If we focus on the physical theory itself, it is obvious that the physical theory does not exist. There is no concrete thing called Newtonian physics in this world, and there is no thing called electrodynamics in this world. Physical theory is indeed the product of human spirit. Or to put it another way, physical theory was invented by human beings! So back to our mathematics, mathematical theories are all conceived by people and the product of the so-called human spirit.
In fact, this has answered the question of the subject, but I guess the subject is actually asking what is the connection between mathematical theory and the real world, is it like physics? If that's what you think in your heart, I can talk about my views on mathematics again, from the perspective of a doctoral student in mathematics. I will intersperse some of my views on physics, but I don't know physics. Please kindly point out any mistakes.
I have seen a lot of discussions about the relationship between mathematical theory and reality. I feel that most of the debates are actually purely philosophical discussions, which are very ungrounded. Being ungrounded means that they don't really know the state of mathematicians in real life. What are they doing? What are they obsessed with? What are they celebrating? What did they win the grand prize for? The answers to these questions are the best angle to understand the relationship between mathematical theory and reality. I can't answer every question, but I hope to start with "What are they doing?" To illustrate my point: pure mathematics is not directly related to reality, but its theoretical motivation may come from reality.
The biggest difference between mathematics and physics is that physics always has two parts, one is theory and the other is reality. The ultimate goal of physicists is to explain a physical phenomenon. As long as a physical phenomenon can be successfully explained, the corresponding theory is a good theory. No matter how beautiful the theory is, it should be abandoned as long as it can't explain some phenomenon in reality. If a physicist claims that he has a theory to explain the formation mechanism of the "green hole" (a celestial body I invented, similar to a black hole), then his colleagues will definitely not accept it, because we have never found the "green hole" at all. So physicists all have the same goal: to construct a theory to explain a phenomenon, or to test whether a theory is wrong within their research interests (because there is no way to know whether this theory is correct). The former corresponds to a theoretical physicist and the latter to an experimental physicist. For physicists, the happiest thing is that a prediction of their own theory is confirmed, or an experimental result that is inconsistent with the theory is found, and there are more opportunities to create new theories. But remember, theory is theory, it can explain reality, and how much is the most important thing.
(The following word mathematics is equivalent to pure mathematics. As for applied mathematics, I will mention it. ) and mathematics is not like this at all! Mathematics is only part of my understanding, that is, mathematical theory. So that we generally don't use this word, just use mathematics itself instead. Due to the lack of popular science in mathematics, many people's understanding of mathematics actually stopped when it came to calculus. Calculus was really invented to solve physical problems, so many people think that there is something unclear between mathematics and physics. In addition, engineering mathematics is what most science students need to master, so many people have the impression that mathematics is just calculation. Unfortunately, the face of real mathematics is completely different.
The development of mathematics is mainly driven by problems, and the problems themselves can be driven for many reasons. Sometimes problems come from the needs of physics, and sometimes problems come from other branches of mathematics itself. In a word, all questions that can interest mathematicians are good questions. This also leads to the fact that what mathematicians think cannot be directly related to the real world. A good mathematical theory should be to answer a mathematical question as systematically as possible. In the process of searching, people will gradually develop a new theory, or abstract a theory to make it not limited to the original problem. It is no exaggeration to say that basically mathematicians have created any mathematical theory relatively "arbitrarily" in the process of being free and rambling. This arbitrariness seems to make some people feel very uneasy, thinking that mathematics seems to be completely free to create. I give my opinion here: there is no difference between mathematics and art, both of which are free to create, and the creative motives are varied. However, not all creation is called good creation. Good creation is often interesting, insightful, systematic and powerful. Let me give you a few examples to illustrate.
(The following example may not be true in the real historical sense. But there is basically no problem in discussing my point of view. Welcome to correct me)
Example 1: The origin of group theory comes from a very simple question: Does quintic equation have the same radical solution as quadratic equation? As for why to answer this question, the answer is that mathematicians can find the root solution of quartic polynomial equation, but they can't find it five times. So why on earth should we find a fundamental solution? Mathematicians began to think about the solution of quadratic equation because they found that linear equation is very useful in life and also found the general solution of linear equation. After solving it twice, they found it interesting three times, so they continued to find ways to find it three times. Finally, I looked for it four times, but I couldn't find more than four times. So many mathematicians want to prove that there is no general root solution for quintic equation.
The problem itself may come from life or physics, but once it enters the mathematician's sight, continuing research is basically driven by interest itself. )
The general radical solution of quintic equation seems to be 100 years or more (? ) did not give a satisfactory answer. Some mathematicians have given proof, but it is not systematic enough. Until Galois considered one thing: take out all the roots of a given quintic equation (that is, five), and then consider all the rotations of these five roots. It uses the word group to describe the set formed by all rotations. He found that when this set meets certain conditions, this quintic equation has no root solution. At that time, the concept of group was still in its infancy, not as abstract as today's definition. Mathematicians found this concept very interesting and began to study the rotation of the roots of general equations. Mathematicians immediately realized that a rotation followed by a rotation is itself a rotation. In addition, you can turn around or turn back. So they are not satisfied with studying the rotation of roots, and begin to study general sets that meet such conditions. In this way, they completely abstract a new concept called group, which stipulates that a group consists of a set and binary operations on the set. This binary operation corresponds to the concept of "immediately following" in the original rotation group. In addition, these binary operations must satisfy some basic properties. In this way, the concept of group is studied by most people, because people have found entities with this concept in other places.
(So mathematicians create something because they need to solve problems, because they find new things interesting, because they want to abstract the research object as much as possible and make the means to solve a problem as powerful and useful as possible. )
EXAMPLE 2: The concepts of point set topology, algebraic topology, homology algebra and category topology originated from Euler's Seven Bridges in Ginsburg. There are seven bridges in Fort Genes, and local people are wondering whether they can visit each bridge without going back.
Research motivation may come from real life, but that has nothing to do with reality.
In order to solve this problem, Euler replaced the bridge with a straight line and the island connected by the bridge with a point, and then found that the problem was whether each point could be drawn without repetition. Euler naturally gave the answer. But people found a very important thing, the length of the line segment is not important, and the size of the point is not important. What matters is how the points are linked. This was a new geometry at that time. In the past, people studied things with specific properties, such as length, size and area. But topology does not have these things. In topology, any two objects can be obtained from each other through continuous deformation without distinction. What people want to study is the invariance under continuous deformation. For example, in the case of seven bridges in gneiss, the length of the bridges is not important. The important thing is that no matter how we shrink each bridge, it is impossible for us not to walk through each bridge repeatedly. This attribute is a topological attribute.
By this time, people have begun to ignore reality and only think about some new discoveries.
After discovering this new geometry, everyone began to study it. Many people gradually realize that an interesting conclusion is that the topological properties of a surface are completely determined by how many holes it has. For example, a circle and a disk are not equivalent in topological sense. Because what is surrounded by a circle is a "hole" and there is no hole in the disk, the disk can actually shrink to its center continuously, but if the circle shrinks, it will inevitably tear, which is not allowed. But how to describe this hole? A hole is something that is not on that object, but you only have this object. How do you describe what is not on it in mathematical language? Poincare gave the answer. He gives each geometric object a heap number (Betti number), and then geometric objects with the same topological properties have the same number. It uses this number to describe holes in different dimensions. For example, the Betti number of a circle is 1, and all of them are 0 after 1, which means that there is a "0-dimensional" hole and a "1 dimensional" hole. Later generations put a group for each geometric object along this line of thought, which is called homology group. One of the properties of homology group is this heap number. Algebraic topology was born. As the name implies, algebraic topology is to study topology by algebraic method. The group mentioned before, people at that time had studied a lot, so many conclusions were just used by chance.
So I personally think that the development of mathematics here has nothing to do with the problem of the Seven Bridges at that time. Starting from the "Seven Bridges" problem, people discovered many new problems and developed new tools. These new problems are all interesting problems in mathematics itself, not from reality.
Naturally, everyone began to study algebraic topology again. However, at that time, people used very inaccurate language and many ambiguous languages to describe the geometric objects they studied. In order to make everything basic, formal and more reliable, point set topology came into being. Of course, point set topology also comes from other research motives, so I won't talk about it here. The study of algebraic topology is also a very hot field now, but this field is becoming more and more abstract and complex, and naturally its universality is also growing. In the process of research, people have found some phenomena: many relationships between homology groups and geometric objects are purely algebraic problems. And the tools derived from this can actually be used in the background without geometry! So people abstract this method, independent of geometry, and derive a new algebra called homology algebra. In homology algebra, although there is an origin in the algebraic topology behind each definition, it actually exists outside the algebraic topology. Moreover, such abstraction is not abstract for the sake of abstraction. Later, people found that homology algebra itself is a powerful tool, which can be used to study other algebraic fields, such as group theory, commutative algebra, algebraic geometry and so on. In short, it is widely used.
(So mathematicians' abstraction is not abstract for the sake of abstraction. Although it seems that they create theories at will, every time the facts prove that this abstraction is not only more interesting, but also provides new ways of thinking and new tools. The motivation of these creations may come from a real problem a hundred years ago, and then more from mathematicians' own thinking about mathematical structure.
Then, people realize a new phenomenon: in algebraic topology, not only geometric objects have homology groups, but if there is a relationship between geometric objects (continuous mapping), there is also a corresponding relationship between corresponding homology groups (group homomorphism). In order to study this phenomenon, people invented the category theory. For example, the collection of all objects with geometry forms a category, which is called a topological category. The category formed by all groups is called group category, and so on. (Of course, category theory obviously doesn't put things together so foolishly. There are many things in it, so I won't explain them in detail here. With the study of category theory, people find that almost all structures, phenomena and properties in mathematics can be described by the language of category theory. In other words, we don't need to prove the same conclusion in essence but in different forms repeatedly in every category, just once. Now this language has become a necessary language for every mathematician who does algebra and geometry, and even a way of thinking.
Summary: How closely mathematics relates to reality depends on what mathematicians are doing. If we want to understand it more clearly, it is more necessary to learn modern mathematics. After all, the popular science above is rough and not rigorous. Generally speaking, the development of pure mathematics itself is not directly related to reality. Mathematicians mostly look for problems they are interested in, then find solutions, and create new mathematical structures or even new branches of mathematics on the way. At the same time, mathematicians like to keep abstracting, make their tools more powerful, or let interesting properties be studied independently. The abstracted tools will be applied to other fields. In this way, mathematicians continue to develop new theories, which continue to cross to form new branches of mathematics. However, the vast majority of research motivation actually comes from mathematics itself, not reality. This is why it is so difficult to popularize mathematics, because an essential content of popular science, in my opinion, is the research motivation and the value of research itself. Because these motives and values are not in this world but mostly in mathematics itself, it is difficult for outsiders to understand and do popular science.
More and more abstract mathematics means wider application, so people outside mathematics apply these research results to other branches. Here, physics is the main audience, they constantly influence mathematics and constantly use new mathematical conclusions. Others tried to optimize some mathematical results and let him solve some practical problems. So these people developed so-called applied mathematics.
Generally speaking, mathematics is mathematics and physics is physics. There is no direct connection between mathematics and reality. In any case, I believe that most mathematicians are not concerned with reality, but with beautiful and interesting problems and mathematical structures. And physics is a subject that always cares about reality.