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After reading some papers about special kinds of algebras and rings like Gorenstein rings, Dickson algebras, Cayley-Dickson construction, i want to ask do examples of general abstract objects in math like groups, rings, manifolds, abstract curvatures, modular forms that help us understand helps us know where the directions of math go to?

For example, the n-manifold in n dimentions is defined and as special cases we have curves and surfaces, another example is the groups , where we have the set of real numbers equipped with addition and gives us the abelian group $(R,+)$, or with the complex numbers. Then we learn other examples of groups like the group of matrices equipped with addition or multiplication where the first is abelian group and the second nonabelian respectively.

Questions: Does this way help us understand generally things in math?By considering special cases? Do you know cases where this perhaps is not applied? If it is not applied could it be applied by analogy?If i try to read math by considering special cases of objects to help me is this valid and okay?

Thank you.

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    $\begingroup$ I’m voting to close this question because I think it would be more appropriate for you to find a local trusted advisor to ask these questions than asking them in this website. $\endgroup$
    – Steven Gubkin
    Commented Jan 20, 2022 at 12:16
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    $\begingroup$ Pretty much all abstract definitions come from extracting out important and useful features of (more) concrete examples that were already of interest. So understanding these can be helpful. "Special" cases, maybe not so much. e.g. Is the empty set a manifold? You might occasionally have technical reason to care, but usually this is a pointless (pun-intended) question. $\endgroup$
    – Adam
    Commented Jan 20, 2022 at 13:31

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I think that constructing examples which satisfy and do not satisfy the definitions you are learning is essential. In fact, I include this as an essential learning outcome tied to every definition in the courses I teach.

You seem to be worried about whether you are learning math "the right way" (or perhaps "the most efficient way").

There is no royal road to geometry. Everyone must clear their own path. I remember an interview with Terry Tao (iirc) where he describes wriggling in bed, thinking of his body as a 2 dimensional wave. This helped him achieve a crucial insight in a problem he was working on which concerned Fourier analysis.

This example just highlights that people are going to engage in all kinds of interesting physical, mental, and social behaviors when they are learning anything, or solving any problem. As long as you are not hurting anyone, you are okay!

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    $\begingroup$ There is also the Paul Halmos quote about "a good stock of examples": math.stackexchange.com/q/3502648/118539 $\endgroup$
    – J W
    Commented Jan 20, 2022 at 12:15
  • $\begingroup$ Can these examples lead to better understanding of theorems also that contain those objects that have these examples? Perhaps of proofs of theorems as well? Thank you. $\endgroup$
    – plants
    Commented Jan 20, 2022 at 13:15
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    $\begingroup$ @plants Yes. Most proofs of most interesting theorems will contain lots of explicit calculations with explicit examples. Even if not, having robust and interesting collections of examples are essential for developing intuition. $\endgroup$
    – Steven Gubkin
    Commented Jan 20, 2022 at 13:33
  • $\begingroup$ Steven, can these collections of examples help in formulating conjectures and solving problems? If so,how? Can you provide examples? $\endgroup$
    – plants
    Commented Jan 20, 2022 at 13:50
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    $\begingroup$ @plants Yes, they can. I am not going to continue engaging in this conversation though. I think the answers to all of your questions must come from experience: just keep doing math! $\endgroup$
    – Steven Gubkin
    Commented Jan 20, 2022 at 13:59

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