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沒有王者之路《幾何原本》



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《原本》成書於西元前三百年左右,距離今天兩千三百年,《原本》的作者是亞歷山卓的歐基里得(Euclid of Alexandria),他的生卒年根據推測大概是公元前330~260年,正是馬其頓英主亞歷山大開始發展勢力,開創希臘化文化的初期。《原本》是一本數學著作,章節安排有著嚴謹的結構,全書由定義、公設、設準、命題(定理)、證明,以及符號和圖像所構成,全書共十三卷。
《原本》其實是歐基里得將古希臘數學集大成的著作,包括了希臘科學數學家:泰利斯、畢達哥拉斯、希波克拉提斯等人的成果。導讀者翁秉仁教授認為《原本》之所以是經典,是因為歐基里得採用了非常特殊的編纂法,就是推理的方法或邏輯。歐基里得的原創性不是表現四百多個命題的敘述,因為許多命題在當時是已知的知識。歐基里得的天才表現在他有精準深刻的眼光,選擇恰當的公設,又有驚人的推理能力,可以一步步將這許多命題整合成一個體系。引用笛卡兒的譬喻,歐基里得不是只找出一條鐵鍊,而是將許多條推理的長練,編織成一張鐵鍊網,將所有的《原本》命題都固定在五個牢靠的首環上──亦即五個公設上,包括著名的「平行公設」。
歐基里得運用希臘文明的資源,將過去零散的經驗知識,整理成不容置疑的真理。他展示的這種發展人類知識的新方法,日後稱為「公設演繹法」,讓人類得以在這個變化、偶然、不可知的世界上,平地一聲雷,突然有了一個典範,可以協助人類分析、判定並取得確定的真理,這在人類的思想史上,是一個了不起的天才成就。

歐基里得

古希臘數學家,被譽為「幾何之父」。早年曾在亞歷山卓圖書館工作,此時也是他思想活躍的黃金時期。代表著作是成為歐洲數學基礎的《原本》,另有《光學》等作品。後世對他的了解甚,僅能從之後的數學家如帕普斯與普洛克勒斯的作品確定其生平。

翁秉仁

民國49年生,80年美國加州大學博士。目前為國立台灣大學數學系副教授,研究興趣為雙曲群論、結論、低維拓樸。

Akibo

藝術家、設計師。畢業於文化大學美術系,台北藝術大學美術系創作研究所碩士。曾任實踐大學應用美術系講師,現任國立台灣師範大學駐校藝術家,Akibo Works 負責人。重要畫展包括:2010年台北燈節巨型投影秀藝術總監、2010年BIGPOW音響機器人、2009年金剛想談戀愛個展、2009年高雄世運開幕秀美術總監等。


《原本》在歷史上的挑戰

《原本》無疑是今日數學與科學書寫的源頭與典範,但它畢竟是時代的產物,從二十世紀的邏輯標準,羅素跟德國數學家希爾伯特都提到《原本》並不夠嚴格,希爾伯特還因此重新寫出一本《幾何基礎》,修補《原本》的邏輯缺陷。

不過這還不是《原本》真正的挑戰。自古以來,許多思想家視《原本》的內容─歐氏幾何─為絕對真理,康德更為此發展了他的哲學系統,說明這是一種先驗綜合式的真理。但是歐基里得一直擔憂的第五公設,也就是平行公設,後來卻走出了一條令人意想不到的大道。

簡而言之,平行公設一直是數學家的挑戰。他們希望找到平行公設更「自明」的敘述方式;或者希望從其他四個公設推論出平行公設,證明平行公設是多餘的;或者希望證明否定平行公設可以導出矛盾,因此平行公設必須是正確的。這些努力都是為了鞏固歐氏幾何的真理地位,不受可疑的平行公設影響。

沒想到最後在十九世紀初發現,否定平行公設不但沒有矛盾,反而發展出非歐幾何學。卸下歐氏幾何是真理的桎梏後,整個幾何思想的飛躍不可以道里計,最重要的成就是十九世紀數學家黎曼所倡議的黎曼幾何,這種更廣義的非歐幾何成為愛因斯坦廣義相對論的基礎。

另外,在十九世紀這段歐氏幾何和非歐幾何的攻防歲月裡,發展出許多檢討公設系統性質的方法,在日後二十世紀的數學基礎論戰中開花結果,讓人類對於理性思考的深度與限度,有了更深的理解。

不過到了二十世紀末,已經可以為《原本》的歐氏幾何做新的辯護。康德將歐氏幾何視為宇宙空間的真理,在非歐幾何出現後似乎成為笑話。但是康德論證的方式仍然有其價值,他認為人類的經驗並不是素樸的感官經驗,而是先透過心靈加工後才呈現的,其中呈現經驗時必須預設時間和空間的直觀。康德昔日缺乏證據的哲學玄想,但今天透過認知科學已經知道,人類感受到的經驗其實有很多層級,通常必須經過各腦區的化約整理,才能做有效率的回應。換句話說,大腦最後面對的「經驗」,其實是加工過的經驗。就這點而言,反而康德的綱領才是正確的,因此認知科學分外重視康德的思想。

例如視覺經驗絕對不只是可見光感應的直接傳遞,為了在大自然中求取生存,人類與動物的大腦會透過不同腦區加強平行、垂直、對稱的處理,這就是為什麼我們從小對這些幾何「概念」特別敏銳的原因。換句話說,即使我們生存的空間不是歐氏空間,但是我們的大腦卻是用歐氏幾何的形式來逼近、重組我們的視覺經驗,所以我們才會特別覺得平行公設是直觀自明的,這也解釋了為什麼歐基里得的《原本》是人類第一個幾何理論。



《原本》濃厚的幾何氣味

《原本》不只是時代的產物,也是文化的產物,由於古希臘人以幾何思想為王道,因此《原本》帶有很強的幾何氣味,徐光啟將譯本稱為《幾何原本》,其實不能算是誤譯。

一般來說,數就是數,地球上每一個文明都會使用數,也熟習數和幾何交匯的問題,例如測量兩地的距離、城牆的高長、田地的面積等。但只有古希臘人將幾何視為更根本的概念,即使在處理數的材料如質數、公因數、公倍數時,仍然採用幾何方法。譬如說他們的整數是用線段來表示,也就是用線段長度來表示數。

希臘人並非對代數知識無知,但是《原本》中的代數式都是用幾何圖形來表現。這並不是歐基里得的獨到見解,而是希臘文化的特點,在其他文化沒有見到這麼徹底的使用。也因此,希臘人在處理無理數如 (根號2)時,才會發展出「不可公度量」的概念。

《原本》的材料並不只是幾何學,但它的內容從頭到尾都混合了文字與圖象,其中每個命題的證明也都是文字與圖形的共同運用,圖形並不只有附屬的說明或裝飾功能。古希臘人在人類文明的早期,恰巧善用了人類先天兩個非常重要的認知資源─語言與視覺,這也是為什麼認知考古學者重視古希臘數學的原因。正如前述,人類特別擅長處理視覺的東西,因此老師教幾何時,一味強調文字或符號的重要性,其實不是理想的方式。如果將《原本》裡面的圖全部拿掉,也許它不再是最成功的教科書了。


 
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