non-existent, and it is impossible not only to lie about him, but even to think and talk about him.

• “One and the same thing cannot have some property and not have it. Self-support implies independence, interest and responsibility. Interest is obviously not responsibility, and responsibility is not independence. It turns out, contrary to what was said at the beginning, that cost accounting includes independence and lack of independence, responsibility and irresponsibility.

• “The joint-stock company, which once received a loan from the state, now no longer owes it, since it has become different: none of those who asked for a loan remained in its board.”

• "The subject of mathematics is so serious that it is good to miss opportunities to make it a little entertaining."
• B. Pascal

• Topic of the lesson
• "Mathematical Sophisms"

• Purpose of the lesson:
• Deepen your knowledge of mathematics. It is interesting and organized to test the knowledge of those present in mathematics.
• 2. Develop logic, imagination, creativity.
• 3. Influence the cognitive activity of colleagues in the direction of its intensification.

• Sophism - proof of a false statement, and the error in the proof is skillfully disguised

• Sophism is a word of Greek origin and in translation means a puzzle, an ingenious invention. Mathematical sophisms are examples of such errors in mathematical reasoning, when, with an obvious incorrectness of the result, the error leading to it is well disguised.

• Sophisms include proof that Achilles, running 10 times faster than a tortoise, will not be able to catch up with it.
• Let the tortoise be 100 meters ahead of Achilles.
• Then Achilles will run these 100 meters, the tortoise will be 10 meters ahead of him.
• Achilles will run these 10 m, and the tortoise will be 1 m ahead, and so on.
• The distance between them will shrink, but will never go to zero. So Achilles will never catch up with the tortoise

• Sophists are a group of ancient Greek philosophers of the 4th-5th centuries. BC, who achieved great skill in logic.
• In the history of mathematics sophistry
• played a significant role, they contributed to a deeper understanding of the concepts and methods of mathematics.

• Academician Ivan Petrovich Pavlov said that "a mistake correctly understood is the path to revelation." Clarification of errors in mathematical reasoning often contributed to the development of mathematics. In this regard, the story of Euclid's axiom of parallel lines is especially instructive.

• Examples
• If the halves are equal, then the whole ones are equal.
• Half-full is the same as half-empty, full is the same as empty

• Find errors in the following reasoning:
• Task number 1.
• Four times four is twenty-five.
• Proof:
• 16:16=25:25
• 16 (1:1)=25(1:1)
• 4*4=25
• Answer: The error lies in the fact that the distributive law of multiplication is automatically transferred to division, which is incorrect.

• Task #2
• With rub.=10000 With kop.
• Proof:
• From rub. = 100 C kop.
• 1 rub. = 100 kop.
• Answer: It is impossible to multiply C rubles by 1 ruble, since there are no “square rubles” and “square kopecks”

• Practical task
• After the new year, the price of goods increased twice by 20%. By what percent did the price of the commodity rise after two successive increases?
• Solution: the cost of goods - and rub.
• after 1 increase - 1.2 and rubles.
• after 2 increases - 1.44 a rub.
• Conclusion: the price of the goods increased by 44%.

• Any two equalities can be multiplied term by term. Applying this statement to the equalities written above, we obtain new equalities
• From rub. = 10000 Cop

• Answer: The question should be asked: “Do you live in this city?”
• Answer: "Yes" - regardless of who answers - a resident of city A or a resident of city B means that you are in city A. Answer: "No" under any conditions will mean that you are in city B.

• Logic puzzle - joke:
• Two cities A and B are located next to each other. Residents of both cities often visit each other. It is known that all the inhabitants of city A always tell only the truth, and the inhabitants of city B always lie.
• What question should be asked to a resident whom you meet in one of the cities (you don’t know which one), so that by his answer “Yes” or “No” you can immediately determine which city you are in.

• Mathematical sophistry can be very useful. The analysis of sophisms develops logical thinking, helps the conscious assimilation of the material being taught, brings up thoughtfulness, observation, and a critical attitude to what is being studied. In addition, the analysis of sophisms is fascinating. Students perceive sophisms with great interest, and the more difficult the sophism, the more satisfying it is to analyze it.

• Of particular interest, this work can be put in additional classes for high school students. Knowledge of mathematics in the primary and secondary levels is still small. However, in additional classes, students can be introduced to simple mathematical sophisms based on violation of the laws of action. At the same time, if we take into account that primary and secondary school students tend to react emotionally to the absurdity of statements, the strength of assimilation of a mathematical fact increases significantly.

• In pedagogical terms, mathematical sophisms should be used not so much to prevent errors, but to check the degree of consciousness of assimilation of the material. It is necessary to start with the simplest sophisms, accessible to the understanding of students, gradually complicating the tasks as students accumulate mathematical knowledge.
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Aims and Objectives The purpose of our project is a comprehensive analysis of the concept of "sophism", establishing a connection between sophistry and mathematics, the influence of sophisms on the development of logic. We have set ourselves the following tasks: 1. Find out: what is sophism? how to find an error in apparently infallible reasoning? criteria for classifying sophisms. 2. Compile a collection of problems for sophisms in various sections of mathematics for grades 6-10.

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What is sophism? Sophism is a deliberate mistake made with the aim of confusing the opponent and passing off a false judgment as true.

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A bit from the history of sophism Sophisms exist and are discussed for more than two millennia, and the sharpness of their discussion does not decrease over the years.

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A bit from the history of sophism The emergence of sophisms is usually associated with the philosophy of the sophists, which substantiated and justified them. The term "sophism" was first introduced by Aristotle, who described sophistry as imaginary, not real wisdom.

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Sophism "Honey" - Tell me, - the sophist addresses the young lover of disputes, - can one and the same thing have some kind of property and not have it? - Obviously not. - Let's see. Is honey sweet? - Yes. - And yellow too? - Yes, honey is sweet and yellow. But what of it? - So honey is sweet and yellow at the same time. But yellow is sweet or not? - Of course not. Yellow is yellow, not sweet. - So, yellow is not sweet? - Of course. - You said about honey that it is sweet and yellow, and then you agreed that yellow means not sweet, and therefore, as it were, you said that honey is sweet and not sweet at the same time. But in the beginning you firmly said that not a single thing can both possess and not possess some property.

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Sophism "Study" The more you study, the more you know The more you know, the more you forget The more you forget, the less you know The less you know, the less you forget The less you forget, the more you know So why study?

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Logical errors Since usually the conclusion can be expressed in a syllogistic form, then any sophism can be reduced to a violation of the rules of the syllogism.

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Terminological errors Inaccurate or incorrect word usage and construction of a phrase, more complex sophisms stem from the incorrect construction of a whole complex course of evidence, where logical errors are disguised inaccuracies in external expression.

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Psychological errors The plausibility of sophism depends on the skill of the one who defends it, and the pliability of the opponent, and these properties depend on the various psychological characteristics of both individuals.

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The formula for the success of sophism The success of sophism is determined by the following formula: a + b + c + d + e + f, where (a + c + e) ​​is an indicator of the strength of the dialectician, (b + d + f) is an indicator of the weakness of his victim. a - negative qualities of the face (lack of development of the ability to control attention). b - positive qualities of a person (the ability to think actively) c - an affective element in the soul of a skilled dialectician d - qualities that awaken in the soul of the sophist's victim and cloud the clarity of thinking in it e - categorical tone that does not allow objection, a certain facial expression f - passivity of the listener

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“The subject of mathematics is so serious that it is useful not to miss an opportunity, to make it a little entertaining,” wrote Blaise Pascal, an outstanding scientist of the 17th century.

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Collection of problems Algebraic sophisms Geometric sophisms Trigonometric sophisms

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Algebraic sophisms All numbers are equal Let's prove that 5=6. Let's write down the equality: 35+10-45=42+12-54 Let's take out the common factors: 5∙(7+2-9)=6∙(7+2-9). Let's divide both parts of this equality by a common factor (it is enclosed in brackets): 5∙(7+2-9)=6∙(7+2-9). So 5=6.

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Geometric sophisms Consider the triangle ABC. Draw a line MN parallel to AB as shown in the figure. Now, for any point L of side AB, draw a line CL that intersects MN at point K. Thus, we establish a one-to-one correspondence between the segments AB and MN, i.e. they both contain the same number of points. So they have the same length.

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Conclusion Having considered sophisms, we have learned a lot from the world of logic. Even a small idea of ​​sophisms greatly expands the horizons. Many things that seem inexplicable at first look completely different. It is a pity that the basics of logic are not studied in the school course of mathematics. Logical thinking is the key to understanding what is happening, its lack affects everything.