不完備定理的威力 -戈德爾 Kurt Friedrich Gödel 之二

in cn •  6 months ago  (edited)

source: see reference

前幾天說了待續這一篇:我的新偶像-戈德爾 Kurt Friedrich Gödel

今天就繼續接著講。

上次介紹了人,談了他的「不完備定理」(Incompleteness Theorem)。今天繼續說說,這不完備定理在很多方面的應用跟影響...

真要牽的話,連我們今天在這裡發文章都能扯到跟這定理有關呢~~~

我們先簡單回顧一下這定理的內容,剛好看到一個比較白話一點的版本,是英文的,我把它翻譯一下:

不完備定理

For any consistent axiomatic formal system that can express facts about basic arithmetic:

對於任何具一致性而且能夠表述基本算術邏輯的公設系統

  1. There are true statements that are unprovable within the system.
  1. 系統內必然存在正確但無法被證明的陳述
  1. The system’s consistency cannot be proven within the system.
  1. 系統的一致性無法在系統內被證明

額,好像還是沒有很白話,那我大膽用自己的話說說看好了,我想應該是錯的 lol...

任何體系都不能用邏輯方式證明體系內所有的論述

感覺就是任何知識都是有預設知識存在的,不可能存在一套知識系統是完全只需要系統內的知識就可以完全支持的。

這是不是有一點像是「神」的存在....

我向來有一種想法就是,人類無法透過自我來理解人類的存在,因此有「神」,有信仰,那就像是體系外的「預設知識」... 戈德爾先生,請不要打我..... >_<

對數學與哲學的影響

地球人類有太多的問題無法解決,有各式各樣的「危機」,如能源危機、道德危機、人口爆炸危機等等,而常有無力感,但在二十世紀初,人類卻充滿了盼望與信心,面對未來都很樂觀,特別對「理智」的信心非常強,相信憑著理智所有的問題都可解決,數學就是一個明顯的例子

十八、十九世紀數學的成就是驚人的,如從希臘時代就留下來的所謂「幾何三大難題」,竟然一次就都被解決了,也難怪希伯特說:「我們必須知道,我們將會知道」,自然交出它所有的問題,而人類必將所有的問題一一克服,這是當時的樂觀看法...

因此,當不完備定理出來,對許多人來說彷如晴天霹靂,人們一度認為找到了數學的基礎,卻發現這個基礎是海市蜃樓,而且不完備定理似乎告訴人們,我們將永遠無法找到這個基礎,連數學這號稱最精確的科學尚且如此,其他所有的知識又如何自處呢?

不完備定理告訴我們,有些事情是真的,但我們無法證明它,若是如此,人要如何面對沒有被證明的事?既無法全部接受,亦不該全部否決,如何決定取捨呢?這似乎已經是哲學層次的思辨了...

對電腦發明的影響

戈德爾1931年發表了不完備定理時,還沒有今日所謂的電腦,對於電腦如何發明的,至今仍眾說紛紜,但在此可以引述普林斯頓高等研究院1978-1979年度報告中所摘錄曾任美國國家科學院副院長的Saunders Mac Lane的一段話:

戈德爾偉大而抽象的邏輯工作,有個令人驚異的結果。在分析戈德爾所描述的何者可被一步步程序所得的正式方法中, 年輕而聰明的英國邏輯家Alan Turing 定出了這程序所得的結果,即一般遞歸函數 (general recursive functions),這也正是一台機器所可能計算的,藉著這個分析,及其在 John Von Neumann 等人身上的作用,以致現代計算機的理論觀念及分析得以開展,直至今日,對於何者可被計算的理論描述,及至更深入的分析,我們可正確的說,仍然根植於戈德爾於1931年所發表的數理邏輯論文中。

你看看,你看看,咱們今天在Steem上發文,是不是可以追溯到戈德爾這位偉大的人物呢?沒電腦還玩什麼區塊鏈嘛!

我只能說,要是中本聰碰上戈德爾,只怕也只配幫他倒茶了吧!(圖零大概有資格跟他聊天,聊天空檔,中本聰也許可以跟圖靈說個兩句,但茶還是得接著倒的..... lol)

參考資料

暫時結束..........

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One of my favorite mathematicians

I thought he is much more than a mathematician. :)

Loved this video about him and his incompleteness theorem