What does it mean to "do mathematics"?

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Homework Statement:: What does the following quote by Mathematician Paul R. Halmos mean?

"The only way to learn mathematics is to do mathematics."
Relevant Equations:: What dose he mean by "doing mathematics"?

Is it solving puzzles or studying theorams or reading about mathematics?

Thanks
 

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Homework Statement:: What does the following quote by Mathematician Paul R. Halmos mean?

"The only way to learn mathematics is to do mathematics."
Relevant Equations:: What dose he mean by "doing mathematics"?

Is it solving puzzles or studying theorams or reading about mathematics?

Thanks
It's not reading about mathematics -- he means solving equations or puzzles. I supposed studying theorems would qualify provided that you can use the theorems in subsequent problem solving.
 
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It means actively reading a textbook, filling in all the gaps in proofs that the author left out, making up your own questions.

Examples:

Is this statement true? Can I ommit this hypothesis from the theorem? Can I maybe weaken the hypothesis? Can I find counterexamples for certain claims?

And maybe most importantly, work many examples and do many exercises.

Another quote is that someone's expertise is determined by how many examples that person worked out.
 
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PeroK
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Homework Statement:: What does the following quote by Mathematician Paul R. Halmos mean?

"The only way to learn mathematics is to do mathematics."
Relevant Equations:: What dose he mean by "doing mathematics"?

Is it solving puzzles or studying theorams or reading about mathematics?

Thanks

To quote or misquote Louis Armstrong: if you have to ask, you'll never know.
 
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bigfooted
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The only way to learn soccer is to play soccer.
The only way to learn french is to speak french.
 
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phinds
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What dose he mean by "doing mathematics"?
Have you ever tried to learn how to ride a bicycle by reading about riding bicycles?
 
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etotheipi
To quote or misquote Louis Armstrong: if you have to ask, you'll never know.

...I thought this quote was from Harry Potter, stared at it for a solid minute going "I'm pretty sure Louis Armstrong wasn't in those movies...?"
 
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There's a Chinese proverb that covers this well:

I hear and I forget, I see and I remember, I do and I understand.

Mathematicians are self-motivated to do and to understand whereas as students hear and forget then see and remember but seldom do and understand.

Sometimes mathematicians work in isolation but more often they work in collaboration investigating a problem and extending mathematical knowledge into new areas of research.

A fun quote from the 1900's was that there were three great English mathematicians, GH Hardy, JE Littlewood, and Hardy-Littlewood and they, in turn, collaborated with Srinivasa Ramanujan.

https://en.wikipedia.org/wiki/G._H._Hardy

https://en.wikipedia.org/wiki/John_Edensor_Littlewood

https://en.wikipedia.org/wiki/Srinivasa_Ramanujan

In contrast, Ramanujan, for the most part, worked in total isolation using textbooks that were considered second rate. It was only after his papers got to Hardy did he have a chance to collaborate with the Hardy-Littlewood team at Oxford.

Hardy wrote a book called The Mathematicians Apology which is a great read for aspiring mathematicians as is Bell's Men of Mathematics book on the lives of mathematicians which sadly doesn't cover Ramanujan.

https://plus.maths.org/content/node/6725

https://en.wikipedia.org/wiki/Men_of_Mathematics

One last book would be The Man Who Knew Infinity about Ramanujan and his life before and after being a part of the Cambridge mathematical world.

https://en.wikipedia.org/wiki/The_Man_Who_Knew_Infinity
 
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Have you ever tried to learn how to ride a bicycle by reading about riding bicycles?
1. I could be wrong here, but I think I learned more about mathematics by reading books than I learned in school. In school I learned how to use formulae like a robot to get the answer and hated it.
I stared enjoying mathematics and solving problems only when I came across some books that showed me how awesome this field is. Here are some of them.

The Art of Infinite by Robert Kaplan
Why do buses come in threes
Fermat's Last Theorem

2. Also found this quote by a professor explaining"doing mathematics".
“Doing mathematics” means a lot more than writing a solution to a math problem - it means thinking deeply about math, struggling with math, communicating about math, practicing math skills, and trying to figure out new mathematical ideas."
 
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phinds
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In school I learned how to use formulae like a robot to get the answer and hated it.
As I think you now know, that is NOT a description of "learning math" it's just a description of rote memory learning and how to pass tests.
I stared enjoying mathematics and solving problems only when I came across some books that showed me how awesome this field is.
Now your cookin'
 
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