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sage

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My personal experience with mathematics

Here is some of my personal experience with mathematics. It reflects some of
my personal opinions. I am sure it is not a good example for everybody.

some years ago, during the full power days of string theory (or should I call
it geometrical engineering), it seems that smartness is
measured by the amount of mathematics one knows. You always hear things like
this fiber, that cycle, this category, that functor, during PHYSICS talks.
Finally, I decided that I had enough of it. I was going to take a serious
math course (maybe more),taught by a real mathematician, and learn all those
stuff once and for all.

I signed up for a graduate course in algebraic topology in math department.
Let me not go into details about how I struggled through it. I came away
with the conviction that I won't do that again, for the following reasons

1) For a physicist like me, I am far more interested in conclusions than the
intermediate steps. I actually don't care there are at least 5 systematical
ways of calculating the homotopy group. I just want to know what they are.
Mathematicains are exactly the opposite. They seems care much less
about the answer than building up complicated machineries to compute. One has
to prove 20 theorems before one can say the fundamental group of a circle is Z.

Anything I said from my intuition is considered to be wrong. One has to invoke
this extension theorem, that separation theorem in order to do anything. I did
pretty well on the final exam since I finally learned their way of saying
things.

2) Because this overwhelming emphasis on the formal proof, I did not learn
that much things that is actually useful. We spent a lot of time on completely
useless spaces like Hawaiian ear ring (it seems to be the counter example of
every wrong theoerems). By the end of the semester, I could compute homology
groups using 3 methods and so on. However, I still did not learn nearly as
much of the useful result as later when I just quickly flipped through the
book by Nakahara.

Even after that, I still held the belief that really good mathematicians must
could somehow branch through both physics and mathematics (as I was told so so
many times). I even had William Fulton on my thesis committee.
My belief was shaken a little bit when I met Michael Atiyah. He
is promised to be one of the foremost person who did something very important
both in mathematics and physics. He gave us 3 lectures, ,not about his famous
index theorem, but about K-theory. He promised to teach us K-theory from a
very physical point of view. Instead, he spent 3 lectures give 3 different
proofs of Bott periodicity theorem. the only physics is the first 15 minutes
of the first lecture on Coulomb's law. Unfortunately, I didn't learn K-theory
from him (I did get some ideas of it later on from d-brane annhilations).

Anyway, I decided that I am just too stupid to become a grand master of both
mathematics and physics. And actually, I am not too depressed either. I have
since noticed that almost all the good ideas in physics are indeed very
physical and not mathematical at all. Mathematics comes in handy in
formalizing them. It is of course true that knowing a lot of math will help and
one cannot do anything purely without math. However, these days, I often excuse
myself of being stupid on mathematics for the reason that I think one can still
be a good physicist without thinking like a mathematician.

发表时间：2005-03-17, 22:50:13

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No-go

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Re: My personal experience with mathematics

Good article!

Maths guys always go a long way to prove something obvious, but in physics the
most useful maths, I think, is the technique to calculate a process, though this or that technique may involve the maths stuff such as group, Lie algebra, etc.

发表时间：2005-03-18, 00:19:07

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HPC

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Re: My personal experience with mathematics

I think I have the same experience as you. Every time I seems to be involved into mathematical formulation, I warn myself I am a person doing physics rather than mathematics. The key point is I am a person smart enough to do mathematics.

Faith, Fashion and Fancy.

Welcome to 我的域名：http://hongbaozhang.blog.edu.cn

发表时间：2005-03-18, 00:20:39

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Milnor

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Re: My personal experience with mathematics

I agree with to some extent.
Mathematicians are very care about developing a systematic way of doing the work. Because these formalism can tell us what is the simplest way to do the computation. Anyway, as a Physicist only need to remember such main theorem, which are enough to work on physics. For example, you only need to remember the long exact sequence of Homolgy group and M-V sequence etc. and Chern root formula for characteristic class. That will be okay. Do not need to care much about the steps in math.

发表时间：2005-03-18, 01:50:24

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Milnor

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Re: My personal experience with mathematics

Oh, sorry for my poor grammar!

发表时间：2005-03-18, 01:52:27

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轩轩

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Re: My personal experience with mathematics

为善不同同归于美

好文章我觉得数学，象幻方它很好懂但它为什么存在这样的问题真是很吸引人

http://zhangxuanzhong.blogone.net
我的主页
(2004-06-01 13:58:27) 轩轩
http://dzh.mop.com/topic/readSub.jsp?sid=5229875#

发表时间：2005-03-18, 09:04:59

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星空浩淼

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Re: My personal experience with mathematics

昨天我回了一个很长的帖子，正要发送，结果外面在折腾网络，网络断了，我也白写了。这里简要写部分：

对于学物理的，不妨把数学当作物理一样理解着学，理解它背后的前因后果，意图、目的和意义，充分理解它的思想。对于定理的证明，除了巩固理解和掌握之外，可以跳过。

如果sage兄中小学时数学不差，那么就没有理由怀疑自己的数学能力。你说的那些课程，如果象赶时髦似的企图一下子掌握，结果可能对谁都差不多，就是效果不好。我觉得，需要的东西，循序渐进比较好，先了解它的基本思想和来源，在物理学中可以用来干什么，然后再去学比较好。目的是为了掌握，不必一开始迷信大师编的书或上的课，有了一定的基础，再去听大师讲课，可能不一样。人的特征是，自己以为简单的，就以为对别人也不言而喻，殊不知自己先前有很多认知背景和摸索过程作为后盾的。也许大师们上课就有些这样吧。

我本人学数学感觉就象学物理一样。我的哲学是：数学和物理是用不同的语言、从不同的角度、以不同的方式来阐述同一个世界。数学之于物理，不仅仅有方法论上的工具意义，也有认识论和本体论上的参考意义。我高中时，搞出罗贝塔法则（不好意思，好象说过几次），完全是凭物理思辩给出的，然后动笔验证果真如此。我是物理地学数学，不知道那些数学家又是怎样去想象数学的？

如果我从本科到博士都是学理论物理（我连应用物理都不是，而且我们学校从本科到博士，数理基础基本在本科水平上重复，我是靠自己的爱好而自学），我相信我能把那些数学掌握得不错，但这并不意味着我如果象sage兄那样的学习数学经历会更好些——我相信我会跟sage兄感觉一样。我觉得听课和自学结合是好办法，但首先是自己急需要用时带着目的带着课题去学，而不是盲目地为学而学。

最后，我非常同意sage兄的一点是：Idea是最高标准！

唯有与时间赛跑，方可维持一息尚存

发表时间：2005-03-18, 19:40:04

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yinhow

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Re: My personal experience with mathematics

不怕任何数学, 用它, 基础打好.

发表时间：2005-03-19, 01:15:56

作者资料

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Re: My personal experience with mathematics

证明数学定理和解绎数学定理是两种不完全重合的能力，前者更具灵活开阔性，后者更具深入合谐感。可不可以这样说，舞要跳得很美，不一定身体的灵活柔韧性要有多么好。