If philosophy studies the movement and development of objects and phenomena, then mathematics examines the objects and their immutable properties. This indicates that mathematics and philosophy are closely interconnected.
“Matter refers to the objective reality presented to humans through sensations, which our senses record, capture, reflect, and exist independently of sensation.” All mathematical objects possess such characteristics. The world of mathematics resembles a miniature material world where mathematical objects mirror material entities, while properties in mathematics resemble phenomena. If philosophy investigates the movement and evolution of objects and phenomena, mathematics explores the objects and their immutable properties. This highlights the close relationship between mathematics and philosophy, as follows:
1) Mathematics as a Material World
According to materialism, matter precedes consciousness, and matter determines consciousness. Similarly, in mathematics, all mathematical objects exist prior to and objectively, independent of human sensations. All mathematical objects predate those who discover them. For instance, functions-graphs, number sets, equations, cubes… all have existed in practice. Indeed, we have:
+ Functions – graphs: all relationships in practice correspond to a “function” (in a narrow sense, a “function”). For example, each house has an address, each person has an identification number, and each internet connection has an IP address… The fluctuations in gold prices, temperature changes, weather, etc., represent graphs.
+ Number sets: a classroom with 40 students, a box of pens containing 12 pens… the numbers 40 and 12 exist objectively even if humans do not discover them; they simply have not yet been labeled as “40-12.” Thus, before humans found those numbers, they still existed objectively… Humans explore, or more accurately, rediscover.
+ Equations: they are inherently available in practice, arising from situations or problems that require identifying certain objects…
+ Cubes: in practice, cubes exist whether humans discover them or not, and they will always be cubes.
Humans abstract these objects from practical research… The only difference is that initially, these objects were not named “functions – graphs,” “number sets,” “equations,” “cubes”… All these objects, as philosophy states, “exist objectively, independent of human consciousness; no one creates them, and no one can destroy them.”
In philosophy, dialectical methodology examines objects and phenomena in their interconnections, within their continuous movement and development. All mathematical proofs are dialectical methodologies. When proving, mathematicians rely on the interconnections between these objects (here, mathematical objects) and their continuous movement. For example, when proving an inequality, the numbers a, b, c in that proof either belong to R together or are all positive… This interconnection may also involve conditions accompanying the inequality. Regarding the proof of the properties of cubic equations, it involves a movement (development) toward a new number set, namely the complex number set.
All objects in mathematics possess dialectical relationships. For example:
+ The operation “1+1=2”: in this addition, the three numbers 1, 1, and 2 have dialectical relationships. More broadly, all formulas in mathematics express dialectical relationships.
+ “Two opposite angles are equal”: the dialectical relationship between two opposite angles. All theorems and properties reflect dialectical relationships within them.
+ Variables and functions
+ Propositions P=>, P Q.
In philosophy, “the material world precedes; dialectics reflects it as the subsequent development. The material world is always in motion and develops according to its objective laws.” Indeed, the mathematical world (comprising all objects and properties of those objects) precedes, while all mathematical proofs follow. Humans are capable of recognizing the laws governing these objects. This recognition comes from the above-mentioned dialectical methodology. Therefore, mathematics and dialectical methodology are inseparable; they must be closely intertwined.
2) The Objective Existence of the Material World
“Human consciousness (through activity) may influence the existence and development of the natural world, but the existence and development of the natural world still adhere to its own laws, and humans cannot decide or change these laws according to their subjective will.” In mathematics, through mathematical activities (exploring objects, proving mathematical properties), the “mathematical world” has developed increasingly; however, mathematics still develops according to overarching objective laws independent of humans, and humans cannot alter those laws. If “two distinct lines parallel to a third line are parallel to each other,” this will remain true indefinitely. It is an undeniable truth, whether discovered or not; it cannot be changed by humans. Even Lobachevsky’s alteration of the axioms of Euclidean geometry to create non-Euclidean geometry naturally aligns with objective laws. Within the new axiomatic system, new laws in non-Euclidean geometry, such as “the sum of the angles in a triangle is not 180°,” also represent inherent laws. Here, we should not assume that non-Euclidean geometry negates Euclidean geometry, as both are built on different axioms. All these laws are not created by some mysterious force; they are natural laws.
“Humans cannot create the natural world, but they can comprehend and transform it.” All mathematical objects and their immutable properties have their own laws. However, humans possess the ability to understand, influence, and uncover these laws early to serve humanity. It remains possible that during the development of mathematics, humans may misinterpret, but sometimes such misunderstandings pave the way for mathematical advancement. These misunderstandings drive humans to seek truth. Understanding mathematics has also enhanced human comprehension of the material world, elevating their worldview and dialectical methodology.
3) The Movement and Development of the Material World
The material world is always in motion and development. This movement and development can manifest as internal changes within mathematical knowledge. For example:
+ Graph translations, trigonometric angles, transformations in geometry, trajectories and point sets, families of curves with parameters, limits of functions, continuity of functions, trigonometric angles…
+ Broadly speaking, movement also manifests in equations and inequalities containing parameters; as the parameters change, so do the equations and inequalities… It is crucial to examine equations and inequalities within a non-rigid state of movement to avoid errors. For instance, if the parameter is m, we must clarify the cases of a=0 and a≠0.
+ Conditional inequalities also reflect movement. Ignoring the conditions can lead to mistakes in proving inequalities.
+ Natural numbers => integers => rational numbers => real numbers => complex numbers.
+ Numbers => addition => multiplication => exponentiation => logarithms.
This movement and development also encompass the overall progression of mathematical knowledge. All mathematical knowledge evolves daily, even hourly. Looking back, humans initially only knew how to solve linear equations, but later they learned to solve quadratic, cubic, quartic equations, and even proved that quintic equations have no general solution. Not only does mathematical theory evolve, but so do problem-solving tools. Through the following examples:
+ While geometry initially relied solely on synthetic methods, new, more effective tools for solving problems have emerged, such as vector methods and analytical methods…
+ Graphing has evolved from using algebraic tools (substituting points) to analytical tools (using variation tables).
+ For word problems, using conventional arithmetic operations can be inconvenient and less efficient than solving with equations. For example, the “chickens and dogs” problem…
+ Examining signs from binomials to trinomials.
All of this shows that the new emerges to replace the old, and progress comes to replace the outdated. However, this replacement does not completely deny the old but is based on the inheritance of previous knowledge. For example, some special cubic and quartic equations can be solved by transforming them into quadratic equations; in a geometry problem, we sometimes need to combine various methods, such as vector methods and analytical methods. All this development is inevitable in mathematics, and because of this inevitability, when considering mathematical knowledge, we must support the new and avoid a conservative attitude. Specifically, when examining the signs of a quadratic trinomial, we must apply the sign examination of the quadratic trinomial to solve the problem rather than separating it into the product of two linear binomials. Sometimes, we think that examining the sign of a binomial is easier, and since we are accustomed to doing it that way, we resist innovating through the method of examining the sign of a trinomial. This is indeed a conservative mindset, a bias against the new and progress.
All this development and movement are closely linked to the evolution of mathematicians’ thought processes. The continuous development of mathematics has led to advancements in applying mathematics to other scientific disciplines and to real-life situations. As mathematics continues to advance, its applicability in practice becomes increasingly significant.
4) The Origin of Movement and Development of Objects and Phenomena
Contradiction is a unity in which two opposing sides both unify and struggle against each other. In mathematics, these opposing sides include negative and positive numbers (within the set of real numbers), even and odd numbers (within the set of natural numbers), increasing and decreasing functions (within the set of functions), propositions and the negation of those propositions (within the set of propositions), sets and their complements, equality and inequality, exact numbers and approximate numbers, axes Ox and Oy, circumcircle and incircle, etc. These opposing aspects are closely interconnected and serve as the foundation for each other’s existence. Philosophy refers to this as the unity of opposites. Indeed, positive real numbers and negative real numbers do not exist separately; without positive real numbers, negative real numbers cannot exist, and vice versa, the set of real numbers cannot exist without both.
5) The Methods of Movement and Development of Objects and Phenomena
A change in quality leads to a change in quantity; the newly formed quality encompasses a corresponding new quantity.
+ Consider the following sum S=a+b
+ The rule of proportions
+ Functions
+ Examine the sign of the expression f(x)=6x+7: as x gradually approaches a limit point, the sign of the expression also changes
+ Consider a polynomial equation. If it is a quadratic equation, its properties regarding roots can be: no roots, a repeated root, or two distinct roots; if it is a cubic equation, its properties can be: one root, two roots, or three distinct roots.
Some questions for further exploration:
1) Provide three examples of movement in mathematics. Philosophy suggests that when examining objects and phenomena, they must be considered in their continuous state of movement. How can this be applied to solving mathematical problems?
2) Cognition consists of two types: sensory cognition and rational cognition. The ancient philosopher Plato criticized sensory cognition in the pursuit of knowledge about the world. They argued that true knowledge cannot be attained through sensory cognition alone; only through rational cognition (specifically human thought) can true knowledge be discovered. What are your thoughts on this issue?
3) In Marxist philosophy, it is stated that “the process of scientific cognition is from vivid intuition to abstract thought and then back to reality.” How is this understood?
4) The transformation of quantity leads to a transformation of quality as a gradual process. How can this be applied to teaching and learning mathematics?
5) In philosophy, “the transformation of quantity leads to a transformation of quality, and simultaneously, the new quality will encompass a new corresponding quantity.” Based on the chapter on “inequalities” that has been studied, clarify this statement.
6) The French mathematician Descartes, the father of the coordinate system, once said a famous quote: “I think, therefore I am.” Does his statement reflect a materialist or idealist philosophical viewpoint? Why?
7) Clarify the relationship between quantity and quality in the following objects:
a) The quadratic equation ax2+bx+c=0 (a≠0) and Delta=b2-4ac
b) In mathematics, it is said that “a circle is a regular polygon with an infinite number of sides.”
c) In the coordinate system Oxy, consider the two points M1(x1,1), M(x2,2)
d) The chapter on inequalities.
Some suggestions:
4) To excel in mathematics and achieve high scores, one must be diligent and patient, avoiding haste and impatience. Start with the smallest tasks (to allow gradual changes in quantity), work carefully, read the problem thoroughly, consider the conditions, and discuss all cases. If necessary, do rough work before finalizing. Remember, “small contributions accumulate into great results,” and “many winds can become a storm.”
5) Through many inequalities and using equivalent transformation methods, the Cauchy inequality for three numbers has been generalized, and from there, it has become an essential tool for proving numerous inequality problems.
7d) Initially, proving inequalities is done through equivalent methods. From some proofs using equivalent transformations (quantitative changes), the Cauchy inequality has been generalized. At this point, the Cauchy inequality is used to prove (qualitative changes). With this method, a significant number of inequalities have been proven (encompassing the corresponding quantities).
