What is mathematics without numbers?

[Guest]

We could for example consider algebra:

"In the new post-Galois world, algebra was longer just about using symbols for numbers in equations. It was about the deep structure of equations--not numbers but processes, transformations, symmetries. These radical innovations changed the face of mathematics. They made it more abstract, but also more general, and more powerful... -- Ian Stewart, Why Beauty is Truth

[Guest]

"That's a fascinating question! While numbers are a fundamental part of many areas of mathematics, there's a rich landscape of mathematical concepts that exist independently of numerical values. Here are a few key examples:

Topology: This branch of mathematics focuses on the properties of shapes and spaces that remain unchanged under continuous deformations (like stretching or bending, but not tearing or gluing).
Key Concepts: Connectedness, compactness, dimension, and the study of different types of spaces (like manifolds).
Example: Imagine a coffee cup and a donut. Topologically, they are equivalent because you can continuously deform one into the other.

Set Theory: Deals with collections of objects (which could be anything, not just numbers).
Key Concepts: Sets, subsets, operations on sets (union, intersection), cardinality (the size of a set), and different types of infinity.

Graph Theory: Studies networks of interconnected points (vertices) and lines (edges).
Key Concepts: Connectivity, cycles, paths, trees, and applications in areas like social networks, computer science, and chemistry.

Abstract Algebra: Explores abstract structures like groups, rings, and fields, which have their own sets of rules and operations.
Key Concepts: Groups (sets with an operation like addition or multiplication that satisfies certain properties), rings (sets with two operations), and fields (sets where division is defined for all non-zero elements).

These are just a few examples of how mathematics can exist without a direct focus on numbers. These areas explore fundamental concepts of structure, relationships, and patterns, revealing a deeper and more abstract understanding of the world around us."
-- Gemini (AI Chat Bot)

[Guest]

Elements of Mathematical Reasoning:

[Guest]

Elements of Mathematical Reasoning: