Mathematics, simply put, is the science of studying numbers and shapes (or simply numbers and shapes). What are numbers and shapes? Why is mathematics useful? And what are the uses of mathematics?
What are number and form? Number and form are the foundation of all things. Form, derived from the shape of things, is an abstraction of the spatial form upon which humankind depends for survival. It can express the form of things, represent their structure, and the relationships between different things; all things coexist. Number, derived from counting, can record the order of things and measure their size or scale; all things share this. Number and form are two sides of the same coin, interconnected and unified in their opposition. Why is mathematics useful? The simplest and most direct reason is that our survival requires space and time. Space involves both form (shape) and quantity (scale), while the fundamental characteristic of time is quantity (sequence, duration), which is the basic reference for all changes. In fact, everything in the world must exist in space in a certain form and be constrained by various quantities such as length, area, volume, mass, concentration, temperature, and color. Therefore, mathematics, which studies "number" and "form," is inevitably closely related to the occurrence and development of all things. In fact, the world is material, matter is in motion, and motion is interconnected. These interconnected material movements can largely be abstracted by mathematicians into mathematical models characterized by the changing relationships between quantities and spatial structural forms. Therefore, mathematics is widely applicable.
So, what exactly is the use of mathematics? As early as 1959, the famous mathematician Hua Luogeng pointed out: From the vastness of the universe to the minuteness of particles, from the speed of rockets to the ingenuity of chemical engineering, from the changes of the earth to the mysteries of biology, and from the complexities of daily life, mathematics is used everywhere.
In real life, numbers appear in numerous contexts such as street signs, house numbers, postal codes, telephone numbers, QQ numbers, bank card numbers, and various passwords. They are not only used to record and calculate our resources and assets, but also to help us quantify, statistically analyze, compare, judge, and predict. Having a clear understanding of things is essential for improving efficiency and saving costs. Quantifying things ensures reliable comparisons and conclusions. In science and technology, "quantity" permeates all scientific fields, and mathematics is inevitably applied to all scientific fields. Anything involving quantity, quantitative relationships, changes in quantity, changes in quantitative relationships, and the relationships between changes in quantity can be described and explained using mathematics. Mathematics is the mother of science, nurturing and driving the emergence of other sciences, such as computer science, information science, systems science, scientific computing, mathematical physics, biomathematics, and financial mathematics. Mathematics is also the servant of science, helping and promoting the development of other sciences, and has become one of the main tools for developing high technologies, such as information transmission and information security, image processing, medical diagnosis, drug testing, data processing, and network information retrieval.
A fundamental means for humankind to understand and transform the world is by establishing mathematical models. Many problems in the real world that seem unrelated to mathematics can, in fact, be perfectly solved using mathematical models: first, ignore the secondary factors, relationships, and processes of the practical problem, and extract its main factors, relationships, and processes; after some reasonable simplification and assumptions, find the correspondence between the problem to be studied and a certain mathematical structure, transforming the practical problem into a mathematical problem; finally, carry out mathematical derivation and calculation on this model to form an understanding, judgment, and prediction of the problem. The application of mathematics in fields such as engineering technology, geological exploration, and disease control can not only significantly save manpower, material resources, and financial resources, but also effectively improve work quality and achieve theoretical optimization.
Number and form are not unique to nature; they also exist in human society and the spiritual world, in the macrocosm and the microcosm, exhibiting quantitative determinacy and structural relationships. Mathematics not only provides scientific tools and methods for studying nature but also for all research involving quantitative determinacy and structural relationships. Today, humanity has entered the information age, and information technology is digital technology. Digital technology unifies the description of all things, their relationships, and their development and changes using numbers, reducing the study of various problems to data storage and data processing.
Data transmission and processing: Its memory capacity far exceeds that of the human brain, and its transmission speed rivals the speed of light. Digital technology has greatly improved work efficiency and quality in various fields, its power is immeasurable, and its impact is exceptionally astonishing. Mathematics is not only the foundation of all natural sciences and engineering technologies, but it has also permeated many research fields in the humanities and social sciences, such as economics, education, demography, psychology, linguistics, literature, and history, becoming a cornerstone of contemporary human civilization.