Mathematics is widely applied in all areas of human society, not only because its objects are closely related to everything, but also because its conclusions are reliable. Why are mathematical conclusions reliable? This depends on how they are derived. Simply put, mathematical premises are undoubtedly sound, and its methods are rigorous and reliable.
For example, the algebra and geometry we learn in primary and secondary schools are derived from a few simple and clear facts (axioms and rules) through rigorous deductive reasoning.
Algebra is built upon ten rules: the commutative law of addition; the associative law of addition; the commutative law of multiplication; the associative law of multiplication; the distributive law of multiplication over addition; adding a number to both sides of an equation does not change the equation; multiplying both sides of an equation by a non-zero number does not change the equation; when multiplying powers with the same base, the base remains the same and the exponents are added; the power of an exponent is equal to the product of exponents with the base unchanged; the power of a product is equal to the product of powers.
Geometry (referring to plane geometry) is based on ten axioms: things that are equal to each other are equal to one another; when equal quantities are added together, the total is equal; when equal quantities are subtracted from each other, the remainder is equal; things that coincide are equal; the whole is greater than the part; two points define a line; it is possible to extend a finite line continuously along it; a circle can be drawn with a point as its center and a specified length as its radius; all right angles are equal to one another; through a point outside a line, one and only one parallel line can be drawn.
The validity of these fundamental rules or axioms is obvious or undeniable; they form the basis or fundamental premises of algebra and geometry, and are the foundation of the reliability of mathematics as a whole. Based on this, a series of mathematical conclusions at different levels are obtained using the following deductive reasoning methods. Deductive reasoning is the guarantee of the reliability of mathematical theories.
The general structure of deductive reasoning is the following syllogism:
| General Structure of Deductive Reasoning | Example | |
|---|---|---|
| Major Premise | A General and Universal Law | Humans are mortal |
| Minor premise | Judgment of a specific object | John is a person |
| Conclusion | Conclusion regarding this specific object | John is going to die |
According to syllogism, a complete reasoning process is as follows: Since anything that satisfies condition A has property C (major premise), and thing B satisfies condition A (minor premise), therefore thing B has property C (conclusion).
The major and minor premises here refer to the existing true judgments used in the reasoning process, which must be guaranteed or assumed to be true. Under these premises, the conclusions drawn are undoubtedly...
It is correct and absolutely reliable.
Mathematics employs various methods of reasoning, but the establishment of mathematical conclusions relies solely on deductive reasoning. Conclusions derived from simple induction, analogy, exemplification, experimentation, simulation, or conjecture can only be used to explain or support conclusions, or to provide useful insights, but cannot serve as the basis for establishing mathematical conclusions.