星期日, 4月 03, 2005

佛曰:不可說

Wir müssen wissen, Wir werden wissen,這是十九世紀末偉大的數學家 Hilbert David 在 1930 年接受 Konigberg 贈與榮譽市民時演講的最後兩句話,同時也是他的墓誌銘。其意思是,"我們一定要知道,我們終將會知道",樂觀的宣稱數學裡所有問題都可以由一些不言自明公理系統出發,經過有限步驟的邏輯推理證明出來,象徵著數學的嚴謹與不容質疑的正確性。

兩個星期前在五樓D桑的blog看到了沒有邏輯,沒有真相這篇文章,讓我想起了這段關於數學家挫敗的故事。

亞里士多德的三段論證法,在邏輯學上屹立數千年之久,其中歸納及演繹的邏輯手法為數學提供了強而有力的後台,讓千年以來的數學家無後顧之憂的探索未知的荒土,到了十九世紀末的數學公理化運動達到了最高峰。然而也在同一時期荷蘭的數學家 Luitzen Egbertus Jan Brouwer 對傳統邏輯提出強烈的質疑,尤其是排中律 (the law of excluded middles),即對任一命題「A」,A或A之否定命題必有一為真,他認為我們不可無條件的接受,Brouwer 堅持有其他的可能性,若接受了此一說法,連帶的,數學中許多的證明將不再被接受,特別是所謂存在性的證明。例如,要證明某一微分方程式有解,則必須給出一個方法,把這個解找出來,而不可僅證明「若無解會導致矛盾」,而這卻是一般數學家們所常用的方法,可想而知這對數學家來說是莫大的災難。

其實早在西元前六世紀,希臘哲學家 Epimenides 便提出了著名的 Epimenides 悖論,身為克里特島人的他提出了下面的命題:

所有克里特島人說的話都是謊話


如果這句話是真的,代表 Epimenides 說的是實話,所以這句話就變成假的了。我們再用大家熟悉的例子來說明,假設某位台灣人宣稱以下的命題:

(A) 某人說 "全台灣人都謊稱天邊一朵雲是部 A片"

(B) 全台灣人說的都不是真話


如果命題「A」是對的,那命題「B」就是對的,代表某人說謊,所以命題「A」就是錯的。換句話說天邊一朵雲是部A片有可能是實話。如果有耐性的各位看到了這邊,我們再以形式邏輯的論點來看這個悖論:

(A) 命題「B」為真

(B) 命題「A」為假


如果命題「A」是對的,那命題「B」告訴我們命題「A」是錯的!!反之,若命題「A」是錯的,則命題「B」是錯的,則我們推得命題「A」是對的!!不管那個方向都會得到矛盾。由此可看出邏輯運用在我們的語言上是多麼的無力。

的確,數學不允許矛盾,二十世紀初的數學家致力要消除這樣的悖論,於是有了數理邏輯的誕生,但是於1931年一位年僅25歲的天才數學家 Kurt Gödel 發表了一篇名為不完備定理的論文,徹底的粉碎 Hilbert 的美夢,其中的第一不完備定理(First Incompleteness Theorem)說:

任何一個足夠強的一致公理系統,必定是不完備的


換句話說一定存在無法證明其真假的命題。

不完備定理徹底的擊垮了自傲的數學家,現今除數理邏輯學家仍奮力不懈的向不完備定理挑戰外,大部分的數學家都採取掩耳盜鈴的心態,認為不完備定理是屬於邏輯學的範疇(1978年已發現了在數學中存在無法被證明其敘述為真的命題)。除此之外,對電腦科學也產生很大的影響,尤其是在人工智慧是否會受到限制的議題上。

十九世紀普遍的數學家都相信,沒有邏輯,沒有真相。但實際的情況是,即使有了邏輯,但真相是什麼?不完備定理告訴我們,再嚴謹的邏輯系統都無法完全的掌握真相,也許正如禪門公案裡面所描述的,佛曰:不可說,不可說。

繼續閱讀...

Hilbert
Gödel

深入閱讀:
希爾伯特的23個數學問題
數學巨人哥德爾
數學

37 Comments:

Blogger Photoman said...

我那篇blog在編寫的時候,就覺得很不舒服,文字果然不太適合解說邏輯,怎麼改都有漏洞,符號比較適合啊。而且,有些人看了還會有情緒反應。我說,你上數學課會生氣嗎 ? 我佛慈悲啊。

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Blogger 冰點下的熱反應 said...

我在寫的時候也覺得很難用中文說清楚我想表達的意思,我想是因為我們太習慣用英文來表達邏輯了。不過我倒是覺得文字才是我們思想的精華,其實文字,不管是任何語言,都是符號的一種,由於太過複雜,我們只好再利用更"簡單"的符號來描述邏輯而已。
這麼說吧,如果我們的語言像是大學微積分,符號邏輯大概只有到建構是式數學程度小學生而已,差得遠咧。

BTW,什麼樣的人看邏輯會起情緒反應啊??數學躁鬱症嗎?

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