Is mathematics invention or discovery?

Author | Atiya (Sir Michael Francis Atiyah)


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“Mathematics and Physics The Frontier”

Sir Michael Francis Atiyah (April 22, 1929 to January 11, 2019), British mathematicians, graduated from Sanyi College of Cambridge University. One of the greatest mathematicians of the century.

Michael Attia was born in London, England. He spent his childhood in the Middle East. He moved to the United Kingdom with his family in 1945. After that, the previous three results were admitted to the Sanyi College of Cambridge University. He obtained a doctorate in 1955. After graduation, he studied and taught at the Princeton High Research Institute, Pengbrock College of Cambridge University, and Oxford University. In 1962, he became a member of the Royal Society of British.

The main research area of ​​Sir Michael Atia is geometry, and by the 1970s, he transferred his focus to mathematical physics. In the 1960s, he established a cooperative relationship with Eshador Singh, which jointly proved the theorem of Atia Singh, which played an important role in some areas of mathematics. Therefore, he won the Phils Award in 1966, and in 2004, he won the Abel Award with Aishadel Singh. In addition, Sir Michael Atia also achieved outstanding achievements in the fields of topology, differential equations, mathematical physics, algebraic algebra. In 1983, he was awarded his subordinate jazz by the British royal family. In 1992, he won the British Merit Medal.

“Mathematics, is it invent or discovered?” This is a very philosophical topic. This topic is not designed for mathematicians, but is suitable for wider readers. What we really care about is the relationship between mathematics and real world and other things.

Mathematics can be said to be in the discipline between art and science, and the relationship between mathematics and science is almost well -known. For example, the theories proposed by Newton, James Clerk Maxwell and Einstein are based on a solid mathematical basis. In turn, the observation and theoretical verification of scientists has a great impact on the development of mathematics. However, the relationship between mathematics and art is not so obvious.

Mathematics is art and science

It is believed that art and mathematics are two yards. In fact, there is a connection between the two. First of all, mathematics is a logical -based on logical thinking. In the field of philosophy, it also pays great attention to logic and thinking analysis. For example, the two great philosophers in different ages, Plato in ancient Greece and Russell of modern times, they are also mathematicians, because both of them use a large number of mathematical languages, there is no doubt, and logic is part of philosophy.

At the same time, logic is also the basis for constructing art.

When we talk about some art projects, such as painting, we will use many perspective methods, that is, the three -dimensional view in the space; the perspective circle method is also regarded as a major discovery of the development of painting art. For example, music, music uses notes as the basis, but the chords are actually a very precise and wonderful mathematical form, which also shows the relationship between art and mathematics. Architecture pursues the beauty of buildings, depending on the proportion and scale of the building itself; whether it is geometric or geometric architecture, it is a very important part of architecture.

The relationship between mathematics and a large number of art projects can be described as inexhaustible. In short, art is the subjective beauty. In physics, the pursuit of beauty is the basic concept of art. Similarly, the pursuit of beauty is also an important link in mathematics; therefore, it can be said that mathematics is both art and science.

The difference between science and art can be explained in this way -scientists are tirelessly studying and discovered the world in front of him. Through science, we can find the truth and the rules of the nature of the natural world. This discovery is based on the arguments from observation. On the other hand, art is a kind of creation of human beings. In thinking, people have obtained important discoveries and feelings. The foundation of this discovery is not based on rational thinking, but emotionally emotionally.

On the surface, art and science are fundamental reversal, but this is not the case.

Mathematical model comes from the concept of thinking

Let me go back to the history of human development thousands of years ago, what is the relationship between the real world and humans. What is the real world? What does it mean? How did humans think about this world at that time? We do not understand the real world, we do not understand our thinking, let alone the relationship between them.

This is one of the important philosophical issues. Of course, philosophical questions do not necessarily have an exact answer. We can obtain wisdom from middle school by asking questions, but we can never get an exact answer. Therefore, ancient and modern philosophers have been grinded for thousands of years for a philosophical issue. In fact, human beings propose scientific theory to understand, interpret and develop the laws we observed with a model or framework. In this way, to some extent, science theory and mathematical models are very similar. They are all concepts in human thinking. We can add these concepts to things observed in the outside to understand them.

Therefore, to a certain extent, there are two parts of science, including data from external experiments, as well as mathematical models obtained by human internal thinking, and try to integrate the two to get a overall understanding. Similarly, there are two parts of art. First, the creation of artists from internal feelings, 2. The constraints that are physically physically. For example, the perspective points of mathematical structure and building are constrained from the outside, which is unreasonable by the artist.

Therefore, artists apply their creations, but they are also living in various “frames” that have been discovered. To some extent, art and sciences share the same characteristics, that is, they are “discovering” these frameworks and various creations within this framework.

This framework is the data that science is, or the scope of our research capabilities. The artist also faces the framework, but they tried to image their ideas. Reason and sensibility are the foundation of science and the foundation of art, but most people believe that the two are rivers and water, and even the wind and horsesy. However, from the research on human brain in recent years, we have learned some exciting results. It shows the reason and sensibility of the human brain, and each other affects each other, “You have me, you have you.” Perhaps we can know more about this in the future and learn more. Therefore, mathematics is art and science.

Is the Plato world already existed or created?


Let’s try these two aspects from multiple methods. The problem is that what mathematicians discovered in the real world? Or is it the ideal world of Plato? Plato uses his concept to understand mathematics. For example, he believes that circles can be perfect. But the perfect circle never exists in the real world. All the circles we paint will always bring a little edges.

In fact, the perfect circle is just an idea. We now call the Plato -style world, it is indeed an idea, an ideal world. In the real world, the round shape we see is just a kind of thought, reflection and replacement of Plato’s ideal world. But some people believe that the world of Plato does exist. There, all the great mathematics ideas can coexist perfectly and harmoniously. However, there is a mess in the real world, so scientists merge the real world and imagine the world in some ways.

Next, everyone may ask: Does the Plato -style world exist from the beginning, and only to discover it? Or is it purely created by human wisdom? The Plato -style world was invented, and then this ideal world reflected the real world.

This is a question that has existed for thousands of years. This is exactly a question we can debate but get different answers. Generally speaking, we discuss these issues from some basic aspects, which will be more effective than the philosophical discussion at the abstract level. Therefore, I will discuss this question through some simple examples: mathematics is a invention or discovery?

Before further discussion, let me point out. Hong Kong is a business -oriented city. Many people come to college to study or teach, and it is inevitable that money and material issues are inevitable. In this regard, one of the main differences between invention and discovery is that patent rights can be obtained through invention, and money can be made from it, but it is not discovered. For example, Maxwich introduces the theory of electromagnetic theory. If the formula he finds can obtain patents, I am afraid he is as rich as Microsoft’s Gates. However, you cannot obtain patent rights for what you find. The human genome is another example in the near future and has caused many arguments. For example, can we obtain a patent for the discovery of human genes? Because this topic involves huge benefits, it often causes widespread debate. Therefore, this is a problem that is invention or discovered. It is not a philosophical issue, which has also caused a strong business response and argument.

Now, let me use the mathematical concepts of Plato and Greeks as an example of our discussion. One of the problems that Plato is interested in is the famous “positive multi -facial body” and is also known as “Plato three -dimensional”. There are 5 “Zhengdang Body”:


Positive tetrahedral, consisting of 4 triangles;

Positive hexainet, that is, a square;

Active octopus, that is, a double pyramid;

Positive tweets, each side is a positive pentagon;

Twenty body, each side is equal to the edge triangle.

One positive on all sides has 4 top points, 6 edges and 4 planes;

The square has 8 top points, 12 edges and 6 faces;

Twenty body, with 20 triangles, 12 top points, and 30 edges. At the same time, they appear in pairs, that is, the number of noodles and the number of vertices, which can alternate each other, called the pair. The first question is, is it found or invented like a positive twenty -mandarin or other three -dimensional objects?

You can argue: Cube is a thing that is obviously existed around, just like square sugar; and tetraonal bodies may be the same. However, it is very difficult to find a positive twenty -mandarin in nature. I don’t think they exist in any form. Although its existence is attributed to Plato, I found that in the 2000 BC, 4,000 years ago, Scottish, there is a positive twenty -manded body and all 5 positive polygon three -dimensional. This appeared earlier than Plato’s age, at least thousands of years earlier than Plato. These may be a stone of cultural relics, which appeared in the era when Scotland had not yet developed a high degree of civilization, and at the time, some people have developed how to make all five three -dimensional.

The mystery of the polyhedron in the sphere

At first, I guess it was an individual example made from some somewhere, but it seemed more than that, because there were hundreds of such stones there across the entire Scotland. Because of some unknown reasons, some people discovered these stones at the time and loved them, and then received more attention from the entire society. This is a very important observation. So far, in an ancient man who has no known text or civilization, ancient people invented these mathematical objects, which was puzzling. Of course, I don’t know if this is the oldest example. Perhaps in China 4000 years ago, some people have discovered the twenty noodles, but at least so far, Scotland still maintains this record.


To this day, we have discovered some of the mysteries, and there are some simple relationships between these numbers. These three -dimensional mathematical relationships were discovered by Leonhard Euler and called “Euler Formula”.

F+V-E = 2: The number of vertices (V) is reduced to the number of side (E) plus the number of (f) above (f) is equal to 2.


From these five examples, we can easily get this simple relationship. But Euler thinks deeper. The law he observes can not only applies the three -dimensional Plato, but also can be applied to all the three -dimensional connected in the sphere. All these three -dimensional numbers can conform to this relationship. At this point, you may ask, is this a discovery or an invention? I can find more, but how far can I apply this formula?

As a result, it became a very important formula in mathematics. For hundreds of years, this formula has evolved differently, but it is still the core part of mathematics. In fact, the research work of Chinese mathematician Chen Sanzheng is closely related to this formula. If we can find these formulas in nature, we can see these things. If you invented the twenty figure, you have invented this formula.

This is the first example of discovery or invention.

How many numbers are God’s creation?

Let me go back further. What is the most primitive in mathematics? Where does mathematics start? Maybe you will answer: starting with numbers and calculations, 1, 2, 3, 4, 5 ….. a bunch of integers. The famous German mathematician Leopold Kronecker said: “The integer is created by God, and the rest are made by people.” Humans have discovered integers and make everything else.

The first question is, what about 0? 0 is also an integer. Before 1, first 0.0 is undoubtedly an important integer. Is it a invention or a discovery? 0 existed for a long time, or our invention? I really want to say that 0 is a invention, but in fact this is a difficult problem.

Then there is a decisive mark. When we start writing the number, we are organizing the number. At the same time, the decisive marker is a very important part of mathematics. Of course, it did not appear early in the morning. The Romans wrote a complex mathematical system. In the hotel room where I came to Hong Kong to lecture, there was a calendar on the desk with a small number of Chinese printed on it, but I couldn’t understand these numbers. For me, they were like the number of Rome. But in any case, the decimal method is a step of great development. Sometimes I try to think that it is a invention, but you may say that the number has already existed. The world invented by God has long been the case, but we later discovered it.

Further discuss. What do mathematicians do for the integer? Yes, mathematicians sometimes bury them on interesting things. They are not interested in all numbers, but they go to deepen quality numbers. Quality numbers cannot be decomposed into an integer of other factor. I know that 6 is the accumulation of 2 by 3. However, if the numbers that cannot be decomposed into the factor are written in turn, they are 2, 3, 5, 7, 11, and 13, which can be written endlessly; just, it is difficult for mathematicians to find a rule to describe their full picture. Mathematicians who specialize in digital theory like quality numbers, because in the process of analyzing the integer, you can decompose all integers into a quality number. Therefore, quality can be regarded as an integer part of the component. Components are often attractive. If you want to understand the structure of something, you will carefully observe its components and unveil its veil from the component. Atoms are material components, so quality numbers are the components of arithmetic.

What is the square root of 2?

So, what is the length? The length is numbers, for example, 1, 2, 3, 4, and 5, all are numbers. The length is represented by the number, and the number represents the length unit? If you take a ruler, a rope or other, you can mark its length. In other words, the length is an important thing that can be measured in the real world. When the Greeks come into contact with the length, they find that not each length can be presented. For example, we use a unit length as a square edge length. The length of this square diagonal should be a number. According to the Pythagorean theorem, the length of the diagonal should be a square root of 2. Conversely, the square of this number is 1+1.

Is the number of square roots of 2 existing? So far, all the numbers are integer or scores (for example :). If the number of square roots of 2 is not an integer, will it be a score? It is easy to prove that the square root of 2 will not be a score.

Let me make a demonstration for everyone: we use two integer P and Q to form a score. When these two numbers can be scored, we will score them first. In the end, we can assume that these two numbers will not be even. Essence Assuming the square root of 2 =;, this formula can be found, so that P is even. Then set P to twice R, replace the pyramid back in 2R, and obtain it. It is over 2R to get Q = 2R. From here, Q is an even number. This is resistant to the assumptions above, so the number of square roots of 2 cannot be written into a score.

This is one of the most beautiful logical inferences that prove that the square root of 2 is not a rational number. What kind of number will this exist? Can you write it? This is a paradox. 2 The square root of 2 does exist as the number of this length, but it cannot be written like that, so we have to invent some numbers to express this number. One of the methods is to use decimal to write the square root of 2 to 1.4142135623731 … Some extraordinary mathematicians can write all the 100 small numbers of this number without thinking. I can’t do it, I can only sigh.

The real invention is a major progress, but this invention comes from objects in the real world, so we must also build a method to deal with this problem.

What mathematicians will deal with negative numbers next. Numbers can be called positive numbers or negative numbers. For example, when we are on the ruler, the number is written from left to right, which can be called positive number. If the opposite direction, it can be called negative number. It seems that the upper, lower, or accountant’s balance sheet is made. This is a very important thing in the real world. Therefore, negative numbers are very important. Look at it deeper, how do humans deal with negative numbers? If-1 by -1, you will get +1. In other words, if you do not go forward, but go forward in the opposite direction, and then go back to the direction again, you will return to the original position. You can say that the development is based on experience. Like the decimal number method, these are the marking methods that we invented in the length of the distance and distance, and write the numbers; these experiences and inventions are tightly linked in some mode. These are the initial levels of mathematics and let us enter more complicated areas.

Virtual number is an important invention of human beings

The invention of the virtual number has entered a new stage of mathematics. According to the rules mentioned above, when a number is self -righteous, it is always a positive number when it is self -righteous. When the two positive numbers are multiplied, the number can be obtained, and the negative number will produce positive numbers: so no one -piece square is negative. The rules of arithmetic do not allow such numbers to exist and it does not make any significance.


However, if we have to write it down, this number is called virtual numbers. For hundreds of years, mathematicians have continued to argue whether this secret thing can be written and applied rigorously, and in the end it can be applied to the real world, everyone has doubts about these issues. For hundreds of years, this concept has not been accepted or tied up. Fortunately, in the end, mathematicians resolved and accepted virtual numbers. This is the invention of human beings because it does not exist in the world. At the same time, because human beings find the law of virtual numbers, it is very useful in mathematics. Electronic engineers are also willing to use it and even become an important foundation in quantum mechanics.

To some extent, plurality is a combination of real and virtual numbers. Therefore, although the square root of -1 does not exist in the real environment, it is still subtle in some things that exist in reality, and it is also hidden in the theory of relativity. I can say boldly that -1’s square root may be the greatest single knowledge invention of human beings.

This is a bold statement. Does anyone oppose this statement? Everyone agrees that it does not exist. However, we can get amazing and abundant results in mathematics and physics. This is a huge achievement, and it must be the discovery of human transcendence. We cannot see the virtual numbers appearing around, but we have contacted and applied it, and we have achieved great achievements. In fact, it took 300 years to make mathematicians. It was not the invention of a single mathematician, but was created by group strategies. It has been explored for a long time, and even the power of poor life, tirelessly contributed to this without regret.

Let’s look at some exciting stories in mathematics. In mathematics, there are some specific numbers or constants, known as the basic constant of the universe, and it does play an important role in mathematics.


Circular ratio

Is the basic constant of periodic phenomena

First–. Everyone who has entered the school will apply the value of this circular circular and diameter ratio. If you want to calculate it through a formula, you must first draw a circle and connect a polygon in the circle. You can increase the number of edges and measure the perimeter of polygon. This polygonal perimeter will definitely be smaller than the weekly. However, if we continue to increase the number of polygonal edges, that is, to shorten the edge length of this polygon, the long edge length of this polygon will get closer and closer to the week. If we continue and repeat the above steps, the amount of the measurement ratio of this circle to the diameter ratio will get closer and closer. The Greeks and Akimids are very accurate from this method. So, it is very basic. Of course, you can also think that this is just a stupid geometrician. Who cares about the ratio of circular and diameter? But it is actually the most basic constant; it has a pivotal role in mathematics and physics.

The reason is very clear, because it is related to some periodic phenomena, that is, any cycle and duplicate things are in handy. The time of the earth and the sun, the time on the clock, and any vibration things have a periodic model. All those descriptions of periodic phenomena will contain basic constant. It will appear in all the textbooks of all mathematics and physics. It is the most common number you can think of. It can also be said that it is one of the cornerstones of human civilization.

Index function E is the basic constant of calculating the number of growth numbers

The other is almost the same foundation, but it is difficult to understand the base of the index function. E is about 2.718, which is a real number between 2 and 3. The importance of this number has its basic reasons. One is that it is related to the number of species, including people, bacteria or animals. In any generation, one individual is replaced by two, as if the parents have a total of 4 offspring. In this way, the number of it will double, and the next generation will double again. This rapid growth is called “index growth”. If this situation continues and no one dies, the food of the earth will be exhausted long after a long time. In fact, this is really worthy of human vigilance, because the global population in the last century grew almost in index. Therefore, a major problem we face is that when the population surpasses the limit, how can we live? The world’s population has now exceeded 6 billion, and it is expected to reach 9 billion. Some are even more expected. What should I do?

This is a problem of human population, but we can also use the problem of small animals or bacteria that spread diseases, and they can exponently grow and spread the disease.

This mathematical concept can also be applied in the financial world, and the calculation of composite interest rates is based on the calculation method of index function. Suppose you have a bank deposit. At the end of each year, the bank manager gives you an additional proportion, such as x, x can be 5%, or more generously gives you 10%extra money. If they are not given you once a year, but every six months, even if the same annual ratio, you will get more money. Because in the first half of the year you have received an additional amount of extra money, this%in the second half of the year is calculated based on the principal plus the sum of interest that has been obtained in half a year, that is, the interest obtained on the principal plus the interest of half a year. The method of calculating interest is slightly compound interest rate. What if it is calculated for you every season? You will get more. If you calculate it once a day? You will get more. What if they or every millisecond? Will you get more and become richer?

The answer is: this is the upper limit. What you can get at most is. Suppose X is 1, that is, you have a very generous bank manager who gives you 100 % interest. How much can you get? The answer is that you will get 2.718 times the principal at the end of the year. If your deposit in the bank is 100 yuan, if the annual interest given by the bank is 1, calculated at the annual interest, the interest you get is 100 yuan. No matter how large the number of calculations are, the most likely extra interest you get is 71.8 yuan.

Therefore, whether everyone is interested in money, population or the growth of germs, you need to know the index function.

The above and E described above are two basic constants in mathematics.


The most beautiful formula:

There are also basic constants in the real world. When people (or wisdom creatures) communicate between planets, just like the National Space Agency launching rockets to space, when the rocket reaches a distant place, we may want to transmit some information to other smart creatures to prove that we are clever. In this way, what will we want to put on the rocket and make the other party understand? Some of them can be placed, because we believe that the distant civilization will also find these constants. Therefore, we can put E and. But of course we can also put physical constant. If the other party is a good physicist, they may want to calculate the ratio of proton quality and electronic quality, maybe we can also put these ratio in those information. However, mathematics constant is relatively easy to understand. It may be a good proposal to put a positive twenty -mandarin. Let’s see if the aliens know that the twenty -mandarin is positive.

In fact, there are many good things that are outside the language outside of language. But in a suitable model, they will understand these mathematics. If someone asks: In mathematics, which formula is the most beautiful? I think all mathematicians will agree, that is, the following formula.

What is? It’s simple. You can substitute it into X+YI. After the operation, the answer is -1. This is the most wonderful formula of mathematics, because in a single formula, including three most important numbers: -1’s square roots I, and E. This amazing formula contains the most basic meaning. Because of this, it is so wonderful. At the same time, I also regard this formula in the mathematical world, which is comparable to the most famous sentences in the literary world -one of Shakespeare’s “Hamlet”, one of Shakespeare’s four tragedies, “to be, or not to be”. Sentences are short, but these simple sentences contain a deep meaning. Therefore, mathematicians are also beautiful. Mathematics and art are the same, and the two depend on the same concept, which is simple and profound.

The extraordinary contribution of mathematics to physics

Then we talked about geometry. Geometric studies have been immersed by Euclid and other Greek scholars, and German philosopher Kant also repeatedly considers it. In fact, geometric learning is space. Even if our head has long determined what the space looks like, we still use different experiments to study it.

In the 19th century, people who were drunk at Ou’s geometry discovered geometry other than Eushi geometry. We found that there were some space that did not comply with the law of Ou’s, and we called non -European geometry. From the research results of mathematicians such as Janos Bolyai, Nikolai IvanovichLobachevsky, C.FGAUSS, etc., we learned that this geometric science is similar to the Eushi geometry to some extent. It’s just that the space is bent. Then there is a German mathematician Hermann Minkowski, and finally Einstein uses it as the interpretation of the gravity field. The gravity here is based on the idea of ​​bending in time and space. However, there are actually more than one kind of geometry.

Geometry not only involves planes, but also has a bending space, which can eventually lead Einstein’s theory.

I love to quote a sentence in the article “The Orunreasonable Effectiveness in NatureSciences” (TheUnReasonable Effectiveness in NaturalSciences in Nature). He pointed out: In all different methods, mathematics has made extraordinary contributions to physics. Maybe you are weird, why these stupid mathematicians made by mathematics will be applied by the physicists of the talented people, but this is exactly the case. So, we may ask a related question at the same time: Is mathematics a invention or discovery?

If mathematics is discovered, it is obviously derived from the natural world; if it is invented by human brains, why does it operate so beautifully? Even the operation is almost flawless? If you believe that God is the creator of the universe, then it is it that it is God creates the universe, and he establishes the principle of mathematics. When people start to explore the world, they find that they can explain physics, insight into the world, and unveil the veil of the world.

If you are Plato, I believe Plato, you will think that there is a world of Plato -style concepts outside the physical world, and I use it to build the physical world. You can also be the Darwin, and say that the evolution of human minds is the result of evolution. In order to survive in the real world, human minds cannot control the laws of physics outside. Otherwise: We will not be selected by nature, and even in the selection of things in the heavenly selection. In this way, the choice of material competition chose the thinking that conforms to the law of physics.


However, this justification is not convincing, but it is a good starting point. One of the amazing things in recent years is that it is very effective in physics that mathematics does not understand that mathematics is very effective in physics. Many results from physics have also appeared very effective in mathematics. Even the work of mathematicians and physics is the same in time, and they can inspire each other and gain each other. From today’s physical theory, we can find a very noticeable mathematical discovery. This relationship is two -way, and it is more difficult to understand. Modern physics surpassed Einstein’s four -dimensional time and space, and has moved towards high -dimensional time and space.

High dimensions other than human cognition

Now we have higher geometry. In one -dimensional space, we have a straight line or a curve and a parameter. In two -dimensional space, we have planes or membranes. In three -dimensional space, we have space, which is likely to be a curved space. In the four -dimensional space, we have the space and time of Einstein. What about other higher dimensions? 5, 6, 7, 8, 9, 10, 11 dimensions? In mathematics, mathematicians can come up with anything they like. Therefore, why not have five -dimensional space? At a point of three -dimensional space, it is accused of coordinates, which can be available in five -dimensional space. There is no difficulty, just put them together. If you are a physicist, you can imagine it as a degree of freedom, that is, how many exercise methods are. One particle has 3 movements, but when 2 particles exercise independently, they have 6 degrees of freedom. If you want to describe the state of a container of two particles, you will use 6 -dimensional space. So the high dimension also occupied a seat. In the 19th century, mathematicians began to study them, but like the previous virtual numbers, high dimensions were esoteric and not so easy to accept. At that time, British mathematician Sylvester wrote a new word “Inconceivable” for high -dimensional. Because he believes that Gao Wei exceeds what human heads can be conceived.

But Silvist also felt that this was not a problem, because he believed in any fictional things, and he believed I and thought that the virtual number had beyond ideas. Unfortunately, this word has not been popular. This is indeed a good word, but things have developed too quickly. Now that physicists can show that when they accept the existence of high -dimensional space, we can show Essence In modern physics, the string theory in high -energy physics proposes that we are actually more than 4 dimensions. The space of these 4 dimensions is time and space, but there are other dimensions here. This has a total of 10 or 11 dimensions, which is the original 4 -dimensional space and extra 6 dimensions.


These additional dimensions are small and tightly bent to some extent. Unless we have a very powerful magnifying glass, we can’t see it at all. It is like a wire, but it has an additional size and thickness. Unless we observe it carefully in a microscope, we can only see it like a straight line. Imagine today’s physics. The magnifying glass we need is a high -energy accelerator. When studying a small space, we need high energy. This is a picture of modern physics, which has a higher dimension. High -dimensional is that you can’t see it in a normal way, but you can study them at high energy.

Because of Babson, we now have indirect evidence to prove these theoretical models, which are consistent with the results of the experiment. Although we cannot see the higher dimension directly, we can indirectly explore them and detect their impact. We can infer that these dimensions are true, because it is consistent with the experiment. But the problem is always: they exist, or the creation of human heads, so that we can understand nature? We always have the next conclusion. If it is used to understand nature, then this is mathematical correctly in the correct mathematics correctness. What happened. But whether it is true or not in nature, or pretending to be like this, even if we think of it, it is not important, and this is the idea that cannot be verified. In other words, what we observed in the laboratory can laid a more fine image about high -dimensional high -dimensional, which we don’t know.

Now this problem is becoming increasingly unavoidable. The physics of today’s world has become more complicated, and such high -dimensional, string -theory models have become more and more precise in recent years, so the mathematics it affects has become wider and wider. However, the question is: this world is really so complicated, or is it just that we see it so complicated from our own point? For example, we will go to another planet. Will they conceive the extra vitamin and how will they imagine it? So, it is difficult to understand the mathematical model we have established, and whether it will have real physical significance. I have discussed the theory of Darwin’s pushing theory and its composition of human wisdom above above. I want to return to this issue for discussion, because human heads are places to discuss mathematics. When we discuss Plato’s ideal space for the ideal space of Plato Where is the ideal space? This is actually within the collective wisdom of humans. This is a conceptual residence. In addition, we can’t find anywhere. If you put the concept on paper, the paper is not the concept, but it is a picture. In fact, the concept is accepted in human thoughts or deeper.

Abstract concept is the soul of mathematics

Physics uses mathematical models, while mathematics is established in human minds. The human brain is likely to evolve. At the same time, evolution is selected by physical competition, which is based on physics and biochemistry. So to some extent, we will see a cycle. Because of these physical laws, the human brain has become this way; but the human brain uses mathematics to obtain the law of physics, and these two are constantly looping. We can ask a question: How much do we know the structure of the brain? This will be a big problem in the next century.

Using advanced scanning technology, neuropsychologists have discovered a lot of knowledge related to the brain. We found that there are chemical reactions and circuits in the brain, and they have also mastered various steps of scanning. We started to have a preliminary understanding of the operation of the brain. Although we don’t know much, we start to know more, including the brain that is likely to involve abstract operations. You may think that when the brain processing vision and other specific information, the methods they organize will involve abstract signals and become like a password. However, these signals do not directly reflect a certain entity, but have been compiled into some abstract passwords. In fact, mathematics is also based on the concept of abstraction. How can humans gain such abstract concepts?

Mathematics has collected a lot of things. When some things are found together, they trace back and sort out, establish ideas and concepts, and then analyze it carefully. The concept of abstract is the soul of mathematics. The reason why a single mathematical concept can be applied to different places in the real world is because we can have different interpretations in different levels. At the same time, the concept of abstraction seems to reflect the operation of the brain in a form, and the circuit in the brain has a mathematical structure to some extent. Today, neuropsychologists have begun to answer the old questions raised by philosophers from a scientific perspective, as well as issues of ideas and decision -making. In the past, the problems discussed by philosophers were endless and there were no exact answers, but now we have a entry point to understand these issues. Of course, we may not get the answer, but to have more questions.

Discussion here, let me make a brief summary. Traditionally, ideas are separated from material. Ideas belong to human internal; objects belong to external. Our question is, what are the associated between the two sides? Does the material really exist? Or is it just an imagination? The idea of ​​the idea is to realize the object, or it reflects other? But one thing is very clear. Mathematics builds the foundation on ideas, and physics is a research object. The problem between mathematics and physics is a narrow version of the problem between ideas and objects. However, what is the relationship between mathematics and physics?

It can be said that the brain is the physical foundation of ideas. The idea is to emerge from the internal structure of the brain through the unknown method, which makes our problem unsteady. The brain is of course composed of material. It has actual components, and the brain is a place that mathematics and physics are clarified. When we have the theory of mathematics or the concept of geometry, it may be recorded in the brain in some form; and the brain is part of the real world and the most important core of them. The philosopher must consider whether the brain is combined by the material and consider it in some way, otherwise they will not be able to talk about the real details or determine the real problem. Maybe this is the real problem that has troubled humans for thousands of years.

With the advancement of science and research, we will know more about the operation of the brain in the future and the relationship between its ideas and ideas. To a certain extent, the progress of neurophantomians can make questions about mathematics, physics and science more clear.

They are all based on “beauty” as a selection criterion

In the summary, let me use another method to answer questions about discovery or invention. Whether in the real world or in the ideal world, there are tens of millions of possible situations composed of ideas or atoms. If I can use symbols, I have tens of thousands of different combinations. If it is material, I can create different weights at will. If it is a mathematical symbol, I can write a lot of square programs. Some equations may be correct and some are wrong. When we write a theory or when a mathematician builds a theory, he will refine some equations from a large pile of correct equations. He will choose one of them to do research, or to choose a question he is interested in. His choice is not the same as artists or writers.

If the monkey is placed in front of the typo and provides enough time, it can write the entire set of Shakespeare’s works. This is of course possible, but this random process takes a long time; Shakespeare is a shortcut to copy the monkey. ; Mathematicians also select the equation from some standards in the correct equation. Physicists also do things in the same way. They did everything feasible and chose to implement it. The criteria we now use are of course the basis for determining the results.

To some extent, I want to say that the guidelines for mathematicians and writers are beauty.

Please note: When Shakespeare writes, he chooses beautiful words to write. The so -called beauty text covers meaning, sound, and elegance: including all aspects of beauty. Mathematicians choose the research equation: because it is beautiful. Why are they beautiful? Because of the following factors: elegant, simple, profound, and universal. Many things have been related to the United States, and the standard of the United States makes mathematicians choose from many equations. At the same time, Mimi also made mathematicians make the right choice. Human investment and participation are just like the process of invention. The invention is to choose what you want to study from all the possibilities, and all the possibilities are in front of them. This is your invention.

The invention is to start a new thing from scratch. Whether in the real world or the ideal world, we are looking for something that has potential development from all possibilities. This is human activity. We can emphasize from the concept that mathematics, art and science are human activities; the role played by humans is based on the criteria we decide. The standards of this beauty include simple, elegant, meaningful ….. Everything. Science is such a process and has become part of the entire human knowledge structure. We are scientists who try to understand the external world and understand nature.

However, “understanding” is also a profound concept. What does it mean? If you think carefully, you will find that you don’t know the answer. The so -called understanding is that after the mind gets the information, it is handled by the data, so that it can be understood, organized, etc. All of them depend on how the brain works. How does the brain understand order and structure? What is the actual situation? We don’t know. As a mathematician, I want to use my own brain to solve the problem. Physicists use their brains to solve the physical world. But before that, something must be operated in their heads. Therefore, it is important that to realize that mathematics and everything are human activities, it is a spontaneous and automatic process. Some people say: If you have a computer, you will no longer need a mathematician. Because it can handle information and do everything for you. However, all machines need to deal with thousands of theorems and choose which one to choose? In the end, humans need to choose materials and organizations to achieve further development. The same is true of science.

Therefore, in the end, the invention and discovery appeared at the same time, and the invention part was the contribution of human beings.

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