Can abstract math be understood through concrete examples?

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    Abstract
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Discussion Overview

The discussion revolves around the understanding of abstract mathematics, particularly abstract algebra, through concrete examples. Participants explore various approaches to grasping abstract concepts, including the importance of definitions and the role of practice.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that working with concrete examples is essential for understanding abstract algebra, proposing that one should think concretely before generalizing.
  • Others inquire about the specific areas of abstract algebra being discussed, such as Group Theory and Representation Theory.
  • A participant quotes Von Neumann, indicating that understanding may come from familiarity rather than deep comprehension, and questions how to achieve this familiarity.
  • One participant emphasizes the importance of learning definitions precisely, noting that definitions serve as "working" tools in mathematics.
  • Another participant mentions that there are two levels in pure mathematics: understanding definitions and applying them, suggesting that many struggle at the first level with abstract concepts.
  • Some argue that adopting a suitable mindset for abstract mathematics may be more beneficial than relying solely on concrete examples.
  • A participant humorously suggests that becoming a mathematician of high standing, like Terry Tao, is an ideal way to understand abstract mathematics, while also noting Tao's focus on more concrete areas of math.
  • Another participant clarifies that they meant having natural abilities akin to those of exceptional mathematicians, implying that practice is crucial for understanding abstract concepts.

Areas of Agreement / Disagreement

Participants express a variety of views on how best to understand abstract mathematics, with no clear consensus on the most effective approach. Some advocate for concrete examples, while others emphasize the importance of definitions and mindset.

Contextual Notes

Participants acknowledge the complexity of transitioning from definitions to practical application in abstract mathematics, highlighting the challenges faced by those who find abstract concepts difficult.

pivoxa15
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How to go about it? I had abstract algebra in mind.

Is the main thing to do as many solid examples as possible?

So the only way to understand the abstract it is to think concrete then generalise?
 
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what kind of abstract algebra? does that mean Group Theory and Representation theory?
 
As Von Neumann once said, you don't understand it, you just get used to it :).

--------
Assaf
http://www.physicallyincorrect.com/"
 
Last edited by a moderator:
ozymandias said:
As Von Neumann once said, you don't understand it, you just get used to it :).

--------
Assaf
http://www.physicallyincorrect.com/"

That's a very nice quote, possibly my favourite in mathematics as it is not only pleasant but useful.

So how do you get used to abstract maths? By constructing as many concrete examples as possible?
 
Last edited by a moderator:
mjsd said:
what kind of abstract algebra? does that mean Group Theory and Representation theory?

I had abstract maths in general on my mind but I am mostly interested in abstract algebra mainly rings and groups at the moment.
 
The first thing to do is to learn the definitions exactly. In mathematics, to a larger extent than other studies, definitions are "working" definitions- you use the precise words of defintions in proofs or problems.
 
In pure maths I have always known that there are two levels. One is the definition, the second is doing the maths. For the mathematically challenged, they get stuck on the first level especially when it gets more abstract.

What are the steps after the definitions are known well?
 
pivoxa15 said:
In pure maths I have always known that there are two levels. One is the definition, the second is doing the maths. For the mathematically challenged, they get stuck on the first level especially when it gets more abstract.

What are the steps after the definitions are known well?

You said it. Do some math.
 
I defer somewhat, since the best way to understand what is abstract is to adopt a frame of mind suited to it instead of trying to bind it to grounded examples.

Pure mathematics flows my friend in the eternal closure of definition, proposition and proof.
 
  • #10
The easiest way to understand difficult and abstract mathematics is to become Terry Tao or someone of a similar standing. If that is not possible, practice makes perfect.
 
  • #11
Gib Z said:
The easiest way to understand difficult and abstract mathematics is to become Terry Tao or someone of a similar standing. If that is not possible, practice makes perfect.

I thought Terry is on the concrete of abstract maths if that makes sense. i.e His speciality is in analysis and combinatorics which may be more concrete then some of the other branches in algebra or topology.
 
  • #12
I meant it more generally, ie Have natural abilities like him. I'm sure he wouldn't have too much trouble grasping most mathematics given some time, even if he hasn't studied it before.
 

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