A theory is constructed by observing real-world objects and organizing their empirical results, which
is typically based on empiricist methodology, studies called science. Mathematics, on the other hand,
is a priori study that independently establishes a system from the internal structure and logical
thinking of human reason without relying on observations from the outside world.
The truth that the sum of the three angles is 180 degrees is unshakable, even though the lines and
shapes can be distorted when drawing a triangle by hand. Only logical consistency and necessity
justify mathematics, not sensational errors or empirical imperfections. Because of this nature,
mathematics has a unique academic position to explore the essential structure of perception that
makes up the world beyond mere instrumental discipline. From the Pythagorean school to Descartes'
rationalism, this transcendentality and logical rigor had a profound impact on Western philosophy
as a whole and continues to serve as the foundation for the construction of all scientific theories.
Mathematics made it possible to formulate all objects that humans experience and think in a
systematic and logical language, and provided a methodological foundation for precisely analyzing
and predicting the complexity and microscopic changes of natural and social phenomena. Descartes'
coordinate plane opened a new paradigm for algebraic representation of geometric space, and
Newton's established calculus provided a mathematical language essential for describing continuous
motion and change, enabling the scientific revolution and the rapid development of modern science.
Mathematics serves as an essential theoretical foundation for analyzing, modeling, and predicting
natural phenomena and artificial systems in physics, chemistry, biology, economics, engineering,
and medicine in modern times. From Fourier transform used in signal processing, probability theory
essential in data analysis, and optimization theory explaining the learning structure of artificial
intelligence, mathematics is always in the depths of modern civilization.
Furthermore, mathematics functions as an cognitive device that reconstructs objects structurally and
discovers concealed lawfulness, not just as a means of measurement and recording. Statistical
reasoning leads to the inference of order amid uncertainty and errors, and numerical analysis
transforms infinite complexity into a computable form. Mathematics has become a 'language' and
a 'thinking system' that interprets and reconstructs the world beyond simple 'tools'.
In secondary education, it is remarkable that mathematics has a learning structure that is essentially
different from other subjects. After high school, mathematics can never go beyond subsequent
stages if it lacks basic arithmetic skills, while other subjects can partially overcome their deficiencies
in elementary and middle schools. Mathematics is a subject that can only progress by conquering
all of the long chains from elementary arithmetic to sets, functions, differential integrals, algebraic
structure, phase space, and hermeneutics. Even a single basic deficit can destroy the entire
subsequent learning.
However, when mathematics and undergraduate students enter their major mathematics in earnest,
they experience that the hierarchical nature of mathematics learning is partially dismantled. With
the exception of core theories such as set theory, hermeneutics, and linear algebra, many topics can
be directly explored without prior learning, and research is conducted by expanding from specificity
to generality rather than from 'general to individual'. Mathematics has its own flow of thinking that
analyzes intuitive cases, builds abstract structures through them, and sublimates them back to proof
and axiom systems.
This structure is most clearly revealed in categorical thinking. Mathematics no longer looks at objects
in isolation, but views the entire structure through morphism and their interactions between objects.
Topology freely crosses between locality and globality, hermeneutics strictly deals with infinitesimal
structures, and algebraic geometry provides a language that integrates algebraic expressions and
geometric objects. The Khomotopy type theory is elevating the foundation of mathematics to a
new level, and skim theory is moving beyond the limits of classical algebraic geometry.
Mathematics is not just a tool for modeling the real world. It is a language that explores all possible
structures and draws even the non-existent world into the realm of logical possibilities. Mathematics
extends the horizon of possibilities beyond existence when set theory controls infinity, topology
organizes continuity, hermeneutics deals with infinity, and category theory newly views the nature
of structure.