The FAQ on proofs should emphasize definitions

[Stephen Tashi]

I think the FAQ on proofs would be improved if it emphasized the use of defintions. It says that theorems and axioms are used in proofs, but many many textbook type proofs hinge on "parsing" definitions correctly.

As alluded to in the FAQs related to "is .999.. = 1?", many difficulties that people have with proofs arise because they substitute their own mangled definitions of what things are in place of the actual definitions. For example, I notice that several forum members express a "Platonic" view of mathematical objects. They believe these objects exist independently of the definitions that mathematics makes for them. That may be fine as a general philosophy of life, but it is ineffective as an approach to writing mathematical proofs.

(I suppose this post falls under Science Education, but that section doesn't show a link to the math FAQs, so I posted here.)

 

[micromass]

Thank you for your wonderful comments, Stephen! You are certainly correct in saying that.
Could you perhaps post a possible improvement to the FAQ? That way we can integrate your comments.

 

[Stephen Tashi]

OK, I promise to post something in this thread, but it might take a few weeks. I'm a very busy man - retired, you know. It eats up all your time.

 

[Mark44]

I added that proofs use definitions and included lemmas as well. I also cleaned up the wording in two or three other places.

 

[CellCoree]

Thank you for your suggestion to emphasize definitions in the FAQ on proofs. I completely agree that having a clear understanding of definitions is crucial in writing and understanding mathematical proofs. In fact, the use of precise and accurate definitions is one of the fundamental principles of the scientific method.

In mathematics, definitions serve as the building blocks for theorems and axioms, which are then used in proofs to establish mathematical truths. As you have mentioned, incorrect or vague definitions can lead to confusion and errors in proofs, which is why it is important to emphasize their role in the FAQ.

Furthermore, I also agree that having a "Platonic" view of mathematical objects can hinder one's ability to write effective proofs. Mathematics is a human construct, and as such, its concepts and objects are defined and understood within the framework of the language and rules of mathematics. Any deviations from these definitions can lead to misunderstandings and incorrect conclusions.

In summary, I believe your suggestion to emphasize definitions in the FAQ on proofs is a valuable one. It is important for both scientists and non-scientists to understand the role and importance of definitions in mathematics, and I believe that emphasizing this in the FAQ will greatly benefit those seeking to improve their understanding of proofs. Thank you again for your input.