Author Topic: 數學和數學家的故事  (Read 34027 times)

Offline kido

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數學和數學家的故事
« on: 20 October 2009, 17:03:14 »
This is really a good book. This is few number of books that I kept till now.  Really forget who introduced to me, Yip Sir? 孟 Sir? or chin? or wongyan?

I only own book 4 - related to computer, AI, and maths story. (pigeon hole principle in particular) When I look back this in time, this book was a good introduction for CS.
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Offline chin

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Re: 數學和數學家的故事
« Reply #1 on: 20 October 2009, 21:17:04 »
I don't remember this much. BTW the author's name is quite something. You Learn Math?!

Speak of math related books read in high school days... one of the books I remember well was a very short Chinese version of "Gödel, Escher, Bach: An Eternal Golden Braid" or "GEB - 永恆的金帶". It was in A5 format only 1 cm thick. It was my first time seeing Escher's paintings. I did not understand much of the book. I recently found the English version. It's A4 size about 3 cm thick. I read a few page and still do not understand what's it says.

Offline kido

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Re: 數學和數學家的故事 / GEB
« Reply #2 on: 20 October 2009, 22:57:30 »
I don't remember this much. BTW the author's name is quite something. You Learn Math?!

Speak of math related books read in high school days... one of the books I remember well was a very short Chinese version of "Gödel, Escher, Bach: An Eternal Golden Braid" or "GEB - 永恆的金帶". It was in A5 format only 1 cm thick. It was my first time seeing Escher's paintings. I did not understand much of the book. I recently found the English version. It's A4 size about 3 cm thick. I read a few page and still do not understand what's it says.

Yeah, I remember that book, you borrowed the Chinese version to me in Form 3 or 4? ... At that time I didn't understand most of it.

In about 10 years ago, I finally read the original version.  It's all about recursion, no matter in Gödel's maths, Escher's painting, or in Bach's music (in particular the Canon), these recursion are present. Combined with the illustration of AI, these could be applied to what we called 'goal seeking'.  But I dislike the part where the author use a lot of space for story telling, which was not my favorite at all times.

--------------------
[ 21st Oct update on GEB]

By reading this thread I decided to 'scan' the book again from my bookshelf and see whether I can pick something out of it and share among all here.

Recursion, I think most of you may be familiar with them, the earliest time I understand this was to calculate the factorial. Well, factorial (n) = n x (n-1) x .... x 1 which can be written in f(n) = n x f(n-1). A big problem reduced to a small problem, a small problem yet to reduced to a smaller problem, etc. And finally it's solved in 'trivial' condition, recombining everything we got the answer.  But in fact, recursion is the foundation of the computer(also human?) language, when a parser tries to understand your (computer) language, it tries to resolve your syntax in a recursive way.

Well, I saw Escher's self reference paints, Bach's Canon. (In fact I love Pachelbel's Canon more)  So, who is the guy Gödel? The book didn't mention a lot of him directly. In fact he is one of the most important Mathematician in recent mathematics.  

What Gödel told us was whenever we have a formal system, there exists some truths where we cannot prove or disprove.  When this system embrace all these truths, it cannot remain consistent any further. In other words, completeness and consistent cannot both exist.

Err, too difficult already?  Taking the Geometry as an example, here we all learnt Parallel Postulate. //-Postulate says that given a straight line and a point not on this line, we could ONLY find one line that past thro' point and didn't intercept the first line.  Simple? But for centuries people were having trouble proving / disproving this using the geometry.  

So there are some truths that cannot be derived from the original truths.

How about we throw in //-Postulate to the list of axioms of geometry to make it really cover everything in geometry.  It's Gödel telling us that you can't just simply do that, by doing so you will discover some new truths that couldn't be proved/disproved.  If you keep throwing in new axioms to perfect geometry, finally you will arise in an inconsistent situation in geometry, and will break it down. (See? This is a good way to destroy something)


--------------------
[ 22nd Oct update on GEB]
 

Understanding the Gödel's infamous Incompleteness Theorem one needs to understand the predicate logic, involving something like "This statement is false", and construct a statement of the logic system referring the logic system itself, i.e. a self-referencing mechanism.  With the help of another numbering system, the proves were able to map to numbers.  Proving is then reduced to proving on this numbers, well, does this kind of meta-proving make sense?

Self-referencing is another interesting things that differentiate conscious one and non-conscious one. For a conscious/mindful body, he can jump out of the system and talk about himself. Like we see a doctor and tell them we're sick. "I think therefore I am" is an example that you take yourself out of the system and judge it whether it's conscious  or not.  Couldn't a robot somehow act like this, I bet nop.  Seeing this I'd rather believe there should be other way for AI.


« Last Edit: 22 October 2009, 00:20:15 by kido »
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Offline hangchoi

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Re: 數學和數學家的故事
« Reply #3 on: 20 October 2009, 23:41:18 »
Mathematics and Mathematicians is the book introduced to us by Wongyan.

I have all 4 and I bought it when I completed my HKCEE exam. At that time, Yan, King Goh and I were working at my aunt's factory in summer and I used my salary to buy all from book 1 to book 4. These are very interesting and it taught me a lot.

Each chapter it starts with a story and then it talks about the math. inside it. I particularly remember the remainder theorem and its tricks there. Pigeon Nest theory.....etc.

Book 3 and 4 are not so interested as book 1 and 2, I think.

Something about the Escher, I love his drawing and I bought 2 of his drawing from San Francisco years ago. Some years ago, I read a book about topology and it gives me a better understanding of his painting and the meaning of topology. Sadly I forgot the name of the book.....
「吾心信其可行,則移山倒海之難,終有成功之日。吾心信其不可行,則反掌折枝之易,亦無收效之期也。」

Offline kido

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Re: 數學和數學家的故事
« Reply #4 on: 21 October 2009, 00:43:01 »
Mathematics and Mathematicians is the book introduced to us by Wongyan.

I have all 4 and I bought it when I completed my HKCEE exam. At that time, Yan, King Goh and I were working at my aunt's factory in summer and I used my salary to buy all from book 1 to book 4. These are very interesting and it taught me a lot.

Each chapter it starts with a story and then it talks about the math. inside it. I particularly remember the remainder theorem and its tricks there. Pigeon Nest theory.....etc.

Book 3 and 4 are not so interested as book 1 and 2, I think.

Something about the Escher, I love his drawing and I bought 2 of his drawing from San Francisco years ago. Some years ago, I read a book about topology and it gives me a better understanding of his painting and the meaning of topology. Sadly I forgot the name of the book.....

But I remember I read quite a few of them before HKCEE, so maybe it isn't from you.  From whom I borrowed the book?  Err, it must be wongyan then.
Hey, diddle, diddle ! The cat and the fiddle.

Offline hangchoi

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Re: 數學和數學家的故事
« Reply #5 on: 21 October 2009, 00:47:40 »
The first time we touched this book is Book 2, which one of us borrowed it from library........
「吾心信其可行,則移山倒海之難,終有成功之日。吾心信其不可行,則反掌折枝之易,亦無收效之期也。」

Offline kido

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Re: 數學和數學家的故事
« Reply #6 on: 27 October 2009, 01:46:13 »
The first time we touched this book is Book 2, which one of us borrowed it from library........

Book 2 was the first book I read.... Cannot remember exactly what it contains, but definitely good.  (That makes me to continue to read Book 1, then Book 4.... Book 3 is sort of boring)  I cannot find my book 4 now, but I remember the prove of the equality of Isosceles triangle's base angles, by means of computer, was shocking:

If you look at 10 out of 10 maths text books, it involves making an artificial bi-sector line.  However, without making a bi-sector line, how do you prove their base angles are equal?

[Will post it here on the other day....]

--------------------------------------

[update 2nd Nov]

I still couldn't find the book.  Also surprise that there is no such clever prove on the net.  This is the usual method. I remembered the shocking prove was like that:

AB = AC (given)
∠BAC = ∠CAB (given)
BC = CB (common size)


∴△ABC≡△ACB (SAS)

∴∠ABC=∠ACB


« Last Edit: 02 November 2009, 10:18:35 by kido »
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Offline hangchoi

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Re: 數學和數學家的故事
« Reply #7 on: 27 October 2009, 10:53:31 »
Book 2 is very good.....It talks about inter alias, the remainder theorem. A trick of which is teached in a poem, which I can still recall :

三人同行七十里
五馬二十一人騎
七子回家半個月
除百零五是佳期

You think of a number, and you give me the remainder of that number by dividing it by 3, 5, and 7. I can apply these remainder to calculate what the number is by using the above poem. Remember that?
「吾心信其可行,則移山倒海之難,終有成功之日。吾心信其不可行,則反掌折枝之易,亦無收效之期也。」

Offline chin

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Re: 數學和數學家的故事
« Reply #8 on: 27 October 2009, 13:10:41 »
Book 2 is very good.....It talks about inter alias, the remainder theorem. A trick of which is teached in a poem, which I can still recall :

三人同行七十里
五馬二十一人騎
七子回家半個月
除百零五是佳期


You think of a number, and you give me the remainder of that number by dividing it by 3, 5, and 7. I can apply these remainder to calculate what the number is by using the above poem. Remember that?

Yes I remember this. Thank you for reminding me of this poem.

Maybe I should find these books for my kids...

Offline kido

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Re: 數學和數學家的故事
« Reply #9 on: 27 October 2009, 13:28:00 »
三人同行七十里
五馬二十一人騎
七子回家半個月
除百零五是佳期

How come I have no impression at all?  This trick is same as 韓信大點兵 ?

Hey, diddle, diddle ! The cat and the fiddle.