24小时热门版块排行榜

返回列表

YANGZL

金虫 (小有名气)

应助: 4 (幼儿园)
金币: 15240.4
帖子: 253
在线: 489.8小时
虫号: 1422023

[资源] [原创] 什么是“完全的数学证明”？

先贴出汉语的（联合国官方正式使用的6种同等有效语言之一）。有空在翻译成英语。请不要歧视汉语！
（1）所有的证明都是相对的。
Any proof is relative, since it is based on certain unprovable assumptions.
《Encyclopaedia of Mathematics》，http://eom.springer.de/p/p075420.htm

（2）K Godel说过：没有一个在特定分辨率层次上形成的知识系统，能够完全解释那个层次，必须具有一个高层元知识才能完全解释它。然而，当我们着手去构造这个更一般的元知识时，它也要求更高一层的元-元知识去解释它。
（来自：近十年人工智能的进展，《模式识别与人工智能》，1995年 8卷 12月增刊，起止页码：1-9）

（3）实系数的一元二次方程，当根的判别式小于0时，有没有解？
答案：如果在实数域（初中学生的答案），没有；相反，在复数域（大学生的答案），有！

（4）三角形内角和等于180°吗？
答案：在欧几何学（Euclidean Geometry）是；在非欧几何学（Non-Euclidean Geometry）里不是。

（5）在几何学中，“点、线、面、体”谁复杂？
答案：依赖于特定的数学评价指标。例如，若以面积作为评价指标时，“点、线”的面积都是0，而“体”是无穷大（不能定义）。可是，以体积作为评价指标时，“点、线、面”的体积都是0。于是“点、线、面、体”的复杂性就出现了多种合理答案。

完全的数学证明需要3种证明：
所以，某数学命题的证明是依赖“证明所采用的数学系统的”。这样：
Based the "The definition of Proof http://eom.springer.de/P/p075420.htm,
the Mathematical proofs of a proposition should have three cases:
(1) the proposition is valid, under a certain axiomatic system;
(2) the proposition is not valid, under another axiomatic system;
(3) the proposition can not be proved, without the necessary designating axiomatic systems.

Under these criterions, GRIGORI PERELMAN only did the 1/3 of the full proofs of Poincare conjecture.

请对照Godel incompleteness theorem和Chaitin theorem。

_____________________________________________________________________

What is a "Full Proof"

"Proof is a reasoning conducted according to certain rules in order to demonstrate some proposition (statement, theorem); it is based on initial statements (axioms). In practice, however, it may also be based on previously demonstrated propositions. Any proof is relative, since it is based on certain unprovable assumptions." [1]

So, a proposition proved can have three results:
according to different rules, the proposition can be (1) valid/proved; (2) invalid; and (3) undecidable.

The Mathematical proofs of a proposition must give the following three cases:
(1) The proposition is valid, under some certain axiomatic systems;
(2) The proposition is not valid, under other axiomatic systems;
(3) The proposition can not be proved/decided, without the necessary designating axiomatic systems.
A Full Proof requires that the three cases are all identified definitely.

This is the use of Gödel incompleteness theorem and Chaitin theorems in the criterion for Future Mathematical Proof.

Under these criterions, GRIGORI PERELMAN only did the 1/3 of the full proofs of Poincare conjecture.

References:
[1] Proof in Encyclopaedia of Mathematics, http://eom.springer.de/p/p075420.htm
[2] Continuum hypothesis in Encyclopaedia of Mathematics, http://eom.springer.de/C/c025790.htm
[3] Gregory J. Chaitin. Information-Theoretic Computational Complexity. IEEE Transactions on Information Theory, IT-20 (1974), pp. 10-15.
[4] Gödel incompleteness theorem in Encyclopaedia of Mathematics, http://eom.springer.de/G/g044530.htm
[5] Morris Kline. Mathematical Thought from Ancient to Modern Times, New York: Oxford University Press, 1972.
[6] Hilbert's Program in Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/hilbert-program/
[7] http://bbs.sciencenet.cn/showtopic-83926.aspx

[ Last edited by YANGZL on 2011-9-30 at 10:28 ]

回复此楼