Giving definitions, proposing theorems in abstract spaces

[trees and plants]

Hello. Questions: how to give new definitions of things in abstract spaces?What are the criteria? Is it just acceptable to define things that do not contradict with other things on the abstract space?What are the motivations to give definitions of things?Also, what are the motivations for proposing theorems and then proving them in abstract spaces? For a person to propose theorems is it ok to propose anything on abstract spaces as long as it does not contradict other things?

For example, Lie derivatives, immersions, embeddings, hypersurfaces, rigid surfaces, harmonic functions, mean curvature, Gaussian curvature, minimal submanifolds are defined on abstract spaces.

A person told me they should be useful but is this correct?Why should they be useful and for what purposes? Thank you.

 

[FactChecker]

When you talk about abstraction, it is usually because there are many specific examples where you notice the same thing being proven over and over in different contexts. The abstraction consolidates all those into one abstract definition and one proof. So the advice you got that it should be useful is a "done deal". It has already been used repeatedly in several contexts before the abstraction was even made.

 

[fresh_42]

I did not understand the part of your question about theorems. Do you mean naming a theorem, e.g. Schur's lemma, or theorems themselves, e.g. 'All subgroups of Abelian groups are normal.' which do not necessarily have a name?

Definitions as in your examples are abbreviations (immersion), and possibly names to distinguish them from related objects (Lie derivative). Their origin can be historical, grown over time, or immediate in a new paper which has been published.

And like all abbreviations, their main purpose is to shorten text. E.g. if you write a paper about Abelian groups, and you would use 'Let $G$ be a group, such that $ab=ba$ for all $a,b\in G,$ and ...' instead, and each time you talk about an Abelian group, then you would almost naturally invent a name for them; be it Abelian or simply commutative. It simply makes no sense to repeat the definition over and over again. Those definitions may survive and become common knowledge, if others publish papers on Abelian groups, too, or otherwise work with them, or will become forgotten again, if nobody talks about these objects again.

In the end it is the same situation as in common language: we say 'dentist' and not 'doctor, who repairs teeth if necessary'.

 

[FactChecker]

One motivation for abstraction is when there is an important property that you want a space to have. For instance, you might want to consider the closure of something that is not already closed under some operation. That is often important.
A related motivation for abstraction is when there is an important property that you want to NOT require. Then you might consider an abstract space without that property and see what is still provable in that space.

 

[trees and plants]

One motivation for abstraction is when there is an important property that you want a space to have. For instance, you might want to consider the closure of something that is not already closed under some operation. That is often important.
A related motivation for abstraction is when there is an important property that you want to NOT require. Then you might consider an abstract space without that property and see what is still provable in that space.

So, about that property to be considered important, what are the criteria? Are they personal or criteria made from other scientists or the scientific community? Perhaps someone by combining the already known could define things all the time or not?

 

[trees and plants]

I did not understand the part of your question about theorems. Do you mean naming a theorem, e.g. Schur's lemma, or theorems themselves, e.g. 'All subgroups of Abelian groups are normal.' which do not necessarily have a name?

Definitions as in your examples are abbreviations (immersion), and possibly names to distinguish them from related objects (Lie derivative). Their origin can be historical, grown over time, or immediate in a new paper which has been published.

And like all abbreviations, their main purpose is to shorten text. E.g. if you write a paper about Abelian groups, and you would use 'Let $G$ be a group, such that $ab=ba$ for all $a,b\in G,$ and ...' instead, and each time you talk about an Abelian group, then you would almost naturally invent a name for them; be it Abelian or simply commutative. It simply makes no sense to repeat the definition over and over again. Those definitions may survive and become common knowledge, if others publish papers on Abelian groups, too, or otherwise work with them, or will become forgotten again, if nobody talks about these objects again.

In the end it is the same situation as in common language: we say 'dentist' and not 'doctor, who repairs teeth if necessary'.

I meant the statement of the theorem not its proof, like the example you gave 'All subgroups of Abelian groups are normal'.Another question i have is about the definitions are they every time based on things that could be expressed as or represented from mathematical symbols and formulas?

 

[fresh_42]

Another question i have is about the definitions are they every time based on things that could be expressed as or represented from mathematical symbols and formulas?

You can define whatever you want. E.g. define a 'Tree-and-Plants-group' as a group whose elements except $e$ are all of order $6$. Now you can start to prove theorems about Tree-and-Plants-groups. One of the first to do, will be whether such groups exist at all. What about other orders? Are there groups in case we replace $6$ by a prime? Shouldn't we call our groups 'n-Tree-and-Plants-groups' then, with the order 'n' instead of $6$? And wouldn't we achieve more acceptance if we'd call them 'uniform-order-groups (UOG)' instead? Or will such a definition turn out to be useless, since all such groups exist only for primes $n=p$ and are commutative, in which case they are simply the finite fields $\mathbb{F}_p$?

You see that naming an object has very much to do whether it makes sense or not. People named certain classes of rings Noetherian and Artinian because it turned out that they have important properties. One could call such rings as well a.c.c. and d.c.c. rings, but the names honoring the two great mathematicians made the race. There is no "one principle fits all" answer to your question.

I think you should better learn your lectures instead of thinking about such unimportant formal questions. After learning some hundred names for theorems, lemmata, and definitions, you will see (or not) which follow a pattern, and which do not. In any case: it is not of interest, unless you are interested in a certain one and its history.

 

[FactChecker]

So, about that property to be considered important, what are the criteria? Are they personal or criteria made from other scientists or the scientific community?

Definitely not personal. I can only say that if you study abstract math long enough, you will develop an intuition about what is important.

Perhaps someone by combining the already known could define things all the time or not?

Abstraction must be motivated by something based on mathematical value. Experience is the only way to know what has value. (Maybe genius is another way, but I wouldn't know anything about that.)

 

[trees and plants]

I think you should better learn your lectures instead of thinking about such unimportant formal questions. After learning some hundred names for theorems, lemmata, and definitions, you will see (or not) which follow a pattern, and which do not. In any case: it is not of interest, unless you are interested in a certain one and its history.

It is good to try to pass my exams and then have a degree. I get stuck at problem solving during the exams. Perhaps i should better try solving problems or exercises? Which exercises should i prefer? Those associated with the lectures?Sorry for being a little off topic.

I do not know but definitions take my attention very much. I also read and try to learn theorems but this thing with definitions happens. What should i do to fix it?

Since i read about Riemannian manifolds and manifolds generally, i wanted to find other abstract spaces or generalise or try to make analogues of mathematical space.

 

[Mark44]

I think you should better learn your lectures instead of thinking about such unimportant formal questions.

Strongly agree.

It is good to try to pass my exams and then have a degree.

Yes, absolutely.

I get stuck at problem solving during the exams. Perhaps i should better try solving problems or exercises?

Yes again, If you take the advice given by fresh_42, maybe you will do better on your exams.

 

[FactChecker]

The lectures undoubtedly cover the most important facts. They may not include a lot of examples and worked exercises. I always liked the Schaums Outlines series because they include a lot of worked examples and exercises with worked answers.