計算 - 城邦阅读花园






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	目錄導論 ....................................................................................................................................1 第一部分計算的誕生第 1 章畢達哥拉斯的困惑..............................................................................................24 數的計算 .......................................................................................................................24 從數覺到計數 .......................................................................................................24 文明古國的計算 ...................................................................................................28 畢達哥拉斯學派 ...........................................................................................................30 柏拉圖的理想世界 .......................................................................................................40 第一次數學危機 ...........................................................................................................44 無理數的發現 .......................................................................................................44 芝諾悖論：無窮之辯 ...........................................................................................46 演繹推理：邏輯學和幾何學 .......................................................................................51 亞里斯多德的邏輯學 ...........................................................................................51 歐幾裡得的《幾何原本》 ...................................................................................55 悖論：推理的暗面 ...............................................................................................59 第 2 章計算之術 ............................................................................................................62 代數：字元的計算 .......................................................................................................62 符號與代數 ...................................................................................................................63 零的誕生 ...............................................................................................................63 言辭代數 ...............................................................................................................65 未知量的表示 .......................................................................................................68 還原與對消 ...........................................................................................................70 代數符號 ...............................................................................................................73 求解多項式方程 ...........................................................................................................77 從數值解到代數解 ...............................................................................................77 三次方程的求根公式 ...........................................................................................81 不可約：複數的發現 ...........................................................................................84 數系的擴張 ...........................................................................................................89 代數基本定理 .......................................................................................................92 代數的結構 ...................................................................................................................94 求解一元五次方程 ...............................................................................................94 方程根的結構 .......................................................................................................95 伽羅瓦的遺珠 .....................................................................................................101 計算工具 .....................................................................................................................108 人類計算員 .........................................................................................................109 面向機器的計算思維 .........................................................................................111 第 3 章萊布尼茨的計算之夢 ........................................................................................116 數理邏輯的創立 .........................................................................................................117 人類思想字母表 .........................................................................................................120 思想的大衍術 .....................................................................................................121 計算之夢 .....................................................................................................................125 思維規律的研究 .........................................................................................................127 19 世紀數理邏輯的復興 ....................................................................................127 布林的邏輯代數 .................................................................................................129 第二部分計算的數學基礎第 4 章數學的基礎.......................................................................................................136 第二次數學危機 .........................................................................................................136 微積分的發明 .....................................................................................................136 消失的鬼魂：貝克萊悖論 .................................................................................139 分析的嚴格化 .....................................................................................................140 集合論的誕生 .............................................................................................................142 無窮大有多大 .....................................................................................................142 對角線方法 .........................................................................................................146 康托爾的超窮數 .........................................................................................................148 超窮基數與超窮序數 .........................................................................................148 連續統假設 .........................................................................................................152 算術的邏輯化 .............................................................................................................156 弗雷格的“概念文字” .....................................................................................156 自然數的定義 .....................................................................................................159 第 5 章第三次數學危機 .........................................................................................163 危機：羅素悖論 .........................................................................................................163 集合論悖論 .........................................................................................................163 自我指涉 .............................................................................................................165 悖論的解決方法 .................................................................................................168 邏輯主義進路 .............................................................................................................169