導(dǎo)論 ....................................................................................................................................1
第一部分 計算的誕生
第 1 章 畢達(dá)哥拉斯的困惑..............................................................................................24
數(shù)的計算 .......................................................................................................................24
從數(shù)覺到計數(shù) .......................................................................................................24
文明古國的計算 ...................................................................................................28
畢達(dá)哥拉斯學(xué)派 ...........................................................................................................30
柏拉圖的理想世界 .......................................................................................................40
第一次數(shù)學(xué)危機 ...........................................................................................................44
無理數(shù)的發(fā)現(xiàn) .......................................................................................................44
芝諾悖論：無窮之辯 ...........................................................................................46
演繹推理：邏輯學(xué)和幾何學(xué) .......................................................................................51
亞里士多德的邏輯學(xué) ...........................................................................................51
歐幾里得的《幾何原本》 ...................................................................................55
悖論：推理的暗面 ...............................................................................................59
第 2 章 計算之術(shù) ............................................................................................................62
代數(shù)：字符的計算 .......................................................................................................62
符號與代數(shù) ...................................................................................................................63
零的誕生 ...............................................................................................................63
言辭代數(shù) ...............................................................................................................65
未知量的表示 .......................................................................................................68
還原與對消 ...........................................................................................................70
代數(shù)符號 ...............................................................................................................73
求解多項式方程 ...........................................................................................................77
從數(shù)值解到代數(shù)解 ...............................................................................................77
三次方程的求根公式 ...........................................................................................81
不可約：復(fù)數(shù)的發(fā)現(xiàn) ...........................................................................................84
數(shù)系的擴張 ...........................................................................................................89
代數(shù)基本定理 .......................................................................................................92
代數(shù)的結(jié)構(gòu) ...................................................................................................................94
求解一元五次方程 ...............................................................................................94
方程根的結(jié)構(gòu) .......................................................................................................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(shù)理邏輯的創(chuàng)立 .........................................................................................................117
人類思想字母表 .........................................................................................................120
思想的大衍術(shù) .....................................................................................................121
計算之夢 .....................................................................................................................125
思維規(guī)律的研究 .........................................................................................................127
19 世紀(jì)數(shù)理邏輯的復(fù)興 ....................................................................................127
布爾的邏輯代數(shù) .................................................................................................129
第二部分 計算的數(shù)學(xué)基礎(chǔ)
第 4 章 數(shù)學(xué)的基礎(chǔ).......................................................................................................136
第二次數(shù)學(xué)危機 .........................................................................................................136
微積分的發(fā)明 .....................................................................................................136
消失的鬼魂：貝克萊悖論 .................................................................................139
分析的嚴(yán)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(shù) .........................................................................................................148
超窮基數(shù)與超窮序數(shù) .........................................................................................148
連續(xù)統(tǒng)假設(shè) .........................................................................................................152
算術(shù)的邏輯化 .............................................................................................................156
弗雷格的“概念文字” .....................................................................................156
自然數(shù)的定義 .....................................................................................................159
第 5 章 第三次數(shù)學(xué)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(jìn)路 .............................................................................................................169
直覺主義進(jìn)路 ...................................................................................................