Create? The source is innocence. Remember when you were in kindergarten, you associated an apple with the number "1" and then counted the puzzles that the apple encountered when calculating "1+ 1=2"? This confusion may not be accidental. The mathematics education we receive is often a process of abstracting concepts and ignoring specific differences, but our way of thinking is amazing. It is a kind of category research, which focuses on the difference between an apple and another apple and the relationship between things.
Eighty years ago, physicists introduced group theory in mathematics into physics in order to study the symmetry of quantum systems. Today, in order to study quantum entanglement in complex systems, physicists have begun to study the category of mathematics. This kind of quantum entanglement is the key to understand the origin of quantum state properties, the origin of elementary particles and even the origin of time and space from the perspective of evolution. In reductionism and performance? In today's competition, classification is becoming more and more important. It can be said how important calculus is to the physics of reductionism, and how are the categories opposed? How important it is to talk about physics. Classification has brought grandeur beyond mathematicians' imagination. Is it conceptually unified? Analysis and algebra, unification? Discrete and continuous. Today we are curious 1+ 1=2? Enter the wonderful world of classification!
Author | Kong Liang (Shenzhen Institute of Quantum Science and Engineering, south university of science and technology of china, China)
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This one? Does Zhang really mean that you are young? Landscape architecture 1+ 1=2 can be seen without any mathematical background, as long as you are curious about 1+ 1=2? . But our motivation is to introduce the basic spirit of mathematics, so it is necessary to mention it briefly first? Next category, no? Can readers skip the quotation directly? .
Categorization originated from algebraic topology and was put forward by Samuel Eilenberg (19 13- 1998) and Sanders McLean (1909-2005) in the 1940s. ? From six o'clock? Alexander Grothendieck (1928-20 14)? Classification language? What is the basis of reconstruction algebra since it appeared in mathematics? Classification has replaced set theory as the new foundation of mathematics. This trend not only intensified in the field of mathematics, but also attracted attention in the 1990s? The new strong man? Moving? : physics. ? Scientists have found that describing two-dimensional rational quantities? Mathematical language of field theory and arbitrary dimensional topological order? It is also a kind. Of course, this is nothing strange. Do you know the classification? Won't be surprised. Because of the changes brought about by classification? Is so low, it fundamentally changed our view of mathematics (even? It is the basic paradigm of other disciplines. ? Don't know the category? May have a lot to do with this sentence? Conflict. This is also normal. Without a real understanding of categorization, it is hard to imagine that categorization is possible. Maybe after reading this? Your resistance will be slightly reduced? Some. I think classification is a continuation? Newton's calculus? After life, a second language? what's up Life, in fact, is the category itself? The new calculus. She? Quantity is reflected in many aspects, such as? A simple category formula can be completed? A complicated quantity? Construction of field theory or simultaneous calculation? How much difference? Fusion of field theory; Many complicated physical and mathematical structures? However? However, it is absolute; More importantly, many mathematical or physical facts beyond set theory can only be stated and understood in the sense of category. Just the last one? Tell us, is there? New mathematics beyond set theory? Lu is waiting for us to discover and explore. Now there is even crazier speculation, that is, how quality is the classification? Tangle and quantity? Quote? The foundation of.
Category change? How is it possible to be so low-level Have questions? What about all scientific fields? , including logic, mathematics, physics, computer science, language? Science, sociology, economics and so on. So let's go beyond math? It is meaningful to know it. Ben. Just throw a brick? Frankly, this change? How about the bottom? Does the bottom need us to keep returning until each one? Answer? Who initiated the enlightenment of mathematics? Carving.
1. 1+ 1 = 2?
I believe in each of us. Answer? All our mathematics education begins with 1+ 1=2. Where do we start? From the beginning, we opened it? What about "de-categorization" in the journey of domain abstract mathematics? Field regression.
Create? The source is innocence.
Alexander Grothendieck
I hope? Home and me? Back to preschool? The state of being a child. Only in this way can you see the essence of the problem clearly.
1+ 1=2 is hard to understand. Do we really understand? Maybe you don't think there is any difficulty, only when you pay? A preschool that has never heard of 1+ 1=2? Can you understand when the child explains? How difficult this question is.
First one? The difficulty is: What is "1"?
First one? The difficulty is: What is "+"?
The third difficulty is: What is "="?
The fourth difficulty is: What is "2"?
What is "1"? Are you sure you know what "1" is? What about you? 1 has it passed? ? Friends don't know what "1" is. In order to make children understand numbers, what is the teacher's teaching method? In kind? Like what? With magnets? Pigs? Ducks, apples, Bananas, wait, suck them into your belt? Genus? On the blackboard. Is it true?/You don't say. East? It is the basic experience of our existence, and nothing else is reliable.
Let me do it. The symbol o stands for apple and j stands for apple? Bananas. Shall we play it again? Are there any apples in it? Then go up. The following formula appears on the blackboard:
O+O=OO，( 1)
All right. we? I have eaten apples, so there is nothing wrong with O, but what is "+"? What is "="?
Really? Friends? Generally acceptable (1), the way of acceptance is to ignore "+". Isn't (1) "OO=OO"? "+"is hard to understand. I'll skip first and say "=" first. In fact, this is even harder!
"=" (equal to) is? A hard thing to understand? . In the real world? We hardly have? Pass two completely? What kind of East? . Are the apples on both sides of "OO=OO" actually not? Sort of. The truth is often, maybe their faces? Some differences, or the attraction of magnets? Some differences, and so on. Then "=" is hard to understand. Are you online? Live broadcast? Right? When we don't say equality, we say "? Sample. " So what do the "OO" on the left and the "OO" on the right mean? Like "?
Can you let me talk? A story that shocked me. I was the first? The second time I gave a report at Shenzhen Middle School, I said I didn't know what "? "Please middle school? For me? A "preschool"? Children "explained. And then what? A brave classmate came over. He took out the OO on the left and the OO on the right respectively. An apple, and then put these two apples in? Get up and put the remaining two apples in? I'm sorry, he said it was "? Sample. " In fact, he gave both left OO and right OO? Correspondence can be represented by graphs:
This is already? It's wonderful enough, but the wonderful is yet to come. . The question is, why are you doing this? We are studying? There are many puzzles in my career, and I often don't know whether some choices are guided by the underlying principles or random. Actually, son? The child is very keen. ? There must be a child who looks stupid? I wonder, why? At that time, after this classmate explained, I asked? "Home here? What does the definition of "sample" mean? ? Then there is a lot of learning? Question this. ? First of all, is this a definition? Natural and reasonable? Secondly, this definition is not unique? Yes? If there is still a choice? This defines "? Example:
Say something? Honestly, I'm shocked. I'll hide in the back, okay? East? All by middle school? Got it. ? Did The Economist see the problem?
There may not be two easts in the real world? Completely? For example (please let me ignore the measurement? The same particle? We're still young? Garden, not heard of). ? General? Do you want to see two oriental styles at home? Sample, just compare the two east? Go down. But these two east? Can't be completely? Sample, so? When you get old, you ignore it? In an extreme case, we ignore all the internal structure and attributes of an apple and regard it as something with neither internal structure nor additional attributes (that is, elements in set theory). In this case, one-to-one correspondence is a good definition of "the same". If we accept this definition "? "Is it a sample? Yes, is it? "Correspondence" to define? Sample. " Then the question is coming:
Are there two kinds? Like "? "Or a sample? Like what?
Is this true in modern mathematics or category? A basic question has been thought deeply. Modern mathematics or category view is: are there two kinds? Like "? "Not a sample? Samples, except? what's up Answer? Another one? Have another drink. However. ? The apple on the left? Which one is red? One is green; Is that right, too? The red one? The green one. ? Answer? However "? Sample "is to save face? "? Sample. "But no face? Before this additional "structure", we have two kinds of no? Like "? Sample ",actually not? Sample.
For those readers who have a foundation in linear algebra? Paragraph:
This question seems simple, but it's not? A nuclear bomb? Question, in mathematics? The ubiquitous appearance has caused confusion to many beginners. ? For example, many textbooks in China teach linear algebra into matrix algebra. Lots of knowledge? A linear space, just? Let's move. Put it on? Vector basis. Is (linear space+given basis) completely different from linear space? A mathematical structure! Unknown? so this is it？ How to distinguish linear space from dual space? Why? It also confuses the difference between tangent space and cotangent space. ? What is a finite dimensional linear space and its dual space? Numbers are linearly isomorphic, not? who is it? Of course. But? What is the dual space from a finite dimensional linear space to its dual space? Answer? Isomorphism of nature.
Have we noticed? Recurring "? The word "Ran". ? The origin of categorization is Allen Berg-McClane's attempt to define what is? However, "which one triggered"? The concept of "natural transformation" to define "? However, does "transformation" require leadership? "Letter?" The concept of (functor) to define letter? ,? Do you need to quote? The concept of "category".
Ben. Don't you want to? These conceptual details, but we want to show them? The basic spirit of subcategories. Roughly speaking: all apples can be regarded as? A "category" all? Bananas are another one? A "category", they can all put it in? Have another drink. What's it called? Fruit "category.
We want to say, abstract from OO? What is the concept of a "2"? It's often difficult. And often requires violence? Clinic. ? The teacher is leading? Before "2", would you put two more in order to deepen your understanding? Bananas. Why don't we? J to represent? Bananas. So what? The following formula appears on the board:
J + J = JJ，(2)
But the same problem will still haunt us. More brands? What is puzzling is that. Is the teacher still here sometimes? Do more things that confuse us on the way to "2". If these are all "2", there may be such a formula for persistence:
OO=JJ。
Isn't there one here? Is a "category" okay? Like "? Very careless, if there are not enough apples? , may also appear temporarily? The formula is:
O + J = J + J。
Crazy, Apple and? Can I add bananas? Apple and? Bananas are not here? How to add a category? In fact, we can say? An apple is "1". Answer? Bananas are also "1". They are all representatives of "1", but what is the concept of "1" abstracted from these representatives? This is often difficult. Maybe those children who don't even know 1+ 1=2? Not stupid? Have you got it? Some deep and primitive East? .
Let's take a look at how to interpret 1+ 1 in the category.
Second, the concept of category: the essence of everything
Is the view on category the same as our most naive view? Sample? An apple is "1". Answer? Bananas are also "1". Are representatives of "1". Since they are only representatives, does that mean that they are not "1"? So what exactly is "1"?
"1" should reflect the "* * * nature" of all these different representatives of "1". Mathematicians gave this "* * * a property"? A formal name is "universal property". How to write the * * * properties of these different representatives "1"?
Classified offer? A new perspective? . No? "? A research object ""What is the East? " Like this? Regular set theory or reductionism? Typing to see the problem? Based on the relationship between objects and other objects? Typing comprehension? Object. This? Is the formula really more fundamental for us to understand the world? Method,? If you want to know? An unknown "existence" (such as a grain? , materials, etc. ), what would you do? Would you? Are you familiar with Dong? Go in and see what you can measure. Physicists can measure? The emission spectrum and absorption spectrum of a new material, X-ray diffraction; Mathematicians can put the ball? Throw it in? An unknown space to measure, or see if you can make it? Group work? Go up, wait. ? Isn't the cloud chamber of the accelerator measuring particles? The trajectory? Is it grain? There are other east? Interaction? The trajectory of. No interaction? , measure also? Start with. what's up You can say for sure:
Not in this world? Interrelation or interaction? More basic existence.
In that case, can we have a try? Relationship to define what is "1".
Shall we review it first? A concept: mapping between sets. Is it a rally? A "collection" of heap elements, hehe. But it is worth pointing out that the empty set is also? Is that a collection? There is no collection of elements. So what is the mapping between two sets A and B? ? Consider two sets x = {a, b}, y = {1, 2, 3},? Mapping from x to y, written as
or
This is actually? Allocation rule: give each x? Elements are only assigned to elements in y? For example, f (a) = 1, f (b) = 1 Yes? Reasonable mapping, g(a)=2, g(b)=3? A map. But you can't assign two elements in y to a! If the set X has no elements (empty set), it is equal to the allocation rule? Once defined, this allocation rule that does not need to allocate anything is called null mapping.
With these preparations, what can we give? Definition.
Definition: 1 Is that right? Do you have a collection? It has all sets and only one mapping [1]. Do we? A concise chart to record this definition: for any set x, we have
This? "
It means' being'! "It means" Wei? " [ 1]。 Also note that "for any set x" in the definition is also true? Often very important! You are kidding! Special collection? It's all assembled
Have you seen this definition? Speaking of the relationship between "1" and all sets, this matter is quite important. But first? It may be more important to see this reader next time. Yes, why "1"? Reasonable definition? Let's have a look. Is Apple's collection full of discontent? This definition? ? Answer? What about the banana collection? Or, zero or three? What about bananas? Or all of China? A batch?
If you are willing to try, you will soon find zero? Bananas, right? Because it breaks the mapping "existence" condition in the definition. "Three? "Not a banana? Because it destroys the uniqueness of the mapping? Make love. "What conference is ok? Is that all? A collection of elements. Like what? A batch of apples, one? A bunch of bananas? A bunch of eggs, one? Set and so on, can they guarantee existence and uniqueness? Make love.
So the definition of "1" is not unique? , this seems to be? A flaw, but a beautiful place? Yes, there is only one possible "1"? Kind? Type corresponds to each other, which is made up of "existence" and "only? Sex ",and the specific content of" 1 "? Close. Like teaching? When you are friends, you can have an apple representative 1 or an apple? Banana stands for 1. We know how to equate them! Can you believe it, young man? Although the garden is hard? Seriously teach "decategorized" mathematics, but teach? What is the law? Classification of legal evasion! Because that's what happened.
This definition is also called "universality" of "1". That is to say, we? 1 "to define" 1 "? No? What does "1" mean? To define "1". All mathematical concepts are ok? Its "universal nature" to define, I said "all", yes, you heard me right!
Great, if you can keep up with me, we will come again? A.
Definition: 0 That's it? A set, it is for any? All sets exist and there is only one mapping. That is to say, for any set x, we have:
What is this for? Do exercises at home. -Do you have any popular science homework? Never heard of it, hehe. But think about it? After learning this exercise, on welfare, can it be ruined? Friends and theirs. Ha ha.
Third, how to interpret 1+ 1 in the category?
Well, the real challenge or destruction is coming. We can finally see what 1+ 1 is. And "1"? Sample, "1+ 1" will also have many different representatives. Like two apples or two? Bananas, wait. So what should "1+ 1" be? It should be the nature of all these representatives, that is, the nature of everything. Next? We will reveal the universality of "1+ 1". I'll wipe my sweat first.
It's time to use your head.
This is the universality and definition of "1+ 1". If we ... Figure to record? The definition or universality of 1+ 1, so what is it? This is the so-called "exchange map" of the East? .
The so-called graphic exchange means:
. Why is this definition called 1+ 1? It's really hard to explain. What's the main point? A lot. Can you stay and exercise? Don't we still have many advantages? Readers? It's up to you. Am I devastated? Come on, you need a rest. A rest? what's up .
Rest, rest? under
What I want to point out is:
(1) Although we don't specify what A and B must be, the universality makes it impossible for A and B to choose at will. Different A and B are full? The universal choice will be regarded as the representative of different 1+ 1! That is to say, the definition of 1+ 1 needs three east? :( 1+ 1，a，b).
(2) full? Is the definition set of 1+ 1 unique? Yes (all representatives), but existence and uniqueness? Gender makes a singularity between any two of their representatives. what's up Kind? Type correspondence. This? It's an important point, but I don't really want to expand it. Maybe I need readers' contact, too To interpret.
(3) What is the essence of all things? Wonderful land? Yes, it not only defines the concept, but also tells you what it is. Yes, is it? To construct an arrow that only exists! ? Is this the only one? what's up Fa! This? Point? Nothing? Did you pass? It's hard to understand. This set defines and corresponds to? Yu? Physical characteristics also strongly prove this point? Good definition.
Job 1: Have we defined 2 or "+"?
How to define Job 2: 1+ 1+ 1?
How to define job 3: 1x 1? (Tip: Invert all arrows that define 1+ 1 in the diagram).
Homework 3 is quite interesting. Can the arrow be reversed? In fact, this means that multiplication is the dual concept of addition. Classified research? Forced landing? You mean math? All concepts are like this! Math? There are only two concepts:? This species is called Extreme (? For example, 1 x 1)? This species is called the "residual limit" (? Such as 1+ 1). Nothing else. Ha ha. So classification puts all the concepts in mathematics into it? A system? Look at the frame.
Readers may wonder, how can this be called "limit" instead of limit? Answer? Finite (approximate) process? Actually? The so-called "limit" that family members are familiar with is just one? Is the finite node exchange graph the same as 1+ 1 or 1 x 1? Type to define the concept? It's already started. ? For example, for any real number x that makes up the following figure, we have:
Where is the ordinary "1"? A real number (not to be understood as a set), and the arrow means "≤" (? Is equal to) (not a mapping between sets! )。 What does this picture show? The limit of 0.9, 0.99 and 0.999, ... is 1. ? Classification language? That is to say, 1 is the complementary limit of 0.9→0.99→0.999 → ...
what do you think? X is like this? Real numbers, each sequence is 0.9, 0.99, 0.999, ... up to x? Arrow, which means the sequence 0.9, 0.99, 0.999, ... Everyone ≤ X? 1 Is that right? Number? And the most? That? A is that right?
So what's the classification? There is calculus you are familiar with, but she can do more! In fact, categorization is unified in concept? Analysis and algebra, unification? There is no essential difference between 1+ 1 and the limit in the traditional sense in terms of discretization and continuity, but what aspects are involved in the exchange graph? You do.
Assignment 4: If the meaning of the arrow is changed to "≥", it means that the arrow in the above figure is reversed. In the sense of categorization, we get "1". The limit of the chart.
In addition, you should also note that the arrow in the category can be any possible relationship rather than a mapping. ? If "≤", then? Like all over China? I have nothing to do with the category of composition, but what if we all pursue the same thing? Answer? Children? In this way, we have the relationship of rival in love, or we can study it within the scope. This example? You can imagine, but many times, the so-called "correlation" in category science is very strange, even? Is it beyond imagination? Yes
Four. Category, physics and computing
I think? There must be? I feel crazy. 1+ 1 It's so complicated. I want to emphasize that this story is not "complicated". Is the original of 1+ 1. However, readers can also object to the existence of "de-categorization"? 1+ 1 If it is so complicated, there is no way to calculate it easily. So how can we understand 1+ 1 in this way, even if it is original? And then what?
Is the definition of 1+ 1 really complicated and untrue? But this is? Kill the chicken, of course you can't see it? Have you measured it? Now you can kill people? Cut the crowd? . Actually, classification is research? Where are the powerful tools of finite dimensional mathematical structure? She? The quantity can really be revealed. ? If you are studying quantity? What is the energy gap when using a multi-body system? The relationship between the boundary and the interior of a multibody system can be derived from? This universal property is defined [2].
What does this picture mean? It doesn't matter what you mean. The important thing is, did you find out? Really? Poor? In the complex physical system of dimensions, there is no relationship between the boundary and the interior. The relationship between 1+ 1 and 1 is complicated! Is this because classification is feasible? Combine finite dimensional mathematics with? A mathematical system with finite dimensions? In the same place? Processing under the framework. Is it worth it? I said, go? The relationship revealed by the chart also depicts the duality of open string and closed string in string theory! These are all? Duality between finite dimensional mathematical structures. If you really write down the mathematical structural elements on both sides of the dual and their relationships, will it be complicated to death? Yes Ha ha.
Play? In today's PK reductionism, categorization is becoming more and more important. Is this because the classification is for acting? About preparation. Look at "1+ 1", isn't it all in the collection? This concept? Similarly, all mathematical concepts are included? In a sense, the "all" objects in the diagram? The object of. What? Classification emphasizes giving up reductionism, don't ask? The elements of the collection. Depending on the mapping, the latter is more abundant. ? Like what? A set of x? This element is actually 1 x? A map
Isn't this view the principle of accelerator? Want to know about particles? What is the East? , just take the other east? , what? It's "it" Go knock it. I want to know how important calculus is to the physics of reductionism, and how does category play its role? How important it is to talk about physics. What is the basis of categorization and what is the physicist's understanding of nature? Methods and principles are completely consistent, they all emphasize:
No? Interrelation or interaction? More basic existence, everything else is acting? Yes
The relationship between categorization and physics is certainly worth it? Specialized books, many cutting-edge theories? Has been talking about this relationship. This? Are we there yet? Although I'm not finished, I just wish I could throw a brick? , induced? Family interests.
What if the category is in mathematics? Very basic, so should it be the basis of physics or other disciplines? The reality is, in physics? There may be only a few types of upper class. Answer? A common topic. Why is this? Is this temporary or? Dragon? Will category society bring new calculus to describe physics? What impact does classification have on computer science in the future? Bit(0 and 1) is used for Turing calculation, and qubit(2-dimensional linear space) is used for quantum calculation. The road of classification is from number to linear space, to 1 order category, to second-order category and to higher-order category. So can we imagine that there will be 1 class calculation and 2-class calculation in the future? I hope to have the opportunity to interpret these problems in the future.
Do you like classification? Are you welcome? Home comes to the wonderful world of categories.
appendix
At the end of this chapter, we will talk about the process of learning categorization. Confusion and misunderstanding.
Is it a lot? (including some mathematicians) complain about the abstraction of classification. I hope before? Is the discussion helpful? Do economists realize the days when we think? This is a classification study. Quote? Abstract concepts such as "1" and "2" are anti-? It is "de-categorization". Where does our mathematics education come from? The beginning is "de-categorization". General calculus can be regarded as a classic of "de-categorization". The end result is, we? Majority? First one? Learning taxonomy next time will make you feel very abstract, hehe. Perhaps it is because the "de-categorization" mathematics education has made us lose our innocence. I remember there was? The second time I gave a math report, some listeners complained that the classification was too abstract. I said, what is abstraction? A meaningless concept, but your so-called non-abstract East? What is this? He replied? Same tune as above. God, is homology abstract? Okay, I can bear it. Seriously, why do you think homology is not abstract? He said, because it can count. I said, it can be counted, but it is not abstract. In this case, categorization is not abstract, because it can also be counted. But it doesn't matter, because this statement itself is ridiculous. If homology can be calculated, it is not better than 1+ 1=2, right? Then what is "1"? What we usually call "non-abstraction" or "abstraction" is actually "familiarity" or "unfamiliarity". Categorization is "abstract" because we are moving towards "decategorization"? It's been a long time, and it's not that easy to get back. It's hard to let go of the burden.
I remember there was? I had lunch with physicist Michael Levin. He said that he spent a lot of time reading classification research, but he always felt that classification research was empty, as if there was nothing. He thinks it's right. Of course he's not the only one? Answer? Complain like this. In fact, category and set theory, right? What is the oriental bottom? . Just like you went to see set theory? Samples, except? Some forms of definition, as if nothing. For physicists, see set theory? Hardly? Place, really? My major is calculus and linear algebra. So only by seeing the "calculus" and "linear algebra" of the category can we understand its power? . I think Grothendieck's algebra? What can be roughly regarded as? The new calculus? Tensor category theory can be regarded as? The category of a new kind of "linear algebra" namely "calculus" (or "linear algebra") is not unique. Yes? Ever-changing Are you most interested in physics? "Calculus" and "linear algebra" may not be born yet? . Different from set theory, for physicists, set theory can be completely ignored and jump directly to calculus and linear algebra, because the language of set theory? And the language of basic functionalism? Covered. But for classification, want to skip her basic language? : category, letter? 、? It is impossible to learn her "calculus" and "linear algebra" directly with natural transformation and Yoneda lemma. Unfortunately, want it? Before? , not yet? A classification book suitable for physicists.
What else is there? The misunderstanding is that the classification has been established? Okay, have you learned? What should this kind of math books do in physics? It may be enough. If you take this? State, then you are doomed to disappointment. ? First, put any (no matter how beautiful) math set? Everything from physical concepts to fate? Beg? Clinic. Only from physical experiments or physical images? The discovered mathematics is meaningful to physics. If this math happened to be discovered by mathematicians, it was just an accident? It's already started. There is no ready-made key to the unknown. What kind of study does physics need? Most of them don't exist yet. Do you want us to go? Develop mathematics while developing physics. In this case, there is no difference between new physics and new mathematics, and they are both natural hidden structures. Classification is still in the primary stage of development. Can calculus develop? A hundred years later, the category? Do you need it? A hundred years. ? My practice over the years tells me that physics can bring us new things beyond mathematicians' imagination? The category of science is really magnificent.
Express gratitude/gratitude
Thanks to Gottingen, Germany? Learn Zhu? Teacher, Tsinghua? Study? Wang zhong of the institute? The math teacher and Qiu Chengtong? what's up Hey? Division, South Division? Quantity? Science and? Wu Yongshi from Cheng Research Institute? What about the teacher and Hao Zheng? Massachusetts organization. Those people in college? Teacher Cao Zexian, Institute of Physics, Chinese Academy of Sciences? Teachers and Stanford? Have you ever studied xiaoliangxiao? Many valuable ideas put forward by the teacher? .
To annotate ...
[1] Are we still there? Here we go. "Wei? This concept seems to be a circular definition of "1". In fact, it is technically avoidable. For example, we can say that the set composed of mappings (making graphs commutative) in everything has bijectivity to the set {O}, or that if h and h' make graphs commutative, then h=h' (thanks to Zhihu netizen Wang Yingjie and Ning Yan). Our place? I don't want to discuss the basics of mathematics. Is it a show? A new interpretation of 1 and 1+ 1 But from the nature of all things, constantly? "Yes" and "Yes?" As you can see, in? "Wei?" in the Philosophical Sense It may be the same basic concept as "existence".