Modern mathematics is mostly based on mathematical logic and axiomatic set theory. They are all described by a clear set of axioms.
The basic rules of logical calculus are described by mathematical logic. It describes a set of axioms such as (in layman's terms) "If both A and B are right, then A is right" and so on.
Axiomatic set theory usually refers to the set theory defined by the famous ZFC(Zemelo-Fraenkel axiom plus [axiom of choice]) axiomatic system. It describes a set of axioms, such as (in layman's terms) "the same set of two elements is equal" and so on.
With the above axiomatic system and proper definition and reasoning, most of the contents of algebra can be deduced.
From a certain point of view, all mathematical definitions are axioms, because the definition stipulates some properties of the research object-and the definition can't even point out the existence of the research object.
A common example is Euclidean geometry, which is the geometry in middle school textbooks. It can be said that it is derived from a set of axioms, but it can also be said that a set of geometric axioms defines what geometry is and what geometric objects such as points, lines and surfaces are. Of course, the axiomatic system used in middle school textbooks is not perfect. For the needs of teaching, some redundant axioms have been added (such as the axiom about triangle congruence, which was originally just a theorem), but some axioms that are difficult to understand in middle school (such as the axiom of continuity, which requires understanding the structure of real numbers) have been omitted.
There is another example that people often ask: What is a natural number?
In fact, the strict definition of natural numbers in mathematics is defined by a set of axioms, namely Piano's axiom. Its strict expression is more complicated, you can refer to Baidu Encyclopedia (that explanation is actually not very good, just let it be).
Peano's axiom is that a natural number must have1; Then 1, followed by a 2 and only a 2, and so on; Then there is induction, or an infinite sequence starting from 1 must form a set.
This set of axioms does not explain the existence of natural numbers, but we can regard a set containing only one empty set and one element as 1, then regard 1 and the empty set as a set of two elements as 2, and so on, and construct a set that does have such a natural number.
On the basis of axioms, we can also define addition operations and prove their operational properties. (By the way, you will find that the "1+ 1 = 2" that many people are tired of asking is directly guaranteed by the addition definition.