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Don't delude yourself with this Protestant hard-work-is-necessary mentality. Better representations are possible and they can save us a lot of work. It's easier

Don't delude yourself with this Protestant hard-work-is-necessary mentality. Better representations are possible and they can save us a lot of work. It's easier to perform multiplication using Hindu-Arab numerals than it is with Roman numerals. It's easier to program with Python than Assembly. Difficulty is a property of the representation, not of some underlying 'abstract thing' that is being represented.

deleted 15y ago
[deleted]
Someone 15y ago
Correction: Difficulty is not necessarily a property of a problem, but can also be caused by a poor choice of representation. I agree that modern technology may help provide better representations, but in the case of most of abstract mathematics, I am not convinced it currently does, or even that it can. There is some truth in the statement "a picture lies more than a thousand words"
scott_s 15y ago
Difficulty is a property of the representation Not necessarily. There are such things as inherent complexity and accidental complexity. forrestthewoods's point was that some things are inherently complex, which I agree with. Accidental complexity happens when we use a poor representation - such as roman numerals.
forrestthewoods 15y ago
That's not my meaning. Higher level math requires abstract thinking. Abstract thinking is hard. If you can learn to think abstractly then it will allow (or help) you solve a large class of abstract problems. If math is broken down such that it does not require abstract thinking and a person does not learn to think abstractly then they will not be able to solve that large class of abstract problems. Or so my line of likely incorrect train of thought goes.
- surrealize 15y ago
  Or maybe interactive visualizations can help people learn to think abstractly, by showing how one level of abstraction relates to another? http://worrydream.com/#!/LadderOfAbstraction http://worrydream.com/#!/LadderOfAbstraction

deleted 15y ago

[deleted]

Someone 15y ago

Correction: Difficulty is not necessarily a property of a problem, but can also be caused by a poor choice of representation. I agree that modern technology may help provide better representations, but in the case of most of abstract mathematics, I am not convinced it currently does, or even that it can. There is some truth in the statement "a picture lies more than a thousand words"

scott_s 15y ago

Difficulty is a property of the representation Not necessarily. There are such things as inherent complexity and accidental complexity. forrestthewoods's point was that some things are inherently complex, which I agree with. Accidental complexity happens when we use a poor representation - such as roman numerals.

forrestthewoods 15y ago

That's not my meaning. Higher level math requires abstract thinking. Abstract thinking is hard. If you can learn to think abstractly then it will allow (or help) you solve a large class of abstract problems. If math is broken down such that it does not require abstract thinking and a person does not learn to think abstractly then they will not be able to solve that large class of abstract problems. Or so my line of likely incorrect train of thought goes.

surrealize 15y ago
Or maybe interactive visualizations can help people learn to think abstractly, by showing how one level of abstraction relates to another? http://worrydream.com/#!/LadderOfAbstraction http://worrydream.com/#!/LadderOfAbstraction