What is Mathematics?
Mathematics is the study of universal truths through the analysis of definitions and conditional statements. It is a subject which at its core focuses on an understanding of substance and structure, and can be thought of as the study of absolute certainty.
The Rich History of Mathematics
Mathematics has a rich history, dating back to the dawn of humans (and perhaps before that). In that time, mathematics has gone from its early oral traditions to development in Africa, Egypt, and Babylon, on to the Greeks and traders through the Fall of Alexandria, further developed in China, India, and the Middle East, then finally more rigorously formalized throughout Eurasia, the Americas, and the rest of the world over the last 500 years.
It is a subject which has been strengthened and refined by the contributions of all cultures (from the quipu of the ancient Inca to the etched bones of Sub-Saharan Africa), and in examining this history and the great accomplishments of a united world, we can begin to understand our role in it.
The Mathematics Method
Ultimately, Science and Religion seek to answer the same questions; the great mysteries of the universe. They have distinctly different approaches, though. Where Science uses experimentation and the Scientific Method to learn more about the universe, Religion seeks to explore the words of those who came before us to better understand the unknown. While fundamentally opposite approaches, in mathematics we can find value in both.
The Mathematics Method is perhaps a more logical and refined approach to divining universal truth. Rather than seeking understanding through experimentation or through stories of prior generations, it focuses more on an understanding and exploration of well-defined objects through rigorous proofs. At its core, mathematics is the study of universal truths through the analysis of definitions and conditional statements, and in exploring different objects we can gain an understanding about questions regarding values, morality, and the universe around us.
Is Mathematics Discovered or Invented?
Both.
Mathematics is the underlying nature of the universe; we live in a universe which appears to utilize a Euclidean norm, but fundamentally it is an n-dimensional manifold which is built around systems of differential equations (both ODEs and PDEs), utilizes vector spaces, measures, and the underlying principles of quantity and space. Furthermore, the universe itself is rich in mathematical objects and structures. In this respect, the mathematics is already out there, we are just exploring it.
At the same time, the mathematics tools which we utilize to explore the unknown are not things which the universe inherently knows or cares about. Things like the algebraic toolset and other mathematical objects are created by us; the Quadratic Formula and Pythagorean Theorem may be useful formulas, but they are meaningless objects when you consider the motions of the stars. They are instead philosophically described and derived tools which we can use to better understand the natural world. There is no such thing as a perfect square in the real world, but we can develop the idea of a square to better understand square-like objects we encounter. In this respect, we are developing the tools to explore the universe, and as such creating mathematics independent of nature.
Ultimately, Mathematics describes both the nature of all things and the toolset with which we explore all things, and in doing so it provides us a way to better understand the world around us and our place within it.