[ Editor's note: Of all the methods of reasoning, analogy is the most intuitive. If induction and deduction are mapping, then analogy may be scattering, diffusing, and insinuating. This article by Mr. Xiong does There is a taste of formal consciousness or thinking! Worth the aftertaste! 】
Oh? Algebraic structure? What is algebraic structure? What is it for? To put it simply, it is to establish a descriptive analogy, making the analogy a computable algebraic structure.
Wait, wait, it doesn't seem to be very clear.
Ok. This needs to be said carefully. Let's have fun first, and talk about the same cage with chickens and rabbits.
It has been widely circulated on the Internet, and among the endless laughter topics, there is a solution to the chicken and the rabbit in the same cage.
Open WeChat, search for it, and see a lot of them. Choose a few. This one is from 2015:
"The Thinking of Mathematical Olympiad-"Brothers on the Run" Bao Beier chicken and rabbit raised their legs in the same cage, God's logic brushes the sense of existence" says:
In fact, Bauber’s algorithm is simpler: assuming that the chickens and rabbits raise both feet, 35*2=70 feet should be lost. The chickens are all sitting on the ground, and there are 24 feet left on the ground, all of which are 24 feet. For rabbits, each rabbit has two feet left. Dividing by 2 is 12 rabbits, 35-12=23 chickens.
Attention, raise the leg!
The following is an article by a teacher (2017):
"Children can't learn in the same cage, you must have missed an important method!" 》
It said: The
chicken and rabbit cage is the worst lesson I taught when I first started as a teacher, and it is also the most troublesome problem when most parents teach their children math. I was particularly puzzled at that time, because I had already explained this question very carefully, first taught it with the orthodox hypothetical method, and then taught it again with the leg raising method. Assuming that the effect of the method is very poor, the children in the improvement class are all looking at you with watery eyes, and I will not be the teacher. When teaching the leg-lifting method, although the child can follow you to report the answer, when you do it yourself, ten Eight or nine of them are holding pencils to draw circles. This class also made me experience the complete frustration when I first became a teacher. (Friendly reminder: elementary school students are not recommended to teach equations)
Attention, it's the leg raising method again! What orthodox hypothesis is there. There are equations!
However, if you are more interested in mathematics, I recommend you to read this article (2018) written by Supermodel Jun:
"Chicken and rabbit in the same cage, a big problem in the elementary school era, I have seen the super brains of the ancient Chinese"
This article is good, fortunately, it points out the super brain hole, that is, to understand this problem, the ancient Chinese thoughtfully opened a super brain hole-the method of raising the leg: let the rabbit and the chicken raise their legs at the same time, haha !
In short, this topic will be circulated on the Internet after a period of time. Make a facelift, what domineering bosses, beautiful girls, jailbreak geniuses, stupid pigs, all kinds of messy flowers, dazzling people, but never change from the sect-the method of raising the legs.
In fact, the so-called leg raising method, after learning the linear equations, knows that it is actually the elimination method. However, the ancients did not have algebraic methods and could not explicitly express linear equations. Everything was described in words and then stirred in their minds. Therefore, when the ancients solved such a mathematical problem, it was necessary to use extreme brain power and come up with extreme methods, such as the method of raising the leg, in order to solve it. In ancient times, it was not easy to be able to open this brain hole, and its difficulty may be even greater than the difficulty of proving a big theorem in modern times.
This is because the ancients did not have a proper algebraic structure to describe this problem. In the ancients, only words can be used to describe, without precise mathematical methods, it is difficult to think, and it is even more difficult to solve, and it is even more difficult to generalize the problem to more variables. Although the linear equation is very simple in our opinion, before the linear equation is used to express the chicken and rabbit in the same cage, the expression of X, Y and the corresponding operation rules must be imported. Leaping from arithmetic to algebra, history has taken thousands of years.
The enlightenment that the chicken and the rabbit in the same cage give us is that there are two ways to deal with a mathematical problem: one is to rack your brains to get a wonderful solution; the other is to find a suitable algebra Expressions, and algebraic expressions can make solving a more mechanical routine. The leg raising method is the former, and the establishment of algebraic expression is the latter. Compared with brilliant brains, the establishment of appropriate algebraic expressions is a greater progress in mathematics. Of course, it is impossible to establish algebra if it is not in the right time. This is not limited by the individual's intelligence, but by the time.
Now, we understand the meaning of algebraic structure.
So, what kind of structure does this era allow us to build? Our time is urgently calling for the establishment of algebraic structures for describing analogies. Moreover, we are willing to predict that this is still a lucky time, that is, it is indeed possible to establish this algebraic structure.
Now let’s talk about the analogy. We can look at some simple examples first. Let us first copy a section from the popular science book "Mathematics Thinking" by Zheng Lejun:
我们来思考一下这个数字集合：
{1，2，3}
你觉得以下哪个集合是和它类似的集合？
1.{2，3，4}
2.{2，4，6}
3.{-1，-2，-3}
4.{11，12，13}
5.{101，102，103}
6.{100，200，300}
7.{13，28，42}
8.{猫，狗，香蕉}
这就是类比。可以说，类比地看，上面的8个集合，和{1，2，3}这个集合都可以类似，不同的类似。例如，{1，2，3}和{2，3，4}类似，因为后者是前者每个数加1。又例如，{1，2，3}和{100，200，300}类似，因为后者是前者每个数乘100。再例如，{1，2，3}和{猫，狗，香蕉}类似，因为在读者心目中，猫狗香蕉的重要程度排序就是1，2，3。而且从情形1到8，集合都有3个元素，这就不仅和{1，2，3}类似，而且是集合的基数相等了。
其实还可以再稍加变动一下：
1.{3，3，4}
2.{6，4，2}
3.{-1，-2，-3，-4}
4.{11，12，13，14，15，16}
5.{。，。。，。。。}
6.{口，哭，品}
7.{13，28，42}
8.{猫，狗，香蕉，金鱼}
上面的8个集合，和{1，2，3}这个集合也都可以类似，只不过，有的类似隐藏得更深一些。
如上的类比，其实仅触及了类比的皮毛。还有更多更深更广更隐藏的类比。在诗词、对联、谜语、脑筋急转弯、笑话段子、政治***、···，中间，类比都无处不在。要证明数学定理，探索物理规律，设计生物实验，编写计算机程序，肯定时时刻刻在使用类比。
Readers must be very easy to experience these similarities, and it is not difficult to write down the similarities that they have experienced. But is there a mathematical expression to express the similarities we experience? This is the problem! This is the algebraic structure we hope to find, which can be used to express the algebraic structure of analogy in our thinking. Hou Shida, the author of "GEB", co-authored a book: "Appearance and Essence: On Analogy as the Fuel and Flame of Thinking", in which analogy is regarded as the most essential activity of thinking. If we can find an algebraic structure that describes analogies, we can describe thinking, and even computational thinking, in a more precise way. If this can be done, it is completely believed that it will greatly promote the research of artificial intelligence.
As far as we know, there is currently no such algebraic structure. Therefore, we list the problems here: to find an algebraic structure that can be used to describe analogies, which can be written as mathematical expressions, and that we can also calculate analogies.
If you are interested in this issue, welcome to participate in the discussion and welcome to contact us.
The problem is of course not easy. Everything is difficult at first, and we have taken a very small step. Interested netizens, welcome to read our article: "Some Preliminary Discussions on the Five Elements", DOI: 10.13140/RG.2.2.16892.69762
Few people understand that there is a more fundamental obstacle. As far as our observation is concerned, the whole itself is never established. They are without exception the construction of our nature. They are not established facts, nor are we automatically regarded as the same objective material of the same kind because of their same natural attributes.