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This article comes from the official account of Wechat: back to Park (ID:fanpu2019), author: Shire Eressman (Charles Ehresmann), translated by Ye Lingyuan

"Mathematics is a creative process that will never be completed."

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This article is a speech by Charles Eressman (Charles Ehresmann,1905-1979) at the Honorary Dinner of the Department of Mathematics at the University of Kansas on April 25, 1966. It was published in Cahiers de topologie et g é m é trie diffé rentielle cat é goriques magazine entitled "Trends toward unity in mathematics" in the same year. Eresmann, a French mathematician born in Germany, was one of the early members of the Burbaki school and did important work in the fields of differential topology and category theory.

In this article, Charles Eresman briefly reviews the development of mathematics from classical times to modern times, and expounds his idea of unifying different branches of mathematics by using the language of category theory. It was no more than 21 years since the original concept of category officially appeared in the mathematical literature. Up to now, nearly 60 years have passed, and category theory has made great progress both in itself and in unified mathematics, so the blueprints and questions raised by Eresman's article may not still be applicable in the current context. However, the achievements in the development of category theory have not received the attention it deserves in the wider group of mathematicians. The purpose of translating this article, which is almost 60 years ago, is to go back with the reader to rethink the nature of mathematics and understand the way of thinking of modern mathematics. The translation as a whole complies with the original text and modifies individual details, and the notes and bold are added by the translator.

In the eyes of ordinary people, mathematical conclusions are often regarded as eternal truths, but mathematics is not composed of immutable theorems; it not only produces a large number of exercises, but also has a wide range of applications in other sciences. Mathematics is a vivid science, which is developing continuously and rapidly. We live in an era of rapid expansion of mathematics, and now there is an important force driving mathematics towards unity.

The same development has led to the emergence of new literature-novels no longer have plots; abstract music, sometimes written by computers; and abstract sculptures and paintings, which are not designed to present the general appearance of real things. This same abstract process has also developed a new kind of mathematics, whose motivation is not to find possible applications, but based on our strong desire to know the nature of each problem and the overall structure on which it depends. This consistency is not surprising. After all, mathematics is very similar to art: mathematical theory not only requires rigor, but also satisfies our pursuit of simplicity, harmony and beauty; a beautiful theory is like a work of art. They are all the creation of human inspiration.

For the Platonists among mathematicians, the motivation for their work is to find real structures in a given situation and abstract representations of these structures. For the more pragmatic mathematician, the goal of his efforts is to use all the means at his disposal to solve pre-given problems in pure or applied mathematics; in the process, he will try his best to avoid introducing new general concepts. All mathematicians will agree that if a mathematical work can stimulate new research, then its value in mathematics can be best proved; the most important application of mathematics should be mathematics itself.

Until recently, most philosophers, and even Bergson, described mathematics as a science related to numbers and quantities in everyday space. This description more or less corresponds to ancient Greek mathematics, but modern mathematics is no longer the case.

For the Greeks, mathematics represented Arithmetics and Geometry. The former is the science of natural numbers, while the latter studies the proportion of shapes and geometric quantities in everyday space. Although their geometry is an axiomatic system, they believe that these axioms are imposed by "evidence". In fact, there are more assumptions implied in their reasoning than explicitly stated axioms. Surprisingly, they never introduced the concept of real numbers, although there was no essential difference between the proportion theory of Odoxos [2] and the definition of real numbers given by Dedkin [3] more than 20 centuries later. This abstract process of taking a previously known class of objects-- in this case a kind of rational numbers-- as a new object is completely strange to their minds. Even Archimedes, who pioneered new fields such as Statics and Hydrodynamics and opened the way for integral theory, is reluctant to define real numbers abstractly. After him, the impulse to create seemed exhausted, and mathematics slept throughout the Middle Ages.

The revival of mathematical creativity is also due to the new numbers introduced by Italian mathematicians in the 16th century, including negative numbers and imaginary numbers, as well as the algebraic symbols introduced by Veda [4] in the same period. The Greeks also had a geometry-based algebra, but they did not introduce any algebraic symbols, making their works difficult to read.

Descartes and Fermat also brought new impetus to mathematics, and their analytic geometry unified algebra and geometry. Although the problem of defining the tangent of a curve and how to find the tangent of a curve have been solved in very special cases (such as Archimedes' helix), but now we can study them in an effective way, which directly led to the invention of calculus by Newton and Leibniz. Leibniz seems to have guessed many future developments in mathematics. He not only explicitly introduced functions as objects into mathematics, thus laying the foundation for functional analysis, but also in his unrealized universal ideographic (universal characteristics) theory, he dreamed of revealing the algebraic structure of everything and constructing a universal algorithm for expression and reasoning. Therefore, he is not satisfied with Descartes' analytic geometry because it depends on the choice of coordinate system. Perhaps in bewilderment, he foresaw that geometry must have an intrinsic algebraic structure, and linear algebra and Glassman algebra could be said to have partially realized his dream. Unfortunately, he lived in an era that could not accept his overly advanced ideas, and he did not have enough followers to develop the path he had envisioned. However, his work on calculus has been widely used, especially the symbols of differential and integral that he created. Calculus has also become a major field of mathematics for a long time.

Non-Euclidean geometry independently discovered by Robachevsky and Janos in the 19th century is another advance. By that time, all the boundaries set for mathematics in the classical era had been broken: (Euclid) Geometry is no longer imposed on us by perceptual experience, but relies on human creation based on axioms; we can imagine different axiom systems to study different geometry. Kant's emphasis on our "a priori" of the concept of space has thus become outdated [7]. So what is the essence of geometry? At that time, a space with transitive group action was regarded as the concept of geometric unity, for example, the transitive group action of Euclidean space is actually Euclidean translation transformation. Therefore, geometry becomes the invariant and covariant theory of group action. But in fact, this definition only applies to geometry in homogeneous space, while other types of geometry have been found, and people feel that it is necessary to generalize the concepts of geometry and space. This finally leads to the definition of topological space, which is the appropriate context to answer all the questions about continuity, limit and approximation, and reveals many common structures in the field of analysis and geometry.

At the same time, Cantor's set theory appeared and increasingly became the unified basic theory of all branches of mathematics. This is a new way of abstraction in mathematics. As Cantor said, "the development of mathematics has been completely free" since then, and the concepts in set theory "only require no contradiction and can be related to the concepts introduced before through precise definitions". Although some paradoxes endangering Cantor's set theory were discovered soon, thus endangering the whole mathematical building, Cantor's masterpiece opened the way of modern mathematical thinking.

Since the beginning of this century [9], the freedom created in mathematical theory has led people to consider many new mathematical structures in sets. In addition to various types of algebraic structures (such as groups, rings, fields, Semigroups, modules, algebras, lie algebras, etc.), there are many measure and probability model structures and refinement of various topologies: uniform structures, metric spaces, topological manifolds, differentiable or analytical manifolds with various differential structures, such as Riemannian manifolds and their connections, algebraic manifolds and so on. Considering different structures on the same set, we can construct new mathematical objects, such as lie group, topological vector space, Banach space, Hilbert space, normed algebra, and so on. The introduction of these structures is mainly to meet the needs of the development of pure mathematics, and once they are understood by more people, their applications in other fields will naturally become more and more, and more and more people will use mathematical theories.

After the introduction of all these different types of mathematical structures, people deeply feel the need for unity; after a period of rapid expansion, if there is no unified theory to connect the various fields, then an unstoppable trend is that different mathematicians, like the builders of the Tower of Babel, will use different and incompatible mathematical languages to develop their own fields.

Considering the similarity of these theories, we can get some unity through the concept of structure, or rather, the general definition of a particular kind of structure on the set. This idea is developed from the Bulbaki School [10], and is also the basis for the arrangement of the content order in their series of textbooks, É l é ments de Math é matique. The two structures of integer and Euclidean space, which were widely considered at the beginning of mathematical research, once axiomatically defined, they certainly correspond to a certain structure on the set, that is, all objects that satisfy this kind of structure are isomorphic. However, different structural categories (such as groups or topologies) on sets introduced by modern mathematics do not have this uniqueness.

The theory of general structure on sets can be more generally axiomatized by the concepts of categories and functors, and the development of category theory seems to be the most characteristic unified trend in mathematics today. I think it will soon be taught in other basic fields, such as linear algebra and topology, in the early days of the university. [11]

A category is composed of a family of elements and composite operations defined on top of them, and composition needs to satisfy certain rules (axioms). For example, each group is a special category, and the compound operation is consistent with the multiplication of the group, which makes each element reversible and has only one unit element under this compound operation; but the most typical example is the category of functions between all sets, one of which is a mapping between two sets, and the compound operation is consistent with the composition of usual functions. The axiomatization of abstract categories is based on this category, which is composed of mappings between sets. We call an element in the category a morphism instead of a function. We can think of it as an arrow from one object (the source of the morphism) to another (the target of the morphism). Therefore, the general concept of morphism in category theory is the generalization of the concept of function, and function is regarded by Dedkin as the basic tool of mathematics.

A functor is a mapping that maintains compound operations between categories. They once again constitute a category, that is, the category of functors. For some mathematical structure attached to the set that we usually consider, the homomorphism between them also constitutes a category. For all of these categories, we can naturally define a functor, which is mapped to the category formed by the functions between the previously mentioned sets [13]; this is often called the forgotten functor (forgetful functor), that is, under the action of this functor, we forget the other structures on the set and only retain the information of the most basic set. For example, the category of continuous mappings between all topological spaces, or the category of all group homomorphisms, has forgetting functors as mentioned above.

Now, we can consider any functor p from category H to category C more abstractly. According to the above discussion, in this context, we can regard any object S of H as a structure relative to the functor p, or rather a p-structure on the object p (S) in category C. Therefore, H can be regarded as a category composed of p-structures on C. Surprisingly, many theories and constructions about a particular structure on a set can be unified by the general theory of p-structure as mentioned above. Under this framework, we can define substructures, quotient structures, free structures, Cartesian products, the sum of a family of objects, or the limits and co-limits of any functor, and so on. At present, I believe that the current mathematical research will pay less attention to the properties of a single p-structure, or even to the properties of a particular functor p; on the contrary, the goal of mathematics now should be to study the properties of a certain family of functors. so that the theorem that once held for a particular functor p and its corresponding p-structure is now true for any functor in this family. Once we understand the real reason why this theorem is valid, we will generally find that there are only a few conditions (assumptions) that are really necessary to prove this theorem. Therefore, the proof of the original theorem can now be extended to a very wide class of functors, not only to the original p functors. In particular, this theorem may contain many known functors that can be applied to fields we have never thought of. For example, the compactness of topological spaces, the completion of uniform spaces, free groups, free modules or, more generally, the construction of free algebras generated by a set, can be regarded as corollaries of some abstract existence theorems of free structures of functors.

Of course, the above unified plan for mathematics is too sketchy. In fact, only the creativity of mathematicians can continue to discover new and interesting functor classes. As we can see, in mathematics, one of the characteristics of the creative process is to recognize the previously defined object as a new mathematical object. When we begin to study the classification and properties of different functors to sort out and unify the existing mathematical theories, do we face the same problem at this higher level? Once this new theory matures and becomes complex and entangled again, is it necessary for us to develop a higher degree of unified theory? We don't try to answer that question. However, we are more and more aware that mathematics is a creative process that will never be completed, and its existence does not need to be proved by its importance or expanding scope of application; its significance is far more than just acting as a "bulldozer of physics". Mathematics is the key to understanding the whole universe, unifying all human thinking from science to philosophy to metaphysics. Therefore, the great ideal of Plato and Leibniz, which makes mathematics the essence of all knowledge, may eventually be realized.

Annotation

[1] Henri Bergson Bergson (1859-1941), a French philosopher and litterateur, won the Nobel Prize for Literature in 1927 for his rich and dynamic thought and language.

[2] Eudoxus,408 B.C.-355 B.C., an ancient Greek mathematician and astronomer, most of the contents in Euclid's Geometry are likely to come from Edoxus. Some people think that he is the most outstanding mathematician in ancient Greece.

[3] Richard Dedkin (Richard Dedekind,1831-1916), a German mathematician, made important contributions in the fields of number theory, abstract algebra (especially ring theory) and the axiomatization of arithmetic.

[4] Francois Veda (Fran ç ois Vi è te,1540-1603), a French mathematician and junior high school student, is familiar with the Veda theorem from him.

[5] Universal ideographic characters (characteristica universalis in Latin) is a universal formal language conceived by Leibniz, which can express mathematical, scientific and metaphysical concepts and supports a general logical calculus.

[6] Nikolai Robachevsky (Nikolai Lobachevsky,1792-1856), Russian mathematician; Boye Janos (J á nos Bolyai, 1802-1860), Hungarian mathematician, they lived at the same time as Gauss. Both of them independently made important contributions to non-Euclidean geometry, especially hyperbolic geometry.

[7] Kant believes that human understanding of time and space is not accomplished by conceptualisation, they are pure forms of our sensory intuition (pure form of sensible intuition). Very roughly speaking, the former involves the operation of understanding, while the latter forms a priori knowledge.

[8] Georg Cantor (Georg Cantor,1845-1918), a German mathematician born in Russia, founded modern set theory, which is the strict definition of real numbers and the theoretical basis of the whole calculus system, and has made outstanding contributions to the basis of mathematics (foundation of mathematics).

[9] this refers to the time when this article was written, that is, the 20th century.

[10] Nicolas Bourbaki is the common pseudonym of a group of French mathematicians in the 20th century. They began to write a series of books on modern higher mathematics since 1935, with the aim of building all mathematics on a solid foundation of set theory. In this process, they devoted themselves to the generalization and rigor of mathematical concepts as much as possible, which had a profound impact on the development of mathematics after the 20th century.

[11] Category Theory originated from Eilenberg and MacLane's paper entitled General Theory of Natural Equivalences in 1945, and quickly participated in the development of various mathematical branches as a mathematical language and tool in the following decades. Unfortunately, Eresman's conjecture has not come true in most universities until many years later.

[12] in modern category theory languages, categories are generally defined as mathematical objects composed of two categories of elements, namely, objects and morphisms between them. however, a category can only be understood as a family of morphisms plus the compound operations defined above, because the objects and unit morphisms in the category correspond to each other one by one. In other words, the information of a morphism contains the information of an object. This paper adopts the latter understanding of the category.

[13] that is, a homomorphism is regarded as a function between its corresponding sets.

This article is translated from Ehresmann Charles. "Trends toward unity in mathematics." Cahiers de Topologie et G é om é trie Diff é rentielle Cat é goriques 8 (1966): 1-7.

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