definition

Aristotle defined mathematics as "quantitative mathematics" until the 18th century. From the 19th century onwards, the study of mathematics became more rigorous, beginning to involve abstract topics such as group theory and projective geometry, which had no clear relationship with quantities and measures, and mathematicians and philosophers began to come up with various new definitions. Some of these definitions emphasize the deductive nature of a large amount of mathematics, some emphasize its abstract nature, and some emphasize certain topics in mathematics. Even among professionals, there is no consensus on the definition of mathematics. There is not even agreement on whether math is an art or a science. [8] Many professional mathematicians are not interested in the definition of mathematics or consider it undefinable. Some are just, "Math is done by mathematicians." ”

The three main types of mathematical definitions are known as logicians, intuitionists, and formalists, each reflecting a different school of philosophical thought. There are serious problems that no one universally accepts, and no reconciliation seems feasible.

An early definition of mathematical logic was Benjamin Peirce's "The Science of Drawing the Necessary Conclusions" (1870). In Principia Mathematica, Bertrand Russell and Alfred North Whitehead proposed a philosophical procedure known as logicism and tried to prove that all mathematical concepts, statements, and principles could be defined and proven with symbolic logic. The logical definition of mathematics is Russell's "All mathematics is symbolic logic" (1903).

Intuitionism defines, from the mathematician .****rouer, to identify mathematics with certain mental phenomena. An example of the definition of intuitionism is "mathematics is a mental activity that is constructed one after the other". Intuitiveism is characterized by its rejection of some mathematical ideas that are considered valid according to other definitions. In particular, while other mathematical philosophies allow objects that can be proven to exist, even if they cannot be constructed, intuitionism only allows mathematical objects that can actually be constructed.

Formalism defines mathematics with its symbols and rules of operation. Haskell Curry defines mathematics simply as "the science of formal systems." [33] The formal system is a set of symbols, or tokens, and there are rules that tell how tokens are combined into formulas. In the formal system, the word axiom has a special meaning, which is different from the ordinary meaning of "self-evident truth". In a formal system, an axiom is a combination of tokens contained in a given formal system without the need to use the system's rules for derivation. [2]

structure

Many mathematical objects such as numbers, functions, geometry, etc., reflect the internal structure of the successive operations or relationships in which they are defined. Mathematics is the study of the properties of these structures, for example, number theory is the study of how integers are represented in arithmetic operations. In addition, it is not uncommon for different structures to have similar properties, which makes it possible to describe their state by further abstraction and then by axioms on a class of structures, and it is necessary to find the structures that satisfy these axioms among all the structures. Thus, we can study groups, rings, fields, and other abstract systems and put these studies (through structures defined by algebraic operations) into fields of abstract algebra. Due to its great versatility, abstract algebra can often be applied to seemingly unrelated problems, such as the ancient problem of ruler diagramming, which is finally solved using Galois theory, which involves domain theory and group theory. Another example of algebraic theory is linear algebra, which makes a general study of vector spaces in which elements are quantitative and directional. These phenomena show that geometry and algebra, which were previously thought to be unrelated, actually have strong correlations. Combinatorics is the study of methods for enumerating number objects that satisfy a given structure.

space

The study of space originated from Euclidean geometric trigonometry, which combines space and numbers, and includes the very famous Pythagorean theorem, trigonometric functions, etc. Nowadays, the study of space has been extended to higher-dimensional geometry, non-Euclidean geometry, and topology. Numbers and spaces play important roles in analytical, differential, and algebraic geometry. In differential geometry, there are concepts such as computation on fiber bundles and manifolds. In algebraic geometry, there are descriptions of geometric objects such as the set of solutions to polynomial equations, combining the concepts of number and space; There is also the study of topological groups, which combine structure and space. Li Qun is used to study space, structure, and change

Main article: Fundamentals of mathematics

In order to clarify the mathematical foundations, fields such as mathematical logic and set theory were developed. The German mathematician Cantor (1845-1918) pioneered set theory and boldly marched towards "infinity", in order to provide a solid foundation for all branches of mathematics, and its own content is also quite rich, put forward the idea of real infinity, and made immeasurable contributions to the development of mathematics in the future.

At the beginning of the 20th century, set theory gradually penetrated into all branches of mathematics and became an indispensable tool in analytical theory, measurement theory, topology, and mathematical science. At the beginning of the 20th century, the mathematician Hilbert disseminated Cantor's ideas in Germany, calling set theory "the paradise of mathematicians" and "the most astonishing product of mathematical thought", and the British philosopher Russell praised Cantor's work as "the greatest work that can be boasted of in this era".

logic

Main article: Mathematical logic

Mathematical logic focuses on placing mathematics on a solid axiomatic framework and studying the results of this framework. For its part, it is the source of Gödel's second incompleteness theorem, which is perhaps the most widely circulated result of logic, and modern logic is divided into recursive theory, model theory, and proof theory, and is closely related to theoretical computer science.

symbol

Main article: Mathematical symbols

Perhaps China's ancient arithmetic is one of the earliest symbols used in the world, originating from divination in the Shang Dynasty.

Most of the mathematical symbols we use today were invented after the 16th century. Before that, mathematics was written in words, a painstaking procedure that would limit the development of mathematics. Today's notation makes math easier for people to manipulate, but beginners often get intimidated by it. It is extremely compressed: a small number of symbols contain a lot of information. Like musical symbols, today's mathematical symbols have a clear grammar and encoding of messages that are difficult to write in other ways.

Rigour

Mathematical language is also difficult for beginners, how to make these words have more precise meanings than everyday language, and it is also troublesome for beginners, such as open and field words have special meanings in mathematics, and mathematical terms also include proper nouns such as embryo and integrability, but there is a reason for using these special symbols and terms: mathematics requires more precision than everyday language, mathematicians call this requirement for linguistic and logical precision "rigor"

Rigor is an important and fundamental part of mathematical proofs, and mathematicians want their theorems to be deduced according to axioms in a systematic way, in order to avoid relying on unreliable intuitions and thus deriving false "theorems" or "proofs", and there have been many examples of this in history. The degree of rigor expected in mathematics varies from time to time: the Greeks expected careful arguments, but in Newton's time, the methods used were less rigorous than Newton's definitions of problem-solving, and it was not until the nineteenth century that mathematicians used rigorous analysis and formal proofs to deal with them. Mathematicians continue to debate the rigor of computer-aided proofs: when a large number of calculations are difficult to verify, their proofs are hardly effectively rigorous

quantity

The learning of quantities begins with numbers, beginning with the familiar natural numbers and integers and the rational and irrational numbers described in arithmetic

Another area of research was its largeness, which led to the cardinality and later another concept of infinity: the Alev number, which allows meaningful comparisons between infinite sets of large ones