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How is Math Abstract?

Published in Mathematical Abstraction 3 mins read

Math is abstract because it focuses on universal concepts, structures, and patterns that are removed from specific physical objects or real-world contexts.

Understanding Mathematical Abstraction

Abstraction in mathematics is fundamentally the process by which mathematical concepts are formed and developed. As the definition states, it involves:

"...extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract..."

This means that instead of studying three specific apples, math studies the concept of 'three' itself. This concept then applies equally to three oranges, three ideas, or three dimensions, without being tied to any single physical manifestation.

Key Characteristics of Abstract Math

The abstract nature of mathematics stems from several key characteristics:

  1. Extraction of Core Ideas: Math identifies the fundamental properties or relationships in various situations. For example, counting applies to many different collections of things.
  2. Detachment from Reality: Mathematical objects (like numbers, sets, functions, geometric shapes) exist as concepts independent of physical existence. A perfect circle doesn't exist physically; it's an abstract mathematical idea defined by properties.
  3. Generalization: Concepts are developed to be as broad as possible, applying to a wide range of cases or even entirely theoretical scenarios. This allows for powerful theorems and principles that hold universally within the defined mathematical system.
  4. Focus on Structure and Relations: Abstract math often studies the relationships and structures between these abstract objects (e.g., how numbers combine, how sets are related, how geometric shapes transform), rather than the objects themselves as physical entities.

Examples Illustrating Abstraction

Let's look at simple examples:

  • Numbers: The number 5 isn't tied to five fingers, five stones, or five anything specific. It's an abstract quantity representing 'fiveness', which can be applied to count any group of five items. This is a step of removing dependence on real-world objects.
  • Geometry: A geometric concept like a triangle is defined by its properties (three sides, three angles summing to 180 degrees on a flat plane). It doesn't matter if the triangle is drawn on paper, formed by stars in the sky, or exists only as points in a coordinate system; the abstract rules and properties apply. This shows generalization and focus on structure.
  • Algebra: Variables like x or y represent unknown or general quantities. The equation x + 2 = 5 deals with an abstract unknown number, allowing us to solve for it regardless of what physical quantity x might represent (dollars, meters, etc.). This demonstrates wider application through abstraction.

Why Abstraction is Powerful

The abstract nature of mathematics is precisely what makes it a powerful and universal tool. By dealing with abstract concepts and structures, math can:

  • Describe patterns found in diverse fields (science, engineering, economics, art).
  • Develop general theories that apply broadly.
  • Explore purely theoretical possibilities that may or may not have current real-world counterparts.
  • Serve as a universal language for describing complex systems.

In essence, math is abstract because it distills the complexity of reality into fundamental, generalized, and detached concepts and rules, enabling deeper understanding and wider application.