1. Many aspects of mathematical quality

We all believe that mathematicians should strive to create good mathematics. But how do you define “good math”? Should we even venture to try to define it? Let us consider the former question first. Almost immediately we realize that there are many different kinds of mathematics that can be called "good". For example, "good mathematics" can refer to (in no particular order):

Good solutions to mathematical problems (such as a major breakthrough in an important mathematical problem);

Good mathematics Skills (such as masterful application of existing methods, or the development of new tools);

Good mathematical theory (such as conceptual frameworks or symbolic choices that systematically unify or generalize a series of existing results);

Good mathematical insight (such as an important conceptual simplification, or the realization of a unifying principle, inspiration, analogy or theme);

Good mathematical discovery (such as the realization of a Revealing unexpected and fascinating new mathematical phenomena, correlations or counterexamples);

Good mathematical applications (such as applying to important problems in the fields of physics, engineering, computer science, statistics, etc., or transforming a mathematical field the results are applied to another area of ??mathematics);

Good mathematical presentation (such as a detailed and broad overview of recent mathematical topics, or a clear and well-motivated argument);

Good mathematics teaching (such as lecture notes or writing style that enable others to learn and study mathematics more effectively, or contributions to mathematics education);

Good mathematics vision (such as productive long-term plans or conjecture);

Good mathematical taste (e.g., research goals that are interesting in themselves and have an impact on important topics, topics, or problems);

Good mathematical public relations (e.g., to non-mathematicians or mathematicians in another field effectively demonstrate mathematical achievements);

Good metamathematics (such as advances in mathematical foundations, philosophy, history, scholarship, or practice); [Translator's Note: Here" "Meta-mathematics" is translated from "meta-mathematics", but some of the content mentioned here, such as history, practice, etc., usually do not belong to the category of meta-mathematics. ]

Rigorous mathematics (all details are given correctly, meticulously and completely);

Beautiful mathematics (such as Ramanujan’s amazing identities; simple and beautiful statements, proofs but very difficult results);

Beautiful mathematics (such as Paul Erd?s's "Proof from the Book of Heaven" concept; difficult results obtained with the least effort); [Translator's Note: "Proof from the Book of Heaven" Proofs" is translated from "proofs from the Book". Paul Erd?s likes to say that the most beautiful mathematical proofs come from "The Book" (I translate it as "The Book"). He has this famous saying: You don't have to believe in God, but you should believe in "The Book" . The third year after Erd?s's death, in 1998, Martin Aigner and Günter M. Ziegler published a book titled "Proofs from the Book of Heaven", which included dozens of beautiful mathematical proofs in memory of Erd? ?s. ]

Creative mathematics (such as original techniques, ideas or various results that are novel in nature);

Useful mathematics (such as ones that will be repeated in future work in a certain field) lemmas or methods used);

Strong mathematics (such as a sharp result that matches a known counterexample, or deriving a surprisingly strong hypothesis from a seemingly weak hypothesis) conclusion);

Profound mathematics (such as an obviously non-trivial result, such as understanding a subtle phenomenon that cannot be approached by more elementary methods);

Intuitive mathematics (such as A natural and easily visualized argument);

Explicit mathematics (such as the classification of all objects of a certain type; the conclusion of a mathematical topic);

Others[ Note 1].

As mentioned above, the concept of mathematical quality is a high-dimensional concept, and there is no obvious standard ranking [Note 2]. I believe this is because mathematics is inherently complex and high-dimensional, and evolves in self-adjusting and unpredictable ways; each of these qualities represents a different way in which we as a community can improve our understanding and application of mathematics. . There appears to be no universal knowledge as to the relative importance or weighting of the above qualities. This is partly due to technical considerations: the development of a certain field of mathematics at a particular period may be more receptive to a particular method; partly due to cultural considerations: any particular field or school of mathematics tends to Attract mathematicians who think similarly and like similar methods.

It also reflects the diversity of mathematical abilities: different mathematicians tend to specialize in different styles and thus adapt to different types of mathematical challenges.

I believe that this diversity and difference in "good mathematics" is very healthy for mathematics as a whole, because it allows us to pursue more mathematical progress and better understanding of mathematics. ***Take many different approaches to the same goal and develop many different mathematical talents. While each of these qualities is generally accepted as required in mathematics, pursuing one or two in isolation at the expense of all others can become a disservice to a field. Consider the following hypothetical (somewhat exaggerated) situation:

A field becomes more and more eccentric, in which individual results are promoted for the sake of promotion, and refined for the sake of refinement, while the entire field becomes more and more eccentric. Drifting haphazardly without any clear purpose or sense of progress.

A field becomes overrun with frightening conjectures, with no hope of making serious progress on any of them.

A field becomes dominated by ad hoc approaches to solving a cluster of unrelated problems without a unifying theme, connection, or purpose.

A field becomes too boring and theoretical, constantly recasting and unifying previous results with more and more technically formal frameworks, but the consequence is that it does not produce any exciting new breakthroughs.

A field advocates classic results and constantly gives shorter, simpler and more elegant proofs of these results, but does not produce any truly original new results beyond the classic works.

In each of these cases, the field will see a lot of work and progress in the short term, but in the long term risks being marginalized and failing to attract younger mathematicians. Fortunately, when a field is challenged and revitalized by its connections to other areas of mathematics (or related disciplines), or is nurtured by a culture that respects and respects a variety of “good mathematics,” it is less likely to end up with declined in this way. These self-correcting mechanisms help keep mathematics balanced, unified, productive, and active.

Let us now turn to consider another question raised earlier, namely whether we should attempt to define “good mathematics” at all. Definitions run the risk of making us arrogant; in particular, we risk overlooking a singular instance of genuine mathematical progress because it does not satisfy the prevailing definition. On the other hand, the opposite view - that all methods are equally suitable and deserve the same resources in any field of mathematical research [Note 4], or that all mathematical contributions are equally important - is also risky. That view may be admirable in terms of its idealism, but it erodes the sense of direction and purpose of mathematics, and may also lead to an unreasonable allocation of mathematical resources [Note 5]. The real situation lies somewhere in between, where for each area of ??mathematics existing results, tradition, intuition, and experience (or their lack thereof) indicate which approaches are likely to be fruitful and thus should receive the most resources; that The approach is more heuristic, so that perhaps only a few independent-minded mathematicians can investigate to avoid omissions. For example, in a mature field, it may be more reasonable to pursue a systematic approach, develop general theories in a rigorous way, and steadily apply effective methods and established intuitions; while in newer and less stable fields, areas where perhaps more emphasis should be placed on formulating and solving conjectures, trying different approaches, and relying to a certain extent on loose revelations and analogies. It would therefore make strategic sense to conduct at least a partial (but evolving) survey in each field of what qualities should be most encouraged in mathematical progress, so that each field would be All stages of development can most effectively develop and advance the field. For example, a certain field may be in urgent need of solving some pressing problems; another field may be waiting for a theoretical framework that can sort out a large number of existing results, or a grand plan or a series of conjectures to stimulate new results; others Fields may benefit greatly from new, simpler, and more conceptual proofs of key theorems; while more fields may need greater openness and a thorough introduction to their topics to attract more interest and participation. Therefore, the determination of what constitutes good mathematics will and should be highly dependent on the status of a field itself. Such determinations should be constantly updated and debated, both within the field and from outside observers. As mentioned earlier, surveys about how a field should develop, if not tested and corrected in a timely manner, are likely to lead to imbalances in the field.

The above discussion seems to indicate that assessing mathematical quality, although important, is a hopelessly complex matter, especially since many good mathematical achievements may score very high on some of the above qualities, and in This is not true of other qualities; at the same time, many of these qualities are subjective and difficult to measure accurately (except with the benefit of hindsight).

However, an eye-catching phenomenon is [Note 6]: Good mathematics in the above sense often tends to lead to good mathematics in many other senses, which leads to a tentative guess that high-quality mathematics The general idea may exist after all, and all of the specific measures mentioned above represent different paths to the discovery of new mathematics, or different stages or aspects in the development of a mathematical story.