• Emphasizes the Role of Axioms: The definition correctly highlights the fundamental role of axioms in mathematics. Every mathematical system, from basic arithmetic to complex topology, starts with a set of unproven assumptions (axioms) that serve as the foundation for all subsequent deductions.
  
• Highlights the Compartmentalization: The idea of “closed knowledge compartments” captures the fact that different areas of mathematics often operate within their own specific sets of axioms and focus on particular types of objects and relationships. For example, Euclidean geometry operates under different axioms than non-Euclidean geometries, leading to distinct sets of theorems and concepts.
  
• Focuses on Internal Consistency: The definition implicitly suggests that within each “compartment,” knowledge is derived logically and consistently from the axioms. This internal consistency is a hallmark of mathematical rigor.
  

• “Closed” Might Be Too Restrictive: While different areas of mathematics can be seen as distinct, the boundaries are not always rigid or “closed.” There are often deep and surprising connections between seemingly disparate fields. For example, the development of analytic geometry by Descartes bridged algebra and geometry, leading to powerful new tools and insights. Modern mathematics often thrives on interdisciplinary approaches.
  
• Doesn’t Fully Capture the Dynamic Nature of Mathematics: Mathematics is not a static collection of “closed compartments.” It is a constantly evolving field with new areas of study emerging, existing areas being refined, and new connections being discovered. This definition might underemphasize this dynamic and creative aspect.
  
• Neglects the Role of Discovery and Intuition: While axioms provide the foundation, the process of developing mathematics also involves intuition, exploration, and the discovery of patterns and relationships that might not be immediately obvious from the axioms themselves. The definition focuses more on the deductive structure than the inductive and creative aspects.
  
• Doesn’t Explicitly Mention Logic: While implied, the definition doesn’t explicitly state the crucial role of logic in moving from axioms to theorems within each “compartment.” Logic is the engine that drives mathematical reasoning.
  
• May Not Encompass Applied Mathematics: The definition primarily focuses on pure mathematics, where the emphasis is on internal consistency and logical deduction from axioms. Applied mathematics, while relying on mathematical principles, is also driven by the need to model and solve problems in the real world. The “closed compartment” idea might not fully capture this external orientation.
  
• Could Imply Isolation: The term “delimited” might suggest an isolation between different areas of mathematics that doesn’t always exist. Mathematical concepts and techniques often cross boundaries and enrich different fields.
  

Your definition offers a valuable perspective by emphasizing the axiomatic foundation and the existence of distinct areas within mathematics. However, the term “closed” might be too strong and could misrepresent the interconnected and evolving nature of the field.

A more comprehensive definition might acknowledge the following aspects:

• Axiomatic Foundation: The starting point of mathematical systems.
  
• Logical Deduction: The method of deriving new knowledge from axioms.
  
• Interconnectedness: The often surprising and fruitful relationships between different areas.
  
• Dynamic Evolution: The continuous development of new concepts, theories, and fields.
  
• Problem-Solving and Modeling: The application of mathematical principles to understand and solve problems in the real world.
  
• Abstract Structures and Patterns: The study of abstract structures and the identification of underlying patterns.
  

In conclusion, while your definition captures some important aspects of mathematics, it might be too narrow to fully encompass the breadth, depth, and dynamism of the field. It’s a good starting point for discussion, though!