Analogy is one of our oldest ways of thinking, and it plays an important role in inspiring human beings to make new discoveries. We often think that we are familiar with analogical reasoning, but sometimes we refuse to accept obvious analogy conclusions due to our own cognitive limitations.
Humans have been staring at the moon since ancient times. For a long time, people believed that the moon was perfect and the domain of the gods. If the balance of the moon’s halo was broken and a lunar eclipse occurred, humans would need to use various actions to correct it: the Chinese threw mirrors, the Incas yelled, and the Romans waved torches frantically. It wasn't until Galileo turned his telescope on the moon in 1609 that he discovered that it was not smooth and perfect, but a mess of cliffs and craters.
Galileo, like some other astronomers, saw in a telescope that there were points of light in the dark part of the moon, which gradually grew larger and brighter, and finally merged with other bright parts. Galileo felt that this phenomenon is very similar to the morning sun shining on a mountain. The higher the sun climbs, the smaller the shadow of the valley shrinks, and finally the whole mountain is bathed in sunlight. He believed that shadows and other optical phenomena should be the same on Earth and on the Moon. Therefore, Galileo concluded that the surface of the moon must not be smooth, but high and low, like the earth, with mountains and valleys.
This is the power of analogy.
So what is an analogy?
An analogy compares two things, highlighting aspects of them that are considered similar, and its main purpose is to explain something unfamiliar to us in terms of something we are familiar with. Analogy reasoning is a way of thinking based on analogy. It deduces that another attribute of two (or two types) things is the same or similar based on the same or similar attributes. It is a special to special reasoning process.
A story about Zou Ji is recorded in the famous ancient Chinese book "Warring States Policy", which can be regarded as an excellent case of analogical reasoning.
Zou Ji has more than eight feet, but his appearance is beautiful. Dressed in court clothes and looking into the mirror, he said to his wife: "Who am I as beautiful as Xu Gong in the north of the city?" His wife said: "The king is so beautiful, how can Xu Gong be as good as the king?" Avoid not being confident, but ask his concubine again: "Which one is more beautiful than Mr. Xu?" The concubine said: "How can Mr. Xu be as good as the king?" When I go to bed at night, I think about it, saying: "My wife who is beautiful to me is selfish; the concubine who is beautiful to me is afraid of me; the guest who is beautiful to me is asking for me."
So he went to the court to meet King Wei, and said: "I know that I am not as beautiful as Xu Gong. The wife of the minister is private, the concubine of the minister is afraid of the minister, and the guest of the minister wants to ask the minister. They are all beautiful than Xu Gong. Today, there are thousands of miles in Qi, and there are hundreds of cities. There are no court ladies who are private to the king, and all the officials in the court are not afraid of the king, and there is no one in the four realms who does not seek the king. From this point of view, the king is very concealed."
In the above story, Zou Ji analyzed the reasons why his wives and concubines and guests did not tell the truth to him, and applied this set of reasoning to similar scenes (that is, the relationship between King Qiwei and court ladies, courtiers, and other vassal states), and came to the conclusion that "the king's concealment is too much", which finally made King Qiwei vigorously accept advice and achieve great achievements.
In the previous story of Galileo, based on a fact (shading and optical phenomena do not change due to the earth and the moon) and similar observational phenomena, Galileo speculated that the moon should have the same mountain and valley as the earth.
Reasoning by analogy is the basis of human thought and, arguably, some non-human animals as well. From the perspective of human development history, as an auxiliary means to generate new discoveries, analogy is widely believed to play an important heuristic role. The pioneer of chemistry, Joseph Priestley, believed that analogy was the best guide to all inquiry, with the help of which all non-accidental discoveries were made.
In elementary school mathematics textbooks, there is a good analogy case, that is, the analogy between the derivation method of the area of a circle and the derivation method of a cylinder. When deriving the area of a circle, the circle is cut into many small sectors, and finally assembled into an approximate rectangle. Similarly, when deriving the volume of a cylinder, the cylinder is cut into small cylinders, and finally assembled into an approximate cuboid.
From the object of analogy, analogy can be divided into: analogy of concept and operation, analogy of conclusion, analogy of method.
A classic example of a conceptual analogy is the definition of a circle and a sphere. We know that the definition of a circle is a set (a curve) of all points with equal distances from the center of the circle, then the definition of a sphere in three dimensions should be a set of points with equal distances from the center of the sphere (a curved surface).
Through analogy, we can often discover some new conclusions.
For example, suppose we know the following proposition: Of all rectangles with a fixed perimeter, a square has the largest area.
Can this conclusion be extended to three-dimensional space by analogy? To this end, we first have to make an analogy between the two-dimensional and three-dimensional concepts involved in this proposition, as shown in the following table.
The concept of two-dimensional space |
The concept of three-dimensional space |
rectangle |
cuboid |
square |
Cube |
perimeter |
surface area |
area |
volume |
From this, can we deduce the three-dimensional analogy conclusion of the above proposition: among all cuboids with fixed surface area, the cube has the largest volume.
Below, I will give a few typical cases of reasoning by analogy.
Nondecimal to Decimal Analogy
We know that in the decimal system, the characteristic of a number divisible by 9 is that the sum of its digits can be divisible by 9. The reasoning process is based on the bit value representation of the number, for example:
297=2×102+9×10+7
=2×(99+1)+9×(9+1)+7
=2×99+9×9+2+9+7
Therefore, 297 is divisible by 9 if and only if the sum of its digits, 2+9+7, is divisible by 9.
So, can we extend this conclusion and reasoning process to other systems? For example, in base 7, what conclusions can we draw?
Here, 9 in decimal should correspond to 6 in base 7. Therefore, we can boldly propose the following analogy conclusion: In base 7, the characteristic of a number divisible by 6 is that the sum of the digits can be divisible by 6.
Whether this conclusion is correct or not, we can also verify it by analogy with the decimal reasoning process.
435(7)=4×100(7)+3×10(7)+5
=4×(66(7)+1)+3×(6(7)+1)+5
=4×66(7)+3×6(7)+4+3+5
Therefore, 435 (7) is divisible by 6, which is equivalent to 4+3+5 being divisible by 6.
We also know that decimal cyclic decimal fractions have the following conclusions:
If the sum of infinite series is not used, the derivation process is as follows:
For other bases, such as base 7, similar conclusions can be drawn by analogy:
Its reasoning method can also be compared to decimal reasoning:
ancestral principle
Zuo’s principle is a well-known proposition involving geometric quadrature. It says this: If the power potential is the same, the product cannot be different.” “Power” is the cross-sectional area, and “potential” is the height of the solid. It means that if two solids bounded between two parallel planes are cut by any plane parallel to these two planes, if the areas of the two cross-sections are equal, then the volumes of the two solids are equal.
If this principle is applied to a two-dimensional plane, what conclusions can be drawn?
To do this, first we have to make a conceptual analogy between 3D and 2D:
Concepts in 3D |
Concepts in 2D |
three-dimensional |
Graphics |
volume |
area |
section |
intercept segment |
high |
high |
parallel plane |
parallel lines |
flat |
straight line |
area |
length |
From this, we can get the two-dimensional analogy conclusion of Zuo’s principle: if two plane figures bounded by two parallel lines are cut by any straight line parallel to these two parallel lines, if the lengths of the two sections of the cut lines are equal, then the areas of the two plane figures are equal.
Center of gravity of triangles and tetrahedrons
We can find out the center of gravity of the triangle as follows: the three midlines of the triangle intersect at one point, which is the center of gravity of the triangle (point G as shown in the figure below).
For a tetrahedron, is it possible to find its center of gravity similarly?
For this, we also need to involve conceptual analogies.
Concepts in 2D |
Concepts in 3D |
triangle |
tetrahedron |
midline |
middle surface |
intersection point of straight line |
plane-to-plane intersection |
Based on this, we can make an analogy: the triangle connecting one edge of a tetrahedron with the midpoint of the opposite edge is called the midplane of the tetrahedron, then there are 6 midplanes in total, and these 6 midplanes intersect at one point, which is the center of gravity of the tetrahedron.
Although we are not now proving the conclusion whether the six mid-planes intersect at one point, at least we have a hunch that the conclusion drawn from this analogy should be correct.
Calculus Finding Area and Volume
In calculating the area of calculus, we regard a plane figure as a figure composed of countless small rectangles. (Note: Don’t be intimidated by the name of calculus. In fact, elementary school students can generally understand the principles of calculus. The area of a circle is calculated using calculus.)
By analogy, we can apply this method to find the volume of an object in three-dimensional space. At this time, we regard the three-dimensional figure as a figure composed of infinitely many cylinders. Here, a rectangle in two dimensions is analogous to a cylinder in three dimensions.
infinite division
The square in the figure below has a side length of 1. First, it is divided into four equal squares, and the upper left corner is colored. Then, the square at the lower right corner is divided into four, and the upper left corner is colored. If we continue this process, how much of the total area is the last part that gets painted?
The most direct approach to this problem is to use infinite series sums that elementary school students cannot understand. If you don’t need to sum the infinite series, you can think about it this way: After removing the 1/4 block in the lower right corner, the remaining part is 1/3 of which is colored (as shown on the left in the figure below).
In the remaining 1/4 block, we remove the lower right corner of this 1/4 block, then the painted part still occupies 1/3 of the entire area (as shown in the right picture above). By analogy, every time a small piece in the lower right corner is cut out, the area of the painted part is 1/3 of the entire area on different scales, so the overall area of the painted part is 1/3 of the entire square area.
Based on this idea, can we solve the following problem by analogy?
In the yellow regular triangle ABC below, take the midpoints D, E, and F of the three sides and connect them respectively, then take the midpoints H, I, and G of the three sides DE, EF, and DF respectively, and paint ΔDGH, ΔEHI, and ΔGIF in blue. Next, repeat the same operation above for the small triangle GHI in the middle. If this operation continues forever, what fraction of the area of the whole equilateral triangle is the area of the part colored in yellow in the figure?
In this problem, the triangle corresponds to the square in the previous problem. In the solution to the above problem, we deduct a piece of the enlarged part that is the same as the original figure (that is, the 1/4 of the lower right corner) from the square. Correspondingly, we also find out the enlarged part of the figure that is the same as the original figure, which is obviously the triangle corresponding to GHI. After deducting it, the remaining part, the proportion of the yellow area is 12/15=4/5. Therefore, the proportion of the entire area is also 4/5.
split item
Split terms are a common technique in elementary and middle school mathematics. Generally, what is taught in textbooks is the simplest fraction split term, and the numerator only has two terms multiplied together. But behind it lies the basic idea of splitting: turning one item into two and subtracting them, so that the front and back can be canceled.
Then, by analogy, we can also split terms in fractions whose denominators are multiplied by three, four, or more arithmetically different numbers.
In addition to fractional split terms, integer products can also split terms:
1×2+2×3+...+n(n+1)
We can multiply two natural numbers and divide the term into the difference of two products, each product is the product of 3 natural numbers, as follows:
n(n+1)=[n(n+1)(n+2)-(n-1)n(n+1)]/3
What if instead of multiplying two natural numbers, three or more natural numbers are multiplied? for example:
1×2×3+2×3×4+...+n(n+1)(n+2)
Using the analogy of the fractional split term and the above integer split term, it is not difficult to obtain the following split term:
n(n+1)(n+2)=[n(n+1)(n+2)(n+3)-(n-1)n(n+1)(n+2)]/4
Straight line divides plane and plane divides space
Let's look at a classic question again:
How many parts can n straight lines divide the plane into?
Using the idea of recursion, we know that the number of blocks that n straight lines divide the plane at most is the number of blocks that n-1 straight lines divide the plane at most plus n. Thus, n straight lines divide the plane into 1+1+2+3+⋯+n=n(n+1)/2+1 blocks. Check the calculation: when n=1, 2, 3, they are 2, 4, 7 respectively, which is satisfied.
Solving the problem here is of course not the end, the core problem has not been solved yet.
The induction we just made is just a guess, and we still need to prove its correctness. Why is the maximum number of blocks divided by n straight lines plus n on the basis of the maximum number of blocks divided by n-1 straight lines? This involves the relationship between " line-intersection-line segment-plane ".
We know that if there are n points on a straight line, then these points will divide the straight line into n+1 segments.
Originally there were n-1 straight lines. After adding the nth straight line, the nth straight line has at most n-1 intersection points with the previous n-1 straight lines. These intersection points will divide the nth straight line into n segments. And each section will divide the original area into two, so there are n more blocks. The figure below gives a schematic diagram of n=3.
We can completely apply this reasoning method to the problem of dividing planes by non-linear plane graphics, such as the following problem:
How many parts can n circles divide the plane into at most?
We can continue to use the previous recursive thinking and the analysis method of "intersection-line segment-plane". If a circle is added on the basis of n-1 circles, then this circle has at most 2(n-1) intersection points with the previous n-1 circles (as shown in the figure below, after the fourth red circle is added, it will intersect with the previous 3 circles at most, and there are 6 red intersection points in total). ) blocks.
According to this, n circles divide the plane into at most:
1+1+2+4+…+2(n-1)=2+n(n-1) (n≥1)
If you add a dimension, the problem becomes:
How many parts can n planes divide the space at most?
If we derive from the idea that points are divided into straight lines into line segments, and line segments into planes into regions, then we will find that the solution to the problem of dividing a plane into a space can also be compared to the method of dividing a line into a plane. First, we have to make some conceptual and operational analogies.
Concepts and operations in 2D |
Concepts and operations in 3D |
straight line |
flat |
Intersection |
intersection line |
Point to Line into Line Segment |
line segments into planes |
Line segment divides plane into plane area |
Plane divides space into space regions |
In the problem of dividing a plane by a straight line, we use the extra line segment to analyze the number of planes that are divided after adding a straight line; then in the problem of dividing a space by a plane, can we also use the extra plane to analyze the number of spaces that are divided after adding a plane?
We still use recursive thinking. We know that 3 planes divide the plane into 8 pieces at most. Add 1 plane on the basis of 3 planes. The first 3 planes have at most 3 intersection lines with this plane. These three intersection lines will divide the 4th plane into 7 parts at most (obtained from the conclusion of dividing the plane by a straight line). Each part will divide the original space into two, so 7 more blocks are divided on the basis of 8 blocks.
Generally speaking, the nth plane will have n-1 intersection lines with the previous n-1 planes. According to the straight line to divide the plane, these n-1 intersection lines can divide the nth plane into n(n-1)/2+1 areas at most, so that n(n-1)/2+1 more spaces can be divided than n-1 planes.
Therefore, the plane subspace satisfies the following recurrence relation:
lack of analogy
However, because the logical basis of analogical reasoning is insufficient, probable, and speculative, their conclusions are only supported to varying degrees, and they are not necessarily completely reliable. Therefore, analogy can only be used as an auxiliary means of discovery, not as a strict mathematical method. For the conclusions drawn through analogical reasoning, rigorous demonstrations are required to confirm the correctness of the guessed conclusions.
For example: "This novel is only 1,000 words long and the writing is smooth. This novel won an award. The novel you wrote is also 1,000 words and the writing is smooth, so you will definitely win the award." Such an analogy will undoubtedly lead to wrong conclusions.
Historically, Thomas Reid's arguments for the existence of life on other planets in the 18th century were also based on reasoning by analogy. Reid pointed out many similarities between Earth and the other planets in the solar system: all orbit and are illuminated by the sun; several planets have moons; all rotate on their axes. Therefore, he concludes, "it is not unreasonable to think that these planets may, like our Earth, be habitats for all kinds of life". Ultimately, modern science has disproved this analogy wrong. However, even so, humans still hope to use analogies to find alien terrestrial planets suitable for life.
Not everyone really understands analogies
At this point, many people may think that analogy is already a way of thinking that we master every day. But is this really the case?
The novel "Flat Country" is a pioneering work in the 19th century imagining four-dimensional space. In the second half of the novel, the author uses analogy reasoning extensively.
The protagonist who has been living in the flat country once dreamed of the straight line country and tried to explain to the king of the straight line country what a two-dimensional plane is, but no matter how he explained it, he could not succeed and had to give up.
Once, when the protagonist was explaining the knowledge of geometry and arithmetic to his hexagonal grandson, his wise grandson asked a question: If you move a point 3 inches, you can get a 3-inch line segment, which can be recorded as 3; ), you must be able to get another figure (I don't know what figure it is)-the length of each side of this figure is also 3 inches, and this figure must be recorded as a cube of 3.
Although the protagonist's grandson has been living in the flat country and has never seen a cube, he has foreseen the existence of a cube through some pure thinking reasoning. This reasoning method is analogy reasoning.
It is a pity that the protagonist cannot break through the confinement of the two-dimensional world. In his eyes, the cube of 3 has only numerical meaning, not geometrical meaning. What a fool, he thought, the boy.
But at this moment, a visitor from a space country arrived. Here's how the visitor described himself: From a certain angle, I am indeed a circle. I am a more perfect circle than any circle in Flatland. More precisely, I am a circle composed of many circles.
That's right, this visitor is the ball.
In order for the stubborn protagonist in the book to understand what the third dimension is, Ball also picked up the powerful weapon of analogy.
Ball's analogical reasoning seems so natural to us, he is so persuasive:
1. A point moves to the north and leaves a glowing track. This track is called a line segment, and a line segment has two endpoints.
2. This north-south line segment moves in parallel in the east-west direction, so each point on the line segment will leave a linear trajectory in the east-west direction. Assuming the line segment is moved a distance equal to the original length of the line segment, then you get a square. A square has four sides and four corners.
To this place, the protagonists of Flatland can fully understand.
But next, the ball wants the protagonist to use his imagination, imagine a square in the flat country moving upwards in parallel, that is, moving outside the flat country, so that each point in the square will not pass through the places that other points once occupied. The trajectory left by each point is a line segment that belongs only to itself.
Honestly, it's a pretty good analogy. But helplessly, the protagonist living in the flat country cannot imagine what it is to move outside the flat country, and he is on the verge of being driven crazy. But he still suppressed his impatience, and thus had the following conversation.
Protagonist: Since you said that moving the square "upward" can produce a new figure, what kind of figure is this figure? I assume you can always describe this figure in the language of Flatland.
Ball: This shape is so simple that it can only be deduced by strict analogy—only, by the way, you cannot call this new shape a "figure" because it is a three-dimensional shape. But I can describe this new shape to you. I can't describe it exactly, but the analogy will give you an idea of what this new shape looks like.
First, suppose we have a point. Since it is a point, of course he has only one vertex.
By moving a point, a line segment can be obtained. A line segment has 2 vertices.
By moving a line segment, a square can be obtained, and a square has 4 vertices.
You can answer the following questions yourself: 1, 2, 4, this is obviously a geometric progression. So what is the next number in the series?
Protagonist: 8.
Ball: Exactly. So, by moving a square, a new shape can be produced. Now you don't know the name of this shape, but people in our space country call it "cube". A cube has 8 vertices. Now do you believe what I say?
Protagonist: Since you said that this new shape has a "vertex", it must be what we call a "corner". So does this new thing have sides too?
Ball: Of course there are sides. This can be deduced by analogy. However, the sides of this new shape are not what you would call "sides," but what we would call "sides." A side is equivalent to a solid figure in the plane country.
Protagonist: So, how many solid shapes, or how many sides, does this new shape have?
Ball: Why do you still ask me? Are you not a mathematician? With all due respect, let me put it this way: any shape can be seen as surrounded by some 'side elements', and the dimension of each 'side element' is always 1 smaller than the dimension of the shape. Since 1 point is zero-dimensional, points have no 'side elements'; and so on, line segments have 2 side points, and squares have 4 sides; 0, 2, 4, what do you call this series?
Protagonist: Arithmetic progression.
Ball: So what's the next number in this series?
Protagonist: 6.
Ball: Exactly. You see, you can answer that question yourself. The cube created by moving the square is bounded by 6 sides, that is to say by 6 insides of you. Now you get it all, right?
However, the protagonist does not understand. He was going mad, calling "monster," and saying, "Whether you're a liar, a wizard, a nightmare, or a devil, I can't stand your pranks any longer."
In the end, the ball had no choice but to take action, pulling the square protagonist out of the flat country and into the space country. Seeing is believing. After experiencing a huge shock, the protagonist finally understands that the space country does exist. At this time, the protagonist who has gained new knowledge seems to be in heaven. His mind was completely ignited, and he could no longer tolerate some dictatorial people restricting the dimension to two, three, or any dimension less than infinite.
Therefore, just when the ball wanted to continue imparting the knowledge of the rules and three-dimensional shapes to the protagonist, the protagonist summoned up the courage to interrupt the ball and started the following dialogue.
Protagonist: Sir, it is your wisdom that opens my mind and ignites my desire. You make me think that beyond you there is something greater, more beautiful, more perfect. You are a shape composed of many circles, and you are higher than all the inhabitants of our flat country; then there is no doubt that above you, there is a supreme being composed of many spheres, who will be higher than any three-dimensional form of the space country. We are now overlooking everything in the flat country in the space country, and we can have a panoramic view of the interior of all plane figures; since this is the case, there must be a higher and purer space above us-you must also plan to take me there to see it. Ah, no matter what dimension you are in, you will always be my philosophical teacher and my friend. Let us go to a higher place together, and overlook all this from a wider space and a deeper dimension. From there, we can see the inside of the cube, and even your gut, and the guts of other spheres like you, will become unobstructed.
Unexpectedly, this time it was Qiu's turn to be dumbfounded.
Ball: But where is the four-dimensional space you mentioned?
Protagonist: I don't know. But mentor, you must know.
Ball: I don't know. There is no such place at all. Your idea is completely unimaginable.
Protagonist : We can use analogy to demonstrate the existence of four-dimensional shapes.
Ball: Analogy! Nonsense! What an analogy!
Protagonist: Your Excellency must be testing me to see if I still remember the apocalypse you passed on to me.
In one-dimensional space, wouldn't moving a point produce a line segment with two endpoints?
In two-dimensional space, wouldn't moving a line segment produce a square with 4 vertices?
In three-dimensional space, wouldn't moving a square produce a divine creature with 8 vertices—a cube?
So in four dimensions, if you move a cube, wouldn't it produce a more divine creature with 16 vertices?
You see, the law of sequence can never go wrong: 2, 4, 8, 16, isn't this a geometric progression? Isn't this the conclusion that 'only by strict analogy can be deduced'?
A line segment has 2 endpoints, and a square has 4 sides, so a cube must have 6 sides. Isn’t this what you taught me? Look at the law of this number sequence: 2, 4, 6, isn't it an arithmetic progression? Then in the next step, we must be able to draw such a conclusion: in the four-dimensional space, the more sacred offspring born from the sacred cube must have 8 sides, isn't it so? Isn't this what you taught me, the conclusion that 'it can only be deduced through strict analogy'?
In four-dimensional space, a cube is moving in some new direction. By strict analogy, every point in the cube passes through a new space, leaving non-overlapping trajectories—this creates a more perfect shape than a cube. This shape has 16 vertices, 16 solid angles, and is surrounded by 8 side cubes.
The following table summarizes the conclusions of the protagonist's analogy.
dimension |
name |
vertex |
side element shape |
Number of side elements |
1 |
line segment |
2 |
point |
2 |
2 |
square |
4 |
line segment |
4 |
3 |
cube |
8 |
square |
6 |
4 |
hypercube |
16 |
Cube |
8 |
The protagonist didn't expect that this time the ball was completely unacceptable, and he roared again and again to tell the protagonist to shut up. In the end, he kicked the protagonist back to Flatland.
After reading this, I remembered a sentence: Summer insects cannot speak ice, and well frogs cannot speak sea.
We laugh at well frogs, but in fact each of us is a well frog. While we all know analogies, not everyone really understands and uses them. We can use analogies to explain what we understand to a newcomer within the scope of our own cognition, but it is difficult to accept analogy conclusions beyond the limits of our own cognition or the environment.
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Author: 昍 Dad, 昍 Mom
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