- #31
andytoh
- 357
- 3
complexPHILOSOPHY said:
say what?
How to FULLY understand a definition?
- Context: Graduate
- Thread starter andytoh
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- Definition
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Discussion Overview
The discussion revolves around the challenges of fully understanding mathematical definitions, particularly in the context of concepts like tensors and the fundamental theorem of calculus (FTC). Participants explore the nature of understanding in mathematics, questioning whether memorization and problem-solving equate to true comprehension.
Discussion Character
- Exploratory
- Debate/contested
- Conceptual clarification
Main Points Raised
- Some participants express that despite years of study and problem-solving, they still feel they do not fully grasp certain mathematical concepts, such as tensors and the FTC.
- One participant suggests that true understanding might come from the ability to explain a concept to someone else, though they humorously note personal limitations in this regard.
- Another participant reflects on the disconnect between proving mathematical statements and genuinely understanding their implications, citing specific examples like the equivalence of the axiom of choice and related theorems.
- A quote from Von Neumann is shared, emphasizing that in mathematics, understanding often comes from familiarity rather than deep comprehension.
- Some participants discuss the FTC, with differing views on its abstraction and the intuitive grasp of its implications regarding area and derivatives.
- One participant mentions the importance of historical context in understanding mathematical definitions and concepts, suggesting that time and familiarity can aid comprehension.
- There is a discussion about the rigorousness of mathematical definitions and whether assumptions are often overlooked in teaching and learning.
Areas of Agreement / Disagreement
Participants generally agree on the difficulty of achieving full understanding of mathematical definitions, but multiple competing views remain regarding what constitutes true understanding and how it can be achieved. The discussion remains unresolved, with various perspectives on the nature of comprehension in mathematics.
Contextual Notes
Participants express uncertainty about the completeness of their understanding and highlight the potential for missing assumptions or deeper insights that are not explicitly stated in definitions or proofs.
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