Abstract Math Explained: How to Use Abstract Mathematics

Abstract math, a subgroup of mathematics, encompasses the constructions of math problems, formats, and equations to further deepen math concepts for practical use in real-world scenarios.

Abstract math is a higher-level field of study that can include abstract algebra, geometry, calculus, and other abstract concepts. Abstract math is useful because it connects different parts of math. For example, results in a given area can lead to conjectures in another, and mathematicians might generalize and apply patterns from one mathematical object to other fields.

Abstract math is an ever-evolving field of study in which mathematicians refine and create new concepts within math. Consider the development of these different areas of abstract math:

• Euclidean math: Attributed to the ancient Greek mathematician Euclid, Euclidean math is perhaps the earliest documentation of abstract math, which noted during ancient times the axioms of plane geometry.
• Cartesian coordinates: In the seventeenth century, René Descartes introduced Cartesian coordinates, progressing the use of complex numbers and analytic geometry.
• Geometry: European mathematicians such as Carl Friedrich Gauss, János Bolyai, and Bernhard Riemann advanced the field in the 1700s and 1800s. In the nineteenth century, experts generalized geometry even further through the application of n dimensions, projective geometry, and affine geometry.

In mathematical studies and high school math classes, students learn about matrices, linear algebra, topology, polynomials, and other mathematical objects. This helps them understand geometric and algebraic structures to solve theoretical or real-world problems. Abstract math problems might include finding the area of a plot of land during a land survey, solving for the mean price of goods to match consumer spending habits, or visualizing symmetric lines for roadways as a part of urban planning.

Australian mathematician Terence Tao says, “There is a perception that mathematics is some sort of sorcery.” Abstract math, however, is a highly creative tool that can unlock new modes of thought. “It's a way of thinking,” he says. In collaboration with mathematician Ben Green, Terence used creative thinking to solve a problem related to the Twin Prime Conjecture. The pair proved it’s possible to always find a progression of prime numbers of any length and of equal spacing in the infinity of integers.

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