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    《易經》與畢達哥拉斯數學美學比較(2)

    發布時間: 2019/7/11 0:44:37 被閱覽數: 次 來源: 論文聯盟
    文字 〖 〗 )
    再看錯卦:   
      從上表可以看出,每對錯卦的相對數的和都不相等,其排列順序及其各相對數亦完全相錯。如乾卦之數量和為“6”,其相對的坤卦的數量和為“12”;兩卦數字序列與相對數都是“1”和“2”相交錯。這種數量編碼運動,沒有整齊,只有參差;沒有對稱,只有對立,更顯示出一種典型的參差變化的美學特征。  

    總之,從《易經》單卦、重卦和陰陽二爻的元結構運動觀之,《易經》數學美學體系的總體特征,都是于整齊中求參差、于對稱中求錯綜、于穩定中求變化。如前所述,《易經》是中國傳統文化的源頭,它的文化意義不可低估;作為中國最早的數學美學體系,《易經》的這種數學美學觀念,無疑對我國文藝的影響也是非常之大的。我們認為,《易經》數量編碼運動的這種參差變化趨勢,直接導致了中國文藝“寓變化于整齊”美學原則的形成,成為中國文藝講求參差錯綜美的一個觀念源頭。   
         
      二   
         
      無獨有偶,大約于公元前6世紀,在希臘人祭祀天后赫拉(即宙斯的合法妻子)的薩摩斯島上,誕生了另一個數學美學體系。希臘數學的第一位天才和奠基人畢達哥拉斯便出在這個美麗的島上。那個與《易經》截然不同的數學美學體系,就是由畢達哥拉斯及其追隨者們創造的。   
      畢達哥拉斯學派大都是些數學家、物理學家或天文學家。因此他們特別重視數學,認為數是宇宙萬物的本原,自然界是受數字支配的,由數字依次產生點、線、面、體和水、土、火、金諸元素,最后形成世界。他們觀察藝術,也是從數的觀點出發的。他們認為,美即數的和諧。   
      或許是受薩摩斯島如音樂般的天籟之聲的感染,畢達哥拉斯首先從音樂入手研究數學美學。他發現,在音樂的和聲與數字之間,有著一種基本的聯系。聲音的質的差別都是由發音體(如琴弦)的長短及振動頻率等因素所決定的。如果發音體長,聲音就長;振動頻率大,聲音就高,反之,則低。畢達哥拉斯發現在弦的張力相同時弦長之間成比例才能發生和諧悅耳的聲音。其具體作法是把弦的長度準確地分為兩等份、三等份、四等份,如此等等。撥一撥或拉一拉空弦,這就是基音;將波節(琴弦的靜止點)移至弦的中心點,這就是高出基音八度的音;將波節移至弦的1/3處,這就是高出基音第五度的音;將波節移至弦的1/4處,這就是高出基音的第四度音。在這個不斷對琴弦進行數量分割的過程中,“畢達哥拉斯發現,悅耳動聽——悅西方人之耳——的和聲,與用整數對琴弦的劃分相對應”。(注4)并且,這個比率數目愈小,音愈和諧:   
      從表中可以看出,琴弦自身同一(1:1)即保持基音不變為最美、最和諧;其次是將基音弦對等劃分即2:1為美、和諧。這種保持自身“一”不變、以整數對等劃分為美的觀點,是畢達哥拉斯及其學派的基本觀點。   
      然而,畢達哥拉斯為什么認為保持“一”不變是最美的呢?   
      原來,畢達哥拉斯認為,“一”乃萬物之源,是一切美的總和,其他事物的美都是由“一”“分享”出來的。“一”而生出“多”,變出他物,美的完整性即被“分享”,被破壞,因而只有保持“一”不變,才是最美的。   
      從這個認識出發,畢達哥拉斯反對“多”,(注意,畢達哥拉斯所說的“偶”數是指“多”,而不是指“雙”)。“多”分享了“一”的完美,故畢達哥拉斯認為它是惡的、不美的。然而事物總是要發生變化,要從“一”分為“多”。故畢達哥拉斯力求在紛亂的雜多中達到相對的同一(最大可能地接近原“一”本身)。這個相對同一就是“對稱”。也就是音樂的“高八度”(將波節移至琴弦的1/2處所得到的諧音)。“高八度”雖然“高”出基音八度,但它是在音的螺旋式升變中所達到的與基音最近似的同一,因而,它也是美的。   
      對于畢達哥拉斯來說,他們對琴弦線段數量美學關系的發現,具有某種神奇的力量。“自然與數之間的這種和諧一致竟是如此具有說服力,以致他們完全相信,不僅自然界的各種聲音,就連自然界所特有的各種維和度,都肯定是一些都表現這種和諧的簡單數”。(注5)從這樣的認識出發,畢達哥拉斯把音樂中和諧的理論推廣到建筑、天文學等研究中去。得出了著名的“勾股弦定理”、“黃金分割率”以及“天體圓”等數學美學原則和理論。   
      畢達哥拉斯強調立體中球形最美、平面中圓形最美,與他強調數字中“一”最美,線段中2:1的對稱最美,在內在本質上都是同一的。畢達哥拉斯這種觀點對西方的“把雜多導致統一,把不協調導致協調”的“寓整齊于變化”的美學原則的形成,開了先河。(注6)   
         
      三   
         
      從上面的論述中我們已經看到,畢達哥拉斯數學美學體系與《易經》數學美學體系在基本原則上存在很大分歧:一個是于參差錯綜中求整齊一律(即“寓整齊于變化”);一個是于整齊一律中求參差錯綜(即“寓變化于整齊”)。這種分歧在具體的“數”的追求上,表現為兩大差異:   
      一、畢達哥拉斯數學美學體系追求數的圓滿,以圓滿的數“十”作為最美的數。畢達哥拉斯認為,天體有十個,相應的道德有十種,事物的對立面有十項……連他獻祭給繆斯女神的公牛數目都是十個“10”(“100”)!《易經》數學美學體系對數的追求則與此完全不同,它不追求數的圓滿,而追求數的不圓滿,以“九”作為最美的數。在《易經》中,“九”為陽數至尊,陽爻的最高位稱為“上九”;至“九”以上,則復歸于“一”。根據《易經》的觀點,“九”在中國傳統文化中亦是一個至善至美的數。在中國傳統文化中,天有“九重”(稱“九天”),地有“九州”,官有“九品”,禮有“九儀”,法有“九刑”,路有“九軌”,音樂有“九歌”、“九聲”。古代帝王亦好以“九”為附會,天子稱“九五”之尊,其服飾稱“九章”,其宴請上公的儀節稱“九獻”,國之寶器稱“九鼎”。畢達哥拉斯所推崇的“十”,在《易經》中則是居于下流的。屬于“陰”、“地”,為陰爻之最,且在具體推演中并不出現(陰爻的最高位為“上六”)。   
      在中國傳統文化中,“十”的地位遠不如“九”。在《易經》數學美學體系中“圓滿”是不長久的。乾卦“上九”屬于最高位,面臨“圓滿”(“不長久”)的危險,故《傳》云:“亢龍有悔,盈不可久”。   
      二、畢達哥拉斯數學美學體系追求數的穩定性即不變性(“和諧”亦即穩定)。如上所述,畢達哥拉斯強調數字中“一”為美、線段中2:1的對稱為美、立體中球形最美、平面中圓形最美,都是從強調數和美的不變性出發的。再如畢達哥拉斯把“公正”這一美的道德規范說成是“同次相等的數”即數的平方,所以公正是永遠與自身相同一的東西,也就是說保持現狀維護舊秩序就是公正(注7)。   
      而《易經》數學美學體系則追求數的不穩定性即變化性。“變”,是《易經》數學美學的命脈和精魂。   
         
      四   
         
      《易經》數學美學體系的這種不求數的整齊圓滿、追求數的參差變化的審美原則,對中國文藝的數量
    編輯:秋痕

    《易經》與畢達哥拉斯數學美學比較(1)
    《易經》與畢達哥拉斯數學美學比較(3)

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